Politecnico Di Milano

POLITECNICO DI MILANO

Scuola dell’Ingegneria Industriale e dell’Informazione Corso di Laurea Magistrale in Ingegneria Biomedica

VIRTUAL TESTING OF PROSTHETIC SOCKETS FOR TRANSTIBIAL AMPUTEES BASED ON THE ISO 10328 STANDARD

Relatore: Prof. Carlo Albino Frigo

Correlatori: Ing. Andrea Giovanni Cutti, PhD, TO Ing. Francesco Pisu

Tesi di laurea magistrale di: Francesca Gariboldi Matricola: 899857

Anno accademico 2018/2019

Table of contents

Table of contents ...... I

Abstract ...... V

Sommario ...... XIV

1 Introduction ...... 1

1.1 Objective of the master’s thesis ...... 1

1.2 INAIL Prostheses Center ...... 1

1.3 Lower limb amputations and prostheses ...... 2

1.3.1 Transtibial prosthesis ...... 3

2 Background ...... 4

2.1 Traditional method for socket production ...... 4

2.2 Innovative method for socket production ...... 5

2.3 3D printing ...... 8

2.3.1 FFF 3D printing technology ...... 8

2.3.2 Slicer ...... 9

2.3.3 3D printer ...... 13

2.3.4 Printing the socket ...... 14

2.4 Static test ...... 16

2.4.1 Socket geometry ...... 16

2.4.2 Test setup ...... 17

2.4.3 Materials ...... 29

2.4.4 Static test results ...... 31

2.5 Possible causes of failure ...... 33

2.6 Objective of this study ...... 34

3 Materials and Methods – Material characterization ...... 36

3.1 Introduction ...... 36

3.2 Material...... 37

3.3 Filament dehumidification ...... 37

3.4 Specimen design ...... 38

3.5 Specimen 3D printing ...... 40

3.5.1 Dimensional control ...... 43

3.6 Specimen testing ...... 44

3.6.1 Test setup ...... 44

3.6.2 Test parameters ...... 48

3.6.3 Measured parameters ...... 48

3.6.4 Data analysis ...... 50

3.6.5 Plasticity data extrapolation for Abaqus ...... 50

4 Materials and Methods – CAD/FEM process ...... 54

4.1 Introduction ...... 54

4.2 Abaqus ...... 55

4.3 Reference model ...... 56

4.3.1 Parts ...... 62

4.3.2 Assembly of the parts ...... 68

4.4 Mesh ...... 69

4.4.1 Element type ...... 70

4.4.2 Meshing technique ...... 72

4.4.3 Mesh validation ...... 73

4.4.4 Accuracy calculation ...... 75

4.4.5 Fillets ...... 77

4.4.6 Partitions and Mesh generation ...... 79

4.4.7 Mesh verification ...... 87

4.5 Material property assignment ...... 88

4.6 Interactions ...... 92

4.6.1 Contact interactions ...... 93

4.6.2 Constraints ...... 97

4.7 Loading ...... 98

4.8 Static analysis ...... 101

4.8.1 First analysis ...... 101

4.8.2 Second analysis...... 101 II

4.8.3 Third analysis ...... 103

4.8.4 Data analysis ...... 104

4.9 Alternative models to enhance mechanical properties ...... 107

4.9.1 Model 1 ...... 107

4.9.2 Model 2 ...... 108

4.9.3 Model 3 ...... 109

4.9.4 Model 4 ...... 111

4.9.5 Model 5 ...... 112

5 Results – Material characterization ...... 114

5.1 3D printed specimens ...... 114

5.2 Dimensional control results ...... 115

5.3 Dimensional control over time ...... 116

5.4 Static testing results ...... 116

5.4.1 Mechanical properties ...... 117

5.4.2 Plasticity data extrapolation ...... 119

6 Results – CAD/FEM process ...... 122

6.1 Mesh ...... 122

6.1.1 Mesh validation ...... 122

6.1.2 Mesh accuracy ...... 123

6.1.3 Mesh verification ...... 124

6.2 Static analysis results ...... 128

6.2.1 Reference Model...... 128

6.2.2 Model 1 ...... 130

6.2.3 Model 2 ...... 132

6.2.4 Model 3 ...... 134

6.2.5 Model 4 ...... 136

6.2.6 Model 5 ...... 138

6.2.7 Models comparison...... 141

7 Discussion – Material characterization ...... 142

7.1 Dimensional control ...... 142 III

7.2 Dimensional control over time ...... 142

7.3 Static testing ...... 142

7.3.1 Mechanical properties ...... 142

7.3.2 Plasticity data ...... 143

8 Discussion – CAD/FEM process ...... 144

8.1 Mesh ...... 144

8.1.1 Mesh verification ...... 144

8.2 Static analysis ...... 144

8.2.1 General conclusions ...... 144

8.2.2 Models comparison...... 146

9 Conclusions and Future perspectives ...... 148

Appendix A ...... 149

Appendix B ...... 157

Appendix C ...... 159

Bibliography ...... 174

Acknowledgements ...... 177

Abstract

Introduction The study described in this master’s thesis has been carried out in collaboration with INAIL Prostheses Center in Vigorso di Budrio (BO), which is a center of research and application of prostheses and orthoses.

Background Transtibial (below-knee) sockets are traditionally manufactured using a heavily manual production process, consisting of eight steps (four of which are manual) and requiring specialized prosthetists.

Recently, the Prostheses Center developed an innovative method which exploits CAD and 3D printing (FDM), aimed to replace three of the four manual phases. This process requires to embed an aluminum connector into the distal part of the 3D printed socket, as the fastening component with the remaining modules (pylon and foot) of the prosthesis (Figure 1).

Figure 1 – Components of a modular transtibial prosthesis. The socket realized with the innovative method is attached to the pylon through the connector embedded inside its distal part.

Sockets of different materials were subjected to static testing according to test loading condition II of the ISO 10328 standard using a setup built by the Ohio Willow Wood Company. Moreover, to obtain unified and comparable results, sockets were printed starting from the same shape of residual limb, which is representative of the 98th percentile of the US male population of transtibial amputees. This socket has enlarged dimensions to represent a worst-case scenario during static testing. In addition, all of the sockets were printed using the same printing parameters, and in particular the same infill pattern (honeycomb) and percentage (50%). The standard prescribed different levels of loading, based on locomotion data of amputees of different body weights: P3 (2790N), P4 (3623N), P5 (4025N), P6 (4425N), P7 (4840N) and P8 (5250N). Tests were V considered successful, if sockets resisted P6 loading level (4425N). Unfortunately, none of the sockets manufactured with this method passed static testing, and all of them failed before reaching P3 loading level (2790N).

In every test, failure took place in the distal part of the socket, due to the interaction between the printed part and the aluminum connector (Figure 2).

Figure 2 – In all experimental tests, fracture took place at the distal interface between the connector and the printed socket.

It was evident that the production process required some improvement. Since the physical test of sockets is expensive and time consuming, we decided to use Finite Element Modelling to improve the design. The aim of this study was to create a FE model in Abaqus able to analyze the stress distribution under the same loading conditions prescribed by the standard, and use it to compare:

- reference geometry with different material properties; - improved geometries of the distal part of the socket; - improved geometries of the connector.

Materials and Methods The materials and methods required to reach these aims are summarized in the visual abstract provided in Figure 3. As it is possible to see from Figure 3, there are three main building blocks highlighted in different colors. In blue, we reported the steps of the innovative method that we want to improve (background information). In green, we show the materials and methods that we used to characterize the strongest material among the ones used during experimental socket testing. In orange, we show materials and methods that we used to implement virtual socket testing in Abaqus.

The steps of materials and method that we describe below refer to the blocks numbering of Figure 3.

Figure 3 – Visual abstract. In blue: steps of the innovative method that needs improvement (background information). In green: materials and methods for material characterization. In orange: materials and method for implementation of socket testing in Abaqus.

VII

Materials and Methods – Material characterization Among the materials used to print the sockets, we selected the one that showed the best test results, and we characterized it according to the ISO 527-1 and ISO 527-2 standards. This material turned out to be Carbonium Nylon, a thermoplastic composite material made of Nylon (PA12) reinforced with carbon fibers (CF) (1).

To obtain unified results, we designed dumb-bell shaped specimens having dimensions in accordance with ISO 527-2 (2).

Given the hygroscopic nature of PA12, we dehumidified the material filament before printing half of the specimens, so that we could compare specimens printed with dehumidified and with non-dehumidified material (3). We printed the specimens using a FDM 3D printer (DELTA WASP 4070 Industrial). To study the influence of infill percentage, we printed half of the specimens with 50% infill honeycomb (same condition as 3D printed sockets) and half of them with 100% infill honeycomb (4).

Then, according to ISO 527-1, we subjected the specimens to static uniaxial tensile testing (5). Finally, according to both ISO 527-1 and 2, we calculated the mechanical properties of the tested specimens (6) and we compared them (7).

Materials and Methods – CAD/FEM process We used Abaqus to set up a method to carry out a Finite Element Analysis on transtibial sockets, according to test loading condition II of the ISO 10328 standard.

Since we had limited computational resources (16GB of RAM), modelling optimization has been the main theme of the entire process. For this reason, we confined the simulation only to the critical area, i.e. the distal part of the socket and its embedded aluminum connector.

In order to allow an easy and efficient geometry and material modification, we used Abaqus’ CAD environment (Sketch module) to draw both the distal part of the socket and the connector as well as the Willow Wood setup (8).

Figure 4 – Distal part of the socket, embedded connector and Willow Wood test setup. The very distal part of the socket that hosts the connector is also referred to as “housing”.

VIII

Then, we meshed the model with reduced integration linear hexahedral elements. Given the complexity of the geometry, we made extensive use of partitions (9). To find the optimal mesh density, we performed a mesh convergence study on a representative simplified version of our model. This model also allowed us to compare the computational results (stresses) with the analytical ones, and thus to calculate their accuracy (10).

We assigned material properties to each region of the model (11). In particular:

- distal socket: isotropic elastoplastic material based on the material characterization of Carbonium Nylon; to spare computational resources, we made the assumption of isotropy, even if the honeycomb pattern is strongly anisotropic and presents varying percentages of empty spaces. - connector and setup: isotropic elastic material based on literature mechanical properties of Al-6061- T6; we neglected its plastic behavior, as this Al-6061-T6 is much stiffer and stronger than Carbonium Nylon.

Then, we defined the interactions between the distal socket and the connector (12) and we assigned the loading and boundary condition to faithfully reproduce test loading condition II of the standard (13). In particular, we applied increasing levels of loading corresponding to test loading levels P3 (2790N), P4 (3623N), P5 (4025N), P6 (4425N). P6 represents the threshold for passing static testing. Finally, we carried out a static analysis on different models (14), namely (Table 1):

Table 1 – Different models realized in Abaqus.

Geometry Model Material Distal socket Connector Reference dehumidified 100% Reference Reference Model infill larger thickness in the anteroposterior direction Model 1 Reference Same as above (direction of maximum bending moment) uniform and larger thickness Model 2 Reference Same as above in all directions Reference with larger fillet radius Model 3 Reference Same as above (0.3mm instead of 0.1mm). New geometry Model 4 Reference with very large Same as above fillet radius (3mm) dehumidified 50% Model 5 Reference Reference infill

Figure 5 – Geometry of the very distal part of the socket (housing) for the Reference Model, Model 1, Model 2 and Model 4. Model 3 has the same geometry of Reference Model with larger fillet radius of the connector. Model 5 has the same geometry of the Reference Model.

After performing FE simulations, we analyzed the stress distributions at the different loading levels (P3, P4, P5 and P6) in the region of interest, which we defined as the groove inside the distal socket. We analyzed both the Von Mises as well as Maximum Principal stress distribution, because the material that was assigned to the distal socket could not be categorized neither as purely ductile nor as purely brittle (15).

Results – Material characterization From the results of material characterization, we found that:

- Specimens printed with non-dehumidified material underwent a marked dilation that caused their dimensions to exceed the ones prescribed by the ISO 527-2 standard. This led us to discard these specimens from static testing. - Specimens printed with dehumidified material maintained steady dimensions in the long term (up to two months post printing). - Both of specimens’ set of dehumidified 100% infill and 50% infill showed an elastoplastic behavior that cannot be categorized neither as purely ductile nor as purely brittle, as the plastic strain ranges around 3% in both cases. - (Dehumidified) 100% infill specimens were stiffer and stronger than (dehumidified) 50% infill specimens.

Results – CAD/FEM process Mesh The mesh that we generated showed some element distortion in the region of interest (groove inside the distal socket) that could not be eliminated with the present mesh density.

Static analysis By analyzing the stress distribution in the region of interest at P6 loading level, we observed that none of the models displayed stress values higher than the ultimate tensile strength. Nevertheless, all models except for Model 4 displayed a few localized nodal regions for which the Von Mises stress overcame the yield stress. Results are summarized in the two tables and compared to the values of the Reference Model and with the material limit stress.

Table 2 – Von Mises stress distribution in the region of interest of the six different models at P6 loading levels.

Nodes over material Von Mises Stress yield stress Yield stress of Model % of % of % of material [MPa] Maximum % of Reference yield Reference value [MPa] nodes Model stress Model Reference 37.48 51.86 - 138% 0.18 - Model Model 1 37.48 51.89 100% 138% 0.18 100% Model 2 37.48 49.71 96% 133% 0.29 161% Model 3 37.48 44.9 87% 120% 0.13 72% Model 4 37.48 29.76 57% 79% 0 0% Model 5 23.23 42.58 n.a. 114% 1.96 1089%

Table 3 – Maximum Principal stress distribution in the region of interest of the six different models at P6 loading levels.

Nodes over material Ultimate Maximum Principal stress ultimate tensile tensile stress Model stress of % of % of % of material Maximum % of Reference ultimate Reference [MPa] value [MPa] nodes Model tensile stress Model Reference 58.34 28.08 - 48% 0 - Model Model 1 58.34 27.54 98% 47% 0 n.a. Model 2 58.34 36.66 131% 63% 0 n.a. Model 3 58.34 32.8 117% 56% 0 n.a. Model 4 58.34 26.67 95% 46% 0 n.a. Model 5 42.35 25.96 n.a. 44% 0 n.a.

Discussions and Conclusions – Material characterization From the results of material characterization, we can conclude that:

- It is essential to dehumidify the filament coil of Carbonium Nylon before printing a part. - The effects of dehumidification remain unchanged in the long term. - A higher infill percentage leads to higher mechanical properties, in accordance with literature data.

Discussions and Conclusions – CAD/FEM process Mesh The distortion of the mesh in the area of interest does not seem to represent a problem, in fact, in this location the stress distribution has smooth contours.

Static analysis General conclusions:

- The stress distribution of both Von Mises and Maximum Principal stress is skewed. In particular the Maximum Principal stress distribution can be best fitted by a Poisson’s distribution, meaning that the bending moment in this region is not pure. - For both distributions, only a few nodes inside the region of interest show high peak values, whereas the majority of the nodes displays values close to the mean stress. - The Von Mises stress display higher values than the Maximum Principal stress. This means that there are non-negligible compressive contributions in the area of interest, probably due to the connector indenting into the distal socket when the distal socket is pulled.

Model-specific considerations:

- In the Reference Model, the percentage of nodes for which the Von Mises stress overcomes the yield stress is negligible (below 0.2%). Therefore, yield is unlikely to take place. Even localized yielding is unlikely, because the possible localized plastic nodes can rely on the elastic support of neighboring nodes, which could well represent the case of 100% infill. An 100% means that the honeycomb pattern of the printed part has a material density of 100%, without empty spaces that would allow crack propagation. - Model 1 and Model 2 did not show any actual improvement from the reference model, neither in terms of stress value nor stress concentration. Moreover, for Model 2, the stress distributions changes producing lower compressive stresses but higher tensile stresses. Thus, while Model 1 can be considered equivalent to the Reference Model, Model 2 produces worse results and should be discarded. Therefore, modifications of the geometry of the distal socket that do not include modifying the groove do not improve the stress distribution. - In Model 3, the small reduction in Von Mises stresses confirms the positive impact of a larger fillet radius in reducing the notch effect. However, the increase of Maximum Principal stress denotes an increase in tensile stresses. Therefore, we decided to redesign the connector. - In Model 4, the behavior has substantially improved, showing very low stress distributions always below the limits of the material. In particular: o the maximum value of Von Mises stress dropped to 57% of the reference value and 79% of yield stress; o the maximum value of Maximum Principal stress has decreased to 95% of the reference and 46% of the ultimate tensile stress. We can conclude that yielding is not possible. - In Model 5 we simulated another material with the reference geometry. The Von Mises stress exceeds by 14% the yield stress in almost 2% of the nodes. The risk of yielding cannot be neglected, also because the plastic nodes cannot rely on the support of the neighboring elastic nodes, since the voids in the 50% infill part could lead to crack propagation and failure of the structure. This behavior is enforced by the fact that 50% infill sockets failed the experimental testing.

XII

Figure 6 – Distribution of Von Mises (left column) and Maximum Principal (right column) stress at P6 loading level for Reference Model (first row) and for Model 4 (second row).

Overall Conclusions and Future Perspectives This study proposes a methodology to improve distal socket and connector design based on parametric modelling, FE analysis and material conditioning. From our results, we can conclude that the reference geometry of the distal part of the socket and of the connector could withstand the prescribed loading level only using dehumidified 100% infill Carbonium Nylon. An improved geometry of the connector (Model 4) would reduce the stress to 57% of the reference.

Further steps should focus, first, on validating the computational results by means of experimental testing and, second, on optimizing the infill percentage for the improved geometry (Model 4).

XIII

Sommario

Introduzione Lo studio descritto in questa tesi magistrale è stato svolto in collaborazione con il Centro Protesi dell’INAIL di Vigorso di Budrio (BO), che è un centro per la sperimentazione ed applicazione di protesi e presidi ortopedici. Contesto Le invasature transtibiali (sotto al ginocchio) vengono tradizionalmente realizzate con un processo produttivo fortemente manuale in otto fasi (quattro delle quali manuali), che necessita di tecnici specializzati. Recentemente, il Centro Protesi ha sviluppato un metodo innovativo che sfrutta la tecnologia CAD e la stampa 3D a Fabbricazione e Fusione di Filamento (FFF), e si prefigge di sostituire tre delle quattro fasi manuali. Questo nuovo processo prevede di inglobare una placca di alluminio all’interno della parte distale dell’invasatura stampata in 3D, come elemento di collegamento con le altre parti della protesi (pilone e piede protesico) (Figura 1).

Figura 1 – Componenti di una protesi transtibiale (sotto il ginocchio). L’invasatura realizzata con il metodo innovativo è collegata al pilone tramite un connettore di alluminio inglobato all’interno della parte distale dell’invasatura stessa.

Con questo metodo sono state stampate invasature di diversi materiali che sono state poi testate staticamente secondo la condizione di test II descritta dalla norma ISO 10328:2016, usando il setup di prova costruito dall’azienda statunitense “the Ohio Willow Wood Company”. Per ottenere risultati unificati e confrontabili, tutte le invasature sono state stampante a partire dalla stessa morfologia di moncone, rappresentativa del 98esimo percentile della popolazione statunitense maschile con amputazione transtibiale. Le invasature risultanti hanno dimensioni maggiorate per rappresentare il caso pessimo durante le prove statiche. Inoltre, tutte le invasature sono state stampante con gli stessi parametri di stampa, in particolare con lo stesso modello

XIV

(nido d’ape) e con la stessa percentuale (50%) di riempimento. La norma prescrive diversi livelli di carico, corrispondenti a diversi dati di locomozione di persone amputate con diverso peso corporeo: P3 (2790N), P4 (3623N), P5 (4025N), P6 (4425N), P7 (4840N) and P8 (5250N). L’esito delle prove è da considerarsi positivo se le invasature superano il livello di carico P6 (4425N). Sfortunatamente, nessuna invasatura realizzata con il metodo tradizionale ha superato le prove statiche, e tutte hanno fallito prima di raggiungere il più basso livello di carico, P3 (2790N).

In tutte le prove sperimentali, la rottura è avvenuta nella parte distale dell’invasatura, a causa dell’interazione tra la parte stampata e la placca di alluminio (Figura 2).

Figura 2 – In tutte le prove sperimentali, la rottura è avvenuta all’interfaccia distale tra placca di alluminio e invasatura stampata in 3D.

Era evidente che il processo produttivo dovesse essere migliorato. Dal momento che le prove sperimentali sono costose e richiedono parecchio tempo, abbiamo deciso di usare un modello agli Elementi Finiti per migliorare il design. Lo scopo di questo studio è stato quello di creare un modello agli elementi finiti in Abaqus che fosse in grado di analizzare la distribuzione degli sforzi sotto le stesse condizioni di carico prescritte dalla norma, e usarlo per confrontare le seguenti situazioni:

- geometria di riferimento con materiali diversi; - geometrie migliorate della parte distale dell’invasatura, a parità di materiali; - geometria migliorata della placca, a parità di materiali.

Materiali e Metodi I materiali e i metodi che si sono resi necessari per raggiungere i nostri obiettivi sono riassunti nello schema di Figura 3 dalla quale si possono individuare tre macro-blocchi evidenziati in diversi colori. In blu, si riportano le fasi principali che costituiscono il metodo innovativo di realizzazione di invasature che ci siamo prefissati di migliorare (informazioni di contesto). In verde, si riportano i materiali e i metodi che abbiamo usato per caratterizzare il materiale più resistente tra quelli usati durante le prove sperimentali. In arancione, si riportano i materiali e i metodi che abbiamo usato per implementare le prove “virtuali” sulle invasature in Abaqus. Le diverse fasi di materiali e metodi che descriviamo qui sotto si riferiscono alla numerazione dei blocchi di Figura 3Figure 3.

Figura 3 – Abstract visual. In blu: le fasi del metodo innovative che deve essere migliorato. In verde: i materiali e i metodi per la caratterizzazione dei materiali. In arancione: I materiali e I metodi per l’implementazione delle prove sulle invasature in Abaqus.

XVI

Materiali e Metodi – Caratterizzazione del Materiale Tra i materiali usati per stampare le invasature, è stato selezionato quello che dalle prove sperimentali risultava avere le proprietà meccaniche più alte, ed è stato caratterizzato secondo le norme ISO 527- 1 e ISO 527-2. Il materiale migliore si è rivelato il Carbonium Nylon, un composito costituito da una matrice di Nylon (PA12) rinforzata da fibre di carbonio (1).

Per ottenere risultati unificati, i provini sono stati disegnati con la stessa forma a osso di cane con dimensioni in accordo con la norma (2).

A causa della natura igroscopica del Nylon, il filamento di Carbonium Nylon è stato deumidificato prima di stampare metà dei provini, in modo da poter confrontare provini stampati con materiale deumidificato con quelli stampati con materiale non deumidificato (3). I provini sono stati stampati usando una stampante 3D FFF (DELTA WASP 4070 Industrial). Inoltre, per studiare l’influenza della percentuale di riempimento, metà provini sono stati stampati con un riempimento a nido d’ape del 50% (stesse condizioni delle invasature stampate in 3D) e metà con un riempimento a nido d’ape del 100% (4).

Successivamente, secondo la norma ISO 527-1, i provini sono stati sottoposti a prove statiche di trazione uniassiale (5). Infine, secondo le indicazioni di entrambe le norme ISO 527-1 and 2, sono state calcolate (6) e confrontate (7) le proprietà meccaniche dei provini testati.

Materiali e Metodi – Processo CAD/FEM Abaqus è il programma scelto per attuare un metodo per realizzare analisi agli elementi finiti su invasature transtibiali secondo la condizione di prova II della norma ISO 10328:2016.

A causa delle limitate risorse computazionali a disposizione (16GB di RAM), l’ottimizzazione del modello ha rappresentato il filo conduttore di tutto il processo. Per questo motivo, la simulazione è stata confinata solamente alla parte più critica, ovvero la parte distale dell’invasatura e la pacca inglobata al suo interno.

Per permettere di modificare in maniera agevole ed efficiente tanto la geometria quanto il materiale, la parte distale dell’invasatura, la placca e il setup di Willow Wood sono stati disegnati direttamente nell’ambiente CAD di Abaqus (Sketch module) (8).

XVII

Figura 4 – Parte distale dell’invasatura, placca e setup di prova di Willow Wood. La parte dell’invsatura dove viene inglobata la placca è anche detta “alloggiamento”.

Una volta creato il modello, è stata creata anche una mesh del modello usando elementi esaedrici lineari con integrazione ridotta. Data la complessità della geometria, questa scelta ha comportato un largo uso di partizioni (9). Per trovare la densità di mesh ottimale, è stata condotta un’analisi di sensitività su una versione semplificata seppur rappresentativa del modello. Tale modello semplificato ha permesso anche di confrontare i risultati (sforzi) computazionali con quelli analitici, e quindi di calcolarne l’accuratezza (10).

A ciascuna regione del modello sono state assegnate le proprietà di un certo materiale (11):

- alla parte distale dell’invasatura: materiale isotropo elastoplastico basato sulla caratterizzazione del materiale Carbonium Nylon; per salvaguardare risorse computazionali, è stata fatta l’ipotesi di isotropia, seppure il modello di riempimento a nido d’ape sia fortemente anisotropo e presenti percentuali variabili di spazi vuoti. - placca e setup di prova: materiali isotropo elastico basato sulle proprietà meccaniche (da letteratura) dell’alluminio Al-6061-T6; il comportamento plastico, seppur presente nella realtà, è stato trascurato dal momento che questo alluminio è molto più rigido e resistente del Carbonium Nylon.

Sono state definite le interazioni tra la parte distale dell’invasatura e la placca (12) e sono stati applicati i carichi e le condizioni al contorno in modo da riprodurre fedelmente la condizione di prova II descritta dalla norma (13). In particolare, sono stati applicati livelli di carico crescenti corrispondenti ai livelli di carico P3 (2790N), P4 (3623N), P5 (4025N), P6 (4425N). Ricordiamo che P6 rappresenta la soglia che segna il superamento della prova statica. Infine, sono state condotte analisi statiche su diversi modelli (14), riassunti in Tabella 1:

XVIII

Tabella 1 – Diversi modelli realizzati in Abaqus.

Geometria Modello Materiale Invasatura distale Placca Modello di Deumidificato con Riferimento Riferimento Riferimento 100% riempimento Spessore maggiore in direzione antero-posteriore Modello 1 Riferimento Come sopra (direzione di massimo momento flettente) Spessore uniforme e maggiore Modello 2 Riferimento Come sopra in tutte le direzioni Riferimento con raggio di raccordo Modello 3 Riferimento Come sopra aumentato (da 0.1 a 0.3mm) Nuova geometria con raggio di Modello 4 Riferimento Come sopra raccordo molto grande (3mm) Deumidificato con Modello 5 Riferimento Riferimento 50% riempimento

Figura 5 – Geometria della parte distale dell’invasatura che funge da alloggiamento del connettore per diversi modelli. Il Modello 3 ha la stessa geometria del Modello di Riferimento con maggior raggio di raccordo della placca. il Modello 5 ha stessa geometria del Modello di Riferimento.

Dopo aver eseguito delle simulazioni agli elementi finiti, è stata analizzata la distribuzione di sforzo ai diversi livelli di carico (P3, P4, P5 e P6) nella regione di interesse, che è stata individuata come la porzione della parte distale dell’invasatura che alloggia la placca. Sono stati analizzate sia la distribuzione di sforzo di Von Mises che quella dello sforzo Principale Massimo, perché il materiale assegnato alla parte distale dell’invasatura (che si rifà al Carbonium Nylon) non può essere catalogato né come puramente duttile né come puramente fragile (15).

Risultati – Caratterizzazione del Materiale Dai risultati della caratterizzazione del materiale, abbiamo trovato che:

XIX

- I provini stampati con materiale non deumidificato hanno subito una marcata dilatazione, che ha portato le dimensioni dei provini al di fuori delle tolleranze prescritte dalla norma ISO 527-2. A causa di questo comportamento, abbiamo dovuto escludere i suddetti provini dalle prove statiche. - I provini stampati con materiale deumidificato hanno mantenuto dimensioni stabili anche a lungo termine (fino a due mesi dopo la stampa). - I provini stampati con materiale deumidificato sia con riempimento al 100% che con riempimento al 50% hanno mostrato un comportamento elastoplastico che non può essere catalogato né come puramente duttile, né come puramente fragile, dal momento che campo di deformazione plastica si aggira intorno al 3% in entrambi i casi. - I provini stampati con materiale deumidificato e riempimento al 100% sono risultati più rigidi e resistenti di quelli stampati con 50% di riempimento.

Risultati – Processo CAD/FEM Mesh La mesh che abbiamo creato ha mostrato alcuni elementi distorti nella regione di interesse che non possono essere eliminati con la presente densità di mesh.

Analisi statica Analizzando la distribuzione di sforzo nella regione di interesse al livello di carico P5, abbiamo osservato che nessun modello ha mostrato valori di sforzo più alti dello sforzo di rottura. Ciononostante, tutti i modelli, eccetto il Modello 4, hanno mostrato un numero limitato di nodi per cui lo sforzo di Von Mises ha superato lo sforzo di snervamento. I risultati sono riassunti nelle due tabelle e confrontati con i valori del Modello di Riferimento e dei valori di sforzo limite del materiale.

Tabella 2 – Distribuzione degli sforzi di Von Mises nella regione di interesse al livello di carico P6.

Nodi al di sopra Sforzo di Sforzo di Von Mises dello snervamento snervamento del materiale Modello del materiale Valore % dal % dallo % dal % di [MPa] massimo Modello di snervamento Modello di nodi [MPa] Riferimento Riferimento Modello di 37.48 51.86 100% 138% 0.18 100% Riferimento Modello 1 37.48 51.89 100% 138% 0.18 100% Modello 2 37.48 49.71 96% 133% 0.29 161% Modello 3 37.48 44.9 87% 120% 0.13 72% Modello 4 37.48 29.76 57% 79% 0 0% Modello 5 23.23 42.58 n.a. 114% 1.96 1089%

Tabella 3 – Distribuzione degli sforzi Principali Massimi nella regione di interesse al livello di carico P6.

Nodi al di sopra dello Sforzo di Sforzo Principale Massimo sforzo di rottura del rottura materiale Modello del Valore % dal % dallo % dal materiale % di massimo Modello di sforzo di Modello di [MPa] nodi [MPa] Riferimento rottura Riferimento Modello di 58.34 28.08 100% 48% 0 n.a. Riferimento Modello 1 58.34 27.54 98% 47% 0 n.a. Modello 2 58.34 36.66 131% 63% 0 n.a. Modello 3 58.34 32.8 117% 56% 0 n.a. Modello 4 58.34 26.67 95% 46% 0 n.a. Modello 5 42.35 25.96 n.a. 44% 0 n.a.

Discussioni e Conclusioni – Caratterizzazione del Materiale Dai risultati della caratterizzazione meccanica del materiale, possiamo concludere che:

- È essenziale deumidificare il filamento di Carbonium Nylon prima di stampare la parte. - Gli effetti della deumidificazione si mantengono inalterati nel lungo termine. - Una percentuale di riempimento maggiore comporta proprietà meccaniche più alte, in accordo con i dati in letteratura.

Discussioni e Conclusioni – Processo CAD/FEM Mesh La distorsione della mesh nella regione di interesse non sembra rappresentare un problema, infatti in questa zona la distribuzione di sforzo ha contorni omogenei.

Static analysis Conclusioni generali:

- La distribuzione di sforzo sia di Von Mises che Principale Massimo è asimmetrica. In particolare, quella dello sforzo Principale Massimo può essere approssimata da una distribuzione di Poisson, per cui il momento flettente agente in questa zona non è puro. - Per entrambe le distribuzioni, esistono pochi nodi all’interno della regione di interesse che mostrano valori di picco, mentre la maggior parte dei nodi mostra valori attorno al valor medio di sforzo. - Lo sforzo di Von Mises mostra valori più alti rispetto a quelli di sforzo Principale Massimo. Questo significa che è presente una componente di compressione non trascurabile nella

XXI

regione di interesse, probabilmente dovuta al fatto che la placca indenta la cavità dell’invasatura quando quest’ultima viene tirata.

Considerazioni specifiche per ciascun modello:

- Nel Modello di Riferimento, la percentuale di nodi per cui lo sforzo di Von Mises supera lo snervamento è trascurabile (sotto lo 0.2%). Di conseguenza, è improbabile che il pezzo si snervi. Anche punti localizzati di snervamento sono improbabili, perché gli eventuali nodi plasticizzati possono fare affidamento sul supporto elastico dato dai nodi delle zone circostanti, il che potrebbe rappresentare in maniera affidabile il caso di materiale al 100% di riempimento. Un riempimento del 100% significa infatti che la parte stampata ha una densità di materiale del 100%, senza spazi vuoti che potrebbero portare alla propagazione di cricche. - Il Modello 1 e il Modello 2 non hanno mostrato alcun miglioramento dal modello di riferimento, né in termini di valore massimo di sforzo né in termini di concentrazione di sforzo. Inoltre, il Modello 2 provoca un cambiamento nella distribuzione degli sforzi che producono minori compressioni ma più alte trazioni. Per cui, se il Modello 1 può essere considerato equivalente al Modello di Riferimento, il Modello 2 è da considerarsi peggiore e deve essere escluso. In ogni caso, in generale possiamo concludere modifiche alla geometria della parte distale dell’invasatura che non comprendono il solco per il connettore non sono producono miglioramenti nella distribuzione di sforzo. - Nel Modello 3, la leggera diminuzione dei valori di sforzo di Von Mises conferma l’impatto positivo dato dal maggior raggio di raccordo nel ridurre l’effetto di intaglio. Tuttavia, l’aumento dello sforzo Principale Massimo denota un aumento degli sforzi a trazione, per cui abbiamo deciso di ridisegnare completamente la placca. - Il Modello 4 ha un comportamento meccanico notevolmente migliore rispetto a tutti gli altri modelli, con valori di sforzo molto bassi e sempre al di sotto dei limiti del materiale. In particolare: o il valore massimo di sforzo di Von Mises è diminuito al 57% del valore di riferimento e al 79% dello snervamento; o il valore massimo di sforzo Principale Massimo è diminuito al 96% del valore di riferimento e al 46% dello sforzo di rottura. Possiamo concludere che lo snervamento non è possibile. - Nel Modello 5 è stato simulato un materiale diverso rispetto a quello di riferimento. Lo sforzo massimo di Von Mises supera del 14% lo snervamento in quasi 2% dei nodi all’interno della regione di interesse. Il rischio di snervamento non può essere trascurato, a maggior ragione considerando che in questo caso i nodi plasticizzati non possono fare affidamento sul

XXII

supporto elastico dei nodi circostanti, dal momento che i vuoti all’interno del modello a nido d’ape con 50% di riempimento possono portare alla propagazione di cricche e quindi al fallimento della struttura. Questo comportamento è confermato dal fatto che le invasature in Carbonium Nylon con 50% di riempimento non hanno superato le prove sperimentali.

Figura 6 – Distribuzione dello sforzo di Von Mises (colonna di sinistra) e dello sforzo Principale Massimo (colonna di destra) al livello di carico P6, per il Modello di Riferimento (prima riga) e per il Modello 4 (seconda riga).

Conclusioni Generali e Spunti Futuri Questo studio propone una metodologia per migliorare il design della parte distale dell’invasatura e della placca di collegamento basandosi su una modellizzazione parametrica, analisi agli elementi finiti e condizionamento del materiale. Dai risultati ottenuti, possiamo concludere che la geometria di riferimento della parte distale dell’invasatura e della placca potrebbero sopportare i carichi prescritti dalla norma (P6) solo utilizzando come materiale Carbonium Nylon deumidificato e con 100% di riempimento. Una geometria migliorata della placca (Modello 4) ridurrebbe gli sforzi al 57% del riferimento.

Tappe future si dovrebbero focalizzare, primo, sulla validazione dei risultati computazionali tramite prove sperimentali e, secondo, sull’ottimizzazione della percentuale di riempimento per la geometria migliorata (Modello 4).

XXIII

1 Introduction

1.1 Objective of the master’s thesis The study described in this master’s thesis has been carried out in collaboration with INAIL Prostheses Center in Vigorso di Budrio (BO), which is a center of research and application of prostheses and orthoses.

Recently, the Prostheses Center developed an innovative method based on Computer-Aided Modelling (CAD) and 3D printing for producing prosthetic transtibial sockets, aimed to replace the traditional method, which is heavily based on manual phases. Nevertheless, the sockets manufactured using this innovative method do not show sufficient mechanical properties, as evidenced by the experimental tests.

The objective of this study is to improve the design of the sockets produced with this new method, by means of a parametric modelling, Finite Element Analysis and material conditioning.

1.2 INAIL Prostheses Center INAIL, the National Institute for Insurance against Accidents at Work, is a public non-profit entity safeguarding workers against physical injuries and occupational diseases. The INAIL Prostheses Center in Vigorso di Budrio (BO)1 was founded in 1961 and it is a company certified ISO 9001:2015. It was defined a “center of research and application of prostheses and orthoses” and operates in three fields:

1. Manufacturing and supplying of prostheses and orthoses; 2. Rehabilitation and training in the use prostheses and orthoses; 3. Research and development of new technologies aimed to manufacture state-of-the-art prostheses.

INAIL Prostheses Center is a very complex and articulated structure that admits patients affected by several kinds of physical disabilities, from mild disabilities to the most severe ones, independently of the cause the generated them (traumatic accidents, diseases, congenital deformities, etc.). This Center assists both patients who were injured on the job as well as patients who were injured elsewhere, independently of their nationality. For the first group of patients (injured on the job), INAIL covers all the expenses related to prostheses manufacturing, supply, rehabilitation and training, whereas for the latter group of patients (injured elsewhere, not on the job), all the expenses

1 https://www.inail.it/cs/internet/attivita/prestazioni/centro-protesi-vigorso-di-budrio.html

1 are covered by National Healthcare System (NHS), which in Italy is called Sistema Sanitario Nazionale (SSN).

The Center is organized in production lines, depending on the type of prosthesis or orthosis that has to be manufactured, namely: upper limb prostheses, lower limb prostheses, which is furtherly divided into transfemoral (above-knee) and transtibial (below-knee) prostheses, silicon prostheses, orthoses and orthopaedic footwear.

We are interested in the production line of lower limb transtibial prostheses, which on average manufactures 1200 prostheses a year.

1.3 Lower limb amputations and prostheses An amputation is the surgical removal of a part of the body. Usually, we distinguish between lower limb and upper limb amputations. An orthopedic prosthesis is a device that replaces lost anatomical parts or functions of the neuroskeletomotor system.

In Italy, according the FIOTO (Federazione Italiana degli Opeatori in Tecniche Ortopediche), the Italian Federation of Prosthetists and Orthotists, each year there are 10000 new amputees, 1000 of which are transtibial amputees [1]. Only in 2005, the Italy registered 621 hospitalizations that required transfemoral amputation and 1143 that led to transtibial amputations [2].

Lower limb amputations may be performed at different levels (Figure 1.1), depending on the medical situation of the patient. The basic principle is to always preserve as much of the limb as possible, because with increasing amputation level the standing surface becomes smaller. In addition, a longer residual limb allows a greater lever arm and a better muscular guidance of the limb during use of the prosthesis [3].

Figure 1.1 – Levels of lower limb amputations.

For the various levels of amputations, there are specific prosthetic devices that are supplied to the patient. In particular, for transtibial (below knee) amputations, the anatomical part that needs to be replaced by a prosthetic device is the ankle-foot complex.

In this work, we will focus on transtibial prostheses.

1.3.1 Transtibial prosthesis A modular transtibial prosthesis consists of the following components (Figure 1.2) [4],[5]:

1. The socket is the rigid frame that contains the stump and connects it to the other components of the prosthesis, allowing the patient to operate the prosthesis, providing load transfer during foot support and suspension during foot lift. Since it has to intimately fit the residual limb, the socket is the only component of the prosthesis that is custom made. For these reasons, it can be considered the fundamental and most critical component of a prosthesis. 2. Usually, between the socket and the stump there is an additional interface called the liner, a sort of sock that tightly wraps the stump and acts as padding. The liner can come in different materials and thicknesses, and it has the effect of reducing skin irritation and pain due to stump-socket friction, to make the interface more comfortable and to provide suspension. 3. The pylon is the metal component that connects the socket to the prosthetic foot (or ankle- foot complex) and can be considered the equivalent of the fibula in nonamputees. It is a hollow tube with a standardized diameter of 34푚푚 usually made of titanium or stainless steel. 4. The foot is the component that provides base of support, transfers the patient’s weight to the ground and adapts to different ground surfaces. Usually, it is designed in a way that its posterior part can absorb impacts during heel-contact, whereas its anterior part can release the energy in the push-off phase of walking. 5. The fastening components are standardized elements that allow to connect the socket to the pylon and the pylon to the foot (or ankle-foot complex). In particular, the socket-to-pylon attachment block consists of a bolted joint: a connector with four M6 tapered holes (socket attachment adapter) is fixed to the socket distal end through a resin foam; a metal connector with four through holes and a pyramidal end (4-hole pyramid adapter) is secured in the 4- hole pyramid receiver adapter of the pylon through four pressure screws; finally, the socket connector and the pylon connector are attached through four M6 countersunk screws.

Figure 1.2 – Modular transtibial (below knee) prosthesis.

2 Background

2.1 Traditional method for socket production Transtibial sockets are traditionally manufactured using a complex production process, that consists of eight steps and it involves specialized prosthetists. In particular, step 4 (plastic lamination), 5 (cleaning), 6 (fastening of the socket attachment block to the socket) and 8 (finishing lamination) re completely manual, and rely on the manual ability of the prosthetists, who are full-fledged skilled craftsmen. Note that these phases require specific equipment and involve the use of toxic agents.

We summarized the total steps of this traditional process in Figure 2.1.

The sockets that are produced using this method are very strong, durable, robust, require inexpensive equipment and represent a great trade-off in terms of mechanical strength, elasticity and lightweight. However, the entire process is very complex, labor and cost intensive as well as time consuming: it can take up to several days to manufacture a socket, due to the many steps and specialized workers involved. In addition, the number of prosthetists is scarce compared to the number of lower limb amputees, contributing to extending the overall manufacturing time [6]. Therefore, an objective and systematic framework for fabricating prosthetic sockets would help improve efﬁciency in prosthetic care, reduce time and cost, and potentially enhance comfort and fit (for example, reducing the overall weight).

Figure 2.1 – Traditional method for socket production

2.2 Innovative method for socket production Recently, the Prostheses Center in collaboration with Politecnico di Milano developed an alternative and innovative method which exploits CAD and 3D printing, aimed to reduce the total number of steps, in particular the manual ones. This study [7] investigated the possibility of producing sockets using the Fused Deposition Modelling (FDM) 3D printing technique, which is a versatile manufacturing technology that could have several advantages over the traditional method. This innovative method is summarized in Figure 2.2.

Figure 2.2 – Innovative method for socket production

The innovative method allows to reduce the total number of steps from eight to only three, and, in particular, to replace three of the four manual phases (plastic lamination, cleaning and finishing lamination) with one expanded CAD modelling phase followed by FDM 3D printing. In addition, this process requires to embed a smaller and lighter aluminum connector into the distal part of the 3D printed socket, as the fastening component with the remaining modules (pylon and foot) of the prosthesis. This innovation allows to partly replace the manual fastening of the socket attachment block to the socket.

The three steps can be summarized as follows:

1. Acquisition of the morphology of the patient’s stump, which can be carried out in the following ways: a. realization of a plaster cast on the stump, followed by a 3D scan of the internal surface of the plaster cast; b. 3D scan of the internal surface of an existing socket. 2. Expanded CAD modelling of the socket, which now consist of two sub-phases: a. modelling of the scanned surface, to reconstruct the internal surface of the socket to best fit the morphology of the stump; b. creation of a groove in the distal part of the socket having the same shape of the connector; the part of the distal socket comprising the groove will be referred to as housing. 3. FDM 3D printing of the final socket with integration of the connector. To incorporate the connector, the printer is paused after it creates the groove to allow manual insertion of the connector, and is then restarted, to continue printing on top of the connector, embedding it inside the socket.

The CAD software programs that are used to model the entire socket are three: Autodesk Fusion 360, Autodesk Meshmixer and Blender.

Figure 2.3 shows above (a,b) the output of the 3D scan, namely the internal surface of the plaster cast or of an existing socket in digital format, and below (c,d) the output of the CAD modelling phase, namely the final socket geometry. Figure 2.4 shows that the distal end of the final socket, that we will call housing, incorporates a groove having a shape complementary to the attachment connector.

Figure 2.3 – Anterior (a) and posterior (b) view of .stl file of internal surface of plaster cast or socket of a right leg stump obtained with a 3D scan; anterior (c) and posterior (d) view of .stl file of socket geometry. The pink frame in figures c and d highlights the area of the housing, that incorporates a groove complementary to the attachment connector.

Figure 2.4 – Socket geometry: (a) the proximal part of the socket comprises the inner socket, which is the hollow part of the socket that hosts the stump; the distal part of the socket is solid, except for the housing, that incorporates a groove complementary to the attachment connector. (b) the bottom surface of the socket has four through holes for accessing the connector that will be embedded within the socket.

The attachment connector that is used in this process is the CD103AF Alignable Connector by CoyoteDesign, that we will simply call Coyote connector or CC for short.

Figure 2.5 – Geometry of coyote connector created in Fusion 360.

Figure 2.6 – Geometry of housing created in Fusion 360.

2.3 3D printing After being modelled in the CAD environment, the socket is manufactured using the FFF (Fused Filament Fabrication), also known as FDM (Fused Deposition Modelling), 3D printing technology.

2.3.1 FFF 3D printing technology In general, 3D printing is a form of additive manufacturing, i.e. creating objects by sequential layering. After creating a 3D model with a CAD program, a printable file is used to create a layer design which is printed afterwards [8].

The Fused Deposition Modelling technique is a relatively new method of rapid prototyping which works by laying down consecutive layers of extruded thermoplastic material at high temperatures, allowing adjacent layers to cool and bond together before the next layer is deposited. 3D models, coming from a CAD environment, are transformed into g-code, essentially a set of instructions which positions the motors precisely and generates the required volume extrusions to create the part [9].

The FDM technique uses a continuous filament of thermoplastic material with an appropriate diameter (filament diameter). The filament is fed from a large filament coil and is led through a moving, heated printer extruder head, thanks to a system comprising a stepper motor and gears; the extruder uses torque and a pinch system to feed and retract the filament by precise amounts. Once in

8 the extruder, the filament is fed through the heater block (hot end), comprising a thermistor or a thermocouple, that melts the filament to a useable temperature. The semi-molten filament is forced out the heated nozzle at a smaller diameter, which depends on the nozzle diameter, and is deposited in the form of a prescribed two-dimensional (X-Y) layer pattern onto the partially constructed part. After finishing a layer, the printer head moves vertically in the Z direction by an increment equal to the fused filament height to begin depositing a new layer on top of the previous one. During the print, the constructed part adheres on a heated platform called the printer plate or bed, that maintains an appropriate temperature to allow efficient bond between the consecutive layers of material. After a period of time, which depends on the volume of printed part, the head will have deposited a full physical representation of the original CAD file. The model is complete and requires no hardening. In fact, as it is laid down, the material quickly solidifies [10],[8],[9],[11],[12]. Once the model is complete, it is removed from the building chamber and cleaned. FDM creates tough parts that are ideal for functional usage.

Figure 2.7 – Extrusion system for FFF 3D printing technology.

2.3.2 Slicer The software that controls the 3D printer, called slicer or slicing software, allows to translate the CAD 3D model (in .stl format) into a g-code containing the set of instructions for the 3D printer motor. These instructions depend on the printing parameters that have been set in the slicer environment.

The slicer used at INAIL Prostheses Center is called Simplify3D. The printing parameters that can be assigned in Simplify3D are described in Table 2.1.

Table 2.1 – Printing parameters that can be assigned in Simplify3D.

Printing parameter Description This value must correspond to the diameter of the nozzle that is Nozzle diameter [mm] mounted on the 3D printer, which most commonly is 0.4푚푚, 0.7푚푚 or 1.2푚푚. It is the width of fused material that is deposited by the nozzle; it has to be equal or greater to the nozzle diameter to improve adhesion, without Extrusion width [mm] being too high. For a nozzle diameter of 0.7푚푚, the extrusion width is usually set around 0.84푚푚. It is the height of fused material that is deposited by the nozzle; this parameter, together with the extrusion width, strongly influences the Layer height [mm] adhesion properties. For a nozzle diameter of 0.7푚푚, the layer height is usually set around 0.2푚푚. It refers to the pattern that is used to deposit the fused material and fill the solid parts of the 3D printed object. The pattern options provided by Simplify3D are: Rectilinear, Grid, Triangular, Wiggle, Fast Infill pattern Honeycomb and Full Honeycomb (Figure 2.8). Two different infill patterns are set, one for the outside perimeter shell and one of the interior area of the printed part. Some infill patterns, as rectilinear, grid or triangular, can be printed with different orientation with respect to the build table cartesian Infill angle [°] reference system. The infill angle refers to the direction of raster relative to the X axis of the build table. It refers to the amount of material deposited in the printed pattern and represents the density of the pattern. It goes from 0% for completely hollow structures to 100% for entirely solid parts. In general, the Infill percentage [%] higher the infill percentage, the stronger the 3D print, but the longer become the printing time and the weight of the printed part. Varying combinations of infill percentage and infill pattern can affect strength, material usage as well as print time. Number of The fused filament is initially deposited along the perimeter of the 3D outline/perimeter printed part, and only after that it fills its internal area. The number of shells perimeter shells can be assigned. Before depositing the layers with the prescribed infill pattern and Number of top and percentage, the printer head deposits a certain number of solid layers, bottom solid layers which are basically rectilinear layers with 100% infill. Their number can be set by the user. It refers to the point on the growing part’s perimeter that will be used by the nozzle as starting point at each layer deposition. If this point is Start points selected randomly and is not always close to the same location, there risk of creating a crack in the part is lower, reducing the probability of delamination.

It is temperature that is reached by the nozzle to allow filament fusion and extrusion. The nozzle is heated by the heater block up to or above Nozzle temperature the thermoplastic material melting temperature. At this temperature, [°C] the filament fuses and becomes molten or semi-molten, transitioning from solid to plastic state, and its shape can be easily modelled by extrusion. It is the temperature of the plate in the build chamber where the fused Plate temperature [°C] material is deposited. As mentioned above, its role is very important to guarantee a correct adhesion between the deposited layers. It controls the power of the cooling system that is used to dissipate the heater block temperature in the barrel to avoid solidification and clog Cooling fan [%] formation in this region of the cartridge. This parameter is very critical, because for some materials the cooling fan could cause solidification in the nozzle as well, which is counterproductive. It is the velocity of fused filament deposition on the plate/growing part. Besides influencing layer adhesion, its value strongly affects the Printing speed [mm/s] precision of the printed part. The fastest the speed, the less the precision of the part. The first layer is the most critical one of the entire 3D printed part, because it is the only layer adhering on the printing plate rather than on First layer settings the growing part; for this reason, there are usually separate settings for this layer in terms of extrusion width, layer height, nozzle and plate temperature (usually higher), and printing speed (usually lower). They are additional perimeter shells of materials that are deposited in the first layer(s) around the printed part and are usually removed once Skirts or brims the print is complete. They are used to improve the adhesion of the first layer. If the geometry of the 3D printed part has regions that are unsupported, Support material in those areas it is possible to print support material, usually at a lower generation density or of a different material, that will be removed once the part is complete. These settings allow to control the movement in the X,Y and Z Retraction settings direction of the printer head in terms of distance and velocity. Machine type It allows to define the machine type as well as its geometrical definition information (build volume and origin offset).

Figure 2.8 – Infill patterns offered by Simplify3D.

Figure 2.9 displays the 3D printing preview of the socket. This preview offers a visual anticipation of the g-code that will be generated by simulating layer by layer the behavior of the print head and the path of the fused filament. Note that the distal end of the socket, the housing, encloses a groove having a shape that is complementary to the attachment connector. Once the groove has been created and its last layer has been deposited (layer 89), the printing process will be paused, allowing the user to manually insert the connector inside the groove. At this point, the print is restarted, and the print head deposits a bridge layer (layer 90) on top of the connector, embedding it inside the groove. Then, it continues printing the consecutive layers to form the socket. As layers are deposited, the socket grows. Around layer 300, the inner part of the socket, that will be in contact with the stump, starts forming. At layer 1512, the socket is complete.

Figure 2.9 – Preview of 3D printing of the distal end of the socket at different layers.

Figure 2.10 – Preview of 3D printing of the socket at different layers.

Before sending it to the 3D printer, the g-code was edited to include instructions that would make the 3D printer pause after creating the groove. The “pausing code” had to be pasted inside the g-code generated by Simplify3D in between the instructions for building layer 89 (last layer of the groove) and layer 90 (bridge layer).

2.3.3 3D printer There 3D printer used at INAIL Prostheses Center is the DELTA WASP 4070 Industrial with the DELTA WASP ZEN Dual Extruder (Figure 2.11) supplied by the Italian company WASP2.

Figure 2.11 – DELTA WASP 4070 Industrial (left) and DELTA WASP ZEN Dual Extruder (right).

This 3D printer uses FDM technology described in chapter 2.3.1. This printer has a very spacious thermally insulated build volume (∅400 × ℎ700푚푚), making it possible to print large volumes. In particular, it comfortably accommodates a socket. The growing part is deposited on a heated platform (printer plate) made of rectified aluminum that can reach up to 120°퐶, contributing to maintaining the chamber environment at an appropriate temperature to allow efficient bond between the fused filament layers of material.

The extrusion system (DELTA WASP ZEN Dual Extruder) heats the filament up to its melting temperature and deposits it in a molten state and with a smaller diameter on the growing part that adheres on the printing plate.

2 https://www.3dwasp.com/

2.3.4 Printing the socket The g-code generated by the slicer was sent to the WASP 3D printer and the socket was printed layer by layer. According to the g-code editing, the printer paused after having deposited layer 89 (last layer of the groove). At this point, the user inserted the connector inside the groove and restarted the printing process, which continued unstopped until the socket was complete. Figure 2.12 displays three moments of the printing process of the housing. Note that the connector lay on the printer plate until it was placed inside the groove. By doing so, once it was placed in the groove, it had the same temperature of the already-deposited part, avoiding temperature drops and allowing a better adhesion with the bridge layer that was deposited on its top.

Figure 2.12 – 3D printing of socket housing with introduction of connector inside the groove. The socket is of Carbonium Nylon.

Figure 2.13 – 3D printing of the socket layer by layer. The socket is in Carbonium Nylon.

Figure 2.14 – Final socket printed with the WASP printer.

Figure 2.15 displays the final 3D printed socket.

Figure 2.15 – Final socket: anterior (a), medial (b), posterior (b), lateral (d) and bottom (e, f) view. From Figure (e) it is possible to see the four through holes in the distal end of the socket that allow accessing the four threaded holes of the Coyote connector (visible in blue). From Figure (f) it is possible to see the leg- pylon connector attached to the Coyote connector through four M6 countersunk screws.

The attachment block that was used in this process is the CD103AF Alignable Connector by CoyoteDesign3, that we will simply call Coyote connector or CC for short (Figure 2.16).

3 https://www.coyotedesign.com/

Figure 2.16 – Coyote Connector (CC), i.e. CD103AF Alignable Connector by CoyoteDesign used instead of wood attachment block for socket-to-leg joint.

2.4 Static test Four sockets were tested:

• three of them were realized with the innovative method, using the three different materials, described in chapter 2.4.3; • one of them was realized with the traditional method used at INAIL Prostheses Center.

The static tests were conducted at the Ohio Willow Wood Company4 using a test setup built by the same company and described in chapter 2.4.2. This setup allowed these sockets to undergo a static structural test according to the ISO 10328:2016 standard.

To obtain unified and comparable results, the sockets were printed with the same geometry, described in chapter 2.4.1. For the same reason, all sockets were printed with the same infill pattern (fast honeycomb) and percentage (50%), as described in described in chapter 2.4.3.

2.4.1 Socket geometry The geometry of the socket that was used for static testing derived from a generic shape of residual limb, which is not an actual 3D scan of a patient’s plaster cast nor of a patient’s existing socket, but rather a generic transtibial residual limb template representing the anthropometric 98th percentile U.S. male population of transtibial amputees, as defined by a U.S. national survey of 2008. From this survey, the Ohio Willow Wood Company extrapolated two measures, i.e. a circumference at the patellar tendon bar of 52.4푐푚 and a length from the patellar tendon bar to the distal end of 19.2푐푚, from which they reconstructed the geometry of the template. From Figure 2.17, we can indeed see that the template does not have the typical anthropological characteristics of a transtibial stump. The increased height and large dimensional socket provided a worst-case scenario for sockets that underwent static testing and were subjected to bending.

4 https://www.willowwoodco.com/

Figure 2.17 – .stl file of internal surface of Willow Wood socket of a left leg stump model. In Figure are displayed the anterior (a), medial (b), posterior (c), lateral (d), top (e) and bottom (f) view. From view (e), it is possible to see that the scan is only a shell, having the same surface morphology of the stump.

Figure 2.18 – Final socket: anterior (a), medial (b), posterior (b), lateral (d), top (e) and bottom (f) view.

2.4.2 Test setup Three sockets were tested according to the international standard BS EN ISO 10328:2016 – Prosthetics — Structural testing of lower-limb prostheses — Requirements and test methods [13]. This standard is suitable for the assessment of the strength conformity of lower limb prosthetic devices or structures, and it specifies procedures for static and cyclic strength tests during the stance phase of walking. The tests apply to several specific types of prostheses, among which transtibial ones, and to many part structures and components of these prostheses. However, this standard does not apply to socket testing.

Since neither this standard nor other standards address directly the matter of socket testing, the Ohio Willow Wood Company (WW) built the standard test setup described in ISO 10328:2016 and made it suitable for socket testing [14].

According to the standard, WW conducted a principal static ultimate strength test. The standard allowed to carry out this test in two different setup configurations that refer to two different loading conditions of the prosthesis. Both configurations relate to the two maxima occurring at different instants during the stance phase of walking, namely heel-contact (test loading condition I) and push-

17 off (test loading condition II), represented by the red arrows in Figure 2.19. These two faces are considered the worst-case scenario in terms of loading acting on the prosthesis, because the entire body weight acts on the prosthetic limb (single support).

For their setup, WW adopted test loading condition II.

Figure 2.19 – Gait cycle of right leg [15]. Test loading conditions I and II are highlighted in red.

Test loading condition II is characterized by a specific test load acting along a specific line of load application and producing compound loading. The different test loading levels that can be applied are displayed in Table 2.2.

Table 2.2 – Test loading levels in test condition I and II. This table combines the following Tables of the standard: Table 8 at page 18 and Table D.2 at page 120.

Test loading level P3 P4 P5 P6 P7 P8 Ultimate static test force [N] 2790 3623 4025 4425 4840 5250 Body mass [kg] 60 80 100 125 150 175

WW considered as test passing criterion the P6 loading level value, namely 4425N. As reported in Table 2.2, this loading level is based on locomotion data from amputees whose body mass is less than 125kg. This body mass is associated with amputees having very large residual limb dimensions,

18 as the ones of the stump template described in chapter 2.4.1 that represents the anthropometric 98th percentile U.S. male model and it is considered the worst-case scenario for pulled sockets.

The setup used for test condition II is displayed in Figure 2.20. The coordinates and distances between the reference points vary for each loading level; the ones corresponding to a P6 loading levels are reported in Table 2.4.

Figure 2.20 – Test loading condition II to left-sided test sample (left prosthetic leg), showing the coordinate

system using 푢퐵 = 0 with reference planes, reference lines, reference points and components of internal loading generated by application of the test force 퐹. This Figure comprises the following Figures of the standard: (left) Figure A.2 at page 113 and (right) Figure 11 at page 49. Legend is reported in Table 2.3.

The coordinate system used by the standard is an orthogonal reference system labeled “f-o-u”. It is equivalent to the much more used cartesian reference system “x-y-z”, where f=x, o=y and u=z. This reference system is in relation to a prosthesis that is standing on the ground in the upright position, meaning that 푢퐵 = 0 and the leg pylon is parallel and adjacent to the u axis.

1. The u-axis extends from the origin 0 of the coordinate system and passes through the effective ankle-joint center (point 3) and the effective knee-joint center (point 5). Its positive direction is upwards (in the proximal direction).

2. The o-axis extends from the origin 0 perpendicular to the u-axis and parallel to the effective

knee-joint centerline (line Ko). Its positive direction is outward (in the lateral direction). Figure 2.20 displays a left leg prosthesis, so the positive direction is directed to the left. 3. The f-axis extends from the origin 0 perpendicular to both the o-axis and u-axis. Its positive direction is forward towards the toe (in the anterior direction).

The effective ankle-joint center (point 3), displayed as a green dot in Figure 2.20 and Figure 2.21, lies: in a vertical plane passing through the longitudinal axis of the foot; in the ankle reference plane located at 푢퐴 = 80푚푚 above the bottom reference plane (horizontal plane containing point B); at a quarter of the length of the foot from the most posterior part of the foot.

Figure 2.21 – Determination of longitudinal axis of the foot (point 6), effective ankle-joint center (point 3),

effective ankle-joint centerline (line 4) and combined bottom offset 푆퐵 for test loading condition II. This Figure corresponds to Figure 4 at page 9 of the standard. Label tags have been modified with respect to the standard for the sake of clarity. Legend is reported in Table 2.3.

Table 2.3 – Legend for Figure 2.20 and Figure 2.21.

Labels Extended label Description 1 Object 1 Left leg 2 Line 2 Load line 3 Point 3 Effective ankle-joint center 4 Line 4 Effective ankle-joint centerline 5 Point 5 Effective knee-joint center 6 Line 6 Longitudinal axis of foot

PT Point T Top (proximal) load application point

PK Point K Knee load reference point

PA Point A Ankle load reference point

PB Point B Bottom (distal) load reference point

Af Line Af Ankle moment (MAf) reference line

Ao Line Ao Ankle moment (MAo) reference line

Kf Line Kf Knee moment (MKf) reference line

Ko Line Ko Knee moment (MKo) reference line and effective knee-joint centerline

hr Segment hr Heel height L Segment L Foot length

Table 2.4 – Values of offsets and lengths for a principal static strength test in test condition II with a test

loading level P6, if the prosthesis is standing on the ground (푢퐵 = 0). This table combines the following Tables of the standard: Table 5 at page 16, Table 6 and Table 7 at page 17.

Point label Offset and lengths Values [mm]

푓푇 55

표푇 -40 Point T 푢푇 650

2 2 푆푇 = √푓푇 + 표푇 68

푓퐾 72

표퐾 -35 Point K 푢퐾 500

2 2 푆퐾 = √푓퐾 + 표퐾 80

푓퐴 120

표퐴 -22 Point A 푢퐴 80

2 2 푆퐴 = √푓퐴 + 표퐴 122

푓퐵 129

표퐵 -19 Point B 푢퐵 0

2 2 푆퐵 = √푓푇 + 표푇 130

The load line is the line that passes through the top (proximal) and bottom (distal) application points, namely point T and B (red dots in Figure 2.20 and Figure 2.21). This line is not parallel to the u-axis but it is inclined in all three bi-dimensional coordinate planes: the tilt angle is 휃푓−푢 ≅ 83.5° in the f- u plane with respect to the f-axis (2.1), 휃표−푢 ≅ 88.1° in the o-u plane with respect to the o-axis (2.2), and 휃푓−표 ≅ 15.8° in the f-o plane with respect to the f-axis (2.3).

−1 푢푇 − 푢퐵 −1 650 − 0 −1 650 휃푓−푢 = tan | | = tan | | = tan ( ) ≅ 83.5° (2.1) 푓푇 − 푓퐵 55 − 129 74

−1 푢푇 − 푢퐵 −1 650 − 0 −1 650 휃표−푢 = tan | | = tan | | = tan ( ) ≅ 88.1° (2.2) 표푇 − 표퐵 −40 + 19 21

−1 표푇 − 표퐵 −1 −40 + 19 −1 21 휃푓−표 = tan | | = tan | | = tan ( ) ≅ 15.8° (2.3) 푓푇 − 푓퐵 55 − 129 74

Note that there is a mistake in Figure 2.20 (Figure A.2, page 113 of the standard): point B should have a negative o-value, according to Table 2.4 (Table 6, page 17 of the standard) but in figure it is displayed on the side of the positive o-axis. A more intuitive drawing of the setup is provided by Graebner et al. [16] and is reported in Figure 2.22 (left). In this picture point B is displayed correctly.

Figure 2.22 – Test loading condition II (left) adopted by Graebner et al. [16] and (right) modified by WillowWood to generate the worst possible moment in the socket.

Since WW uses this setup to tests sockets alone, without the leg pylon, their setup does not use the knee reference measurement (point K is not considered), and the socket is mounted directly on the ankle plane (푢퐴 = 80푚푚), so that its distal end is located in correspondence of the effective ankle- joint center (Point 3); this configuration is displayed in Figure 2.22 on the right. The right

22 configuration generates the worst possible moment in the socket, because the distance between the socket distal end and the bottom loading point (point B) is minimum and corresponds to 푢퐴 − 푢퐵 = 80푚푚. The left configuration generates a lower moment, because the distance between the socket distal end and the bottom loading plate (point B) has increased by an amount equal to the leg pylon length. The reason that causes the moment in the socket to increase with increasing distance from the bottom loading point (point B) depends on the fact that the load line is not parallel to the u-axis (and thus to the socket axis) but is rather tilted and directed along the line connecting the two loading points. Note that the application of a force having this orientation produces compound loadings in the structure that is worth investigating. For the complete analysis of internal actions please refer to Appendix A.

Figure 2.23 to Figure 2.27 display the WW setup in the three-dimensional and bi-dimensional orthogonal reference planes of the standard. This setup was created by WW using SolidWorks, and we imported it in Fusion 360.

Figure 2.23 – WW setup in f-o-u plane.

Figure 2.24 – WW setup in f-u plane (left) and in o-u plane (right).

Figure 2.25 – WW setup in f-o plane. (top) both setup horizontal plates, (bottom-left) bottom plate only, (bottom-right) top plate.

Figure 2.26 – WW setup (left) in plane parallel to setup bottom plate and (right) in plane parallel to setup top plate.

Figure 2.27 – WW setup in f-u plane. Differently from Figure 2.24, here we report measures that are useful for construction.

In WW setup the load is applied at the bottom (distal) loading point, point B, and is measured at the top (proximal) loading point, point T. The transmission of loads in these two points takes place through two spherical joints: each joint consists of a loading cylinder with a spherical head that rotates inside a cup, attached to the (top or bottom) horizontal plate. Therefore, the load lies on the line that connects the two spherical joints, as explained above.

Please note that, for sake of clarity, the two loading cylinders are not displayed in the Figures above. Figure 2.26 gives a side view of each cup attached to its horizontal plate; note that the cup’s diameter is slightly smaller than the cylinder head’s diameter, the first being 41.229푚푚 and the latter 44.450푚푚 for both the top and bottom spherical joints.

The socket was distally attached to the bottom part of the setup using a standard four-hole distal attachment plate. Its measures are reported in Figure 2.25 and Figure 2.27. The loads were transferred by a simulated residual limb model. A cone-shaped aluminum mandrel with a 5° tilt served as the core bone shape of the residual limb model. The mandrel was then casted in a polyurethane block. Finally, a 9푚푚 uniform silicone liner was interposed between the residual limb model and the prosthetic socket.

Figure 2.28 – Willow Wood setup.

푁 During static test, the load was applied with a loading rate of 250 , according to the standard 푠 requirments (Chapter 16.2.2.1.1 at page 55 of the standard). The static failure test resulted in a force- displacement curve for each socket.

2.4.3 Materials The sockets were printed with three different materials were printed. The materials were supplied by TreeD Filaments5 and were:

1. Carbonium Nylon 2. StructuraMa 3. VerumT

The first two, Carbonium Nylon and StructuraMA, are both composite materials of Nylon (Polyamide 12, PA12) reinforced with carbon fibers (CF). The exact percentage of carbon fibers is strictly confidential, but it is around 12%. Differently from Carbonium Nylon, StructuraMA also had a small percentage (2%) of elastomer; for this reason, it is labeled as “healthfil”. They have a glass transition temperature of around 98°C and a melting temperature of 178°C.

Carbon fibers are dispersed uniformly in the PA12 matrix. During the extrusion process, great shear forces are generated, and long carbon fibers are cut into several segments and dispersed in the matrix. Then, during the printing process, a huge shear rate is generated between the composite melt and the inner wall of the nozzle, resulting in the orientation of the carbon fibers along the printing direction. In addition, the carbon fibers are homogeneously distributed in the deposited material. Finally, the microstructure of the orientation state is preserved during the print because of the fast cooling speed of this composite material. Carbon fibers result perfectly embedded in the matrix: while the fibers stiffen and strengthen the composite, the matrix can easily transmit forces to the fibers and stabilizes them [17].

However, polyamides in equilibrium with the environment contain a small amount of moisture. This hygroscopic nature is attributed to the hydrogen bonds formed between water and the amide groups from the polymer chain, which turn them hydrolytically unstable. The probability of hydrolysis occurring in these materials increases with an increase in their molecular weight and processing temperature [18]. Since Carbonium Nylon and StructuraMA are extruded at very high temperatures, i.e. 240°퐶, moisture absorption could be very critical for these materials. In fact, moisture inside polyamides reduces their mechanical properties. In addition, it reduces the adhesion between the layers of fused filament during FDM printing, increasing the risk of delamination.

Differently from Carbonium Nylon and StructuraMA, VerumT is a composite material of medical- use Polycarbonate (PC) reinforced with fiberglass (FG). Its molding temperature is 143°C.

5 http://treedfilaments.com/

The printing parameters that were assigned in the slicer to the socket were different for each material and are reported in Table 2.5.

Table 2.5 – Printing parameters for sockets printed by [7]. Note that the only parameters that change for each material are the ones highlighted in blue; the other ones remain constant.

Material 1 Material 2 Material 3 Printing parameters (Carbonium Nylon) (StructuraMA) (VerumT) Nozzle diameter 0.7mm 0.7mm 0.7mm Extrusion width 0.84mm 0.84mm 0.84mm Layer height 0.2mm 0.2mm 0.2mm Internal infill pattern Fast honeycomb Fast honeycomb Fast honeycomb Internal infill percentage 50% 50% 50% External infill pattern Rectilinear Rectilinear Rectilinear Number of 2 2 2 outline/perimeter shells Number of top and 4 4 4 bottom solid layers Nozzle temperature 240°C 240°C 300°C Plate temperature 90°C 70°C 125°C Cooling fan 0% 0% 0% Printing speed 80mm/s 45mm/s 60mm/s Nozzle temperature: Nozzle temperature: 245°C First layer settings - 255°C Plate temperature: 75°C Skirts or brims 1 layer, 5 outlines, 0 offset from the part Support material No generation Retraction distance of 3mm; Extra retraction distance of 0mm; Retraction settings Retraction vertical lift of 1mm; Retraction speed of 80mm/s. DELTA WASP 4070 Industrial: Delta robot (cylindrical build volume) Machine type definition Build volume: 282.8mm in X and Y direction; 700mm in Z direction; Origin offset: 141.4mm in X and Y direction; 0mm in Z direction.

2.4.4 Static test results After printing these sockets, they were tested according to the ISO 10328:2016 standard, using the test setup described in chapter 2.4.2. These results are summarized in Table 2.6.

Table 2.6 – Static test results.

Socket type Breaking force [N] Mechanical behavior Socket made with traditional method 5738 Elastoplastic Socket made of Carbonium Nylon 2542 Elastoplastic Socket made of StructuraMA 2426 Elastoplastic Socket made of VerumT 2134 Purely elastic

From these results, it is possible to see that only the socket manufactured using the traditional method passed the test. On the contrary, all the 3D printed sockets failed the test, and in addition all of them failed before even reaching P3 loading level.

Among these materials, we decided to furtherly investigate Carbonium Nylon, as it was the socket manufactured with this material showed the highest mechanical properties and displayed an elastoplastic behavior (which ensures a safer behavior in case of failure).

The force-displacement curve for the socket made of Carbonium Nylon is displayed in Figure 2.29. As mentioned in chapter 2.4.2, the test can be considered passed if the force-displacement curve of the socket overcomes P6 loading level, represented by a horizontal line in the graph.

Figure 2.29 – Socket testing for Carbonium Nylon, displayed by the blue curve labeled as “I3”. P6 loading level is displayed by the grey horizontal line.

All the 3D printed sockets failed in the same spot. In Figure 2.31 we reported the broken socket made of Carbonium Nylon.

Figure 2.30 – Static test using WW setup: (a) the load is applied along the vertical direction (red arrow); (b) as the load increases, the bottom loading cylinder moves upward; (c) fractures initiates at the posterior distal region of the socket (highlighted by the pink arrow).

Figure 2.31 – Broken Carbonium Nylon socket.

2.5 Possible causes of failure Looking at Figure 2.31, it is possible to see that fracture took place at the lower (distal) interface between the socket housing and the connector, in the posterior region. The distal posterior region is indeed the most stressed area of the entire socket: in the distal area the bending moment is maximum, and in the posterior area the deriving stresses are of tensile nature, causing the materials fibers to stretch.

The possible causes that may have increased the stress in this area causing an early failure are the following:

1. Housing geometry The geometry of the housing is such that there is the least amount of material in the anterior and posterior sides. However, in the test configuration, the bending moment is maximum along the antero-posterior direction. Having little material in this direction may have increased the stress concentration in these two areas. In addition, this effect might have been emphasized in the posterior region, where the stresses are of tensile nature and cause the material fibers to stretch. 2. Notch effect The edges of the connector, and consequently the edges of the housing’s groove, are very sharp. These sharp edges may have caused stress concentrations due to notch effect, which might have been extremely high at the bottom interface between the housing and the connector, where the stresses were already at their maximum. 3. Connector geometry The geometry of the connector is not optimized: it has a small thickness and many curvatures, that could create problems in situations of high stresses. 4. Hygroscopic nature of polyamides Given the hygroscopic nature of polyamides, the moisture inside the filament of material could decrease the adhesion and mechanical properties the printed filament. 5. Material with 50% infill The 50% infill might have decreased the mechanical properties of the part. In fact, the empty honeycomb cells could have behaved as cracks, triggering or facilitating failure propagation. The honeycomb infill pattern could have its share of blame. In general, the honeycomb structure is very optimized to resist bending, even though it could easily fail by buckling under compressed. Since the socket is subjected to a tri-axial state of stress, there could have been some areas in which the honeycomb failed due to buckling. Nevertheless, this

situation would not be so critical in presence of a 100% infill, in which case the honeycomb cells are so tight that the material is “full”. 6. FDM 3D printing technique Compared with the conventional manufacturing processes, FDM printed part properties depend on structural and process parameters rather than purely on material properties. Due to this process, effects as delamination of the component layers can occur. Additionally, printed components typically have lower elastic properties than injection molded components of the same thermoplastics [10].

2.6 Objective of this study From the results of socket testing and on the considerations we made in the previous chapter (2.5), it is evident that the innovative socket production process needs some improvement. Since the physical test of sockets is expensive and time consuming, we decided to use Finite Element Modelling to improve the design. The aim of this study is to create a Finite Element Model in Abaqus able to analyze the stress distribution under the same loading conditions prescribed by the ISO 10328:2016 standard (test loading condition II), and use it to compare:

- reference geometry with different material properties; - improved geometries of the distal part of the socket; - improved geometries of the connector.

To achieve this goal, we followed two paths:

- Material characterization We selected the strongest elastoplastic material among the ones used in the previous study. This material turned out to be Carbonium Nylon. We performed a mechanical characterization of this material under four conditions: a. 3D printing using dehumidified filament, with 100% infill b. 3D printing using dehumidified filament, with 50% infill c. 3D printing using non-dehumidified filament, with 100% infill d. 3D printing using non-dehumidified filament, with 50% infill to investigate the effect of hygroscopy and infill percentage on the mechanical properties of the 3D printed part. From this study, it emerged the strongest material was Carbonium Nylon printed using dehumidified filament, with 100% infill. - CAD/FEM process In Abaqus FEA, we realized a CAD/FEM process that would allow to perform a Finite Element Analysis of the sockets under test loading condition II of the standard. This CAD/FEM process allowed us to:

a. Find the most stressed regions within the distal part of the socket; b. Improve the geometry of both the distal part of the socket and of the connector, to reduce stress concentration; c. Compare sockets printed with different infill percentages (100% infill versus 50% infill). In particular, we created a Reference Model, in which the geometry of the distal socket and of the connector were equivalent to the ones used during experimental static testing, but the socket was assigned the mechanical properties of the strongest elastoplastic material that resulted from material characterization (dehumidified Carbonium Nylon with 100% infill). Then, we created alternative models with either different geometries or different materials, namely: a. Model 1 and Model 2: different geometries of the housing, with a higher amount of material in the most stressed zone; b. Model 3 and Model 4: different geometries of connector and of housing groove, aimed to reduce the notch effect; c. Model 5: different material, i.e. dehumidified Carbonium Nylon with 50% infill.

In chapter 3 and 4, we described the materials and methods that we used during material characterization and for the CAD/FEM process, respectively. Then, in chapter 5 and 6, we outline the results of the material characterization and the CAD/FEM process, respectively. Finally, in chapter 7 and 8, we discuss the results. Lastly, in chapter 9 we present the conclusions and the future perspectives.

3 Materials and Methods – Material characterization

3.1 Introduction Among the three materials that were used for 3D printing, we decided to furtherly investigate material 1, i.e. Carbonium Nylon, because, from the results of the socket static testing, it turned out that it was the strongest among the 3D printed materials, and that it also displayed an elastoplastic behavior.

To characterize this material we designed, 3D printed and tested different sets of Carbonium Nylon specimens, according to the standards BS EN ISO 527-1:2012 – Plastic — Determination of tensile properties. Part 1: General principles [19] and BS EN ISO 527-2:2012 – Plastic — Determination of tensile properties. Part 2: Test conditions for molding and extrusion plastics [20].

Given the hygroscopic nature of PA12, we wanted to investigate whether material dehumidification would improve layer adhesion during 3D printing and mechanical properties of the specimens. Therefore, for some specimens’ set, we dehumidified the material filament coil before 3D printing, as described in chapter 3.3.

Furthermore, we also wanted to investigate how much the infill percentage would affect tensile strength, so we printed some specimens with a 100% infill and some specimens with 50% infill.

According to ISO 527-2:2012, we designed the specimens’ geometry with a flat dumb-bell shape, described in chapter 3.4. Then, according to ISO 527-1:2012, we 3D printed five sets of specimens, each consisting of five specimens, with the following characteristics:

1. five specimens from dehumidified filament of Carbonium Nylon printed with 100% infill (NC100D); 2. five specimens from dehumidified filament of Carbonium Nylon printed with 50% infill (NC50D); 3. five specimens from non-dehumidified filament of Carbonium Nylon printed with 100% infill (NC100); 4. five specimens from non-dehumidified filament of Carbonium Nylon printed with 50% infill (NC50). 5. five additional specimens from dehumidified filament of Carbonium Nylon printed with 100% infill (NC100D);

Our objective was to test the first four sets statically, according to ISO 527-1:2012, as described in 3.6, and keep the last set to monitor it over time and observe whether hygroscopy would somehow influence in the long term. However, as we will explain in chapter 5.2 of the results, we had to discard the non-dehumidified specimens’ sets, i.e. NC100 and NC50, from static testing, as they did not

36 comply with the dimensional specifications of ISO 527-2:2012. Therefore, we ended up testing only the two dehumidified specimens’ sets, i.e. NC100D and NC50D, and we kept the extra NC100D set for dimensional control over time.

3.2 Material We chose to characterize Carbonium Nylon. The details of this material are provided in chapter 2.4.3. As reported in Table 2.6, the socket made of Carbonium Nylon broke at a force of 2542푁, showing a ductile (elastoplastic) type of fracture.

We chose to investigate the mechanical properties of this material because:

• first, it was the strongest of all 3D printed materials (highest breaking force); • second, it had an elastoplastic behavior. Using a ductile/elastoplastic material for socket realization is very important. In fact, in case of failure, brittle materials break in a sudden and catastrophic way, forming sharp and thus dangerous surfaces. On the contrary, ductile materials have to fully plasticize before they break, giving the amputee a hint that socket is breaking, and thus reducing their risk of falling; in addition, even if they break, they do not create sharp breaking surfaces, preventing the amputee from cutting themselves.

3.3 Filament dehumidification Given the hygroscopic nature of polyamides (see chapter 2.4.3), we decided to dry the filament of Carbonium Nylon before using it to 3D print the specimens.

Figure 3.1 – The filament is dried using a commercial food dryer.

The filament was dehumidified at 85°퐶 for 150푚𝑖푛, as suggested by the supplier of the filament coil, TreeD Filaments, using a commercial food dryer.

The dryer that we used works as a thermal desiccant dehumidifier, in which convective fluxes of hot dry air remove moisture from the moist filament coil; then, hot wet air is discarded in the external environment.

The physical principle that allows material drying is related to the concepts of relative humidity and absolute humidity. Ambient air enters the oven with a certain relative humidity. Here, it is heated, and the temperature increase causes its maximum absolute humidity to increase and consequently its relative humidity to decrease: the air in the oven becomes “dry air”. The filament coil contains moisture and has thus a higher relative humidity than dry air. Once it is placed in the oven environment, its relative humidity tends to decrease and the relative humidity of dry air in the oven tends to increase, to reach an equilibrium. Therefore, water molecules contained in the filament evaporate, exit the filament and pass to the dry air in the oven. The evaporation is speeded up by the increased temperature of the oven. The air in the oven has now a higher relative humidity and becomes “humid air”. Finally, the hot humid air is discarded in the external environment.

The definitions of absolute relative humidity, maximum absolute humidity and relative humidity refer to [21].

3.4 Specimen design The specimen’s geometry was realized according to the standard ISO 527-2:2012 [20]. This standard is suitable for rigid and semi-rigid thermoplastics molding, extrusion and cast materials, including compounds filled and reinforced by, for example, short fibers, small rods, plates or granules but excluding textile fibers. The 3D printed Carbonium Nylon is indeed a thermoplastics extrusion compound material reinforced by short carbon fibers: carbon fibers are short because, as already mentioned, during the extrusion process the initially long carbon fibers uniformly dispersed in the PA12 matrix are cut into several short segments, due to the high shear stresses generated in the nozzle [17].

The geometry that we used for the specimens is the one shown in Figure 3.2, i.e. a flat dumbbell shaped geometry of type 1A, which, according to the standard, is a suitable geometry for specimens obtained by extrusion or molding. Dimensions are given in Table 3.1.

Figure 3.2 – Geometry of a dumbbell shaped specimen of type 1A. This Figure corresponds to Figure 1 at page 4 of the standard [20].

Table 3.1 – Dimensions of type 1A test specimens (displayed in Figure 3.2). This Table derives from Table 1 at page 5 of the standard [20]. All dimensions are given in millimeters [mm].

Dimensions and Label Description tolerances [mm]

푙3 Overall length 170 푙1 Length of narrow parallel-sided portion 80 ± 0.2 푟 Radius 24 ± 1 Distance between broad parallel-sided 푙 109.3 ± 3.2 2 portions

푏2 Width at ends 20.0 ± 0.2 푏1 Width at narrow portion 10.0 ± 0.2 ℎ Preferred thickness 4.0 ± 0.2 6 퐿0 Gauge length (preferred) 75.0 ± 0.5 퐿 Initial distance between grips 115 ± 1

We realized the specimen’s geometry in the CAD software program Autodesk Fusion 360 (Figure 3.3). Note that the sketch nominal dimensions fall within the dimension values prescribed in the standard.

Figure 3.3 – Geometry of the specimen created in Fusion 360.

6 Portion of the specimen being measured by the extensometer

3.5 Specimen 3D printing We printed the specimens using the DELTA WASP 4070 Industrial with the DELTA WASP ZEN Dual Extruder available at INAIL Prostheses Center and described in chapter 2.3.3.

According to the standard ISO 527-2:2012 [19], we had to test a minimum of five test specimens for each of the required directions of testing. In principles, we decided to test four different specimen sets:

To print the specimens, we exported the specimen’s geometry from Fusion 360 as .stl, and we imported it in the slicer, Simplify3D. In Simplify3D, we set the printing parameters of each specimens’ set, as reported in Table 3.2.

Table 3.2 – Printing parameters for specimens printed according to the standards [20]and [19].

Non-dehumidified Dehumidified filaments filaments Printing Set 1 Set 2 Set 3 Set 4 parameters (NC100D) (NC50D) (NC100) (NC50) Nozzle diameter 0.7mm 0.7mm 0.7mm 0.7mm Extrusion width 0.84mm 0.84mm 0.84mm 0.84mm Layer height 0.2mm 0.2mm 0.2mm 0.2mm Internal infill Fast Fast Fast Fast pattern honeycomb honeycomb honeycomb honeycomb Internal infill 100% 50% 100% 50% percentage External infill Rectilinear Rectilinear Rectilinear Rectilinear pattern Number of outline/perimeter 2 2 2 2 shells Number of top and 4 4 4 4 bottom solid layers Nozzle 240°C 240°C 240°C 240°C temperature Plate temperature 90°C 90°C 90°C 90°C Cooling fan 0% 0% 0% 0% Printing speed 80mm/s 80mm/s 80mm/s 80mm/s Nozzle Nozzle Nozzle Nozzle First layer settings temperature: temperature: temperature: temperature 255°C 255°C 255°C : 255°C Skirts or brims 1 layer, 5 outlines, 0 offset from the part Support material No generation Retraction distance of 3mm; Extra retraction distance of Retraction settings 0mm; Retraction vertical lift of 1mm; Retraction speed of 80mm/s. DELTA WASP 4070 Industrial: Delta robot (cylindrical build volume) Machine type Build volume: 282.8mm in X and Y direction; 700mm in Z definition direction; Origin offset: 141.4mm in X and Y direction; 0mm in Z direction.

The five specimens of each specimens’ set had to be printed within the same print. Figure 3.4 (a) shows the positioning of the five specimens on the printer plate. The coordinate with respect to the delta robot coordinate system are shown in Table 3.3 and Figure 3.4 (b).

Figure 3.4 – Position of the specimens on the printer plate in Simplify3D (a) and corresponding position labels (b).

Table 3.3 – Position of the five specimens of each specimen set in the delta robot coordinate system. The position labels are shown in Figure 3.4 (b).

Specimen position Z-coordinate X-coordinate (red) Y-coordinate (green) label (blue) 1 푥 = −80,03 푚푚 푦 = 0 푚푚 푧 = 0 퐴 푥 = −45 푚푚 푦 = 0 푚푚 푧 = 0 2 푥 = −9,08 푚푚 푦 = 0 푚푚 푧 = 0 퐵 푥 = 25 푚푚 푦 = 0 푚푚 푧 = 0 3 푥 = 60,01 푚푚 푦 = 0 푚푚 푧 = 0

Since the FDM technique introduces, by its nature, anisotropies in the printed parts, it is important to specify the printing direction that we used. We printed all specimens in the face-up direction, because this direction did not require using support materials, as would have required the edge-up and especially the up-right direction (Figure 3.5). Printing the specimens in all three directions might have been useful to investigate the anisotropy induced by the FDM technique. In addition, the up- right direction could have been particularly useful for us, because it is the direction that better reproduces the honeycomb distribution in a 3D printed socket.

Figure 3.5 – Possible printing directions of a flat dumbbell specimen: face-up (a), edge-up (b) and up-right (c).

3.5.1 Dimensional control After 3D printing the specimens, we performed a dimensional control, to verify whether the actual dimensions of the printed specimens did fall within the prescribed tolerances. In particular, according to the standard ISO 527-1:2012 [19]:

- for each printed specimen, we recorded three values for the width 푏1 and thickness ℎ, namely

at the center and within each end of the gauge length, and two values for the width 푏2, one at each specimen end;

- for each specimen, we computed the mean value for dimensions 푏1, ℎ and 푏2, as well as the median and the standard deviation.

Then, for each specimen, we calculated the deviation of each measured dimension from its corresponding nominal dimension, and we computed the accuracy (퐴푐푐) as:

푀 − 푁 (3.1) 퐴푐푐 = ∙ 100% 푁 where 퐴푐푐 is the accuracy expressed as a percentage [%], 푀 and 푁 are the measured and nominal dimension value, respectively, expressed in millimeters [푚푚].

According to Table 3.1 and equation (3.1), to comply with the tolerances prescribed by the ISO 527-

2 standard, the accuracy requirments for dimensions 푏1, ℎ and 푏2 are reported in Table 3.4.

Table 3.4 – Accuracy requirments for the most important specimen dimensions.

Nominal dimension Admissible tolerance Accuracy requirement Dimension [mm] [mm] [%]

푏1 10 10.0 ± 0.2 ±2 ℎ 4 4.0 ± 0.2 ±5

푏2 20 20.0 ± 0.2 ±1

For each specimen, we also calculated the coefficient of variation (퐶푉), to study the dispersion of the recorded dimension values from their mean value, independently of the unit of measurement:

휎∗ (3.2) 퐶푉 = ∗ 100 휇

43 where 퐶푉 is the coefficient of variation, 휎∗ the standard deviation and 휇 the mean value.

Finally, for each specimen, we calculated the cross-sectional area at the narrow portion, as:

퐴 = 푏1 ∙ ℎ (3.3) measured in millimeters [푚푚2].

We will use these cross-sectional area values to calculate the engineering stress values starting from the force values recorded during static testing, as described more in detail in chapter 3.5.1.

The tables with the integral results are reported in Appendix C. In chapter 5.2 of the results section, we summarized the results, and for each specimens’ set we only reported the mean values of the accuracy, coefficient of variation and cross-sectional area, obtained by averaging the corresponding values of the five specimens within each specimens’ set.

3.6 Specimen testing We subjected the specimens to uniaxial tensile testing up to fracture, according to the standard ISO 527-1:2012 [19]. Each test specimen was extended along its major longitudinal axis at a constant speed until the specimen fractured. During this procedure, the load sustained by the specimen, the elongation and the mean deformations were measured. The load, denoted by 퐹, is expressed in 푁, the elongation, denoted by ∆퐿, is expressed in 푚푚, and the deformation, denoted by 휖, is expressed 푚푚 in . 푚푚

The test setup is described in chapter 3.6.1, the test parameters that were set according to the standard are reported in chapter 3.6.2 and, finally, the parameters that were measured after the test are reported in chapter 3.6.3.

3.6.1 Test setup We conducted all testing at the University of Padova7 in the machine design laboratory of the department of industrial engineering. To record force and displacement we used the MTS8 858 Mini Bionix II Tabletop Testing System machine, which consists of an MTS load cell (25kN) and MTS 647 Hydraulic Wedge Grips (Figure 3.6).

7 www.unipd.it 8 www.mts.com

While the hydraulic grips firmly grasp and hold the specimen in place during testing, the linear actuator within the load cell applies an axial force to the specimen. This force is measured and recorded by the force transducer.

Figure 3.6 – Axial MTS 858 Mini Bionix II (left) with MTS 647 Hydraulic Wedge Grip (right).

To measure the average strain in tension we used the MTS 634.12 Axial Extensometer (Figure 3.7), i.e. a clip-on contact extensometer that measures only axial strain. Its technical parameters are reported in Table 3.5.

Figure 3.7 – MTS 634.12 Axial Extensometer mounter on a specimen.

Table 3.5 – MTS 634.12 Axial Extensometer parameters

Parameter Value in SI units Gauge length (25 ± 0.5)푚푚 Maximum travel −2.5 푡표 12.5푚푚 Maximum strain −10 푡표 50% Activation force 35푔 Bridge resistance 350Ω Meets or exceeds ASTM E83 Class B1 Accuracy and ISO 9513 Class 0.5 standards.

The contact extensometer is applied to the specimen through its two symmetrical knife edges (grips), at a distance equal to the gauge length. One knife edge is fixed and the other one is movable, and it moves as the specimen elongates. The extensometer measures the mean deformation of the elongated portion of the specimen comprised by the two knife edges (gauge length), namely:

∆퐿 (3.4) 휖 = 퐿0

According to ISO 527-1:2012, the gauge length 퐿0 is defined as the initial distance between the gauge marks on the central portion of the test specimen, and it is expressed in 푚푚. According to ISO 527- 2:2012, the preferred value of the gauge length for a dumbbell shaped type 1A specimen should be 75푚푚 (see Table 3.1), as this value allows to include a larger portion of the specimen narrow sections within the extensometer grips. As we did not have such an extensometer, we used an extensometer with a gauge length of 25푚푚 (see Table 3.7), which complied with the accuracy requirments of ASTM E83 Class B1 and ISO 9513 Class 0.5 standards.

Finally, to measure Poisson’s ratio, we used two KYOWA Strain Gages9, which are foil strain gauges having the technical parameters reported in Table 3.6. We attached both strain gauges on the broader surface of the narrow portion of the specimen, one along the specimen’s longitudinal direction and the other one along its transverse direction, as showed in Figure 3.8. The strain gauges were used together with a data acquisition (DAQ) device, the Somat eDAQ Bridge Layer (EBRG)10.

9 www.kyowa-ei.com 10 https://www.hbm.com/en/2180/somat-edaq-bridge-layer-ebrg-for-rugged-mobile-data-acquisition/

Figure 3.8 – Specimen with the two strain gauges attached on the broader surface of its narrow portion.

Table 3.6 – KYOWA Strain Gages technical parameters.

Parameter Values in SI units Notes Gauge factor 푘 2.08 ± 1.0% At 23°퐶, 50% RH Gauge length 3푚푚 Gauge resistance 119.6Ω ± 0.4% At 23°퐶, 50% RH Transverse sensitivity ratio (0.3 ± 0.3)%

The configuration associated to the strain gauge sensors is the electrical circuit called Wheatstone bridge [22]. Each strain gauge measured a deformation given by (3.5):

∆푈퐸 4 1 휖 = ∙ ∙ (3.5) 푈퐴 푘 푛 where:

∆푈퐸/푈퐴 is the output voltage variation in the signal diagonal, which is proportional to the relative deformation of the Wheatstone bridge.

푘 is the strain gauge factor, i.e. the principal parameter of a strain gauge, also called strain coefficient of resistance, which in our case was is equal to 2.08 according to Table 3.6.

푛 corresponds to the number of strain gauges in each Wheatstone bridge. In our case, 푛 = 1. In fact, each of the two strain gauges were a quarter of Wheatstone bridge each, meaning that in their Wheatstone bridge configuration each strain gauge was one of the four resistances, and the other three were all equal to each other and known.

The strain gauge applied along the longitudinal direction of the specimen measured the axial or longitudinal deformation, 휖푙, parallel to the direction of the elongation, whereas the one applied along the specimen’s transverse direction measured the transverse deformation, 휖푡푟, perpendicular to the direction of elongation.

3.6.2 Test parameters According to the standard ISO 527-1:2012 [19], we set the test parameters as in Table 3.7.

Table 3.7 – Test parameters of the MTS testing machine.

Test parameter Value Gripping force 15푀푃푎 푚푚 Test speed 2 푚𝑖푛 Acquisition frequency 50퐻푧 Gauge length 25푚푚

The gripping force is the force exerted by the hydraulic grips to clamp the specimen in place during the test. The test speed is defined in the standard as the rate of separation of the gripping jaws. The acquisition frequency is the frequency used to record the data (force, displacement and strain).

3.6.3 Measured parameters After recording the force (퐹 [푁]), the axial tensile displacement or elongation (∆퐿 [푚푚]) and the 푚푚 average deformation or strain (휖 [ ]), we calculated some useful parameters. 푚푚

푚푚 First, we computed the stress values 휎[푀푃푎] corresponding to the strain values 휖 [ ] recorded by 푚푚 the extensometer. We computed each stress value as the ratio between the recorded force 퐹[푁] and the specimen cross-section 퐴[푚푚2], namely:

퐹 휎 = (3.6) 퐴

From the stress-strain values we plotted the engineering stress-strain curves for each specimen.

For each stress-strain set of values, we calculated Young’s modulus, the yield stress and the ultimate tensile stress, that are highlighted in each stress-strain curve.

The ultimate tensile stress or stress at break 휎푏[푀푃푎], defined as the stress at which the specimen breaks: it is the highest value of stress in the stress-strain curve directly prior to the separation of the specimen, i.e. directly prior to the load drop caused by crack initiation. We computed it as the maximum value among all the stress 휎 value, since we noticed that the stress-strain curve increased

48 monotonically (thus, there was no maximum value before the break point). alternatively, we could have computed it as the ratio of the force at break to the cross-section:

퐹 휎 = 푏 (3.7) 푏 퐴 where the ultimate tensile force at break, 퐹푏[푁] is found as the maximum value among all the 퐹 recorded values.

The Young’s modulus or elastic tensile modulus 퐸푡[푀푃푎] is a measure of the tensile stiffness of the material and it is calculated as the slope of the stress-strain curve in the elastic field. In the standard ISO 527-1:2012 it is defined as:

휎2 − 휎1 퐸푡 = (3.8) 휖2 − 휖1 where 휎1[푀푃푎] is the stress value measured at the strain value 휖1 = 0.05% = 0.0005 and 휎2[푀푃푎] is the stress value measured at the strain value 휖2 = 0.25% = 0.0025.

Figure 3.9 – Requirments for calculating Young’s modulus depending on the gauge length. This Figure derives from Figure 2 at page 9 of the standard ISO 527-1:2012 [19].

Since all specimens showed a ductile behavior, we also calculated the yield stress 휎푦[푀푃푎], i.e. the stress at which the material begins to deform plastically, exiting the elastic regime and entering the plastic one. The standard ISO 527-1:2012 defines it as the stress value at the yield strain 휖푦, where 푚푚 the yield strain 휖 [ ] is defined as the first occurrence in a tensile test of strain increase without a 푦 푚푚 stress increase. Since this definition could give rise to confusion, we decided to calculate 휎푦 using the “0.2% offset yield stress definition”, that is the definition most often quoted by material suppliers and used by design engineering: the yield stress is defines as the amount of stress that will result in plastic strain of 0.2%. Using this definition, we found the yield stress as the intercept between the stress-strain curve and the line expressed by equation (3.9):

푦 = 퐸푡 ∙ (휖 − 0.002) (3.9)

49 where 푦 is the unknown value of stress, 퐸푡 is Young’s modulus, 휖 is the strain value measured by the extensometer, and 0.002 indicated that this line has an offset of 0.2% = 0.002 from the initial strain value.

Finally, for specimen NC100D labelled as “1”, we were able to calculate Poisson’s ratio, as: 휖 휈 = − 푡푟 (3.10) 휖푙 where 휖푙 and 휖푡푟 are the longitudinal and transverse deformation, respectively, measured using the strain gauges, as described in chapter 3.6.1.

3.6.4 Data analysis Before plotting the stress-strain curves and before computing the measured parameters, we first cleaned the rough data of stress and strain in MATLAB.

First, we applied the function “unique” to the rough strain data, to discard the repeated strain values. Then, for each unique value of strain, we averaged the corresponding values of stress, using the function “accumarray” and “mean”. Then, we applied the filter “medfilt1” to the stress data; medfilt1 applies a third-order median filter to the input array, efficiently removing peaks from the data.

To find the mean stress-strain curve for each specimen’s set, we did the following.

- For each specimen data set we first rescaled the strain data points with a defined step rate of 0.0001 from 0 to the strain value corresponding to the ultimate tensile stress. - Then, we interpolated the stress data over the newly-defined strain axis, using the function “interp1”. In addition, we also applied the filter “sgolayfilt” to the stress data. - At this point, for each specimens’ set we averaged the interpolated and filtered stress values over the new strain axis from 0 to the strain value corresponding to the minimum ultimate tensile stress in the specimens’ set. - Finally, we found the mean stress-strain curve by plotting the average stress over the rescaled strain.

On the contrary, to find the mean values of the measured parameters for each specimens’ set (mean Young’s modulus, mean yield stress, mean ultimate tensile stress, mean ultimate tensile force), we did not derive them from the mean stress-strain curves, but, according to standard ISO 527-1:2012, we averaged the numerical values of the measured parameters for each specimen’s set.

3.6.5 Plasticity data extrapolation for Abaqus We used the results of material characterization to define the material “Carbonium Nylon” in Abaqus and to assign it to the virtual model of the socket. Since this material has an elastoplastic behavior,

50 we had to define both its elastic properties as well as its plastic properties, according to Abaqus definitions [23].

Defining elasticity of a material in Abaqus is simple, as it is sufficient to assign the value of Young’s modulus and Poisson’s ratio.

On the contrary, defining plasticity in Abaqus is not as straightforward. In fact, it is essential to provide a set of stress-strain values from the plastic regime of the material’s stress-strain curve. In particular, it is necessary to use true stress and true strain values. However, the stress-strain curves that we obtained from the data analysis in MATLAB (3.6.4) are engineering, or nominal, values of stress and strain, because of the way we calculated them: the strain was measured by the extensometer, as the average elongation of the gauge portion to the initial gauge length (3.4); the stress was computed as the ratio of the applied force to the original cross-sectional area (3.6). These two definitions correspond to the ones of engineering stress and strain.

In fact, the engineering or nominal stress is defined as the ratio of the applied force to the original (undeformed) cross-sectional area: 퐹 (3.11) 휎푛표푚 = 퐴0

On the contrary, the true stress is defined as the ratio of the applied force to the current (deformed) cross-sectional area:

퐹 (3.12) 휎 = 퐴 where 퐴 is the current cross-sectional area and 퐴0 is the original cross-sectional area.

The strain measures conjugated to the stress measures defined above are the engineering or nominal strain, defined as the ratio of the total elongation to the original (undeformed) length:

푙 − 푙0 (3.13) 휖푛표푚 = 푙0 and the true or logarithmic strain, defined as:

푙 (3.14) 휖 = ln ( ) 푙0 where 푙 is the current length, 푙0 is the original length, and ∆푙 = 푙 − 푙0 is the total elongation [24],[25].

Even though the engineering stress is a useful parameter, easy to calculate, it does not capture the actual behavior of the material, because it does not account for the deformation of the cross-sectional area. The true stress, on the other hand, accurately describes the physical behavior of the material,

51 because it also accounts for the deformation of the cross-sectional area. However, it is not always easy or possible to calculate it, because its calculation implies knowing the entity of area deformation during the entire tensile test.

The difference between engineering and true stress is negligible in the elastic regime but becomes apparent after yield has started (Figure 3.10). After the material yields, it begins to experience a high rate of plastic deformation and both stresses increase (strain hardening), the true stress at a higher pace, as it also accounts for the reduction of the cross-sectional area. After reaching the ultimate tensile strength, the specimen starts to neck-in, and while the engineering stress dramatically decreases due to the reduced load-carrying capability caused by a smaller cross-section, the true stress rapidly increases, because the reduction of the cross-sectional area outpaces the reduction of the load-carrying capability of the specimen.

Figure 3.10 – True-stress strain curve versus engineering stress-strain curve, resulting from an elastoplastic specimen subjected to a uniaxial tensile force, 퐹.

Therefore, since there are very signiﬁcant differences at larger strain values between engineering and true stresses and strain, it is extremely important to provide the proper stress-strain data to Abaqus if the strains in the simulation will be large [23].

To convert the material data from nominal stress-strain values to true stress-strain values, we used the following equations:

휖 = ln(1 + 휖푛표푚) (3.15)

(3.16) 휎 = 휎푛표푚(1 + 휖푛표푚)

These relationships are formed by considering the incompressible nature of the plastic deformation and assuming that the elasticity is also incompressible. Moreover, they are valid only prior to necking [23]. Therefore, they will provide what in Figure 3.10 is called “corrected stress-strain curve”.

Once we obtained the true stress-strain values, we decomposed the (true) total strain values into the elastic and plastic strain components. We obtained the (true) plastic strain by subtracting the elastic strain from the value of total strain, where the elastic strain is defined as the value of true yield stress divided by Young’s modulus.

휖푝푙 = 휖푡 − 휖푒 (3.17)

휎푦 (3.18) 휖푒 = 퐸푡

Because of this definition, the first value of (true) plastic strain, i.e. the value of plastic strain at the yield point, is zero.

Then, we plotted the so-called “true plastic stress-strain curve”, given by the values of true stresses and true plastic strain from the yield point to fracture. Afterwards, from the true plastic stress-strain curves, we plotted the mean true plastic stress-strain curve for both specimens’ set, using the same approach described in chapter 3.6.4. Finally, we applied the MATLAB function “linspace” to the mean true plastic stress-strain values and obtain 10 linearly spaced values of true plastic stress and strain, that we used in Abaqus to define the plastic behavior of Carbonium Nylon, as we will described in chapter 4.5.

The (true) corrected stress-strain curves, the true plastic stress-strain curves and the parameters given as input to Abaqus are summarized in the results section in chapter 5.4.2. Integral results are available in Appendix C.

4 Materials and Methods – CAD/FEM process

4.1 Introduction In Abaqus FEA, we realized a CAD/FEM process that virtually reproduced static testing of the 3D printed sockets. This CAD/FEM process allowed us to:

1. Find the most stressed regions within the distal part of the socket; 2. Improve the geometry of both the distal part of the socket and of the connector, to reduce stress concentration; 3. Compare sockets printed with different infill percentages (100% infill versus 50% infill).

We used the CAD environment of Abaqus FEA to design a model comprising:

- The Willow Wood setup; - The distal part of the socket; - The connector.

Our target was to create the simplest possible model that delivers sufficiently accurate results. In fact, INAIL Prostheses Center does not have access to supercomputers, and to perform all Abaqus simulations we had to rely on the computational resources of a computer with four cores and 16GB or RAM.

We reduced the geometry of the socket to focus on the most interesting parts, i.e. distal part of the socket and connector, and we considered the rest of the assembly through suitable boundary conditions. In particular, we neglected the proximal part of the socket because:

- First, the experimental results show that failure took place in the distal part of the socket at the interface between housing and connector. Thus, we decided to focus the Finite Element Analysis on this part of the socket and on the interactions between the distal part of the socket and the connector. - Second, the distal part is the only piece of the socket that is not customized on the patient and therefore its geometry is the only one that can be modified for structural purposes. - Third, modelling the entire geometry of the socket would have been overly complicated without adding any benefit to the study, with the additional drawback of excessively increase the computational cost of the simulation, probably beyond our computational resources.

As described in chapter 4.3, the first model that we created is a Reference Model, made of the following components:

- A simplified version of the Willow Wood setup;

- The distal part of the socket deriving from the generic transtibial residual limb template provided by the Willow Wood Company (Figure 2.18); this part was assigned the mechanical properties of the dehumidified Carbonium Nylon printed with 100% infill, derived from the material characterization. - The aluminum Coyote connector.

In the consecutive chapters we described how we created the mesh (4.4), assigned the material properties to the components (4.5), defined the contact interactions between the parts (0), applied the loading condition to the model (4.7) and conducted the static analysis (4.8) in the reference model. Then, in chapter 4.9, we present alternative models for comparison, namely:

- Model 1 and Model 2: different geometries of the housing, with a higher amount of material in the most stressed zone; - Model 3 and Model 4: different geometries of the connector and of the housing groove, aimed to reduce notch effect; - Model 5: different material, i.e. dehumidified Carbonium Nylon with 50% infill.

4.2 Abaqus Abaqus is a suite of powerful engineering simulation software, based on the finite element method (FEM).

For this project, we used the Abaqus/CAE to operate the preprocessing and postprocessing stages of the FE analysis and Abaqus/Standard for the simulation. This analysis product solves a system of equations (4.1) implicitly at each solution “increment”, using the Newton-Raphson numerical method [26]. The algebraic equations that need to be solved to find the nodal displacements are summarized in matrix form (4.1):

푲풖 = 푭 (4.1)

where 푲 is the global stiffness matrix, 풖 is the global displacement vector, and 푭 is the global load vector. The word “global” refers to the fact that nodes and elements are numbered with respect to the entire mesh and not with respect to the single element (local).

풖 and 푭 are column vectors of 푁 entries, where 푁 is the total number of nodes in the mesh. 풖 contains the nodal displacements and 푭 contains the values of the loads at each node.

푲 is a square matrix, with dimensions (푁 × 푁푑표푓푠) × (푁 × 푁푑표푓푠), where, again, 푁 is the total

number of nodes in the mesh and 푁푑표푓푠 is the number of degrees of freedom (dofs) per node. It is useful to remind that 퐾 is symmetric, banded and it is nonsingular, and thus invertible, only upon imposing the displacement boundary conditions. 퐾 is a coefficient matrix containing the stiffness

information (Young’s modulus, Poisson’s ratio, spatial derivatives of the shape functions) of the structure at each node [27], [28].

The larger the stiffness matrix, the higher the computational cost, namely the RAM required to run the analysis in reasonable time. For this reason, when meshing a geometry, the number of elements, the element type and number of dofs can strongly affect the computational cost.

In general, the higher the number of elements, the larger the number of nodes, and thus the larger the stiffness matrix, which entails larger amount of data to be hosted by the RAM. However, elements can be linear or quadratic, depending on the order of interpolation of the shape functions. Linear elements have nodes only at their corners. Quadratic elements, besides the corner nodes, have additional mid-edge nodes. Therefore, for the same number of elements in a mesh, choosing quadratic elements can strongly increase the size of the stiffness matrix and thus the computational cost.

In addition, the size of the stiffness matrix also depends on the number of dofs per node. Usually, continuum or solid elements, differently from beams or shells, have only translational dofs but not rotational dofs. Consequently, 1D elements (e.g. beams) have one dof per node, 2D elements (e.g. triangles or quadrilaterals) have two dofs per node and 3D elements (e.g. tetrahedrons or hexahedrons) have three dofs per node, because nodal displacement can take place along one, two or three perpendicular directions, respectively.

The reader that might not be familiar with Abaqus terminology can refer to Appendix B, where we briefly defined some terms that we used in the following chapters, to avoid any misunderstanding. The extensive definitions and descriptions can be found in Abaqus user’s guides [26],[29],[23].

4.3 Reference model To virtually reproduce the static testing of the distal part of the socket, we needed to create a model comprising:

1. Test setup 2. Distal part of the socket 3. Connector

The test setup had the function of guaranteeing the correct loading condition of the distal socket, according to the standard ISO 10328:2016 [13], as we described more in detail in chapter 2.4.2. Even though we were not interested in knowing the stress distribution in the setup, in principle, we decided to completely reproduce its geometry in Abaqus. We made this choice to take into account the load

56 condition in its entirety, and thus to improve the accuracy of the model and the confidence in the results.

However, in retrospect, we reconsidered this choice, and we came to the conclusion that we could have easily substituted the geometry of the setup with a structure made of beams. This option would have not affected the outcome of the simulation, with the advantage of making it less computationally expensive. On the contrary, neglecting the setup by transporting the loading to the distal socket would not have been a valid option. In fact, as we will explain in chapter 4.4, we meshed the entire geometry with continuum (solid) elements, which only have translational degrees of freedom. Applying a transportation moment directly to this structure would have been meaningless. To obtain meaningful results, we should have introduced reference points and couplings.

In principle, as mentioned above, we decided to reproduce the entire geometry of the setup. To reduce the complexity of the problem and the computational cost of the simulation, we created a simplified version of the much more detailed geometry of the Willow Wood setup that is displayed in Figure 2.23 of chapter 2.4.2. The simplification consisted in removing all the details that were unnecessary for the sake of the simulation, i.e. that did not affect the loading condition of the socket, but that would have just increased the computational cost of the simulation. Instead, we did preserve all the dimensions that actually affected the loading conditions of the socket, namely the dimensions listed in Table 2.4 of chapter 2.4.2.

Since our objective was to find the stress distribution within the distal part of the socket, due to its interactions with the connector, it was crucial to faithfully reproduce the geometry of both the distal part of the socket and the connector. Therefore, we just applied some small simplifications to these two components. Hereunder we report the simulations that we perform or did not perform on the two components.

• The geometry of the distal part of the socket is not simplified: it has the same shape and dimensions as in reality and it is completely solid, with the exception of having a groove in its very distal part (in the housing) that has the same shape of the connector. In fact, from Figure 4.1 (b) it is possible to see that only the proximal part of the socket is hollow, because it is the part hosting the stump, whereas the distal part of the socket is solid, with a connector- shaped groove in the housing. As explained above, we neglected the proximal part of the socket and we only reproduced the distal part.

Figure 4.1 – Willow Wood socket: see-through view (a), section view (b) and normal view (c).

• We did not account for the honeycomb pattern introduced during 3D printing in the distal part of the socket. Recreating a honeycomb pattern would have increased the computational cost of the simulation beyond our available resources. Therefore, we modelled this part as completely solid; nevertheless, we did account for the mechanical properties of the honeycomb structure by simply considering the mechanical properties found during material characterization for specimens with 50% infill and 100% infill. Please not that when considering the 100% infill, the honeycomb pattern becomes irrelevant, as the material density reaches 100%. This aspect will be discussed more in detail in chapter 4.5. • In principle, we simplified the geometry of the connector by neglecting the presence of bolted joints: we assumed that its four columns were not tapered but rather solid, and that they were attached to the distal connector of the setup as if they were glued to it, when in reality they are connected to it through four countersunk screws. The actual geometry of the connector (with tapered columns) is displayed in Figure 4.2, while the one with solid columns is displayed in Figure 4.3. We chose this way of modelling for its conservativeness in the probable failure region. In fact, by replacing bolted joints with perfectly stiff constraints, the stress is amplified due to the lack of flexibility that screws or bolts would introduce. This approach leads to a more conservative and simpler assessment of the structure. Moreover, we do not believe that this simplification will strongly affect the stress distribution within the distal socket, because the bolted connection involves only the aluminum connector, which is much stiffer than the material of the distal socket and therefore sustains most of the load transmitted through the bolted connection. Furthermore, the small portion of distal socket clamped between the connector and the setup connection plate is not under compression, because the bottom parts of the four columns of the connector are aligned with the bottom surface of the distal socket and are not sunk in.

• The geometry of the groove within the housing in the distal part of the socket has the same dimensions of the Coyote connector, to allow a perfect incorporation of the latter inside the first. Figure 4.4 displays the geometry of the groove of the housing.

Figure 4.2 – Connector: real geometry and dimensions.

Figure 4.3 – Connector: adapted geometry and dimensions.

Figure 4.4 – Housing (bottom part of the distal socket with a groove complementary to connector).

In conclusion, the model that we had to recreate had to look like the one showed in Figure 4.5 and Figure 4.6, with the dimensions reported in the drawings of Figure 4.7, Figure 4.3 and Figure 4.4.

Figure 4.5 – Setup and distal part of the socket with embedded connector.

Figure 4.6 – Explosion of the distal part of the socket: the distal part of the socket comprises a groove having the shape of the connector, called the housing, that in fact hosts the connector.

Figure 4.7 – Technical drawing of the “setup, distal socket (comprising the housing) and connector” in three different views: f-u (x-z) plane (top left), o-u (y-z) plane (top right) and f-o (x-y) plane (bottom left).

4.3.1 Parts The model that we wanted to reproduce consisted of six components:

1. Top loading cylinder of the setup 2. Bottom loading cylinder of the setup 3. Top part of the setup 4. Bottom part of the setup 5. Socket (comprising the housing) 6. Connector

However, Abaqus has difficulties in dealing with interactions between the parts, as we will explain more in detail in chapter 0. Therefore, to ease the simulation, it is important to reduce as much as possible the number of interactions. One way to do this consists in reducing the number of the parts in the model, by assembling more components within one part. By this logic, we reduced the number of parts from six to two, namely:

1. Part 1: Bottom part of the setup + Connector (Figure 4.8); 2. Part 2: Top part of the setup + Socket (comprising the housing) (Figure 4.9).

Figure 4.8 – Part 1: Bottom setup and Connector.

Figure 4.9 – Part 2: Top setup (in grey) and Socket (in green). Note that the distal part of the socket incorporates a groove having the same geometry of the connector. The bottom part of the distal socket, the housing, incorporating the connector-shaped groove.

Note that we neglected the two loading cylinders. As we will explain more in detail in chapter 4.7, we substituted them with a Reference Point (RP) each, and we appropriately connected it to each setup cup.

To design these two parts, we had two options:

1. Creating the two parts in Abaqus/CAE, using its CAD environment, namely the Sketch module; 2. Creating the two parts using an external CAD software program, such as SolidWorks or Autodesk Fusion 360, and importing them in Abaqus/CAE as .stp or .iges files.

We decided on option 1, so that we could obtain a totally parametric model within Abaqus. This gave us two main advantages:

• First, we were able to quickly and easily modify the geometry of the distal socket and connector within the Abaqus environment, in order to see how these changes would affect stress distribution. This procedure would have been much more complicated and time-

consuming if we had to modify the geometry in a separate CAD software program: in this case, for every geometrical change, we should have modified the geometry in the external CAD program, re-imported the file in Abaqus, and reassigned the mesh, property, interaction and loading conditions to a new model each time. • Second, as long as it did not alter the loading condition of the socket, we were able to modify the geometry of the non-interesting parts of the model (the setup), in order to promote mesh generation and reduce element distortion.

Hereunder, the construction of the model in Abaqus Sketch module is described by means of Figures.

Extrusion height = 10

Extrusion height = 8

Figure 4.10 – Construction of the connector.

Extrusion height = 12.7

Extrusion height = 28.4

Figure 4.11 – Construction of the distal setup attachment plate and column.

Extrusion height = 12.7

Extrusion height = 38.92

Cut revolution = 180°

Figure 4.12 – Construction of bottom part of the setup, including the loading cup.

Extrusion height = 463.08

Loft height = 68

Figure 4.13 – Construction of the column of the top setup and of the distal part of the socket.

Cut extrusion height = 10

Cut extrusion height = 8

Figure 4.14 – Construction of the groove inside the distal part of the socket.

Extrusion height = 12.7

Extrusion height = 38.92

Cut revolution = 180°

Figure 4.15 – Construction of the top part of the setup, including the top loading cup.

4.3.2 Assembly of the parts After creating the two parts, we assembled them in a way that they could assume the configuration of the setup described by the standard ISO 10328:2016 [13] and implemented by the Ohio Willow Wood Company, described in detail in chapter 2.4.2.

Since in Abaqus each part exists in its own coordinate system, we first created instances of the two parts, which are a representation of the part; then, we positioned the part instances relative to each other in a global coordinate system, creating an assembly. We created dependent part instances, which are only a pointer and not a real copy of the original part; since this type of instances cannot be meshed, we meshed the original parts, but we applied the interactions and loadings to the part instances. Note that part instances maintain their association with the original part of the model: if the geometry of the part changes, Abaqus/CAE automatically updates all instances of the part to

68 reflect these changes [29]. This is the reason for which we decided to create the parts directly using Abaqus CAD environment (Sketch module), so that we could have a completely parametric model.

Figure 4.16 shows the assembly of the two part instances correctly positioned according to the Willow Wood setup configuration.

Figure 4.16 – Assembly of Part 1 and Part 2. From the figure on the right, it is possible to see that the connector is completely embedded inside the distal socket. Also note the Reference Points displayed in red.

Finally, since in the Part module we did not create the two loading cylinders, in the Assembly module we created two Reference Points and we positioned them in correspondence of the “virtual” position of the centers of the spherical heads of the loading cylinders (remember that, as explained in chapter 2.4.2, each loading cylinder ends with a spherical head that can rotate inside its conjugated cup of the setup). Then, as we will describe in chapter 0, to simulate their behavior we constrained each Reference Point to the internal surface of its conjugated cup of the setup.

4.4 Mesh After creating the part, it is advisable to immediately create the mesh, before assigning the material properties and loading conditions. In fact, in case we need to change the geometry to facilitate mesh generation, we do not need to redo all the other steps.

Note that, since in the Assembly module we created dependent part instances, we had to directly mesh the parts and not the part instances. Meshing the part instances would be possible only in case of independent part instances, which was not our case.

4.4.1 Element type To mesh our model, we used elements that in Abaqus are labeled as C3D8R, that univocally refer to elements that are:

• deformable, • stress/displacement continuum (i.e. solid), • three-dimensional, • linear hexahedral, • with reduced integration.

Hereunder we define each term and we justify its choice.

• We used deformable elements instead of rigid bodies because: first, the deformations in our model are not negligible, in fact one of the materials that we modelled had also a plastic behavior (non-linear material model); second, we wanted to avoid the risk of gap between the real model and the simulation model, especially in terms of displacements and strains. • Among the deformable elements, we chose elements from the continuum or solid three- dimensional family, which are the standard volume elements in Abaqus. In particular, since we conducted a structural static analysis, we chose the stress/displacement continuum elements, that give as output variables stresses and strains. These elements are denoted by the code C3D (C = stress/displacement continuum, 3D = three-dimensional elements). The other families of elements we could choose among where are reported in Figure 4.17 [23].

Figure 4.17 – Element families in Abaqus.

• Among the possible element shapes of the continuum family (tetrahedra, wedges and hexahedra, as shown in Figure 4.18), we chose hexahedral elements, because they give the best results for the minimum computational cost. In particular, as denoted by number 8 in the label C3D8R, we used linear hexahedral elements instead of quadratic hexahedral

elements to reduce the computational cost of the simulation. In fact, linear elements only have one node per corner (8 in total for hexahedra) and use first-order shape functions to interpolate the displacements at the nodes within the element, as opposed to quadratic elements, that have both corner nodes as well as mid-side nodes at the middle of each edge (20 nodes in total for hexahedra) and use a quadratic interpolation in each direction. Since linear elements use a lower-order interpolation, they provide a worse approximation of the displacement than quadratic elements, but at the same time, having 12 nodes less per element, they allow to drastically reduce the computational cost of the simulation, because the stiffness matrix has much reduced dimensions. Since we had very stringent limitations regarding the computational resources, we opted for linear elements and finer mesh in the key regions.

Figure 4.18 – Three-dimensional continuum element shapes.

• Finally, in the label C3D8R, the letter R refers to the type of integration that we adopted, namely reduced integration, as opposed to full integration. The term integration refers to the type of numerical technique that Abaqus uses to integrate the stiffness matrix and load vector over the volume of each element, which is the Gaussian Quadrature. Differently form fully integrated elements, reduced integration elements use less Gauss points to integrate the polynomial terms in an element’s stiffness matrix. In particular, reduced integration linear hexahedra only have one integration point (Gauss point) at the centroid of the element, as displayed in Figure 4.19.

Figure 4.19 – Full integration versus reduced integration linear hexahedral elements.

We chose linear hexahedral elements with reduced integration because o They do not suffer from shear locking under bending, because they only have one integration point at their centroid; o Even if they suffer from hourglassing, the propagation of this zero-energy mode can be easily avoided by using a sufficiently refined mesh. In particular, according to what suggest the experimental results [30], [23], we used from 3 to 5 elements along the thickness in the planes of bending moments. In fact, the small amount of artificial “hourglass stiffness” introduced by Abaqus to limit the propagation of zero-energy mode is more effective in finer meshes. o They are very tolerant of distortion. In all location where distortion levels may be very high, it is advisable to use a fine mesh of these elements [23].

4.4.2 Meshing technique Despite its apparent simplicity, our geometry was particularly complex, and it was very difficult to mesh it completely with hexahedral elements. In fact, by default Abaqus suggested a bottom-up mesh approach, which differently from the top-down approaches does not allow automatic generation of a hexahedral mesh [29].

Figure 4.20 – When using hexahedra on complex geometry, by default Abaqus suggests a bottom-up mesh and it colors the geometry in orange, meaning that the mesh cannot be generated automatically using hexahedral elements.

To being able to use a top-down mesh technique, we had two options:

1. Switch to another element type, such as tetrahedra or wedges, as these elements have a shape that is more versatile and adapts better to complex geometries. However, we should have used the quadratic version of these elements (C3D10 or C3D15), as the linear version (C3D4 or C3D6) are poor elements and very fine meshes are needed for accurate results [23]. Nonetheless, by using a quadratic mesh of tetrahedra or wedges we would have increased the computational cost excessively. Therefore, we discarded this option. 2. Partition the geometry, in order to divide the two complex part instances into simpler regions that Abaqus/CAE could mesh using top-down hexahedral meshing. We chose this option.

In chapter 4.4.6 we will explain in detail the partitioning strategy, the meshing technique and algorithm that we applied on the reference model. However, to find the optimal mesh size for our reference model, we first had to perform a mesh convergence study (chapter 4.4.3) and evaluate the accuracy of the results found using this mesh (chapter 4.4.4).

4.4.3 Mesh validation To find the optimal mesh density, we performed a mesh convergence study.

The objective of a mesh convergence study is to find a mesh size that is a trade-off between accuracy of the results and computational cost. In fact, both the accuracy of the results and the computational

73 cost increase with increasing number of elements in a mesh, because a larger number of elements translates into a better approximation of the problem but also into a larger stiffness matrix, which needs more computational resources to be solved.

Therefore, the elements of the mesh must have a dimension that ensures that the results of an analysis are not affected by changing the size of the mesh while having a computational cost that is acceptable with respect to the tools that are available for the analysis.

Finding an optimized mesh density was crucial for our study, since we had very stringent limitations regarding the computational resources (four cores and 16GB of RAM).

To conduct this study, we did the following:

• We created a simpler version of our model, with a simplified geometry (only one part, without interactions), the same materials and the same loading condition. In particular, we chose a simplified version of the model that we were able to solve analytically, in order to compare the computational results with the analytical ones. • Then, we progressively refined the mesh of the simplified model. In particular, since De Saint Venant’s Principle enforces that the local stresses in one region do not affect the stresses elsewhere [31] we kept a global mesh size of an appropriate dimension and we performed a local refinement only in the area of interest. This way, we were able to increase the convergence of the solutions, without increasing the size of the overall problem. Nevertheless, to minimize mesh distortion, we gradually transitioned from the coarse regions to the more refined one, through appropriate local seeding, as shown in Figure 4.21. • For each mesh refinement, we evaluated the mean value of a certain quantity (namely the Von Mises Stress) in a region of interest. o Our region of interest is highlighted in green in Figure 4.21, and lies in the same location where the distal socket would lie in the non-simplified mode. At each mesh refinement, we made sure that the number of nodes in the region of interest would at least double. o The quantity of interest that we monitored was the Von Mises stress. • The quantity of interest would converge when two subsequent mesh refinements did not change the result “substantially”. We considered that the quantity converged when two subsequent mesh refinements caused an increase of the mean Von Mises stress that was lower than 1%. • In addition, we compared the computational result with the analytical one to determine its accuracy.

Global size = 6

Mesh 1

Mesh 2

From 6 to 4 From 6 to 4

2.3

Mesh 3

From 6 to 2.3 From 6 to 2.3

1.8

Mesh 4

From 6 to 1.8 From 6 to 1.8

Figure 4.21 – Different mesh densities. In pink we show the distribution of the local seeds, while in black the distribution of global seeds. The part displayed in green represents the location of the socket in the non- simplified model.

Results of the mesh convergence study are reported in chapter 6.1.1 in the results section.

4.4.4 Accuracy calculation After choosing a mesh size that would ensure stable results, we calculated the accuracy of these converged results (i.e. results produced by the mesh that resulted from the convergence study) by comparing them with the analytical results.

To calculate the analytical values of Von Mises stress, we drew the three-dimensional simplified model as an assembly of beams, to which we applied the same boundary conditions and loading of the simplified model. In particular:

- in point T we placed a hinge, which constrained translations in the three orthogonal directions while preserving all rotations; - in point B we placed a roller, which constrained translations in the x and y direction, while preserving translation in the z direction and rotations in all directions;

- in point B we applied a vertical force 퐹푧.

Due to 퐹푧, the roller generated two reaction forces, one directed along the x axis, 퐹푥, and the other directed along the y axis, 퐹푦. Similarly, the hinge reacted with three components of reaction force directed along the three reference axes. By solving the statics equations, we could easily see that the three reaction forces in T (hinge) had the same module, same direction but opposite sign as the three forces in B (roller). The composition of the three reaction forces gave the total component of the force, 퐹푇푂푇, which, as we intuitively predicted, was directed along the line connecting point B to point T, i.e. the roller to the hinge.

Figure 4.22 – Simplified analytical model that we used for accuracy calculation.

퐹푇푂푇 is the total component of the force that the ISO 10328:2016 standard prescribes to fall within certain loading levels (see Table 2.2). During static testing, WW a vertical force to the bottom spherical joint, we rewrote all the force value as functions of the vertical force component 퐹푧.

Afterwards, we calculated the internal actions (i.e. bending moment, torque and the normal forces) of the model as functions of 퐹푧, first in two-dimensional planes and then we reported them in the three-dimensional reference system.

Once we knew the internal actions, we calculated the normal and tangential components of the stress as functions of 퐹푧 at three z-values of the region of interest. After that, we used Von Mises criterion to combine the resulting normal and tangential stress components.

Finally, we evaluated the Von Mises stress for a 퐹푧 value corresponding to a P6 test loading level

(퐹푇푂푇 = 4425푁), at the three z-values in the region of interest.

The diagrams of the internal actions and the stresses within the analytical simplified model are reported in Appendix A, whereas the resulting accuracy of the converged results is reported in chapter 6.1.2 in the results section.

4.4.5 Fillets Since our location of interest (housing) had many sharp edges, we had to accurately model fillets, because otherwise the 90° angle between two surfaces would create a stress singularity in the finite element model, where theoretically the stress is infinite.

In the critical sharp edges, we assigned a value for fillet radius of 0.1푚푚, which was small enough to be considered “sharp” in reality, but large enough to avoid an ideal sharp edge in the finite element model, preventing a stress singularity from taking place in the location of interest.

Figure 4.23 – Sharp edges.

We assigned fillets not only to the housing’s edges but also to the connector’s complementary edges, to allow proper interaction between the two regions. Each region had eight sharp edges (Figure 4.24, Figure 4.25).

Figure 4.24 – Sharp edges in the location of interest of Part 1 (connector).

Figure 4.25 – Sharp edges in the location of interest of Part 2 (housing).

However, since fillets introduce a greater level of complexity during partitioning and mesh generation, we modelled only fillets on the edges labelled as (a), (b), (c) and (d). Among these edges, the most critical one was the outer distal edge (d), as failure initiated right at this edge, in the posterior area of the distal socket. Therefore, creating a fillet in this location was essential to obtain accurate values of stress. In addition, we also modelled fillets on edges (a), (b), and (c) to facilitate partitioning and ultimately mesh generation. On the contrary, we did not model fillets on edges (e), as we judged that failure took place because of the interaction between edge (d) of the connector and edge (d) of the housing, and not because of the interaction among edges (e) of the two counterparts.

In Table 4.1, we summarize the values of fillet radii that we assigned to the four edges of the connector and of the housing.

Table 4.1 – Fillets in the model, for both the connector and the groove in the housing.

Edge a b c d e Inner proximal Outer proximal Inner distal Outer distal Columns’ Description edge edge edge edge edges Fillet Yes Yes Yes Yes No Fillet 0.1푚푚 0.1푚푚 0.1푚푚 0.1푚푚 N/A radius

By way of example, in we show fillets on edge c and d of the connector.

Figure 4.26 – Edge d and c on connector without (left) and with (right) a fillet radius of 0.1mm.

In presence of fillets, generating a top-down hexahedral mesh becomes challenging, especially if the fillet radius is very small (as in this case). For this reason, to guide and facilitate hexahedral mesh generation in these conditions, we adopted the partition pattern displayed in Figure 4.27: for each fillet, we partitioned the component in two directions perpendicular to each other, each passing through one of the two profiles of the fillet, making sure that the two partitions were not tangent to the fillet. In some cases, we did not need to partition the fillet in both directions, as one would be enough.

Figure 4.27 – Partitioning in presence of fillets: the two partitions have to pass through the two fillet profiles, but they cannot be tangent to the fillet. In case of the connector, the wrong partitions is equivalent to not partitioning the geometry; in case of the housing, the wrong partitions will cause distorted elements during meshing.

4.4.6 Partitions and Mesh generation To be able to mesh the two parts of the reference model using a top-down (either swept or structured) meshing technique, we made extensive use of partitions.

Partitioning was quite challenging, because we wanted to create the least amount of partitions that would allow to mesh the part with a top-down meshing technique. To do so, we had to keep in mind the logic behind swept and structured meshing techniques, in order to guide and facilitate mesh generation [29].

For Part 1, we created ten partitions (Figure 4.28), two of which (partition 7 and partition 8) being of the “fillet type” described in the previous chapter (4.4.5). These partitions allowed us to mesh the entire geometry using a swept meshing technique, which creates a 2D mesh on one face of the region (source side), and then copies the nodes of that mesh along an edge (sweep path) until the final face (target side) is reached. This technique is very versatile and allows to mesh even complex geometries.

3 4

1 2

8 7

10 9

Figure 4.28 – Part 1: before partitioning Abaqus suggested a bottom-up meshing (orange); after partitioning, Abaqus allows to create a swept mesh everywhere (yellow). In red we display the dividing surfaces between adjacent regions.

On the other hand, partitioning Part 2 was not as straightforward. We first created fourteen partitions (Figure 4.29), five of which (partition 9 to 14) of the “fillet type” described in the previous chapter. These partitions allowed us to mesh the entire geometry using a swept meshing technique but gave us little control over seed deposition and ultimately over distribution and size of the elements in the mesh. Therefore, we created three additional partitions in the column of the setup (Figure 4.30).

1 2

7 6

4 8

10 9 11

13 12 14

Figure 4.29 – Part 2: before partitioning Abaqus suggested a bottom-up meshing (orange); after partitioning, Abaqus allows to create a swept mesh everywhere (yellow). In red we display the dividing surfaces between adjacent regions.

Figure 4.30 – Part 2: additional partitioning of the column of the setup to allow seeds deposition. In red we display the dividing surfaces between adjacent regions.

After partitioning the two parts, we assigned a mesh size compatible with the solution of the mesh convergence study (chapter 6.1.1 of the results):

- a coarser mesh with a global size of at least 6 in the non-interesting regions (setup); - a more refined mesh with a local size of at least 2.3 in the interesting regions, namely in the connector for Part 1 and in the distal socket for Part 2.

It is very important that these two regions of interest have a sufficiently refined mesh with a good quality, to ensure the accuracy of the results (stress distribution). In addition, these two parts are also the ones in contact, therefore, a coarse or bad quality mesh would not allow the convergence of the contact interactions, which will be discussed in chapter 0.

For Part 1, we assigned a global mesh size of 3 and a local size in the region of the connector of 2.3. The seeded part is shown in Figure 4.31.

Figure 4.31 – Part 1: global (black) and local (pink) seeds.

For Part 2, we had to try different strategies:

- We first applied a global size of 6 and a local size of 2 only in the region of the distal socket. However, we noted that the local seeds of the socket would propagate from bottom (distal socket) to top (entire column of the setup) and create a refined mesh almost everywhere, even in the non-interesting regions, causing the overall number of elements to increase excessively and unnecessarily (125011 elements, 135547 nodes). - To overcome seed propagation, we assigned a global size of 2 and a local size of 6 to all the regions of the setup. Nevertheless, this only inverted the direction of seed propagation, as the seeds would propagate from top (setup) to bottom (distal socket), causing a coarse mesh in this location, which was not admissible, being the region of interest. - To reduce this phenomenon, we assigned a global size of 2, a local size of 6 to all the regions of the setup and a local size of 2 only to the bottom of the distal socket, to counteract seed propagation in this location. We gladly noted that this strategy reduced seed propagation, while only slightly increasing the number of elements (86032 elements, 94201 nodes). - Finally, we introduced local seeds with a bias from 6 to 2 in the more proximal part of the distal socket, to allow a gradual and smooth transition from the coarser mesh regions to the more refined mesh regions. We also noted that these seeds would reduce the total number of elements in the part (60732 elements, 68361 nodes).

By summarizing, the final seed distribution in Part 2 consisted of a global size of 2, a local size of 6 in the setup regions, a local size of 2 on the bottom surface of the distal setup and a local size transitioning from 6 to 2 in the proximal region of the distal socket (Figure 4.32).

Figure 4.32 – Part 2: global (black) and local (pink) seeds.

After assigning the mesh size to the two parts, we generated the mesh. We used the advancing front algorithm, which is one of the two possible algorithms available for swept meshing, and it works by generating quadrilateral elements at the boundary of the region and then by continuously generating quadrilateral elements as it moves systematically to the interior of the region. Since elements produced by advancing front algorithm follow the seeds positioned by the user, this algorithm allows to generate elements of a more uniform size with a more consistent aspect ratio, and it also allows to reduce shear between adjacent regions [29].

For the sake of comparison, we also tried to switch to the other algorithm, the medial axis algorithm, which works by first decomposing the region to be meshed into a group of simpler regions, to which it applies a structured meshing. However, we noted that this latter algorithm would produce meshes of a lower quality.

Sometimes, when using the advancing front algorithm, Abaqus automatically switches to mapped meshing, if it finds it appropriate, which is a structured meshing algorithm applicable to four sided regions. Nevertheless, we prevented Abaqus from doing so, as we noted that, because of the geometry of our part, mapped meshing would only create lower quality meshes.

The resulting mesh of Part 1 is shown in Figure 4.33, whereas the one of Part 2 in Figure 4.34.

Figure 4.33 – Part 1: swept mesh with advancing front meshing but not mapped meshing.

Figure 4.34 – Part 2 – final mesh: swept mesh with advancing front meshing but not mapped meshing.

Also note that when first generating the mesh on Part 2, we noted that the mesh was severely distorted (Figure 4.35, left), due to the presence of constrained seeds automatically positioned in correspondence of construction vertices. To improve the mesh, we used the virtual topology tool to eliminate these vertices. The resulting mesh was no longer distorted (Figure 4.35, right).

Figure 4.35 – Mesh with severely distorted elements (highlighted in pink) due to constrained seeds positioned in correspondence of construction vertices.

In Table 4.2 and Table 4.3, we summarize the mesh parameters for Part 1 and Part 2, and the resulting number of nodes and elements within each mesh.

Table 4.2 – Mesh parameters for Part 1.

Part 1 Control parameter Controlled region Value Global size All regions without local seeds 3 Local seeds Connector 2.3 Swept, advancing front (do not allow Meshing technique Everywhere mapped meshing) Number of nodes - 18596 Number of elements - 14575

Table 4.3 – Mesh parameters for Part 2.

Part 2 – Final mesh Control parameter Controlled region Value Global size All regions without local seeds 2 Local seeds Top part of the setup 6 Proximal part of distal socket 6 to 2 Bottom surface of the housing 2 Swept, advancing front (do not allow Meshing technique Everywhere mapped meshing) Number of nodes - 68361 Number of elements - 60732

4.4.7 Mesh verification Abaqus highlights any element that is distorted, i.e. that largely deviates from its ideal shape. In particular, the ideal shape of a hexahedral element is a regular cube (equal edges and square angles). The criteria to judge whether a hexahedral element is distorted or not are: - Small or large face corner angle: ideally, the angle between two faces should be 90°; its value becomes critical below 10° (small angle) or above 160° (large angle). - Aspect ratio, i.e. the ratio between the longest and the shortest edge; ideally, it should be 1; values of aspect ratio higher than 10 are usually critical. - Geometric deviation factor, i.e. a measure of how much an edge deviates from its original geometry; it is found by dividing the maximum gap between an element edge and its parent

geometric face or edge by the length of the element edge. Critical values of geometric deviation factor are above 0.2. - Short or long edge: Abaqus highlights elements with edges shorter or longer than a specified value.

Results of the mesh verification are reported in chapter 6.1.3 in the results section.

4.5 Material property assignment In the Property module, we assigned the material properties to the two parts. Thanks to the partitions, we were able to assign different materials properties to components that belonged to the same part.

In reality, the materials that formed the model are many, and namely:

1. the setup is made of different materials, but predominantly is made of steel, titanium and aluminum; 2. the connector is made of aluminum, in particular of Al-6061-T6 (i.e. Al-6061 with T6 temper); 3. the socket is made of Carbonium Nylon (dehumidified and 100% infill).

The mechanical properties of Al-6061-T6 and Carbonium Nylon are summarized in Table 4.4 and their stress-strain curves are displayed in Figure 4.36. The properties of Al-6061-T6 derive from the material database of MatWeb11 [32] and from literature data [33], whereas the properties of Carbonium Nylon derive from the mechanical characterization that we performed on the dehumidified 100% infill Carbonium Nylon specimens, for which the results are available in chapter 5.4.1.

Table 4.4 – Mechanical properties of the materials used in the model.

Elastic properties Material Behavior Young’s modulus [MPa] Poisson’s ratio Carbonium Nylon Elastoplastic 4676 0.42 Al-6061-T6 Elastoplastic 68900 0.33

11 http://www.matweb.com/

Figure 4.36 – Stress-strain curves of Carbonium Nylon (dehumidified at 100% and 50% infill) and Al-6061- T6.

In modelling the materials in Abaqus, we made the following assumptions:

- Even though from Table 4.4 and Figure 4.36 we observe that both the Al-6061-T6 and the Carbonium Nylon display an elastoplastic behavior, since the Al-6061-T6 is much stiffer than Carbonium Nylon (by an order of magnitude), we decided to only model Carbonium Nylon as elastoplastic and to model Al-6061-T6 as simply elastic. - We treated both materials as isotropic. This assumption may be accurate for Al-6061-T6, but it is not for Carbonium Nylon, which has a strong anisotropic nature. Nevertheless, we made this choice to simplify the model, in view of reducing the computational cost of the simulation. - For the sake of simplicity, we assumed that the setup was made of the same aluminum of the connector. We made this assumption because the materials of the setup (steel, titanium and aluminum) are much stiffer than Nylon Carbonium and because we are not interested specifically in the stress-distribution within the setup.

Therefore, we only had to create two materials, i.e. Carbonium Nylon and Al-6061-T6, that from now on we will label as Material-1-NC and Material-2-Al, respectively.

We created Material-1-NC as isotropic and elastoplastic, to which we assigned both elasticity data, namely Young’s modulus and Poisson’s ratio (

1. Table 4.5), as well as plasticity data (Table 4.6), derived from the material characterization that we performed on Carbonium Nylon (results are provided in chapter 5.4.1). 2. On the contrary, we created Material-2-Al as isotropic and purely elastic, to which we assigned only elasticity data (Table 4.7) derived from literature data.

Then, as shown in Figure 4.37, we assigned Material-1-NC to all the regions that formed the socket, and Material-2-Al to all the regions of Part 1 (bottom setup and connector) and to all the regions that formed the top setup of Part 2.

Figure 4.37 – Material properties assignment: the regions displayed in grey, namely the setup and the connector, were assigned the aluminum properties (Material-2-Al), whereas the components in green, namely the socket, were assigned the Carbonium Nylon properties (Material-1-NC).

Table 4.5 – Elasticity data for Carbonium Nylon.

Young’s modulus Poisson’s ratio 4676 0.42

Table 4.6 – Plasticity data for Carbonium Nylon.

Post-yield stress Plastic strain 38.08293232 0 44.27649554 0.003159874 48.55633173 0.006319748 51.64779261 0.009479621 54.0669711 0.012639495 55.92181794 0.015799369 57.48899956 0.018959243 58.80768439 0.022119117 59.84605073 0.025278991 60.6140711 0.028438864

Table 4.7 – Elasticity data for Al-6061-T6.

Young’s modulus Poisson’s ratio 68900 0.33

Please note that the plasticity data of Table 4.6 consists in a set of 10 true stress-strain values, in which the true stress is the post yield-stress (from yield point to ultimate tensile strength) and the true strain is the plastic strain (strain purified from its elastic component). These data have been extrapolated from the mean stress-strain curve of NC100D specimens, upon

- performing the conversion from engineering to true stress-strain values, - eliminating the elastic strain component from the total strain, to obtain values of pure plastic strain.

All details are provided in chapter 3.6.5.

From these data, Abaqus (in Abaqus/Standard) interpolates linearly between the given data points to obtain the material’s response, and it assumes that the response is constant outside the range deﬁned by the input data. Any number of points can be used to approximate the actual material behavior, thus, in theory, it is possible to use a very close approximation of the actual material behavior [23]. However, we preferred using only 10 points to save computational expenses.

Modelling the plasticity behavior of Material-1-NC introduced the first source of nonlinearity in the problem.

4.6 Interactions The contact interactions between our parts were probably the most problematic aspect of our simulations. Since Abaqus has many difficulties in dealing with contact interactions in static analyses [23], [26], it was crucial to reduce as much as possible the number of surfaces in contact during the simulation.

As described in chapter 4.3.1, the original model consisted of six parts (top and bottom loading cylinders, top and bottom parts of the setup, distal socket and connector). These parts interacted with one another leading to the following contact interactions (Figure 4.38):

1. Contact interaction between top loading cylinder and top setup: the spherical head of the top loading cylinder can rotate inside the cup of the top setup; 2. Contact interaction between bottom loading cylinder and bottom setup: the spherical head of the bottom loading cylinder can rotate inside the cup of the bottom setup; 3. Contact interaction between top setup and distal socket: the socket is attached to the top part of the setup through a simulated residual limb model described in chapter 2.4.2; 4. Contact interaction between connector and distal socket: the connector is embedded in the distal part of the socket; thus, it is in contact with the entire groove of the housing; 5. Contact interaction between connector and bottom setup: the connector and the socket-to- setup attachment plate are bolted together by four M6 countersunk screw; 6. Contact interaction between distal socket and bottom setup: the bottom surface of the distal socket is in contact with the top surface of the socket-to-setup attachment plate.

Figure 4.38 – Contact interactions in the real model.

By reducing the number of parts in the model to only two parts, namely Part 1 – Bottom part of the setup plus Connector (Figure 4.8), and Part 2 – Top part of the setup plus Socket (comprising the

92 housing) (Figure 4.9), we were able to drastically reduce the number of interactions to the ones listed below:

1. Contact interaction between connector and distal socket: the connector is embedded in the distal part of the socket; thus, it is in contact with the entire groove of the housing (Figure 4.39); 2. Contact interaction between the distal socket and bottom setup: the bottom surface of the distal socket is in contact with the top surface of the socket-to-setup attachment plate (Figure 4.41).

In addition, always in the Interaction module, we constrained each reference point to the internal surface of the corresponding loading cup (Figure 4.43), namely:

A. Coupling constraint between RP-1 and internal surface of top cup; B. Coupling constrained between RP-2 and internal surface of bottom cup.

Contact interactions and constraints are defined differently in Abaqus [29]. We will first describe the contact interactions and then the constraints that we created to simulate the contacts listed above.

4.6.1 Contact interactions Figure 4.39 displays the two contact interactions that we modelled. The description of the contact definition for both interactions is summarized in

Table 4.8.

Figure 4.39 – Contact interactions in the model: (left) contact interaction 1, between the connector and the distal socket; (right) contact interaction 2, between the distal socket and the bottom setup.

Table 4.8 – Contact interactions.

Parameter Value Type Interaction Approach Surface-based (contact pairs) Algorithm Master-slave Discretization (contact formulation) Surface-to-surface Sliding formulation (contact tracking approach) Small sliding Interaction Property Contact > normal behavior > hard contact Adjust Specify tolerance for adjustment zone = 0.001 Creation step Initial step Surface smoothing Automatic

- To create contacts, we used the surface-based approach, which uses surfaces to define contact. This approach is based on the contact pairs algorithm: for each contact interaction, we had to define the pairs of surfaces of the part instances that came in contact during the simulation. - Contact pairs in Abaqus/Standard use a pure master-slave contact algorithm: nodes on one surface (the slave) cannot penetrate the segments that make up the other surface (the master), as shown in Figure 4.40. The algorithm places no restrictions on the master surface, which can penetrate the slave surface between slave nodes.

Figure 4.40 – Pure master-slave contact algorithm: the master surface can penetrate the slave surface.

Due to the strict master-slave formulation, it is crucial to select the slave and master surfaces correctly in order to achieve the best possible contact simulation. Some simple rules to follow are:

1. The master surface should be the one with the coarser mesh; 2. If the mesh densities are similar, the master surface should be the surface with the stiffer underlying material; 3. If mesh densities and stiffnesses of the underlying materials are similar, the master surface should be the more extended one.

In Figure 4.41 and Figure 4.42, we display our choice of master and slave surfaces for the two contact interactions of our model.

Figure 4.41 – Master (displayed in red) and slave (displayed in pink) surfaces for contact interaction 1.

Figure 4.42 – Master (displayed in red) and slave (displayed in pink) surfaces for contact interaction 2.

- As contact formulation, i.e. the discretization method that Abaqus uses to apply conditional constraints at various locations on interacting surfaces to simulate contact conditions, we used the surface-to-surface discretization. The surface-to-surface discretization, differently from the alternative node-to-surface discretization, considers the shape of both the slave and the master surfaces in the region of contact constraint. This formulation enforces contact conditions in an average sense over regions nearby slave nodes rather than only at individual slave nodes. The averaging regions are approximately centered on slave nodes, so each contact constraint will predominantly consider one slave node but will also consider adjacent slave nodes. We chose the surface-to-surface discretization because it is a good formulation to describe contact between two deformable surfaces. In fact:

• It does not allow large, undetected penetrations of master nodes into the slave surface, because the contact conditions are defined in an average sense over finite regions nearby slave nodes. • It provides more accurate stress and pressure results than node-to-surface discretization if the surface geometry is reasonably well represented by the contact surfaces (if the mesh is sufficiently refined). In fact, since node-to-surface discretization simply resists penetration of slave nodes into the master surface, forces tend to concentrate at these slave nodes, leading to spikes and valleys in the distribution of pressure across the surface. On the other hand, surface-to-surface discretization resists penetrations in an average sense over finite regions of the slave surface, which has a smoothing effect. • Contact using surface-to-surface discretization is also less sensitive to master and slave surface designations than node-to-surface contact. - We assigned the same contact interaction property to both contact interactions of our model, namely the normal hard contact, which accounts only for the tangential behavior that arises during contact, but not for the tangential one (friction). In a first analysis, we neglected the tangential behavior for two reasons: • first, because we judged that the normal forces better describe the type of interactions taking place in our model, whereas the contribution of friction was only marginal; • second, because running a simulation without friction allows to solve the contact interactions completely and correctly, and to check whether the contacts have been properly described. On the contrary, shear forces that would arise because of friction, would aid the convergence of the contact simulation and they could lead to convergence even though the contacts are not perfectly described, ultimately leading to inaccurate results. - As tracking approach (sliding formulation) to account for relative motion of the two interacting surfaces in mechanical contact, we chose the small-sliding formulation, which assumes that, although two bodies may undergo large motions, there will be little relative sliding of one surface along the other. We chose the small-sliding formulation, because we judged that the sliding between the parts in contact would be very small (the connector is completely embedded in a groove that is its exact negative counterpart, and large sliding between the socket and the bottom setup is impeded to the presence of the connector), and, most importantly, to reduce the computational cost of the simulation.

4.6.2 Constraints In the interaction module, we used constraints to constrain the reference points to each internal surface of the loading cups, as shown in Figure 4.43.

Figure 4.43 – Coupling constraint between each Reference Point and the corresponding setup cup.

Differently from the constraints defined in the Assembly module, the ones defined in the Interaction module define constraints on the analysis degrees of freedom, and not only to the initial position of the instances.

In particular, to constraint the RPs to the cups’ internal surfaces we used Coupling constraints, which allow to constrain the motion of the nodes on a surface (the coupling nodes) to the motion of a single point (reference node). There are two types of coupling constraints we could choose among, namely kinematic and distributing (either continuum or structural) constraints.

- Kinematic coupling is enforced in a strict master-slave approach; degrees of freedom (dofs) at the coupling nodes are eliminated, and the coupling nodes will be constrained to move with a rigid body motion with the reference node; no relative motion is allowed between the coupling nodes. This type of constraint is equivalent to the Multi Point Constraint (MPC) of beam type. - Distributing coupling, on the other hand, is enforced in an average sense: not all the dofs at the coupling nodes are eliminated, but only the rotational ones. The constraint is enforced by distributing loads such that the resultants of the forces at the coupling nodes are equivalent to the forces and moments at the reference node, and force and moment equilibrium of the distributed loads about the reference point is maintained. Basically, in this type of coupling,

the coupling nodes are not rigidly constrained to the reference node, but they can move relative to each other according to the stiffness properties of the underlying material, and in a way that the equilibrium condition on the distributed loads in maintained.

While the distributing coupling leads to more accurate results, the kinematic coupling is less accurate but more stable. In general, we always used the distributing coupling of type structural. However, for some model (Model 4) where convergence was difficult, we stitched to the kinematic coupling.

4.7 Loading In the Load module, we applied the load and boundary conditions to the model.

As extensively explained in chapter 2.4.2, in the Willow Wood setup, the load is applied at the bottom (distal) loading point, point B, and is measured at the top (proximal) loading point, point T. The transmission of loads in these two points takes place through the two spherical joints (loading cylinder with a spherical head that rotates inside a cup), and, consequently, the load lies on the line that connects the two spherical joints. For the sake of clarity, we reported the setup configuration in Figure 4.44.

Figure 4.44 – Figure on the left is Figure 2.22 and refers to the WW setup configuration; Figure on the right is the simplification of this model used in chapter 4.4.3 to calculate the accuracy of the mesh results.

Since the top spherical joint can rotate in all directions, but it cannot translate in any direction, it can be modelled as a hinge. Differently, since the bottom spherical joint can rotate in all directions and translate along the vertical direction, but not in the two horizontal directions, it can be modelled as a roller that can translate along the vertical direction.

The force that WW applies at the bottom loading point (point B, RP-2) is vertical, but the one measured at the top loading point (point T, RP-1) is tilted, as it is directed along the line that connects the roller to the hinge. The vertical force applied by WW is only a fraction of the total force acting on the system as a result of the two spherical constraints. In Table 4.9, we report all the vertical force values depending on the loading level prescribed by the standard ISO 10328:2016.

Table 4.9 – Vertical forces applied during static testing.

Test loading level P3 P4 P5 P6 P7 P8 Total force (directed along the line connecting the two spherical joints) = ultimate static test force 2790 3623 4025 4425 4840 5250 prescribed by the standard [N] Vertical force applied during static testing (total 2771 3598 3997 4394 4806 5214 force projected along the vertical direction) [N]

The vertical force values for each loading level were found by using the relationships (9.3) and (9.4) of Appendix A, which we report below for convenience:

74 2 21 2 퐹 = √퐹2 + 퐹2 + 퐹2 = 퐹 ∙ √( ) + ( ) + 1 ≅ (1.007) ∙ 퐹 (4.2) 푇푂푇 푥 푦 푧 푧 650 650 푧

퐹푇푂푇 퐹푇푂푇 퐹푧 = ≅ 74 2 21 2 1.007 (4.3) √( ) + ( ) + 1 650 650

Also please note that from equation (4.3), it follows that the angle between the vertical force and the load line is given by:

휃푧 = arccos(퐹푧/퐹푇푂푇) ≅ 6.79° (4.4)

According to WW setup, we applied the vertical force to the bottom reference point (RP-2), which corresponds to point B of the WW setup (red arrow in Figure 4.45).

Then, we applied the boundary conditions to the two RPs, so that each RP – cup’s internal surface compound would behave as a spherical joint. In particular:

- we constrained all translational dofs of RP-1, so that it would behave as a hinge;

- we constrained only the translations along direction x and y of RP-2, so that it would behave as a roller that could translate along the z direction.

As boundary condition property, we used the Displacement/Rotation type, which allows to individually constrain all six possible dofs: translational dofs, i.e. translations along the three reference directions (U1, U2, U3), and rotational dofs, i.e. rotations about the three reference directions (UR1, UR2, UR3), where the numbering 1, 2 and 3 is related to the x, y and z axis, respectively, of the global reference system.

Also note that, for computational reasons, in some simulations we introduced an additional boundary condition, namely BC-add. We will extensively describe its behavior, application and function in the chapter 4.8.2.

Table 4.10 – Boundary conditions applied to the model.

Boundary condition Applied to Degrees of constraint BC-1 RP-1 U1, U2, U3 BC-2 RP-2 U1, U2 BC-add Bottom setup U1, U2, U3

Figure 4.45 – Loading condition of the model.

100

4.8 Static analysis We divided the complete load history of the simulation into a number of steps. Each step is an arbitrary period of “time”, that we set equal to 1, for which Abaqus calculates the response of the model to a particular set of loads and boundary conditions. In particular, we only created general steps, because the response of the model during a general analysis procedure (general step) may be either nonlinear or linear. Since we are dealing with a nonlinear analysis, we only used general steps. The source of nonlinearities of our simulations are primarily introduced by the plasticity of Material- 1-NC, but also by the additional boundary condition, as we will explain in hereunder in chapter 4.8.2.

4.8.1 First analysis In principle, we only created one step. Therefore, the total number of steps was two: Initial Step and Step-1 (static, general).

- We created the interactions (Int-1 and Int-2) and boundary conditions (BC-1 and BC-2) in the Initial Step, and we left them unchanged (they propagated) in Step-1. - We created the load (vertical concentrated force of value 4394푁) in Step-1 and we applied it with a ramp, as displayed in Figure 4.46.

Figure 4.46 – First loading condition of the reference model.

However, the simulation would not converge, because Abaqus had to simultaneously deal with the convergence of the contacts, while applying the load.

4.8.2 Second analysis Since one step in addition to the initial one was not enough to ensure convergence, we created four (static, general) steps in addition to the initial one.

101

- We created the interactions (Int-1 and Int-2) and boundary conditions (BC-1 and BC-2) in the Initial Step, and we left them unchanged (they propagated) in all subsequent steps (Step- 1, 2, 3 and 4). - We introduced the additional boundary condition (BC-add) in the Initial Step, we left it unchanged in Step-1, we modified it in Step-2 and 3, and we deactivated it in Step-4, as displayed in Figure 4.47 (orange curve). - We created the load (vertical force with value 4394푁) in Step-1 and we applied it with a ramp from 0 to its maximum value in Step-1, then we kept its value constant in Step-2, 3 and 4, as displayed in Figure 4.47 (blue curve).

Having more steps, allowed us to switch from force control to displacement control and aid convergence:

- In Step-1, by applying the additional boundary condition, we completely constrained all degrees of freedom of the bottom part of the setup; in this same step, we applied the entire force with a ramp from 0 to its maximum value. This way, the parts could not move, even though the force was present, and Abaqus could focus solely on the convergence of the contacts. - Then, once the contacts had converged, we started to gradually remove the degrees of constraint of the additional boundary condition: in Step-2, we removed the constraint U3=0, in Step-3 we removed the constraint U2=0 and in Step-4 we removed the last constraint U1=0, i.e. we deactivated the boundary condition. In all these steps, the force was constant and equal to its maximum value.

Figure 4.47 – Second loading condition of the reference model.

102

4.8.3 Third analysis Finally, to observe more loading levels within the same simulation, we created several steps, and we observed the resulting stress values at each step. In fact, during static analyses, only the final “time” instants of general steps have a physical significant and deliver physical values of stresses and strains. All the other instants at the different iterations are only mathematical calculations aimed at convergence.

In particular, we created 13 steps. The first five steps, from Initial Step to Step-4, are exactly the same as in the second analysis, with the only difference that we applied a force with a value of 10푁. Their objective is to ensure the convergence of contacts.

Once the contacts had converged, from Step-5 to Step-13, we gradually increased the amplitude of the force from 10 to 4400푁, as displayed in Figure 4.48. In particular, in Step-10 the force had an amplitude of 2770푁, in Step-11 an amplitude of 3600푁, in Step-12 an amplitude of 4000푁 and finally in Step-13 an amplitude of 4400푁. These values roughly correspond to the vertical forces that need to be applied to reproduce a P3, P4, P5, and P6, respectively, loading levels, according to ISO 10328:2016 and as shown in Table 4.9 of chapter 4.7

Figure 4.48 – Third loading condition of the reference model.

We created this loading condition, to be able to see from the same simulation whether the model would yield or break test at each loading levels.

103

4.8.4 Data analysis Once we ran the simulation and obtained the results, we had to judge whether the model had passed or failed the static test.

Using the third static analysis (chapter 4.8.3), we applied discrete values of force, corresponding to the loading levels P3, P4, P5 and P6. According to ISO 10328:2016 and the Ohio Willow Wood Company, the test can be considered passed if the structure can resist a P6 loading level.

Figure 4.49 – Socket testing for Carbonium Nylon, displayed by the blue curve labeled as “I3”. P6 loading level is displayed by the grey horizontal line.

Therefore, for each loading level, we extracted the values of stress from the nodes in our location of interest in the model. Then, we compared these values with the critical stresses of the underlying material that we define in the Property module. As describe in chapter 4.5, the material properties derived from the material characterization that we performed. Results of mechanical characterization of dehumidified Carbonium Nylon (both 100% and 50% infill) are displayed in Figure 4.50 and Table 4.11. For the reference model, we referred to the Carbonium Nylon with 100% infill.

Table 4.11 – Average values of measured parameters for each specimens’ set.

Ultimate Ultimate Young’s Yield tensile Specimens’ Poisson’s tensile modulus stress force at set ratio stress [MPa] [MPa] break [MPa] [N] NC100D 4676 0.42 37.48 58.34 2328 NC50D 3095 - 23.23 42.35 1645

104

Figure 4.50 – Mean stress-strain curves for NC100D specimens and NC50D specimens.

From Figure 4.50, we see that the material has an elastoplastic response. However, since the plastic area is not very extended (plastic deformations around 3%), we cannot categorize this material neither as completely ductile nor as completely brittle [25].

- For completely ductile materials, the failure condition is represented by yielding, therefore the limit stress is the yield stress; - For completely brittle materials, the failure condition is represented by fracture, therefore the limit stress if the ultimate tensile stress.

To judge whether a component fails under a complex three-dimensional state of stress, we need to use failure criteria, which compare an equivalent quantity representative of danger with the limit stress of the material. The equivalent quantity representative of danger is precisely called “equivalent” because it has to translate the complex three-dimensional state of stress in each point of the component into an equivalent single stress value that can be easily compared with the limit stress of the material.

If the material was completely ductile, we would have to use the Von Mises criterion of failure, which provides as equivalent stress a composition of the three principal stresses, namely:

∗ 2 2 2 (4.5) 휎푉푀 = √휎퐼 + 휎퐼퐼 + 휎퐼퐼퐼 − 휎퐼휎퐼퐼 − 휎퐼휎퐼퐼퐼 − 휎퐼퐼휎퐼퐼퐼 where 휎퐼 > 휎퐼퐼 > 휎퐼퐼퐼 are the Maximum, Mid and Minimum Principal stresses, respectively.

Note that, being a square root, the equivalent Von Mises stress is always positive, even if it accounts for both tension and compression.

105

In alternative, we could also use the Guest-Tresca criterion of failure, which provides as equivalent stress the difference between the Maximum and the Minimum Principal stresses, namely:

∗ 휎퐺푇 = 휎퐼 − 휎퐼퐼퐼 (4.6)

This criterion is very similar but more restrictive than the Von Mises criterion.

If the material was completely brittle, we would have to use the Galileo-Rankine-Navier criterion of failure, which provides as equivalent stress the maximum principal stress for traction and the minimum principal stress for compression. Traction is more dangerous than compression, especially for brittle materials, because it causes crack opening, whereas compression causes crack closing. Therefore, we will only consider the maximum principal stress, namely:

∗ 휎퐺푅푁 = 휎퐼 (4.7)

Each equivalent stress is compared with the limit stress of the material. Failure does not take place in a structure, if the equivalent stresses in all the point of the structure are smaller than the limit stress of the material by an arbitrary coefficient, 휂, called the safety factor; the higher 휂, the less likely failure occurs. 휎 휎∗ < lim (4.8) 휂 where 휎∗ is the equivalent stress, i.e. Von Mises or Guest-Tresca for ductile materials and Galileo-

Rankine-Navier for brittle materials; 휎lim is the limit stress of the material, i.e. yield stress for ductile materials and ultimate tensile stress for brittle materials [34],[35].

Since we assessed our material as neither completely ductile and nor completely brittle, we will consider both the Von Mises stress as well as the Galileo-Rankine-Navier (Maximum Principal) stress distribution in all the nodes inside the region of interest of our model. Our region of interest is the distal posterior part of the groove inside the distal part of the socket, as we expect the highest stresses in this location (Figure 4.51).

Figure 4.51 – Region of Interest (ROI) of the reference model.

106

To analyze the stress data, we selected all the nodes inside our region of interest and we separately plotted the Von Mises and Maximum Principal stress distribution over the nodes in the region of interest. Then, for each loading level:

- we found the maximum value of both Von Mises and Maximum Principal stress and we calculated the percentages that they represented of the material yield stress and ultimate tensile stress, respectively; - we calculated the percentage of nodes for which the Von Mises stress overcame the yield stress and the percentage of nodes for which the Maximum Principal stress overcame the ultimate tensile strength.

4.9 Alternative models to enhance mechanical properties As described in the visual abstract (Figure 3), results of the Finite Element Analysis can be used to improve the geometry of the distal socket and to compare different materials. In the chapters below, we illustrate five possible alternatives to the reference model.

For each model, we performed a data analysis as described in chapter 0 for the Reference Model. In addition, we compared these five additional models with the Reference one, in terms of:

- percentage variation of the maximum value of Von Mises stress; - percentage variation of the percentage of nodes for which the Von Mises stress overcame the yield stress and for which the Maximum Principal stress overcame the ultimate tensile stress; - percentage variation of safety factor (if applicable).

4.9.1 Model 1 Model 1 is identical to the Reference Model, except for having the housing rotated by 90° with respect to the longitudinal axis. This way, the thicker edges of the housing were moved from the mediolateral direction to the anteroposterior direction, where the bending moment and consequently of stress were highest.

107

Figure 4.52 – The housing rotated by 90°.

Figure 4.53 – Reference model (left) versus Model 1 with Housing rotated by 90°.

After modifying the geometry of the housing, we generated again the mesh on Part 2, without changing the seed distribution. The resulting mesh is in fact almost identical to the mesh of the reference model.

Table 4.12 – Mesh properties for Part 2 od Model 1.

Part 2 Number of nodes 68426 Number of elements 60800 4.9.2 Model 2 Model 2 differs from the Reference Model only for the geometry of the housing: we preserved the geometry of the groove, while changing the geometry of the housing, passing from a spline shape to a round shape, to have a constant a uniform edge all around the groove. In particular, we chose a

108 bottom diameter of 82푚푚; this way, the edges around the groove were as thick as the thicker ones in the reference model.

Figure 4.54 – Round housing with external bottom diameter of 82mm.

Figure 4.55 – Reference Model (left) versus Model 2 (right) with round housing, with diameter of 82mm.

After modifying the geometry of the housing, we generated again the mesh on Part 2, without changing the seed distribution. The resulting mesh is slightly different to the mesh of the reference model, because overall geometry of the distal socket has slightly increased.

Table 4.13 – Mesh properties for Part 2 od Model 2.

Part 2 Number of nodes 76510 Number of elements 68159 4.9.3 Model 3 Model 3 differs from the Reference Model by the value of fillet radius in the critical edges. In fact, in this model we tried to reduce the notch effect by increasing the fillet radii of the connector and the groove of the housing. We wanted to investigate to what extent a larger fillet radius in the critical

109 locations would reduce stress concentration. However, for geometrical constraints, we could only increase it from 0.1푚푚 to 0.3푚푚.

Figure 4.56 – The fillet radius of both the connector (left) and the groove inside the housing was increased from 0.1 to 0.3mm.

After modifying the geometry of the housing, the part was regenerated. For Part 2, the mesh was automatically regenerated without changing the seed distribution. On the contrary, for Part 1 the mesh could not be automatically regenerated. We had to reduce the seed size in the connector from 2.3 to 2, in order to obtain an acceptable mesh in the fillet areas. The resulting meshes are not very different from the mesh of the reference model.

Table 4.14 – Mesh parameters of Part 1 of Model 3.

Part 1 Control parameter Controlled region Value Global size All regions without local seeds 3 Local seeds Connector 2 Swept, advancing front (do not allow Meshing technique Everywhere mapped meshing) Number of nodes - 22210 Number of elements - 17691

Table 4.15 – Mesh parameters of Part 2 of Model 3.

110

4.9.4 Model 4 Model 4 is the one that differs the most from the Reference Model. In fact, in this model, we created a new geometry for the connector, which consequently affected the geometry of the groove within the housing. This new design is axisymmetric and, more importantly, is modelled with a very large fillet in its outer distal edge, having a radius of 3푚푚. We hoped that such a large fillet radius would drastically reduce the stress concentration due to notch effect.

Figure 4.57 – Geometry of the new connector of model 4.

Figure 4.58 – Geometry of the new housing of model 4.

Since for this model we completely changed the geometry of the connector and of the housing, we could rely on the regeneration features of Abaqus. On the contrary, we had to create new partitions and generate a new mesh. This time, we needed less partitions to pass from bottom-up to top-down meshing, and there were many more regions that could be meshed with a structured technique. This is an indication that the geometry was now more regular, and therefore easier to mesh with more regular patterns.

On both Part 1 and Part 2, we generated a mesh having the same parameters (meshing technique, global and local seeds) of the reference model. Of course, since this time the geometry had

111 significantly changed, the resulting mesh looked different, and had less elements, as shown in Figure 4.59.

Figure 4.59 – Mesh of Part 1 (left) and Part 2 (right) of model 4.

Table 4.16 – Mesh parameters of Part 1 of Model 4.

Table 4.17 – Mesh parameters of Part 2 of Model 4.

4.9.5 Model 5 Model 5 has the same geometry as the Reference model, but the distal socket was assigned a different material, namely a material having the mechanical properties of the dehumidified Carbonium Nylon

112 with 50% infill instead of 100% infill. Everything else is the same as in the Reference Model. The mechanical properties of the dehumidified Carbonium Nylon with 50% infill are displayed in Figure 4.60 and Table 4.18, under label “NC50D”, and derive from the results of material characterization (chapter 5.4.1). This material is less stiff and less mechanically resistant than the 100% infill material.

Figure 4.60 – Mean stress-strain curves for NC100D specimens and NC50D specimens.

Table 4.18 – Average values of measured parameters for each specimens’ set.

Note that the socket that underwent the static testing at the Ohio Willow Wood Company was made of a very similar material, namely a non-dehumidified Carbonium Nylon with 50% infill. We could not virtually reproduce this same test, because we were not able to characterize the non-dehumidified form of Carbonium Nylon, as explained in chapter 3.5.1.

Even if the two materials are not the same, we could still use the results of virtual testing of this model to make some considerations regarding of the experimental results.

113

5 Results – Material characterization

5.1 3D printed specimens We printed five sets of specimens, namely:

By way of example, in Figure 5.1, we display the specimens’ set NC100D after 3D printing. Note that all specimens’ sets look like this specimens’ set, because they have the same geometry and are made of the same material. The color black is due to the presence of carbon fibers in the matrix of nylon.

Figure 5.1 – Carbonium Nylon specimens positioned on the printer plate of the delta robot slicer (a) and of the WASP 4070 Industrial 3D printer.

After 3D printing the specimens, we had to remove the layers of skirt that were used to improve adhesion to the printer plate (see definition of skirt in Table 2.1 of chapter 2.3.2). As specified in

114

Table 3.2 to print the specimens, we used 1 layer of skirt with outlines. After printing the specimens, we removed them using a cutter (Figure 5.2).

Figure 5.2 – Carbonium Nylon specimens on the printer plate (left) and ready for static testing, upon removal of the brim layer (right).

5.2 Dimensional control results The summarized results of the dimensional control analysis are reported in Table 5.1 and Table 5.2. For the integral results, please refer to Appendix C.

Table 5.1 – Mean values of accuracy and coefficient of variation for each 3D printed specimens’ set. The values of accuracy highlighted in red do to not comply with the accuracy requirments of ISO 527-2, i.e. they exceed the limit values of accuracy, also displayed in table.

Mean accuracy [%] Mean coefficient of variation

Specimens’ set 푏1 ℎ 푏2 푏1 ℎ 푏2 NC100D 1.00 −1.17 0.63 0.19 0.01 0.12 NC50D −0.47 −2.42 −0.15 0.31 0.00 0.15 NC100 ퟒ. ퟕퟔ 0.08 ퟒ. ퟐퟕ 0.43 0.00 0.05 NC50 ퟒ. ퟔퟓ −0.17 ퟐ. ퟒퟐ 0.23 0.00 0.02 Limit value of accuracy [%] ±2.00 ±5.00 ±1.00

Differently from the dehumidified specimens, whose actual dimensions fell around the nominal ones (high accuracy), the non-dehumidified specimens underwent a marked dilation, especially along direction x. The dilation caused the actual dimensions of the specimens to overcome the nominal dimensions and, for widths 푏1 and 푏2, to exceed the prescribed tolerances. In fact, their values of accuracy do not comply with the limiting values prescribed by the ISO 527:2 standard. For this reason, we decided to discard the non-dehumidified specimens and to only test the dehumidified ones.

115

Table 5.2 – Mean values of cross-sectional area for each specimens’ set.

Specimens’ set Mean cross-sectional area 퐴 [푚푚2]

NC100D 39.93 NC50D 38.85 NC100 40.26 NC50 40.12 5.3 Dimensional control over time In addition to the four specimens’ set described in the previous chapters, namely NC100D, NC50D, NC100 and NC50, we also printed an additional set of five 100% infill specimens from a dehumidified filament, NC100D specimens, that we did not test, but that we used to verify whether hygroscopy would cause dilation over time.

Table 5.3 – Mean values of accuracy and coefficient of variation for the 3D printed specimens’ set NC100D recorded at different times post printing.

Mean accuracy [%] Mean coefficient of variation

Time post printing 푏1 ℎ 푏2 푏1 ℎ 푏2 1h −0.13 −0.92 −0.15 0.31 0.00 0.05 24h −0.37 −1.33 −0.30 0.19 0.00 0.15 1 week −0.33 −1.00 −0.28 0.32 0.00 0.13 1 month −0.23 −0.50 −0.15 0.14 0.00 0.05 2 months −0.47 −1.00 −0.23 0.22 0.00 0.08 Limit value of accuracy [%] ±2.00 ±5.00 ±1.00

5.4 Static testing results Hereunder we report the results of the static test conducted on the 3D printed specimens. As explained in chapter 5.2, in principle we decided to only test the dehumidified specimens, namely:

• five NC100D specimens • five NC50D specimens

However, due to a technical problem that took place during testing, we had to discard the results relative to the specimen labeled as “NC100D_05 (5)”.

In total, we tested the following specimens:

• four NC100D specimens • five NC50D specimens

116

5.4.1 Mechanical properties To obtain the parameters described in chapter 3.6.3 and to plot the stress-strain curves, we used the methods described in chapter 3.6.4.

For each specimen, we plotted the stress-strain curve and, in addition, the following parameters:

- the slope of the elastic region, which identifies Young’s modulus 퐸푡; - the line defined by equation (3.9), used to find the yield stress; - the stress values corresponding to the values of strain of 0.0005 and 0.0025, used to calculate Young’s modulus; - the yield point, whose coordinates are the yield strain and the yield stress; - the break point, corresponding to the ultimate tensile stress.

By way of example, in Figure 5.3 we display the stress-strain curve for one specimen of the NC100D set.

Figure 5.3 – Stress-strain curve for specimen 1 of the specimens’ set NC100D.

In Figure 5.4 we plotted all the curves for the NC100D specimens in the one graph and all the curves for the NC50D specimens in the other graph.

117

Figure 5.4 – Stress-strain curves for NC100D specimens (left) and NC50D specimens (right).

In addition, Figure 5.5 we plotted the output of the deformations measured by the two strain gauges applied to specimen NC100D labeled as “1”, as well as the resulting Poisson’s ratio.

Figure 5.5 – Output of the strain gauges applied to the NC100D (1) specimen: (top) deformation measured by the axial strain gauge, (middle) deformation measured by the transverse strain gauge, (bottom) Poisson’s ratio, found as the negative ratio of the transverse to the axial deformation. The correct values are calculated at the plateau.

In Figure 5.6 we plotted the mean stress-strain curves for the two specimens’ set, and in Table 5.4 we summarized the mean values of the parameters that were measured for each specimens’ set.

118

Figure 5.6 – Mean stress-strain curves for NC100D specimens and NC50D specimens.

Table 5.4 – Average values of measured parameters for each specimens’ set.

5.4.2 Plasticity data extrapolation In this chapter we summarize the plasticity data for the dehumidified specimens, i.e. the only ones that underwent static testing. By way of example, in Figure 5.7, we display the engineering and true (corrected) stress-strain curves for one specimen of the NC100D specimens’ set, whereas in Figure 5.8 we display the true plastic curve for the same specimen.

Then, in Figure 5.9, we display the mean true plastic curves for both the NC100D specimens and the NC50D specimens, whereas, in Table 5.5, we report the mean values of engineering and true yield stress for both specimens’ set.

Finally, in Table 5.6, we report the plasticity data that we gave as input to Abaqus for the material modelling (both dehumidified Carbonium Nylon with 100% infill as well as 50% infill) described in chapter 4.5.

119

Figure 5.7 – Engineering vs. true stress-strain curve for specimen 1 of the NC100D specimens’ set.

Note that the true yield stress is slightly higher than the engineering yield stress.

Figure 5.8 – True plastic stress-strain curve for specimen 1 of the NC100D specimens’ set.

120

Figure 5.9 – Mean true plastic stress-strain curves for NC100D specimens (left) and for NC50D specimens (right).

Table 5.5 – Mean engineering vs. mean true yield stress for NC100D and NC50D specimens.

Specimens’ set Mean engineering Yield stress [MPa] Mean true Yield stress [MPa] NC100D 37.48 38.08 NC50D 23.23 23.60

In Table 5.6 reports the plasticity data that given as in input to Abaqus for modelling the plastic properties of dehumidified Carbonium Nylon with 100% infill and dehumidified Carbonium Nylon with 50% infill. Each data set consists of 10 linearly spaced stress-strain values obtained from the mean plastic stress-strain curve of NC100D and NC50D, respectively.

Table 5.6 – Plastic data for NC100D specimens

Plastic data for NC100D specimens Plastic data for NC50D specimens Post-yield stress Plastic strain Post-yield stress Plastic strain 38.08293232 0 23.60048512 0 44.27649554 0.003159874 28.61094029 0.003496247 48.55633173 0.006319748 32.2136282 0.006992495 51.64779261 0.009479621 34.99255993 0.010488742 54.0669711 0.012639495 37.22141796 0.013984989 55.92181794 0.015799369 39.13161384 0.017481237 57.48899956 0.018959243 40.74072925 0.020977484 58.80768439 0.022119117 42.14648795 0.024473731 59.84605073 0.025278991 43.32146042 0.027969979 60.6140711 0.028438864 44.16003623 0.031466226

121

6 Results – CAD/FEM process

6.1 Mesh

6.1.1 Mesh validation In this section we report the results of the mesh convergence study. Figure 6.1 shows the Von Mises stress distribution for the four meshes of the simplified model.

Figure 6.1 – Von Mises stress distribution in the simplified model for the four different meshes of the mesh convergence study. The region of interest (ROI) is pointed by the arrow in Mesh 1 but it is the same for all four meshes.

In Table 6.1, for each mesh refinement, we reported the mean value of Von Mises stress in the region of interest, the percentage of increase of mean Von Mises stress that would take place with a consecutive mesh refinement, and the wallclock time as indication of the computational cost. In Figure 6.2, we plotted the mean value of Von Mises stress to the number of nodes at each mesh refinement.

122

Table 6.1 – Mesh convergence study.

Von Mises Stress in Ratio Number of the region of interest Local seeds Total between the nodes in Total Global of the number number of Relative Mesh the wallclock size simplified of nodes in the Mean increase simplified value compared time [s] socket nodes simplified socket [MPa] to coarser socket mesh [%] 1 6 6 14342 1356 3.62 19.79 3.88 32 2 6 4 31977 4902 2.84 20.55 2.11 136 3 6 2.7 67781 13905 3.17 20.99 0.95 606 4 6 1.8 165462 44109 21.19 3641

Figure 6.2 – Mesh convergence study: mean Von Mises stress in the region of interest versus the number of elements in the region of interest.

From Table 6.1 and Figure 6.2, we see that the mesh size that produces converged results is Mesh 3. In fact, by furtherly refining the mesh (and passing from Mesh 3 to Mesh 4), the rate of increase of Von Mises stress keeps below 1%. using this mesh, the results are stable, i.e. they are independent of the mesh density, and, at the same time, the computational cost is reasonable.

6.1.2 Mesh accuracy The diagrams of the internal actions (bending moment, torque and axial forces) are reported in Appendix A, together with the stress components calculations.

Table 6.2 – Accuracy of the results using the mesh chosen from the mesh convergence study.

Mean Von Mises stress [MPa] Accuracy of the Computational value (mesh 3) Analytical value converged result [%] 20.99 23.17 9.41

123

From the results of Appendix A, is turns out the mean value of Von Mises stress in the region of interest of the simplified model is 23.17푀푃푎. By comparing it with the mean Von Mises stress produced by Mesh 3, namely 20.99푀푃푎, we found that the accuracy of the results is around 9%.

6.1.3 Mesh verification In Table 6.3, we report the values of each of the distortion criteria described in chapter 4.4.7, for the mesh of Part 2 of our reference model. From Figure 6.3 to Figure 6.6, we show the location of the distorted elements according to each criterion in Part 2. We do the same for Part 1, in Table 6.4, and from Figure 6.7 to Figure 6.10.

Table 6.3 – Mesh distortion in Part 2 of the reference model.

Part 2 Number of elements Selection Selection criterion Statistics exceeding the criterion limit selection criterion limit Average min angle on quad Min angle on Quad Smaller face faces: 72.94, Faces < 1 0 (0%) corner angle Worst min angle on quad faces:

10.49 Average max angle on quad Larger face faces: 108.16, Max angle on Quad 39 (0.0642166%) corner angle Worst max angle on quad faces: faces > 160 176.72 Average aspect ratio: 3.33, Aspect ratio Aspect ratio > 10 338 (0.556544%) Worst aspect ratio: 25.76 Geometric Average geometric deviation factor: 0.00867, Geometric deviation deviation 336 (0.55325%) Worst geometric deviation factor > 0.2 factor factor: 0.207 Average min edge length: 1.53, Min edge length < 336 (0.55325%) Short edge Shortest edge: 0.141 0.2 Average max edge length: 4.48, Max edge length > 6 2475 (4.07528%) Long edge Longest edge: 15.16 Max edge length > 7 92 (0.151485%)

124

Face corner angle > 160 Face corner angle > 170

Figure 6.3 – Elements with face corner angle greater than the specified limits are highlighted in yellow.

Aspect ratio > 10 Geometric deviation factor > 0.2

Figure 6.4 – Elements with aspect ratio greater than 10 (left) and with geometric deviation factor greater than 0.2 (right) are highlighted in yellow.

Edge length Edge length

Edge length <

Figure 6.5 – Elements with edge length below or above the specified limits are highlighted in yellow.

125

Face corner angle < 1 Face corner angle > 170 Aspect ratio > 25.76 Geometric deviation factor > 0.207 Edge length < 0.1 Edge length > 7

Figure 6.6 – There are no elements in the distal sockets for which the selected criteria exceed the specified limits.

Table 6.4 – Mesh distortion in Part 1 of the reference model

Part 1 Number of elements Selection Selection criterion Statistics exceeding the criterion limit selection criterion limit Average min angle on quad Min angle on Quad Smaller face faces: 72.78, Faces < 1 0 (0%) corner angle Worst min angle on quad faces:

12.43 Average max angle on quad Larger face faces: 109.48 Max angle on Quad 181 (1.23879%) corner angle Worst max angle on quad faces: faces > 160 175.49 Average aspect ratio: 2.52, Aspect ratio Aspect ratio > 10 585 (4.00383%) Worst aspect ratio: 25.65 Geometric Average geometric deviation factor: 0.0148, Geometric deviation deviation 253 (1.73157%) Worst geometric deviation factor > 0.2 factor factor: 0.289 Average min edge length: 2.03, Min edge length < 584 (3.99699%) Shortest edge: 0.1 0.2 Short edge Min edge length < 66 (0.451714%) 0.1 Average max edge length: 3.16, Max edge length > 4 383 (2.62131%) Longest edge: 7.89 Long edge 11540 Max edge length > 3 (78.9816%)

126

Face corner angle > 160

Figure 6.7 – Elements with face corner angle greater than the specified limits are highlighted in yellow.

Geometric deviation factor > 0.2 Aspect ratio > 10

Figure 6.8 – Elements with aspect ratio greater than 10 (left) and with geometric deviation factor greater than 0.2 (right) are highlighted in yellow.

Edge length < 0.2 Edge length > 3 Edge length > 4

Figure 6.9 – Elements with edge length below or above the specified limits are highlighted in yellow.

Face corner angle < 1 Face corner angle > 175.49 Aspect ratio > 25.65 Geometric deviation factor > 0.289 Edge length < 0.1 Edge length > 4

Figure 6.10 – There are no elements in Part 1 for which the selected criteria exceed the specified limits.

127

6.2 Static analysis results In this chapter, we report both the Von Mises as well as the Maximum Principal stress distribution in the housing, for all the models that we subjected to virtual testing. In particular, the histograms representing of the two stress distributions refer to the nodes inside our region of interest, which is the distal posterior part of the groove inside the distal part of the socket, displayed in Figure 6.11.

Figure 6.11 – Region of Interest (ROI) of the reference model.

6.2.1 Reference Model The Von Mises and Maximum Principal stress distribution in the entire housing of the Reference Model are displayed in Figure 6.12, whereas the histograms in Figure 6.13 and Figure 6.14 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level.

Figure 6.12 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of the reference model. The region of interest is highlighted in red.

128

Figure 6.13 – Von Mises stress distribution for Reference Model at P6 loading level.

Figure 6.14 – Maximum Principal Stress distribution for Reference Model at P6 loading level.

In Table 6.5 and Table 6.6, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

129

Table 6.5 – Von Mises stress distribution of the Reference Model at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 37.48 51.86 138% 0.18 P5 37.48 49.59 132% 0.10 P4 37.48 47.08 126% 0.06 P3 37.48 41.29 110% 0.01

Table 6.6 – Maximum Principal stress distribution of the Reference Model at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 58.34 28.08 48% 0 P5 58.34 25.82 44% 0 P4 58.34 23.25 40% 0 P3 58.34 17.85 31% 0

6.2.2 Model 1 The Von Mises and Maximum Principal stress distribution in the entire housing of Model 1 are displayed in Figure 6.15, whereas the histograms Figure 6.17 and Figure 6.17 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level.

Figure 6.15 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of the model with the housing rotated by 90°.

130

Figure 6.16 – Von Mises stress distribution for Model 1 at P6 loading level.

Figure 6.17 – Maximum Principal Stress distribution for Model 1 at P6 loading level.

In Table 6.7 and Table 6.8, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

131

Table 6.7 – Von Mises stress distribution of Model 1 at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 37.48 51.89 138% 0.18 P5 37.48 49.62 132% 0.10 P4 37.48 47.12 126% 0.06 P3 37.48 41.33 110% 0.01

Table 6.8 – Maximum Principal stress distribution of Model 1 at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 58.34 27.54 47% 0 P5 58.34 24.77 42% 0 P4 58.34 22.29 38% 0 P3 58.34 17.06 29% 0

6.2.3 Model 2 The Von Mises and Maximum Principal stress distribution in the entire housing of Model 2 are displayed in Figure 6.18 whereas the histograms in Figure 6.19 and Figure 6.20 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level

Figure 6.18 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of Model 2, with round housing. The region of interest is highlighted in red.

132

Figure 6.19 – Von Mises stress distribution for Model 2 at P6 loading level.

Figure 6.20 – Maximum Principal Stress distribution for Model 2 at P6 loading level.

In Table 6.9 and Table 6.10, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

133

Table 6.9 – Von Mises stress distribution of Model 2 at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 37.48 49.71 133% 0.29 P5 37.48 47.44 127% 0.18 P4 37.48 44.95 120% 0.05 P3 37.48 39.40 105% 0.00

Table 6.10 – Maximum Principal stress distribution of Model 2 at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 58.34 36.66 63% 0 P5 58.34 33.82 58% 0 P4 58.34 30.85 53% 0 P3 58.34 23.74 41% 0

6.2.4 Model 3 The Von Mises and Maximum Principal stress distribution in the entire housing of Model 3 are displayed in Figure 6.21, whereas the histograms in Figure 6.22 and Figure 6.23 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level

Figure 6.21 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of the model with larger fillet radius.

134

Figure 6.22 – Von Mises stress distribution for Model 3 at P6 loading level.

Figure 6.23 – Maximum Principal Stress distribution for Model 3 at P6 loading level.

In Table 6.11 and Table 6.12, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

135

Table 6.11 – Von Mises stress distribution of Model 3 at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 37.48 44.90 120% 0.13 P5 37.48 42.72 114% 0.08 P4 37.48 40.41 108% 0.05 P3 37.48 33.09 88% 0

Table 6.12 – Maximum Principal stress distribution of Model 3 at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 58.34 32.80 56% 0 P5 58.34 30.65 53% 0 P4 58.34 28.17 48% 0 P3 58.34 22.89 39% 0

6.2.5 Model 4 The Von Mises and Maximum Principal stress distribution in the entire housing of Model 4 are displayed in Figure 6.24, whereas the histograms in Figure 6.25 and Figure 6.26 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level

Figure 6.24 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of the model with the new connector.

136

Figure 6.25 – Von Mises stress distribution for Model 4 at P6 loading level.

Figure 6.26 – Maximum Principal Stress distribution for Model 4 at P6 loading level.

In Table 6.13 and Table 6.14, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

137

Table 6.13 – Von Mises stress distribution of Model 4 at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 37.48 29.76 79% 0 P5 37.48 26.78 71% 0 P4 37.48 23.80 64% 0 P3 37.48 17.90 48% 0

Table 6.14 – Maximum Principal stress distribution of Model 4 at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 58.34 26.67 46% 0 P5 58.34 24.11 41% 0 P4 58.34 21.57 37% 0 P3 58.34 16.51 28% 0

6.2.6 Model 5 The Von Mises and Maximum Principal stress distribution in the entire housing of Model 5 are displayed in Figure 6.27, whereas the histograms in Figure 6.28 and Figure 6.29 refer to the nodes in the region of interest. Note that these two stress distributions refer to the P6 loading level

Figure 6.27 – Von Mises (left) and Maximum Principal (right) stress distribution in the region of interest of model with different material.

138

Figure 6.28 – Von Mises stress distribution for Model 5 at P6 loading level.

Figure 6.29 – Maximum Principal Stress distribution for Model 5 at P6 loading level.

In Table 6.15 and Table 6.16, we summarize the percentage of nodes overcoming the yield stress and ultimate tensile stress in both stress distribution, at all loading levels (P6, P5, P4 and P3).

139

Table 6.15 – Von Mises stress distribution of Model 5 at different loading levels.

Nodes overcoming yield Von Mises Stress Loading Yield stress stress level [MPa] Maximum value % of yield % of nodes [MPa] stress P6 23.23 42.58 183% 1.96 P5 23.23 39.17 169% 1.15 P4 23.23 36.87 159% 0.68 P3 23.23 31.74 137% 0.12

Table 6.16 – Maximum Principal stress distribution of Model 5 at different loading levels.

Nodes overcoming Maximum Principal Stress Loading Ultimate tensile ultimate tensile stress level stress [MPa] Maximum % of ultimate % of nodes value [MPa] tensile stress P6 42.35 25.96 61% 0 P5 42.35 23.73 56% 0 P4 42.35 21.53 51% 0 P3 42.35 16.87 40% 0

140

6.2.7 Models comparison In this chapter, we compared the behavior of the five alternative models to the behavior of the reference model, at P6 loading level. In fact, this level marks the threshold for passing static testing. Moreover, the behavior of the models for lower loading levels was not very significant.

In Table 6.17, for each model, we summarized the maximum values of Von Mises stress, the percentage of maximum value of Von Mises stress with respect to the reference value, the percentage of nodes for which the Von Mises stress overcame the yield stress, and the percentage of nodes overcoming the yield stress with respect to the reference value.

Table 6.17 – Comparison of Von Mises stress distributions for all models at P6 loading level.

Nodes over yield Von Mises Stress Yield stress Model stress % of Maximum % of % of % of [MPa] yield value [MPa] reference nodes reference stress Reference 37.48 51.86 100% 138% 0.18 100% Model Model 1 37.48 51.89 100% 138% 0.18 100% Model 2 37.48 49.71 96% 133% 0.29 161% Model 3 37.48 44.9 87% 120% 0.13 72% Model 4 37.48 29.76 57% 79% 0 0% Model 5 23.23 42.58 n.a. 114% 1.96 1089%

In Table 6.18, for each model, we summarized the maximum values of Maximum Principal stress, the percentage of maximum value of Maximum Principal stress with respect to the reference value, the percentage of nodes for which the Maximum Principal stress overcame the ultimate tensile stress, and the percentage of nodes overcoming the ultimate tensile stress with respect to the reference value.

Table 6.18 – Comparison of Maximum Principal stress distribution for all models at P6 loading level.

Nodes over ultimate Ultimate Maximum Principal stress tensile stress tensile Model % of stress Maximum % of % of % of ultimate [MPa] value [MPa] reference nodes reference tensile stress Reference 58.34 28.08 100% 48% 0 n.a. Model Model 1 58.34 27.54 98% 47% 0 n.a. Model 2 58.34 36.66 131% 63% 0 n.a. Model 3 58.34 32.8 117% 56% 0 n.a. Model 4 58.34 26.67 95% 46% 0 n.a. Model 5 42.35 25.96 n.a. 44% 0 n.a.

141

7 Discussions and Conclusions – Material characterization

7.1 Dimensional control From the results of chapter 5.2, we observed that:

• the actual dimensions of the specimens printed with dehumidified filament kept close the nominal ones and never exceeded the prescribed tolerances; • instead, specimens printed with non-dehumidified material underwent a marked dilation that caused their actual dimensions to exceed the prescribed tolerances. For this reason, we had to discard them from static testing.

We conclude that it is essential to dehumidify Carbonium Nylon filaments before 3D printing a part, because the moisture inside the material of the filament (in particular, inside its matrix, PA12) at the high temperature of 3D printing (240°퐶) causes geometrical modifications of the printed part that are not acceptable.

7.2 Dimensional control over time From the results of chapter 5.3, it is evident that the hygroscopic nature of PA12 does not represent a problem in the long term. In fact, all specimens printed with dehumidified filament of Carbonium Nylon showed dimensions that kept steady at different times post printing.

In accordance with what we found in literature [18], we conclude that the probability of hydrolysis occurring in PA12 because of the presence of hydrogen bonds between water and the amide groups increases at higher processing temperatures. In particular, as observed from results of chapter 5.2, at the 3D printing temperature (240°퐶), the enhanced hygroscopy interacts with fused filament deposition, resulting in a marked dilation of the 3D printed part. Nevertheless, after 3D printing the part, the probability of hydrolysis occurring in the material keeps steady, and does not cause geometrical modification of the part, as proven from the results of chapter 5.3.

7.3 Static testing

7.3.1 Mechanical properties From the results of specimen static testing (chapter 5.4.1), we observed that the 100% infill Carbonium Nylon is stiffer and stronger than the 50% infill material. In particular, the 100% material has a Young’s modulus of 4676푀푃푎 and a ultimate tensile strength of around 58푀푃푎, whereas the 50% material has a Young’s modulus of 3059푀푃푎 and a ultimate tensile strength of approximately 42푀푃푎. Therefore, we can agree with the literature data and say that a higher infill percentage leads to higher mechanical properties of the 3D printed part [36].

142

Regarding the plasticity information, the two materials display a very similar plastic strain area, of around 3%.

7.3.2 Plasticity data By analyzing the results of specimens static testing (chapter 5.4.1), we observed that both dehumidified Carbonium Nylon specimens with 100% infill (NC100D) as well as with 50% infill (NC50D), displayed a total plastic deformation of approximately 3%. Therefore, even though it is evident that the two materials display an elastoplastic behavior, it would not be accurate to label them neither as completely ductile nor as completely brittle. In fact, ductility is defined as the ability of a material to undergo significant plastic deformation before strain, where significant denotes about 5% plastic strain [25].

Consequently, in chapter 4.8.4, when we had to choose a failure criteria to asses stress distribution in the virtual model, we decided to use both the Von Mises failure criterion, typically used for ductile materials, as well as the Galileo-Rankine-Navier failure criterion, typically used for brittle materials.

143

8 Discussions and Conclusions – CAD/FEM process

8.1 Mesh

8.1.1 Mesh verification From the mesh verification analysis, we observed that our mesh had a very good quality, as it had a very small percentage of distorted elements. Nevertheless, we noted that most of the distorted elements were in the location of interest, i.e. in the fillet areas of the groove inside the housing.

To locally improve the quality of the mesh in this location, we should have furtherly reduced the size of the mesh, and thus the number of elements. However, with the computational resources available to us (16GB of RAM), we could not increase the number of elements any further. Nevertheless, in the results section (chapter 6.2), we showed that the stresses in this location are not affected by the presence of distorted elements, as the stress distribution displays smooth contours.

8.2 Static analysis

8.2.1 General conclusions By looking at the results of the six virtual models (chapter 6.2), we observe that the stress distribution of both Von Mises as well as Maximum Principal stress is skewed. In particular, the Maximum Principal stress distribution can be best fitted by a Poisson distribution [37]. This means that the bending moment acting on the region of interest is not pure. If the bending moment was pure, the stress distribution would be symmetrical with respect to the mean value of stress and could be best fitted by a Gaussian distribution.

Moreover, for both distributions, we observe that only a few nodes inside the region of interest show high peak values, whereas the majority of the nodes displays values close to the mean stress. In particular:

- For the Reference Model, Model 1, Model 2 and Model 3, the percentage of the nodes showing this trend is already negligible (below 0.2%) at P6 loading level. Therefore, even if in some nodes the Von Mises stress overcomes the yield stress, yield is unlikely to take place, because the behavior of the material is not completely ductile, but it is close to being brittle. Even if some localized points may undergo yielding, this would not severely affect the resistance of the structure, that could still rely on the elastic support of the neighboring nodes. This is possible because the underlying materials of these models is a material with 100% infill, which signifies that the honeycomb pattern of the printed part has a material density of 100%, without empty spaces that would allow crack propagation.

144

- Model 4 does not have any nodes in which the Von Mises stress overcomes the yield stress, thus yielding is not possible at any loading level, including P6. - On the contrary, Model 5, at P6 loading level, presents a non-negligible percentage of nodes in which the Von Mises stress overcomes the yield stress (almost 2%). Therefore, the risk of yielding cannot be neglected. This percentage becomes negligible (below 0.2%) only at P3 loading level. Nevertheless, the plastic nodes cannot rely on the elastic support of neighboring nodes, because the underlying material has a 50% infill, meaning that the honeycomb pattern has a material density of only 50%; thus, 50% of the printed part is made of empty spaces, which can promote crack propagation in case of localized failures. Therefore, yielding cannot be excluded neither at P3 loading level. This behavior is confirmed by the experimental tests, in which sockets with 50% infill Carbonium Nylon failed before reaching P3 loading level. We should also consider that sockets were made of non-dehumidified Carbonium Nylon (with 50% infill), while the material created in Abaqus for Model 5 refers to the dehumidified version of Carbonium Nylon (with 50% infill), which is probably stronger.

By comparing the Von Mises stress distribution with the Maximum Principal stress distribution, we observe that, at all loading levels, the maximum values of Von Mises stress are higher than the maximum values of Maximum Principal stress. In general, this means that there are non-negligible compressive contributions in the area of interest. In fact, the Von Mises stress is always a positive value, obtained from a combination of the three principal stresses (see equation (4.5)), and consequently it comprises information of both traction and compression. On the contrary, the Maximum Principal stress accounts solely for the maximum values of stress in the part, either of tensile (positive values) or compressive nature (negative values).

- In all models expect for Model 4, the maximum values of Von Mises stress are much higher than the maximum values of the Maximum Principal stress, the first almost doubling the latter. This means that, in all models except for Model 4, there are non-negligible compressive contributions in the area of interest. These non-negligible compressive stresses are probably due to the connector indenting into the groove of the distal socket when the distal socket is pulled. In fact, by looking at the Maximum Principal stress distributions for the Reference Model (Figure 6.12), Model 1 (Figure 6.15), Model 2 (Figure 6.18), Model 3 (Figure 6.21) and Model 5 (Figure 6.27), we can observe a dark area denoting compressive stresses in correspondence of the horizontal surface of the groove in the region of interest, while the tensile stresses (showed in green) are on the vertical walls of the groove. - In Model 4, the difference between the maximum values of the two stresses is not as marked (28.76MPa versus 25.96MPa). In fact, the dark area in the Maximum Principal stress

145

distributions (Figure 6.24) is no longer present, replaced by a blue area denoting stress values close to 0. This means that, thanks to the improved geometry, the connector is no longer “pushed” against the groove’s horizontal surface while the distal socket is pulled.

Nevertheless, compression is less dangerous than traction, as it does not cause crack opening. In addition, the maximum tensile stresses (positive values of Maximum Principal stress) remain below the ultimate tensile stress, for all models and at all loading levels. Therefore, failure does happen because of fracture, but (when it happens) because of yielding.

8.2.2 Models comparison By comparing the alternative models with the reference one we can conclude that:

- Model 1 and Model 2 did not show any improvement from the Reference Model, neither in terms of maximum stress values nor in terms of stress concentration. o In particular, Model 1 shows maximum values of stress and stress concentrations that are only slightly lower than the reference values. o On the other hand, Model 2 shows a slight decrease in the maximum value of Von Mises stress (96% of reference value at P6 loading level) but an increase of 61% of the number of nodes overcoming yielding with respect to the reference number, even if the total percentage remains almost negligible (0.29%). In addition, the maximum values of Maximum Principal stress display a non-negligible increase of 31% with respect to the reference value, denoting an increase in the tensile stresses. This means that the stress distributions changes producing lower compressive stresses but higher tensile stresses. Therefore, differently from what we thought, a larger thickness around the groove in the anteroposterior direction (Model 1) or uniformly in all directions (Model 2) does not improve the mechanical behavior of the distal socket, as it does not lower stress distributions nor it reduces the number of nodes overcoming the yield stress. In particular, while Model 1 can be considered equivalent to the Reference Model, Model 2 produces worse results and should be discarded. - In Model 3, the small reduction in Von Mises stresses (reduction to 87% of the reference value at P6 loading level) confirms the positive impact of a larger fillet radius in reducing the notch effect. However, the increase of Maximum Principal stress (to 117% of the reference value) denotes an increase in tensile stresses and thus a dangerous behavior. It is evident that the reference design of the connector is not optimal for being modified. Therefore, we decided to redesign the connector, which led us to Model 4.

146

- Model 4 shows the best results among all models. In fact, it displays the lowest values of both Von Mises and Maximum Principal stress, which never overcome the yield stress and ultimate tensile stress, respectively. In particular, the maximum value of Von Mises stress at P6 loading level reduced to 57% of the reference value and to 79% of the yield stress, while the maximum value of Maximum Principal stress reduced to 95% of the reference value and to 46% of the ultimate tensile stress. The marked reduction in the Von Mises stress compared to the slighter reduction in Maximum Principal stress, confirms that the contribution of compression has drastically decreased, as already discussed in chapter 8.2.1. We can conclude that this new geometry substantially improves the mechanical behavior of the structure. - Model 5 is not directly comparable to the Reference Model (as denoted by label “n.a.” in Table 6.17 and Table 6.18), because it is made of a different material than the Reference Model: the first refers to dehumidified Carbonium Nylon with 100% infill, while the latter to dehumidified Carbonium Nylon with 50% infill.

147

9 Overall Conclusions and Future perspectives

This study proposes a methodology to improve distal socket and connector design based on parametric modelling, FE analysis and material conditioning.

From our results, we can conclude that the reference geometry of the distal part of the socket and of the connector could withstand the prescribed loading level (P6) only using dehumidified 100% infill Carbonium Nylon. While the proposed alternatives geometries of the distal socket (Model 1 and Model 2) did not produce any benefit in the mechanical resistance of the part, an improved geometry of the connector (Model 4) would reduce the stress to 57% of the Reference and to 79% of the yield.

Further steps should focus on these two points:

- first, validating the computational results by means of experimental testing performed on the reference geometry of the socket printed with dehumidified Carbonium Nylon with 100% infill; - second, optimizing the infill percentage for the improved geometry (Model 4), i.e. finding the lowest infill percentage that would still allow successful testing.

In addition, the FE model created in Abaqus could be optimized by means of:

- modelling bolted connections between the Coyote connector and the distal plate of the setup; - replacing the geometry of the setup with a structure made of beams, to spare additional computational resources.

148

Appendix A In this appendix, we report the calculations that lead us to find the accuracy of the converged result in chapter 4.4.4.

Figure A.1 – Analytical simplified model.

Table A.1 – Geometric parameters.

Segment Label Length [mm] BC 푎 80 CD 푏 130

푏푥 129 푏푦 19 DE 푐 531 EG 푑 68

푑푥 55 푑푦 40 GT 푒 39

149

Figure A. 2 – Simplified model in the three cartesian bi-dimensional reference planes.

150

From the statics equations, we can express the reaction forces as functions of the vertical reaction force 퐹푧, that is easily derived from the total force 퐹푇푂푇 prescribe by the ISO 10328 standard:

푎 + 푐 + 푒 74 (9.1) 퐹푥 = 퐹푧 ∙ = 퐹푧 ∙ 푏푥 − 푑푥 650 푎 + 푐 + 푒 21 (9.2) 퐹푦 = 퐹푧 ∙ = 퐹푧 ∙ 푑푦 − 푏푦 650 (9.3) 푎 + 푐 + 푒 2 푎 + 푐 + 푒 2 2 2 2 퐹푇푂푇 = √퐹푥 + 퐹푦 + 퐹푧 = 퐹푧 ∙ √( ) + ( ) + 1 푏푥 − 푑푥 푑푦 − 푏푦

74 2 21 2 = 퐹 ∙ √( ) + ( ) + 1 ≅ 퐹 ∙ (1.00697802) 푧 650 650 푧

퐹푇푂푇 (9.4) 퐹푧 = 푎 + 푐 + 푒 2 푎 + 푐 + 푒 2 √( ) + ( ) + 1 푏푥 − 푑푥 푑푦 − 푏푦 (9.5) 푎 + 푐 + 푒 2 푎 + 푐 + 푒 2 74 2 21 2 2 2 √ 퐹퐻 = √퐹푥 + 퐹푦 = 퐹푧 ∙ √( ) + ( ) = 퐹푧 ∙ ( ) + ( ) 푏푥 − 푑푥 푑푦 − 푏푦 650 650

≅ 퐹푧 ∙ (0.1183415976) 푑푦 40 (9.6) 훼 = arctan ( ) = arctan ( ) ≅ 36.0° 푑푥 55

푏푦 19 (9.7) 훽 = arctan ( ) = arctan ( ) ≅ 8.38° 푏푥 129 퐹 74 (9.8) 휃 = arctan ( 푥) = arctan ( ) ≅ 74.2° 퐹푦 21

퐹||푑 = FH ∙ sin(휃 − 훼) (9.9)

퐹⊥푑 = 퐹퐻 ∙ cos(휃 − 훼) (9.10) 퐹||푏 = 퐹퐻 ∙ sin (휃 − 훽) (9.11)

퐹⊥푏 = 퐹퐻 ∙ cos (휃 − 훽) (9.12)

Table A.2 – Values of reaction force components for P6 different loading level. Force values are expressed in Newton [N].

Loading 퐹 퐹 퐹 퐹 퐹 퐹 퐹 퐹 퐹 level 푇푂푇 푧 푥 푦 퐻 ||푑 ⊥푑 ||푏 ⊥푏 P6 4425 4394 500 142 529 322 411 473 213

151

Figure A.3 – Diagram of bending moments and torque in the three cartesian bi-dimensional reference planes.

152

Figure A.4 – Diagram of bending moments and torque in the planes perpendicular and parallel to beam EG (d) and to beam CD (e).

153

Figure A.5 – Diagram of bending moments and torque. The numerical values at the corner points are expressed in [푁 ∙ 푚푚] and refer to a P6 loading level.

154

At this point, we calculated the stress values in the location of interest (where the socket would lie), at a P6 loading level. From Table A.2, we see that a P6 loading level corresponds to a vertical force component 퐹푧 = 4394푁.

First, we computed the value of internal actions in beam DE (c), at three different z-values: 푧 = 0 (point D), at 푧 = 18 (point K, corresponding to the interface between the housing the and rest of the distal socket) and at 푧 = 68 (point J, corresponding to the interface between the distal socket and the setup column). At these points, we had two contributions of bending moment to combine the two contributions of bending moment (one lying on the xy plane and the other on the yz plane) and an axial compression equal to 퐹푧.

Then, we computed the maximum value of the normal and tangential stress components in these three locations, knowing that the column in the simplified model has diameter 퐷 = 60푚푚 and using the following equations:

푀푏 퐷 32 푀푏 (9.13) 휎푏,푚푎푥 = ∙ = 3 퐼푏 2 휋퐷 퐹 4퐹 (9.14) 휎 = 푧 = 푧 푁 퐴 휋퐷2 푀푡 퐷 16 푀푡 (9.15) 휏푚푎푥 = ∙ = 3 퐼푡 2 휋퐷 where 휎푏,푚푎푥 is the maximum normal components of stress due to bending and 휎푁 the normal component of stress due to axial force; 휏 is the tangential component of stress; 푀푏 is the bending moment, which for every 푧 location is found by combining the bending moments in the reference bi-

휋퐷4 휋퐷4 dimensional planes, e.g. 푀 = √푀 2 + 푀 2 ; 푀 is the torque; 퐼 = and 퐼 = are the 푏 푏푌푍 푏푋푍 푡 푏 64 푡 32 inertia moments for bending and torque, respectively; 퐷 is the diameter of the column DE (c).

155

Figure A.6 – Distribution of normal stress in beam DE (c). Tangential stress is negligible in this location.

Finally, we combined the two stress components using the Von Mises principle.

2 2 (9.16) 휎푉푀 = √휎 + 3휏 where

휎 = 휎푏푒푛푑 − 휎푁 휏 = 휏푚푎푥

We found that the maximum values of Von Mises stress in the three locations of interest were 23.8 푀푃푎, 푎푡 푝표𝑖푛푡 퐷 휎푉푀 = {23.4 푀푃푎, 푎푡 푝표𝑖푛푡 퐾 22.3 푀푝푎, 푎푡 푝표𝑖푛푡 퐽 with an average value of 23.17 푀푃푎.

156

Appendix B Abaqus essential dictionary:

• Reference point A reference point (RP) is a point that can be create in the model and can be positioned anywhere in space (not necessarily on the geometry of the model). It is useful for creating a point where a vertex is not available. Note that, in contrast to a vertex, the RP is ignored in the Mesh module when the mesh is generated. • Mesh density The mesh density refers to the number of elements in a mesh per unit volume (in 3D), area (in 2D) or length (in 1D): the higher the number of elements, the denser the mesh. • Nodes, seeds and vertices Nodes are the points of an elements where displacement is calculated. Seeds are markers that the user places along the edges of a region to indicate where the corner nodes of the elements should be located and thus to specify the target mesh density in that region. Vertices are points of the model’s geometry. They are present where two or more edges cross, or, if the model is crated within Abaqus CAD environment, the Sketcher module creates vertices at the locations that the user “clicks” to define a shape (circumference, square, etc.).

Figure B.1 – (left) vertices on a 2D part; (middle) fully constrained seeds appear at each vertex; (right): nodes appear at each vertex and, if possible, at each seed.

• Global size and local seeds Seeds can be placed either by globally seeding the part instance (assigning a global mesh size) or by seeding directly certain edges (applying local seeds). In the first case, Abaqus/CAE might change the element distribution so that the mesh can be generated successfully. In the latter case, the number of seeds along an edge are constrained, which is equivalent to prescribing the number of elements along the edge, and, to a lesser extent, the precise locations of the nodes. Note that, either way, Abaqus/CAE will automatically place the nodes wherever vertices appear along the model’s edges.

157

• Top-down and bottom-up meshing Top-down and bottom-up meshing are two different types of meshing techniques. In the top- down approach, the mesh is automatically created to conform exactly with the geometry of a region and works down to the element and node positioning. On the contrary, in the bottom- up approach, the geometry of a region is so complex that it is difficult for the mesh to conform exactly with it; therefore it requires the user to adopt a manual and incremental process consisting in defining the element and nodes first, even if they do not conform exactly with the geometry. • Shear locking (Figure B.2, b) is the inability of the element’s displacement ﬁeld to model the kinematics associated with bending. This numerical problem affects fully integrated linear elements, as they only have the corner nodes and lack the mid-side nodes, therefore their edges result too stiff and do not deform correctly under bending: instead of creating bending deformation, strain energy introduces spurious shear stresses at the integration points, producing a nonphysical state of stress and strain in the element. Reduced integration linear hexahedra still have only the corner nods, but do not suffer from shear locking because they only have one integration point, which makes the element more flexible under bending. • Hourglassing or zero-energy mode (Figure B.2, c) is a numerical problem that causes the elements to be too flexible under bending. Reduced integration linear elements suffer from this problem because they have no stiffness in this mode.

Figure B.2 – Figure shows three types of deformation of a linear hexahedral element under bending. The dotted lines cross at the integration points. In a physical deformation (a) the lines that are originally horizontal take on a constant curvature, the ones originally vertical keep perpendicular to the curved ones and the shear stresses at the integration point is zero. In case of shear locking (b), the horizontal lines cannot curve, and the vertical ones assume an angle 훼 different from 90°, introducing nonzero shear stresses at the integration points (nonphysical). In case of hourglassing (c), the element deforms in a way that the doted lines do not change their length, and angle 훼 between them remains 90°: this means that all components of stress at the integration point are zero (nonphysical).

158

Appendix C Hereunder we display the integral results of dimensional control (chapter 5.2) and dimensional control over time (chapter 5.3).

Table C.1 – Dimensional control of width 푏1 for all specimen sets.

푏1[푚푚] Specimen Specimen position on Standard Accuracy Coefficient Nominal Mean Median type label printer plate deviation [%] of variation label 1 10 10.10 10.10 0.00 1.00 0.00 A 10 10.08 10.10 0.02 0.83 0.23 NC100D 2 10 10.07 10.05 0.02 0.67 0.23 B 10 10.13 10.15 0.02 1.33 0.23 3 10 10.12 10.10 0.02 1.17 0.23 1 10 9.95 9.95 0.04 -0.50 0.41 A 10 9.97 9.95 0.02 -0.33 0.24 NC50D 2 10 9.97 9.95 0.02 -0.33 0.24 B 10 9.93 9.95 0.02 -0.67 0.24 3 10 9.95 9.95 0.04 -0.50 0.41 1 9.6 10.07 10.10 0.05 4.86 0.47 A 9.6 10.05 10.05 0.04 4.69 0.41 NC100 2 9.6 10.07 10.10 0.05 4.86 0.47 B 9.6 10.05 10.05 0.04 4.69 0.41 3 9.6 10.05 10.05 0.04 4.69 0.41 1 9.6 10.05 10.05 0.00 4.69 0.00 A 9.6 10.08 10.10 0.02 5.03 0.23 NC50 2 9.6 10.02 10.00 0.02 4.34 0.24 B 9.6 10.02 10.00 0.02 4.34 0.24 3 9.6 10.07 10.10 0.05 4.86 0.47

159

Table 9.1 – Dimensional control of thickness ℎ for all specimen sets.

ℎ[푚푚] Specimen Specimen position on Standard Accuracy Coefficient Nominal Mean Median type label printer plate deviation [%] of variation label 1 4 4.00 4.00 0.00 0.00 0.00 A 4 3.97 4.00 0.05 -0.83 0.01 NC100D 2 4 3.93 3.95 0.02 -1.67 0.01 B 4 3.92 3.90 0.02 -2.08 0.01 3 4 3.95 3.95 0.04 -1.25 0.01 1 4 3.92 3.90 0.02 -2.08 0.01 A 4 3.90 3.90 0.00 -2.50 0.00 NC50D 2 4 3.90 3.90 0.00 -2.50 0.00 B 4 3.90 3.90 0.00 -2.50 0.00 3 4 3.90 3.90 0.00 -2.50 0.00 1 4 4.00 4.00 0.00 0.00 0.00 A 4 4.02 4.00 0.02 0.42 0.01 NC100 2 4 4.00 4.00 0.00 0.00 0.00 B 4 4.00 4.00 0.00 0.00 0.00 3 4 4.00 4.00 0.00 0.00 0.00 1 4 4.03 4.05 0.02 0.83 0.01 A 4 4.02 4.00 0.02 0.42 0.01 NC50 2 4 3.95 3.95 0.00 -1.25 0.00 B 4 4.00 4.00 0.00 0.00 0.00 3 4 3.97 4.00 0.05 -0.83 0.01

160

Table C.2 – Dimensional control of width 푏2 for all specimen sets.

푏2[푚푚] Specimen Specimen position on Standard Accuracy Coefficient Nominal Mean Median type label printer plate deviation [%] of variation label 1 20 20.10 20.10 0.00 0.50 0.00 A 20 20.13 20.13 0.02 0.63 0.12 NC100D 2 20 20.10 20.10 0.05 0.50 0.25 B 20 20.18 20.18 0.03 0.87 0.12 3 20 20.13 20.13 0.02 0.63 0.12 1 20 19.98 19.98 0.03 -0.12 0.13 A 20 19.98 19.98 0.03 -0.12 0.13 NC50D 2 20 19.95 19.95 0.05 -0.25 0.25 B 20 19.98 19.98 0.03 -0.12 0.13 3 20 19.98 19.98 0.03 -0.12 0.13 1 19.2 20.00 20.00 0.00 4.17 0.00 A 19.2 20.03 20.03 0.03 4.30 0.12 NC100 2 19.2 20.05 20.05 0.00 4.43 0.00 B 19.2 20.00 20.00 0.00 4.17 0.00 3 19.2 20.03 20.03 0.03 4.30 0.12 1 19.6 20.10 20.10 0.00 2.55 0.00 A 19.6 20.10 20.10 0.00 2.55 0.00 NC50 2 19.6 20.08 20.08 0.03 2.42 0.12 B 19.6 20.05 20.05 0.00 2.30 0.00 3 19.6 20.05 20.05 0.00 2.30 0.00

161

Table C.3 – Cross-sectional area of the narrow portion of each specimen.

Specimen position ퟐ Specimen type 풃 [풎풎] mean 풉[풎풎] mean 푨 = 풃ퟏ ∙ 풉 [풎풎 ] on printer plate ퟏ label value value label Cross sectional area 1 10.10 4.00 40.40 A 10.08 3.97 40.00 NC100D 2 10.07 3.93 39.60 B 10.13 3.92 39.69 3 10.12 3.95 39.96 1 9.95 3.92 38.97 A 9.97 3.90 38.87 NC50D 2 9.97 3.90 38.87 B 9.93 3.90 38.74 3 9.95 3.90 38.81 1 10.07 4.00 40.27 A 10.05 4.02 40.37 NC100 2 10.07 4.00 40.27 B 10.05 4.00 40.20 3 10.05 4.00 40.20 1 10.05 4.03 40.54 A 10.08 4.02 40.50 NC50 2 10.02 3.95 39.57 B 10.02 4.00 40.07 3 10.07 3.97 39.93

162

Hereunder we display the integral results of dimensional control over time.

Table C.4 – Dimensional control of width 푏1 for NC100D specimens at different times after printing.

푏1[푚푚] Specimen Time post position on Standard Accuracy Coefficient Nominal Mean Median printing printer plate deviation [%] of variation label 1 10 9.98 10.00 0.02 -0.17 0.24 A 10 9.98 10.00 0.02 -0.17 0.24 1h 2 10 10.00 10.00 0.04 0.00 0.41 B 10 9.97 9.95 0.02 -0.33 0.24 3 10 10.00 10.00 0.04 0.00 0.41 1 10 9.93 9.95 0.02 -0.67 0.24 A 10 9.98 10.00 0.02 -0.17 0.24 24h 2 10 9.98 10.00 0.02 -0.17 0.24 B 10 9.97 9.95 0.02 -0.33 0.24 3 10 9.95 9.95 0.00 -0.50 0.00 1 10 9.92 9.90 0.02 -0.83 0.24 A 10 9.97 9.95 0.02 -0.33 0.24 1 week 2 10 10.00 10.00 0.04 0.00 0.41 B 10 9.98 10.00 0.02 -0.17 0.24 3 10 9.97 10.00 0.05 -0.33 0.47 1 10 9.93 9.95 0.02 -0.67 0.24 A 10 9.98 10.00 0.02 -0.17 0.24 1 month 2 10 10.02 10.00 0.02 0.17 0.24 B 10 9.95 9.95 0.00 -0.50 0.00 3 10 10.00 10.00 0.00 0.00 0.00 1 10 9.93 9.95 0.02 -0.67 0.24 A 10 9.97 9.95 0.02 -0.33 0.24 2 months 2 10 9.95 9.95 0.00 -0.50 0.00 B 10 9.95 9.95 0.04 -0.50 0.41 3 10 9.97 9.95 0.02 -0.33 0.24

163

Table C.5 – Dimensional control of thickness ℎ for NC100D specimens at different times after printing.

ℎ [푚푚] Specimen Time post position on Standard Accuracy Coefficient Nominal Mean Median printing printer plate deviation [%] of variation label 1 4 4.00 4.00 0.00 0.00 0.00 A 4 3.98 4.00 0.02 -0.42 0.01 1h 2 4 3.95 3.95 0.00 -1.25 0.00 B 4 3.92 3.90 0.02 -2.08 0.01 3 4 3.97 4.00 0.05 -0.83 0.01 1 4 3.97 3.95 0.02 -0.83 0.01 A 4 3.97 3.95 0.02 -0.83 0.01 24h 2 4 3.95 3.95 0.00 -1.25 0.00 B 4 3.90 3.90 0.00 -2.50 0.00 3 4 3.95 3.95 0.04 -1.25 0.01 1 4 4.00 4.00 0.00 0.00 0.00 A 4 3.95 3.95 0.00 -1.25 0.00 1 week 2 4 3.93 3.95 0.02 -1.67 0.01 B 4 3.95 3.95 0.00 -1.25 0.00 3 4 3.97 4.00 0.05 -0.83 0.01 1 4 4.00 4.00 0.00 0.00 0.00 A 4 4.00 4.00 0.00 0.00 0.00 1 month 2 4 3.95 3.95 0.00 -1.25 0.00 B 4 3.97 3.95 0.02 -0.83 0.01 3 4 3.98 3.95 0.05 -0.42 0.01 1 4 3.98 4.00 0.02 -0.42 0.01 A 4 3.95 3.95 0.00 -1.25 0.00 2 months 2 4 3.95 3.95 0.00 -1.25 0.00 B 4 3.95 3.95 0.00 -1.25 0.00 3 4 3.97 3.95 0.06 -0.83 0.02

164

Table C.6 – Dimensional control of width 푏2 for NC100D specimens at different times after printing.

푏2[푚푚] Specimen Time post position on Standard Accuracy Coefficient Nominal Mean Median printing printer plate deviation [%] of variation label 1 20 19.95 19.95 0.00 -0.25 0.00 A 20 20.03 20.03 0.03 0.12 0.12 1h 2 20 19.95 19.95 0.00 -0.25 0.00 B 20 19.98 19.98 0.03 -0.12 0.13 3 20 19.95 19.95 0.00 -0.25 0.00 1 20 19.93 19.93 0.03 -0.38 0.13 A 20 19.98 19.98 0.03 -0.12 0.13 24h 2 20 19.93 19.93 0.03 -0.38 0.13 B 20 19.93 19.93 0.03 -0.38 0.13 3 20 19.95 19.95 0.05 -0.25 0.25 1 20 19.95 19.95 0.00 -0.25 0.00 A 20 19.98 19.98 0.03 -0.12 0.13 1 week 2 20 19.93 19.93 0.03 -0.38 0.13 B 20 19.93 19.93 0.03 -0.38 0.13 3 20 19.95 19.95 0.05 -0.25 0.25 1 20 19.95 19.95 0.00 -0.25 0.00 A 20 20.05 20.05 0.00 0.25 0.00 1 month 2 20 19.90 19.90 0.00 -0.50 0.00 B 20 20.00 20.00 0.00 0.00 0.00 3 20 19.95 19.95 0.05 -0.25 0.25 1 20 19.95 19.95 0.00 -0.25 0.00 A 20 20.00 20.00 0.00 0.00 0.00 2 months 2 20 19.95 19.95 0.00 -0.25 0.00 B 20 19.95 19.95 0.05 -0.25 0.25 3 20 19.93 19.93 0.03 -0.38 0.13

165

Hereunder we display the integral results of the specimen static testing (chapter 5.4.1) for both dehumidified 100% infill Carbonium Nylon specimens (NC100D) and dehumidified 50% infill Carbonium Nylon specimens (NC50D).

Figure C.1 – Stress-strain curves for each of the specimens of the NC100D specimens’ set.

166

Figure C.2 – Stress-strain curves for each of the specimens of the NC50D specimens’ set.

167

The values of the measured parameters for each specimen are summarized in Table C.7.

Table C.7 – Measured parameters for each specimen.

Ultimate Young’s Yield Ultimate Specimens’ Specimen Poisson’s tensile modulus stress tensile force set label ratio stress [MPa] [MPa] at break [N] [MPa] 1 4328 0.42 35.20 54.56 2204 A 4956 - 38.71 59.50 2380 NC100D 2 4960 - 39.26 61.14 2421 B 4460 - 36.74 58.17 2309 3 - - - - - 1 2922 - 19.44 36.79 1434 A 3165 - 24.09 43.99 1710 NC50D 2 3235 - 25.69 44.94 1747 B 3001 - 23.83 42.88 1661 3 2969 - 23.13 43.13 1674

168

Hereunder we report the integral engineering and true stress-strain curves as well as the true plastic strain curves for each specimen that underwent static testing (chapter 5.4.2), namely dehumidified Carbonium Nylon with 100% infill (NC100D) and 50% infill (NC50D), as well as the values of engineering and true yield stress for each specimen.

Figure C.3 – Engineering and true stress-strain curve for the NC100D specimens.

Table C.8 – Engineering vs. true yield stress for NC100D specimens.

Engineering Specimens’ Specimen True Yield Yield stress set label stress [MPa] [MPa] 1 35.20 35.71 A 38.71 39.23 2 39.26 39.96 NC100D B 36.74 37.52 3 - - Mean 37.48 38.08

169

Figure C.4 – True plastic stress-strain curve for NC100D specimens.

170

Figure C.5 – Engineering and true stress-strain curve for one specimen of the NC50D set.

171

Table C.9 – Engineering vs. true yield stress for NC50D specimens.

True Engineering Specimens’ Specimen Yield Yield stress set label stress [MPa] [MPa] 1 19.44 19.87 A 24.09 24.30 2 25.69 26.12 NC50D B 23.83 24.22 3 23.13 23.49 Mean 23.23 23.60

172

Figure C.6 – True plastic stress-strain curve for one specimen of the NC50D set.

173

Bibliography

[1] “Roadrunnerfoot_Invalidi_civili_dati.” [Online]. Available: http://www.roadrunnerfoot.com/chisiamo/Invalidi_civili.html.

[2] Ministero della Salute, “I rapporti annuali,” www.salute.gov.it. [Online]. Available: http://www.salute.gov.it/portale/temi/p2_6.jsp?lingua=italiano&id=1237&area=ricoveriOsped alieri&menu=vuoto.

[3] C. A. Frigo and E. E. Pavan, “Prosthetic and Orthotic Devices,” p. 67.

[4] E. Di Stanislao, R. Pellegrini, D. Zenardi, and F. Mattogno, “The prosthetic phase: The prosthesis, the prostheses,” G. Ital. Med. Lav. Ergon., vol. 37, 2015.

[5] M. D. Rotta, “ANALISI NUMERICA DELLE PRESSIONI ALL’INTERFACCIA TRA IL MONCONE DI UN AMPUTATO TRANSFEMORALE E TRE DIFFERENTI TIPOLOGIE DI INVASATURA DURANTE LA CALZATA ED IL CAMMINO,” Laurea Magistrale in Ingegneria Biomedica, Politecnico di Milano, Milano, 2014.

[6] M. C. Faustini, R. R. Neptune, R. H. Crawford, W. E. Rogers, and G. Bosker, “An Experimental and Theoretical Framework for Manufacturing Prosthetic Sockets for Transtibial Amputees,” IEEE Trans. Neural Syst. Rehabil. Eng., vol. 14, no. 3, pp. 304–310, Sep. 2006.

[7] I. Tubertini, “Studio di un processo produttivo per la realizzazione di invasature transtibiali mediante stampa 3D FDM,” Laurea Magistrale in Ingegneria Biomedica, Politecnico di Milano, Milano, 2019.

[8] R. Melnikova, A. Ehrmann, and K. Finsterbusch, “3D printing of textile-based structures by Fused Deposition Modelling (FDM) with different polymer materials,” IOP Conf. Ser. Mater. Sci. Eng., vol. 62, p. 012018, Aug. 2014.

[9] “Best Rapid Prototyping Solution? SLA vs. SLS vs. FFF | SD3D Printing.” [Online]. Available: https://www.sd3d.com/fff-vs-sla-vs-sls/.

[10] C. Dudescu and L. Racz, “Effects of Raster Orientation, Infill Rate and Infill Pattern on the Mechanical Properties of 3D Printed Materials,” ACTA Univ. Cibiniensis, vol. 69, no. 1, pp. 23–30, Dec. 2017.

[11] P. Dudek, “FDM 3D Printing Technology in Manufacturing Composite Elements,” Arch. Metall. Mater., vol. 58, no. 4, pp. 1415–1418, Dec. 2013.

[12] MANUFACTUR3D, “How FDM/FFF 3D Printing Technology Works?,” MANUFACTUR3D, 29-Jan-2018. .

[13] ISO/TC 168 Prosthetics and Orthotics, ISO 10328:2016. Prosthetics—Structural testing of lower-limb prostheses—Requirements and test methods., 2nd ed. BSI Standards Publication, 2016.

[14] M. J. Gerschutz, M. L. Haynes, D. Nixon, and J. M. Colvin, “Strength evaluation of prosthetic check sockets, copolymer sockets, and definitive laminated sockets,” J. Rehabil. Res. Dev., vol. 49, no. 3, p. 405, 2012.

[15] C. A. Frigo, Bioingegneria del sistema motorio. Aracne, 2018.

174

[16] R. H. Graebner and T. A. Current, “Relative Strength of Pylon-to-Socket Attachment Systems Used in Transtibial Composite Sockets:,” JPO J. Prosthet. Orthot., vol. 19, no. 3, pp. 67–74, Jul. 2007.

[17] G. Liao et al., “Properties of oriented carbon fiber/polyamide 12 composite parts fabricated by fused deposition modeling,” Mater. Des., vol. 139, pp. 283–292, Feb. 2018.

[18] A. G. Pedroso, L. H. I. Mei, J. A. M. Agnelli, and D. S. Rosa, “The influence of the drying process time on the final properties of recycled glass fiber reinforced polyamide 6,” Polym. Test., vol. 21, no. 2, pp. 229–232, Jan. 2002.

[19] ISO/TC 61/SC 2 Mechanical behavior, ISO 527-1:2012. Plastics: determination of tensile properties - Part 1: Geneal Principles, 2nd ed. British Standards Institution, 2012.

[20] ISO/TC 61/SC 2 Mechanical behavior, ISO 527-2:2012. Plastics — Determination of tensile properties — Part 2: Test conditions for moulding and extrusion plastics, 2nd ed. British Standards Institution, 2012.

[21] Y. A. Çengel, J. M. Cimbala, and R. H. Turner, Elementi di fisica tecnica. Termodinamica applicata meccanica dei fluidi trasmissione del calore. McGraw-Hill Education, 2017.

[22] D. M. Ştefănescu, “Wheatstone Bridge - The Basic Circuit for Strain Gauge Force Transducers,” in Handbook of Force Transducers: Principles and Components, D. M. Ştefănescu, Ed. Berlin, Heidelberg: Springer, 2011, pp. 347–360.

[23] “Getting Started with Abaqus/CAE,” p. 695.

[24] F. P. Beer, Mechanics of materials, Seventh edition. New York, NY: McGraw-Hill Education, 2015.

[25] R. G. Budynas and J. K. Nisbett, Shigley’s mechanical engineering design, 9th ed. New York: McGraw-Hill, 2011.

[26] “Abaqus Analysis User’s Guide, vol5,” p. 946.

[27] T. R. Chandrupatla and A. D. Belegundu, Introduction to Finite Elements in Engineering, 4th ed. Pearson College Div, 2011.

[28] J. Fish and T. Belytschko, A First Course in Finite Elements. Wiley, 2007.

[29] “Abaqus/CAE User’s Guide,” p. 1146.

[30] E. Q. Sun, “Shear Locking and Hourglassing in MSC Nastran, ABAQUS, and ANSYS,” p. 9.

[31] “What is FEA | Finite Element Analysis? — SimScale Documentation.” [Online]. Available: https://www.simscale.com/docs/content/simwiki/fea/whatisfea.html.

[32] “Aluminum 6061-T6; 6061-T651,” MatWeb - Material Property data. [Online]. Available: http://matweb.com/search/DataSheet.aspx?MatGUID=b8d536e0b9b54bd7b69e4124d8f1d20a &ckck=1.

[33] J. S. Weaver, A. Khosravani, A. Castillo, and S. R. Kalidindi, “Tensile and Microindentation Stress-Strain Curves of Al-6061.” Jul-2016.

175

[34] M. Giglio, M. Gobbi, S. Miccoli, and M. Sangirardi, Costruzione di macchine. McGraw-Hill, 2010.

[35] N. E. Dowling, S. L. Kampe, and M. V. Kral, Mechanical Behavior of Materials, 5th ed. Pearson, 2018.

[36] M. Yarwindran, N. A. Sa’aban, M. Ibrahim, and R. Periyasamy, “THERMOPLASTIC ELASTOMER INFILL PATTERN IMPACT ON MECHANICAL PROPERTIES 3D PRINTED CUSTOMIZED ORTHOTIC INSOLE,” vol. 11, no. 10, p. 6, 2016.

[37] J. H. Ahrens and U. Dieter, “Computer methods for sampling from gamma, beta, poisson and bionomial distributio,” Springer-Verl., vol. 12, no. 3, pp. 223–246, 1974.

176

Acknowledgements

Questo progetto di tesi è stato lungo e faticoso, ma allo stesso tempo ricco di soddisfazioni. Certamente, non sarebbe stato possibile senza l’aiuto di alcune persone che vorrei ringraziare.

Ringrazio innanzi tutto il professor Frigo che mi ha dato l’opportunità di svolgere la tesi presso il Centro Protesi di Vigorso di Budrio.

All’interno del Centro Protesi, il mio più grande ringraziamento va all’Ing. Cutti, per avermi sempre permesso di lavorare in totale libertà e autonomia ma senza mai lasciarmi sola. Grazie per avermi sempre incoraggiata e sostenuta, e per la grande fiducia che ha mostrato nelle mie capacità fin dall’inizio.

Un ringraziamento speciale va all’Ing. Pisu, per l’immensa disponibilità che ha mostrato in questi mesi. Non ci sono parole per esprimere la mia gratitudine per la costanza con la quale mi ha seguito. Il suo aiuto e i suoi consigli sono stati preziosissimi.

Un grazie di cuore va poi ad Ilaria, per il bellissimo progetto che mi ha lasciato e per tutto il tempo che mi ha dedicato all’inizio della tesi.

Infine, credo che un ringraziamento (o forse più di uno) debba andare alla mia famiglia, che mi ha sempre supportato, ma soprattutto sopportato, in questi mesi di tesi e anni di studio.

177