Application of Fracture Mechanics to Engineering Design of Complex Structures G

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1st Virtual European Conference on Fracture Application of fracture mechanics to engineering design of complex structures G. Clerc1*, Andreas J. Brunner2, Peter Niemz3, Jan-Willem van de Kuilen1

1Technical University of Munich, Wood Technology Munich, DE-80797 München 2Empa, Swiss Federal Laboratories for Materials Science and Technology, Mechanical Systems Engineering, CH-8600 Dübendorf3ETH Zürich, Institute for building material, Schafmattstr. 6, 8093 Zürich

Abstract

The use of a fracture mechanics approach for design of engineering structures is relatively rare compared to stress or stiffness based approach. Using a fracture mechanics approach, the influence of defects and their propagation can be better described than using a stress or stiffness approach. Also, stress or stiffness approaches are generally conservative compared to the fracture mechanics approach. Despite these advantages, a fracture mechanics approach is relatively rarely used in engineering design due to the difficulty to obtain accurate energy release rates (G-value) and to the derivation of the compliance/crack length equation. In this study, the energy release rate value was derived based on a standard 3 points end notched flexure samples (3-ENF) for a crack propagation in mode II. To investigate whether the determined energy release rate (ERR) value is influenced by the sample size, a new sample geometry was designed with a crack positioned in the middle of the specimen (but without open cracked ends) and compared with the 3-ENF specimen. For this new geometry, the compliance/crack length equation was determined according to the Griffith approach. The specimen compliance was derived according to the Euler-Bernoulli equation. The specimen was tested under quasi-static and fatigue flexure loads. It was shown that the fracture mechanics approach could not simply be used on this new sample geometry. Two possible explanations are discussed, one being the difficulty of obtaining ERR-values which are not influenced by the sample geometry, and the limitation of using a Euler-Bernoulli approach to derive the compliance of specimen. This study presents and discusses a few of the major difficulties which influence the application of fracture mechanics approach to design of engineering structure, such as influence of the sample size on ERR value and the validity of the Griffith equation for complex structures and anisotropic materials.

© 2020 The Authors. Published by Elsevier B.V. This© 2020 is an The open Authors. access article Publis underhed theby ELSEVIERCC BY-NC-ND B.V. license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo

* Corresponding author. E-mail address: [email protected]

2452-3216 © 2020 The Authors. Published by ELSEVIER B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo

2452-3216 © 2020 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0) Peer-review under responsibility of the European Structural Integrity Society (ESIS) ExCo 10.1016/j.prostr.2020.10.152

10.1016/j.prostr.2020.10.152 2452-3216 1762 G. Clerc et al. / Procedia Structural Integrity 28 (2020) 1761–1767 2 Gaspard Clerc/ Structural Integrity Procedia 00 (2020) 000–000 Gaspard Clerc / Structural Integrity Procedia 00 (2019) 000–000 3

Keywords: Fracture Mechanics, Wood Adhesive sample, the crack was placed in the middle of the sample, the sample was loaded under 3-points flexure. The approach to solve the crack equation was based on the Euler-Bernoulli Beam equation as applied for example by Williams, 1987 for central crack sample. The main difference is that a notch was present on the tension side of the sample. To the 1. Introduction author’s knowledge it is the first time, that the 3-CCF geometry is proposed.

Most of the engineering design nowadays is based on a stress/stiffness approach, meaning that the resistance of a 1.1. Application of fracture mechanics to 3-ENF sample structural member is estimated according to the maximum stress occurring in the structural element. To estimate the strength of the structural member its stiffness is obtained which is generally well correlated with the strength. This The first equation of fracture mechanics is the Griffith equation (1): approach is relatively simple and robust. However, it fails to describe accurately the influence of defects such as cracks or delaminations. Indeed, in a stress/stiffness approach the structural member is simplified as a continuum with homogenous properties, however this is rarely the case. To take as example timber, cracks can be present in the timber (1) � due to the drying process and/or delamination can occur due to ambient humidity change. In a stress/stiffness approach 𝑃𝑃 𝑑𝑑𝑑𝑑 those defects are generally considered using a strength reduction factor. This approach cannot accurately describe the where P is the load, b is the width, C the compliance�� and∙ a the crack length. The compliance should be expressed complexity of the presence of defects and fails to predict for example if the size of a given defect is critical or not and 2𝑏𝑏 𝑑𝑑𝑑𝑑 according to the crack length. This is generally done using the Euler-Bernoulli equation: if this defect will propagate or not. Also, it is generally too conservative.

To answer these specific questions, another approach should be used. The theory of fracture mechanics was developed

by Griffith (1921) and can be used to predict the material resistance to failure. Two main approaches can be used, a stress intensity factor approach and an energy based approach. Barret and Foschi (1977) investigated the stress (2) intensity factor approach. In this paper only the energy based approach is investigated as it is more suited to anisotropic � 𝑑𝑑 𝑤𝑤 and heterogenous material such as bonded wood, and as the derivation of the main equation is less dependent on the 𝐸𝐸𝐸𝐸 � � ��𝑥𝑥� sample geometry. 𝑑𝑑𝑥𝑥 Fracture mechanics is often used in material testing with samples having well defined properties. It is however EI is the flexural rigidity, w the deflection and q(x) is the load depending of the length of the sample. rarely used in engineering design with structural members of arbitrary geometry. In timber construction standards, This equation should be solved for the 3-ENF sample geometry: there are few design problems which require a fracture mechanic approach. One example of fracture mechanics in timber construction is found in Eurocode 5 (EC5), where the design of connections loaded perpendicular to the grain are done according to Linear Elastic Fracture Mechanics (LEFM) (Jockwer and Dietsch, 2018). (Gustafsson et al., � 2001) proposed a design approach based on fracture mechanics for the design of glued-in-rods. This approach is 𝐸𝐸�𝐼𝐼 𝑑𝑑 𝑦𝑦 1 however not yet implemented in design standards. Other authors (Bengtsson and Johansson, 2002), (Gustafsson, 1987) ⎧ � �� 𝑃𝑃𝑥𝑥 �� 𝑥𝑥𝑑𝑑� 8 𝑑𝑑𝑑𝑑 4 proposed to use fracture mechanics for the design of bonded glue line instead of using strength/stiffness criteria. The ⎪ � general issue with fracture mechanic based design is that despite a more sound physical basis, it is complex to acquire 𝑑𝑑 𝑦𝑦 1 (3) � � reliable Energy Release Rate values. This is due to the difficulty to determine such values, to the relatively scant 𝐸𝐸 𝐼𝐼 �� 𝑃𝑃𝑥𝑥��𝑑𝑑 �𝑥𝑥𝐿𝐿� ⎨ 𝑑𝑑�𝑑𝑑 2 available literature on the topic compared to strength values and to the larger scatter in material morphology and ⎪ 𝑑𝑑 𝑦𝑦 1 fracture properties. Another, difficulty, which is examined in this paper, is the complexity of deriving the crack-growth 𝐸𝐸�𝐼𝐼 � �� 𝑃𝑃𝑥𝑥 � 𝑃𝑃𝐿𝐿��𝐿𝐿 �𝑥𝑥2𝐿𝐿� equation for arbitrary geometry. These points are summarized in table 1. ⎩ 𝑑𝑑𝑑𝑑 2

Figure 1: Left. 3-ENF sample geometry, Right. System of equation to solve for the 3-ENF sample. according to Yoshihara et al. (2000) Table 1: Comparison between Stress/Stiffness and Fracture Mechanics approach for structure design

Stress/Stiffness Approach Fracture mechanics Approach

Material characteristic values are widely Material values are rarely available and The system of equations can be solved to obtain the deflection v of the beam at the middle point: available more difficult to obtain Simple design approach Application more complicated (derivation of crack growth equation) (4) Difficulty to deal with defects and their Can be used to predict the crack � � 𝑃𝑃�2𝐿𝐿 � �𝑑𝑑 � propagation in material propagation for specific loading situation 𝑣𝑣� From equation 5, the compliance is obtained according12𝐸𝐸 to: �𝐼𝐼

In this paper, the difficulty of applying fracture mechanics to arbitrary sized structural member is investigated. First (5) the main steps to derive the fracture mechanics equation are shown for a well-known sample geometry (3 point End � � 𝑣𝑣 2𝐿𝐿 � �𝑑𝑑 Notched Flexure sample, 3-ENF). Then these main steps are applied to a new sample geometry, the 3 points central 𝑑𝑑� � � crack flexure sample (3-CCF). This geometry was designed to investigate the effect of a specimen upscaling on the This expression can then be inserted in the equation𝑃𝑃 1: 12𝐸𝐸 𝐼𝐼 crack propagation. The size of the sample was however limited by the available testing machine. For the 3-CCF G. Clerc et al. / Procedia Structural Integrity 28 (2020) 1761–1767 1763 2 Gaspard Clerc/ Structural Integrity Procedia 00 (2020) 000–000 Gaspard Clerc / Structural Integrity Procedia 00 (2019) 000–000 3

To answer these specific questions, another approach should be used. The theory of fracture mechanics was developed by Griffith (1921) and can be used to predict the material resistance to failure. Two main approaches can be used, a stress intensity factor approach and an energy based approach. Barret and Foschi (1977) investigated the stress (2) intensity factor approach. In this paper only the energy based approach is investigated as it is more suited to anisotropic � 𝑑𝑑 𝑤𝑤 and heterogenous material such as bonded wood, and as the derivation of the main equation is less dependent on the 𝐸𝐸𝐸𝐸 � � ��𝑥𝑥� sample geometry. 𝑑𝑑𝑥𝑥 Fracture mechanics is often used in material testing with samples having well defined properties. It is however EI is the flexural rigidity, w the deflection and q(x) is the load depending of the length of the sample. rarely used in engineering design with structural members of arbitrary geometry. In timber construction standards, This equation should be solved for the 3-ENF sample geometry: there are few design problems which require a fracture mechanic approach. One example of fracture mechanics in timber construction is found in Eurocode 5 (EC5), where the design of connections loaded perpendicular to the grain are done according to Linear Elastic Fracture Mechanics (LEFM) (Jockwer and Dietsch, 2018). (Gustafsson et al., � 2001) proposed a design approach based on fracture mechanics for the design of glued-in-rods. This approach is 𝐸𝐸�𝐼𝐼 𝑑𝑑 𝑦𝑦 1 however not yet implemented in design standards. Other authors (Bengtsson and Johansson, 2002), (Gustafsson, 1987) ⎧ � �� 𝑃𝑃𝑥𝑥 �� 𝑥𝑥𝑑𝑑� 8 𝑑𝑑𝑑𝑑 4 proposed to use fracture mechanics for the design of bonded glue line instead of using strength/stiffness criteria. The ⎪ � general issue with fracture mechanic based design is that despite a more sound physical basis, it is complex to acquire 𝑑𝑑 𝑦𝑦 1 (3) � � reliable Energy Release Rate values. This is due to the difficulty to determine such values, to the relatively scant 𝐸𝐸 𝐼𝐼 �� 𝑃𝑃𝑥𝑥��𝑑𝑑 �𝑥𝑥𝐿𝐿� ⎨ 𝑑𝑑�𝑑𝑑 2 available literature on the topic compared to strength values and to the larger scatter in material morphology and ⎪ 𝑑𝑑 𝑦𝑦 1 fracture properties. Another, difficulty, which is examined in this paper, is the complexity of deriving the crack-growth 𝐸𝐸�𝐼𝐼 � �� 𝑃𝑃𝑥𝑥 � 𝑃𝑃𝐿𝐿��𝐿𝐿 �𝑥𝑥2𝐿𝐿� equation for arbitrary geometry. These points are summarized in table 1. ⎩ 𝑑𝑑𝑑𝑑 2

Stress/Stiffness Approach Fracture mechanics Approach

In this paper, the difficulty of applying fracture mechanics to arbitrary sized structural member is investigated. First (5) the main steps to derive the fracture mechanics equation are shown for a well-known sample geometry (3 point End � � 𝑣𝑣 2𝐿𝐿 � �𝑑𝑑 Notched Flexure sample, 3-ENF). Then these main steps are applied to a new sample geometry, the 3 points central 𝑑𝑑� � � crack flexure sample (3-CCF). This geometry was designed to investigate the effect of a specimen upscaling on the This expression can then be inserted in the equation𝑃𝑃 1: 12𝐸𝐸 𝐼𝐼 crack propagation. The size of the sample was however limited by the available testing machine. For the 3-CCF 1764 G. Clerc et al. / Procedia Structural Integrity 28 (2020) 1761–1767 4 Gaspard Clerc/ Structural Integrity Procedia 00 (2020) 000–000 Gaspard Clerc / Structural Integrity Procedia 00 (2019) 000–000 5

1.3. Stability of the crack propagation (6) � � 9𝑎𝑎 𝑃𝑃 Once the energy release rate is reached the crack will propagate. If the amount of energy released is higher than the 𝐺𝐺� This final equation describes the energy of the sample dependin24𝐸𝐸𝐸𝐸 g on its flexural rigidity, crack length and load. This amount needed for the crack to propagate, the crack will propagate in an unstable way. This is described in equation value should then be lower than a critical G so that to avoid the further propagation of the crack and 10: ultimately the failure of the sample. �� 𝐺𝐺�𝐺𝐺 (10)

This method was then applied to a new designed sample geometry. For this sample, the crack is present at the middle 𝑑𝑑𝑑𝑑 of the sample, both ends are adhesively bonded as shown in Fig. 2. �0 For the 3-ENF sample (left) and the 3-CCF sample𝑑𝑑𝑑𝑑 (right), the following conditions for a stable crack growth are obtained:

(11)

�𝐿𝐿 𝐿𝐿 𝑑𝑑� � ≅ 0.7𝐿𝐿 𝑑𝑑� √3 2 2. Material and Methods

Beech wood (Fagus sylvatica L.) was used for manufacturing the samples and glued with an 1C-PUR adhesive (Loctite HB110, Henkel). The dimensions of the 3-CCF sample are shown in Fig. 2. Steel supports were used for the 3-CCF to prevent a compression failure of the wood under the supports. A detailed description of the 3-ENF sample Figure 2: Geometry of the 3 point central crack flexure (3-CCF) sample manufacturing method can be found in Clerc et al (2019). The testing was performed on a universal hydraulic testing machine with a 20 kN load sensor with a load and displacement accuracy of at least 1% of the measured value. 1.2. Application of fracture mechanics to 3-CCF sample

For 3-CCF sample geometry a new system of equation should be written as :

(7) � 𝐸𝐸𝐸𝐸 퐸𝐸𝐸 𝑦𝑦 𝑃𝑃 ⎧ � �� 퐸 𝑑𝑑 �𝑑𝑑� 𝐿𝐿� � 𝑎𝑎� 𝑑𝑑𝑑𝑑 � 2 2 ⎨𝐸𝐸𝐸𝐸 퐸𝐸𝐸 𝑦𝑦 𝑃𝑃 � �� 퐸𝑑𝑑𝐿𝐿 � � � 𝑎𝑎 �𝑑𝑑�𝐿𝐿� � ⎩ 8∙𝑑𝑑𝑑𝑑 2 2 2 Solving this system of equations gives the following expression for the beam deflection at the middle point:

(8)

𝑃𝑃 � � � � �� 6𝑎𝑎 �𝐿𝐿 � 84𝐿𝐿𝑎𝑎 � 42𝐿𝐿 𝑎𝑎� 48𝐸𝐸𝐸𝐸

This expression can then be inserted in equations 5 and 6 to obtain equation (9) for the energy release rate of the 3- Figure 3: Difference between the calculated and measured rupture force for the 3-CCF sample with different height CCF: 3. Results (9)

(11)

�𝐿𝐿 𝐿𝐿 𝑑𝑑� � ≅ 0.7𝐿𝐿 𝑑𝑑� √3 2 2. Material and Methods

For 3-CCF sample geometry a new system of equation should be written as :

(8)

𝑃𝑃 � � � � �� 6𝑎𝑎 �𝐿𝐿 � 84𝐿𝐿𝑎𝑎 � 42𝐿𝐿 𝑎𝑎� 48𝐸𝐸𝐸𝐸

� In the figure 3, the measured force needed to break a 3-CCF sample with different height is compared with the rupture 7𝑃𝑃 � 𝐺𝐺�� 𝐿𝐿 � 2𝑎𝑎� force calculated according to equation 9. Only for a sample height of 50 mm did the tested and calculated force 16𝐵𝐵𝐵𝐵𝐵𝐵 coincide. With increasing height, the rupture force increases. This should, however, not be the case as shown in equation 9, with increasing height the rupture force should decrease for constant G-value as depicted in Figure 3. It Gaspard Clerc / Structural Integrity Procedia 00 (2019) 000–000 7 1766 G. Clerc et al. / Procedia Structural Integrity 28 (2020) 1761–1767 6 Gaspard Clerc/ Structural Integrity Procedia 00 (2020) 000–000 shall also be noted that the maximum available force from the testing apparatus was 17 kN. It can be suspected that despite the different type of plot used in Fig. 4 and 5, for the 3-CCF the crack propagation is highly unstable. Once the rupture force would continue to increase, but this should be verified with a better suited force sensor. the rupture force is reached the deformation increases very rapidly, indicating a rapid crack growth. In comparison for the 3-ENF sample the crack propagation remains stable once the critical energy released rate value is reached. 3.1. Consideration about the crack stability between the 3-ENF and the 3-CCF sample 4. Discussion For the 3-ENF, if the crack length is chosen according to equation 11, the following crack growth curves are typically obtained. The results clearly show that the calculation method is not suited to describe the behavior for the 3-CCF sample. Indeed, the rupture force was not satisfyingly calculated. Also, the propagation stability was not achieved. The reasons to explain this are still not understood in detail. The following points should be investigated: 2500 - Due to the central crack and to the position of the loading head (see Fig.2), higher friction effects are present VN 3158 (N=3) compared to the 3-ENF sample. Those are not accounted for in the Euler-Bernoulli equation and should HB 110 (N=4) probably be accounted for a more accurate description of the crack growth. 2000 PRF (N=4) - For the 3-ENF sample, the flexural rigidity of the cracked part of the sample is reduced by a factor of 8. This could be different for the 3-CCF sample as both ends are not free as it is the case for the 3-ENF sample. - The Euler-Bernoulli equation is a simplification of the more complex Timoshenko beam equation. For long

] and slender beams, the differences between both equations are small, but increase for short and high beams.

2 1500 The 3-CCF sample is composed of three relatively short parts, the first glued, the non-glued and the last glued

[J/m part. It could be that the Euler-Bernoulli beam equation is not suited for this type of sample. This is also II

G 1000 suggested by the increasing difference between the calculated and tested rupture force with increasing sample height. To account for these points, a finite element model of the 3-CCF sample should be compared to the test results. Using 500 a FEM model would allow to account for the non-linear effects mentioned above.

5. Conclusion 0 60 70 80 90 100 110 120 130 140 In this example, it was shown that the basic fracture mechanics principle used to characterize 3-ENF samples cannot Crack length [mm] be directly used to design any arbitrary structures. Even for a simple sample geometry as the 3-CCF sample, no satisfying equation was obtained. This illustrates the difficulty of using fracture mechanics for engineering design of Figure 4: Example of crack propagation for the 3-ENF sample more complex and realistic structures. The use of more advanced modeling methods, such as finite element method, should probably be investigated, as it allows to consider non-linear effects such as friction and shear stresses on the In comparison the crack growth of the 3-CCF is shown in figure 5. beam deflection.

6. References

J.G. Williams, 1987, On the calculation of energy releases rates for cracked laminates, International Journal of Fracture 36: 101-119 G. Clerc, A. J. Brunner, S. Josset, P. Niemz, F. Pichelin, J-W. G. van de Kuilen, 2019, Adhesive wood joints under quasi-static and cyclic fatigue fracture Mode II loads, International Journal of Fatigue 123: 40-52 H. Yoshiara and M. Ohta (2000). Measurement of mode II fracture toughness of wood by the end-notched flexure test Journal of Wood Science 46:273-278 Griffith, A. (1921). The phenomena of rupture and flow in solids. Philosphical Transactions of the Royal Society of London, 221:163–198. Jockwer, R. and Dietsch, P. (2018). Review of design approaches and test results on brittle failure modes of connections loaded at an angle to the grain. Engineering and Structures, 171:362–372. Bengtsson, C. and Johansson, C.-J. (2002). GIROD - Glued in Rods for Timber Structures. SP Swedisch National Testing and Research Institute, Report: 2002:26. Gustafsson, P., Serrano, E., Aicher, S., and Johansson, C.-J. (2001). A strength design equation for glued-in rods. Symposium, Joints in timber Structures; 2001; Stuttgart, Germany, RILEM:323–332.

Figure 5: Example of load-deformation curve corresponding to crack propagation for the 3-CCF. Gaspard Clerc / Structural Integrity Procedia 00 (2019) 000–000 7 G. Clerc et al. / Procedia Structural Integrity 28 (2020) 1761–1767 1767 6 Gaspard Clerc/ Structural Integrity Procedia 00 (2020) 000–000 shall also be noted that the maximum available force from the testing apparatus was 17 kN. It can be suspected that despite the different type of plot used in Fig. 4 and 5, for the 3-CCF the crack propagation is highly unstable. Once the rupture force would continue to increase, but this should be verified with a better suited force sensor. the rupture force is reached the deformation increases very rapidly, indicating a rapid crack growth. In comparison for the 3-ENF sample the crack propagation remains stable once the critical energy released rate value is reached. 3.1. Consideration about the crack stability between the 3-ENF and the 3-CCF sample 4. Discussion For the 3-ENF, if the crack length is chosen according to equation 11, the following crack growth curves are typically obtained. The results clearly show that the calculation method is not suited to describe the behavior for the 3-CCF sample. Indeed, the rupture force was not satisfyingly calculated. Also, the propagation stability was not achieved. The reasons to explain this are still not understood in detail. The following points should be investigated: 2500 - Due to the central crack and to the position of the loading head (see Fig.2), higher friction effects are present VN 3158 (N=3) compared to the 3-ENF sample. Those are not accounted for in the Euler-Bernoulli equation and should HB 110 (N=4) probably be accounted for a more accurate description of the crack growth. 2000 PRF (N=4) - For the 3-ENF sample, the flexural rigidity of the cracked part of the sample is reduced by a factor of 8. This could be different for the 3-CCF sample as both ends are not free as it is the case for the 3-ENF sample. - The Euler-Bernoulli equation is a simplification of the more complex Timoshenko beam equation. For long

] and slender beams, the differences between both equations are small, but increase for short and high beams.

2 1500 The 3-CCF sample is composed of three relatively short parts, the first glued, the non-glued and the last glued

[J/m part. It could be that the Euler-Bernoulli beam equation is not suited for this type of sample. This is also II

6. References

Figure 5: Example of load-deformation curve corresponding to crack propagation for the 3-CCF.