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On the Relationship Between Indenation Hardness and Modulus, and the Damage Resistance of Biological Materials

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On the Relationship Between Indenation Hardness and Modulus, and the Damage Resistance of Biological Materials bioRxiv preprint doi: https://doi.org/10.1101/107284; this version posted February 9, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. On the relationship between indenation hardness and modulus, and the damage resistance of biological materials. David Labontea,∗, Anne-Kristin Lenzb, Michelle L. Oyena aThe NanoScience Centre, Department of Engineering, Cambridge, UK bUniversity of Applied Sciences, Bremen, Germany Abstract The remarkable mechanical performance of biological materials is based on intricate structure-function relation- ships. Nanoindentation has become the primary tool for characterising biological materials, as it allows to relate structural changes to variations in mechanical properties on small scales. However, the respective theoretical back- ground and associated interpretation of the parameters measured via indentation derives largely from research on ‘traditional’ engineering materials such as metals or ceramics. Here, we discuss the functional relevance of inden- tation hardness in biological materials by presenting a meta-analysis of its relationship with indentation modulus. Across seven orders of magnitude, indentation hardness was directly proportional to indentation modulus, illus- trating that hardness is not an independent material property. Using a lumped parameter model to deconvolute indentation hardness into components arising from reversible and irreversible deformation, we establish crite- ria which allow to interpret differences in indentation hardness across or within biological materials. The ratio between hardness and modulus arises as a key parameter, which is a proxy for the ratio between irreversible and reversible deformation during indentation, and the material’s yield strength. Indentation hardness generally increases upon material dehydration, however to a larger extend than expected from accompanying changes in indentation modulus, indicating that water acts as a ‘plasticiser’. A detailed discussion of the role of indenta- tion hardness, modulus and toughness in damage control during sharp or blunt indentation yields comprehensive guidelines for a performance-based ranking of biological materials, and suggests that quasi-plastic deformation is a frequent yet poorly understood damage mode, highlighting an important area of future research. Keywords: Structure-function relationships, Biomechanics, Wear, Hydration Introduction performance of the ‘bulk’ materials is a prerequisite for any attempt at replicating their functionality. Hence, In search for inspiration for the design of novel ma- mechanical and structural characterisation methods tra- terials with enhanced properties, biological materials ditionally developed for and applied to engineering ma- receive an increasing amount of attention (e. g. 1–5). terials are increasingly used to study biological mate- Two features of these materials stand in stark contrast to rials. Among these, instrumented indentation has been ‘classic’ engineering materials: first, they are hierarchi- particularly popular, due to the relative ease of use and cal in design, that is they are constructed from multiple its ability to accurately measure material properties on building blocks arranged on characteristic length scales various length scales (6–8). In biological materials, in- ranging from a couple of nanometres to a few millime- strumented indentation is frequently used as a tool to re- tres. Second, the structural arrangement and properties late structural changes, for example degree of minerali- of these components are tailored to specific functional sation, to variations in material properties, such as frac- demands. ture toughness, elastic modulus, and hardness. These A thorough understanding of how the arrangement properties, in turn, determine performance and hence and properties of individual components determine the the adaptive value of the structural changes in ques- Preprint submitted to Elsevier February 9, 2017 bioRxiv preprint doi: https://doi.org/10.1101/107284; this version posted February 9, 2017. The copyright holder for this preprint (which was not certified by peer review) is the author/funder, who has granted bioRxiv a license to display the preprint in perpetuity. It is made available under aCC-BY 4.0 International license. tion, and therefore can shed light on critical structure- indentation modulus, EI, and indentation hardness, HI, function relationships. of diverse biological materials. We investigate the rela- Fracture toughness, T, quantifies a material’s resis- tionship between the two properties, which is then inter- tance to crack initiation and propagation, and is super- preted with a series elastic-plastic deformation model. seding ‘strength’ as the property most associated with This analysis provides guidelines for a careful interpre- high-performance materials (9, 10). The hierarchical tation and comparison of indentation hardness across design and the ensuing complexity of biological mate- and within biological materials, and highlights poten- rials necessitates a careful execution and application of tial pitfalls. Next, we focus more specifically on the the existing indentation methodology (11). However, functional relevance of T, EI and HI for the avoidance instrumented indentation remains an unrivalled method and containment of mechanical damage introduced by for the evaluation of toughness, in particular in biologi- ‘blunt’ or ‘sharp’ contacts,1 and discuss the competition cal materials which are frequently too small or irregular between quasi-plasticity and fracture in biological ma- to allow the fabrication of ‘standardised’ specimen re- terials. Quasi-plasticity emerges as the dominant dam- quired for more conventional fracture tests (8). age mode for most biological materials, which is con- The elastic modulus quantifies the resistance to elas- trolled not by indentation hardness, but by its ratio with tic (reversible) deformation. In contrast to toughness, the indentation modulus. a large modulus is not necessarily desired (for exam- ple, adhesive structures need to be somewhat compli- Analysis & Discussion ant), so that a direct link between its magnitude and performance is often not straightforward. Most studies The correlation between indentation hardness and mod- assume isotropic behaviour of biological materials, i. e. ulus a mechanical response that is independent of the test- Indentation hardness and modulus of various biolog- ing direction. However, it is frequently acknowledged ical materials were assembled from the literature (22– that this assumption is more practical than accurate, and 89, all data are available in the supplemental material). most of the literature hence reports an indentation mod- Across seven orders of magnitude, indentation hardness ulus, E , which is equal to the plain strain modulus, E0, I was directly proportional to indentation modulus (fig- if the modulus of the indenter is much larger than that ure 1), with a proportionality constant of approximately of the material, and if the material is isotropic. 0.05, H = 0:046E1:03 (reduced major axis regression Hardness is probably the most commonly measured, I I on log-transformed data, 95 % confidence intervals (CI) but least understood of the material properties that can [0.043; 0.050] and [1.01; 1.06]). The strength of be assessed by instrumented indentation. Despite an in- the correlation, R2 = 0.96, is somewhat deceptive, as creasing body of literature which has highlighted that the regression was performed on log-transformed data. hardness is a complex property that needs to be inter- Some of the variation in the ratio H /E across well- preted with care (see e. g. 6, 12–17), it is still often I I represented materials is captured in Table 1. The data taken as a proxy for the resistance to ‘plastic’ (i. e. ir- used in the regression include properties measured with reversible) deformation. This interpretation has been different indenter geometries, and in hydrated or dehy- particularly popular in the more biological literature, drated conditions. These factors are discussed further perhaps because it provides a straightforward link to below. functional and hence ecological relevance. Hardness A correlation between hardness and bulk, Young’s has long been a property of considerable importance in and in particular shear modulus has also been reported engineering materials: it is an index of strength (18) for metals, bulk metallic glasses, ceramics and poly- and – in combination with toughness – of brittleness mers (see e. g. 90–99), which can be qualitatively un- (19). Hardness is frequently associated with a mate- rial’s resistance against being penetrated (i. e. elasti- cally deformed), spread (i. e. irreversibly deformed) or 1The distinction between sharp and blunt contacts is somewhat ar- scratched (i. e. fractured) (20). Clearly, the physical bitrary – sharp indenters are those with a steep angle of the tangent at mechanisms involved in these processes differ. What is the point of contact. Irreversible deformation will necessarily precede the functional meaning of indentation hardness? cracking if the contact is sharp, while the deformation can remain dominantly elastic prior to fracture if the indenter is blunt. To a first In this article, this question is addressed by present- approximation, spheres are blunt, and conical,
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