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Mechanisms governing the inelastic deformation of cortical bone and application to trabecular bone
Acta BIOMATERIALIA Acta Biomaterialia 2 (2006) 59–68 www.actamat-journals.com
Mechanisms governing the inelastic deformation of cortical bone and application to trabecular bone C. Mercer
a,*
, M.Y. He a, R. Wang b, A.G. Evans
a
a
b
Materials Department, University of California Santa Barbara, Santa Barbara, CA 93106, USA Department of Materials Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 Received 9 May 2005; received in revised form 28 July 2005; accepted 10 August 2005
Abstract To understand the inelastic response of bone, a three-part investigation has been conducted. In the first, unload/reload tests have been used to characterize the hysteresis and provide insight into the mechanisms causing the strain. The second part devises a model for the stress/strain response, based on understanding developed from the measurements. The model rationalizes the inelastic deformation in tension, as well as the permanent strain and hysteresis. In the third part, a constitutive law representative of the deformation is selected and used to illustrate the coupled buckling and bending of ligaments that arise when trabecular bone is loaded in compression. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Bone; Cortical; Trabecular; Inelastic deformation; Hysteresis
1. Introduction Many studies of both cortical and trabecular bone have demonstrated their propensity to inelastic strain [1–4] (Fig. 1). Parallel to the long axis, cortical bone exhibits inelastic responses that differ in tension and compression [1–6] (Fig. 1a). In tension, it yields, followed by (linear) hardening up to a failure strain of order 2.5%. The inelastic strain has been attributed to the development of diffuse microcrack arrays [7–9]. Simultaneous measurements of the axial and transverse strains have revealed that such deformation involves dilatation [10], consistent with a role of microcracks. The strains have been characterized by means of a damage model with internal state variables [11]. In compression, cortical bone also yields, but at higher stress than in tension. It strain hardens rapidly to a peak, then softens and fails at strains of about 1.5%. The softening has been attributed to the formation of shear bands [7].
*
Corresponding author. Tel.: +1 805893 5930. E-mail address: [email protected] (C. Mercer).
Such behavior is not unique, but akin to the deformation characteristics of nacre [12] and fibrous oxides [13]. When compressed, trabecular bone exhibits extensive inelastic deformation (Fig. 1b), often attaining strains exceeding 60% before failure [4]. A peak stress occurs at strains of 5–10%. The deformations preceding the peak dictate the overall load capacity. Yield surface determinations [14] suggest that inelastic mechanisms are involved. The characteristics of synthetic foams with topology similar to trabecular bone may provide benchmarks. The load capacity of elastomeric/polymer foams is dictated by the elastic buckling of its ligaments [15], while that for metal foams is controlled by yielding [16]. The consequence is that different scaling formulae relate the stress maximum, rmax, to the properties of the ligaments. For isotropic, open cell polymer foams [1]: rmax/E 0.05f2, where E is the YoungÕs modulus for the material in the ligaments and f is the volume concentration of the solid in the foam. For isotropic metal foams [16]: rmax/rY f3/2, where rY is the uniaxial yield strength of the material in the ligaments. The differences in these formulae simply underscore the need to confirm that the mechanism governing the load capacity of trabecular bone is inelastic.
1742-7061/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actbio.2005.08.004
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Fig. 2. Schematic of the instrumented flexure test showing the locations of the strain gages.
Fig. 1. (a) Schematic of the tensile and compressive stress/strain curves for cortical bone along the axis of a long bone [1]. (b) Schematic of a compressive stress/strain curve for trabecular bone [1].
The present study has three parts. (i) A mechanical probe is used to ascertain the hysteresis associated with the inelastic strains, as well as the elastic modulus evolution. (ii) Motivated by these measurements, a model is presented that highlights the relative roles of the constituent phases (collagen and mineral platelets) and of the interfaces between them. (iii) A constitutive law is invoked that incorporates key aspects of the inelastic deformation. The criterion for the choice is that the law already exists within a commercial finite element code such that, upon calibration, it can be applied immediately to large-scale problems, such as the cell level response of trabecular bone [1]. 2. Materials and methods The cortical bone used in this study was obtained from bovine femurs. Cylindrical, mid-diaphyseal sections (40 mm long) were extracted from fresh femurs (exact age not known, presumably between 2 and 7 years old). Rectangular specimens were cut on a longitudinal/radial plane at the posterior sector of the femur, with a low-speed diamond blade saw under continuous irrigation. The four rectangu-
lar surfaces were mechanically ground using silicon carbide media, and then polished with diamond suspensions, followed by 0.05 lm alumina suspension on a precision machine (MULTIPOL 2 Precision Polishing Machine, Ultra Tec, Santa Ana, USA). The final specimens were 38 mm long, 3.85 mm wide (across the thickness of the bone), and 3.05 mm (four specimens) or 1.52 mm (six specimens) thick. The specimens were stored in water at 4 °C for 1–15 days prior to testing. Four-point bend tests (Fig. 2) were conducted using a servo-hydraulic testing machine at a cross-head speed of 0.5 mm/min. The inner loading span was 15 mm and the outer support span 30 mm (Fig. 2). The upper and lower loading surfaces were chosen to be perpendicular to the lamellar boundaries of the cortical bone to minimize the structural variations across the thickness of the specimens and therefore, to allow comparison between the compressive and tensile behavior. To measure the strains on both tensile and compressive surfaces, strain gages (from either Omega Engineering or Vishay Micro-Measurements) were applied to both surfaces (one on each side) using M-Bond 200 adhesive following degreasing and cleaning. In some cases, 0°/90° strain gages were used for measurement of the transverse strains in addition to axial strains. Specimens were either monotonically loaded to failure, or repeatedly unloaded/reloaded at incrementally higher plastic strains, in order to measure hysteresis. The specimens were kept hydrated during testing by using an agarose gel (Sigma–Aldrich, USA). The inelastic deformation has been reproducibly established using six specimens. After mechanical testing, the specimens were examined with a stereomicroscope for macro-scale damage. For this purpose, some specimens were stained with basic fuchsin [17]. The stained specimens were embedded in a mounting medium. Using a low-speed diamond saw, thin slices were cut along the longitudinal direction of the specimens, with each slice either parallel or perpendicular to the compressive (upper loading) surface. The slices were polished to
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80 lm thick and embedded in a polymer medium on glass slides. This procedure caused the damage to be stained pink, and easily discernable in transmitted light (Nikon 600, USA). 3. Flexural measurements A typical response relating the load to the axial strain upon monotonic testing to failure is summarized in Fig. 3a. The corresponding stress/strain curves obtained by deconvolution [12] are plotted in Fig. 4. All tests gave essentially the same inelastic strains and strain hardening. The YoungÕs modulus (E = 25 GPa) is the same in tension and compression, and consistent with the literature [2,3]. The yield strains in tension (0.6%) and compression (1.1%) are also consistent with those reported elsewhere [2,3]. This reproducibility encourages further analysis of the inelastic strains. The simultaneous measurements of the transverse and axial strains, typified by the results presented in Fig. 3b, enable determination of the volumetric strains. The inelastic Poisson ratio (the ratio of the transverse to the axial inelastic
Fig. 4. Tensile and compressive stress/strain curves for cortical bone ascertained from the results presented in Fig. 3. The curves are similar to those reported in the literature [2,3].
strains) has been determined from these measurements and plotted as a function of the inelastic strain (Fig. 5). The results highlight the differences in tension and compression, with implications discussed below. Note that, in tension (unlike plasticity) the strains are not volume conserving [10]. At strains beyond yield, the following features are most notable (Figs. 4–8). (i) In tension, the material behaves as an inelastic solid with linear strain hardening (Fig. 4): tangent modulus, EH 1 GPa. The transverse strain is negligible, signifying an inelastic Poisson ratio, mpl 0 (Fig. 5). That is, the material dilates [10].
Fig. 3. (a) The relationships between load and axial strain measured on cortical bone in flexure. The results obtained by simulation using the Drucker–Prager constitutive law are superposed. (b) Plots of the axial and transverse strains measured on the tension and compression faces, as a function of load.
Fig. 5. Trends in the inelastic Poisson ratio with inelastic strain ascertained from Fig. 3b.
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Fig. 7. Optical images of a specimen of cortical bone fractured in bending: (a) side face; (b) partial view of the compression face (arrows indicate direction of shear band formation).
mpl 0.30.4 (Fig. 5), suggesting that the shear mechanisms governing compressive deformation [7] are nearly volume-conserving.
Fig. 6. (a) Stress–strain curves in tension upon unloading and reloading, showing 12 hysteresis loops. (b) Variation in the width of the hysteresis loops with permanent strain. (c) Variation in initial loading modulus with permanent strain.
(ii) In compression, the peak stress is appreciably larger than in tension, but the strain at failure is smaller (ef 1.6% in compression and ef 2.4% in tension; Fig. 4). Consequently, the load maximum in flexure is compression-dominated, as described elsewhere [11]. The inelastic Poisson ratio is non-zero,
Periodic unloading/reloading tests (Fig. 6a) indicate appreciable stress/strain hysteresis in tension. The measurements, unprecedented on cortical bone, elucidate the underlying inelastic mechanisms. The hysteresis is a measure of the internal friction and its irreversibility [13]. It is quantified by plots of the maximum loop width (unloading to reloading) with permanent strain (Fig. 6b). The initial loading slope (shown in Fig. 6a) is a measure of the current YoungÕs modulus, E [13]. A plot against strain (Fig. 6c) indicates that E does not change significantly. The implication is that straining does not cause cracking of the mineral platelets nor does it cause separations between the mineral phase and the collagen. Taken together, the hysteresis and modulus measurements indicate that the deformation involves internal slip (with associated friction), with the mineral phase and interfaces remaining intact. These findings motivate the mechanism to be described in the following section. The cumulative strain ascertained both on the strain gages and from the final deformed shape (>4%), significantly
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Fig. 8. Shear bands that develop on the compression face: (a) overview; (b) bands nucleating at blood vessel.
exceeds the rupture strain upon monotonic loading. The extra strain appears to accumulate on a loop-by-loop basis, manifest as a strain bias per loop. Namely, higher strains are needed for the reloading curves to attain the load maximum achieved prior to unloading (Fig. 6a). One interpretation is that the sacrificial bonds [18–20] between the collagen and mineral platelets rupture on loading, but re-form upon unloading. Side and top views of fractured specimens (Fig. 7) confirm the differences in failure mode between tension and compression. In tension, a dominant crack develops normal to the surface, penetrating to the neutral plane. In compression, shear bands are evident [7]. The angle between the bands and the loading axis is / = 28 ± 5°. Staining and cross-sectioning establishes that the bands extend across the bone lamellae and are mirror symmetric with respect to the loading axis (Fig. 8a). Higher resolution images reveal that many of the bands nucleate at the blood vessels (Fig. 8b). The similarity of this failure mode with the kinking of fibrous oxides at high temperature (Fig. 9) [21], suggests a methodology for modeling the compressive strength. Similar bands have also been observed in the ligaments of trabecular bone [22].
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Fig. 9. Scanning electron microscope images of the kink bands found in columnar-grained oxides compressed at high temperature (where they deform plastically). Note the abrupt bending of the columns within the band and the fracturing that occurs at the boundaries [21]. It is suggested that the collagen fibrils in bone experience similar deformations when subject to failure in compression.
4. Inelastic deformation mechanism The results in Fig. 6 can only be rationalized if the collagen phase is continuous and the mineral phase discontinuous. Namely, if the brittle mineral phase were continuous it would crack at quite small strains causing the modulus to decrease with increasing inelastic strain [13], contrasting with the actual measurements (Fig. 6c). Moreover, the evolving nature of the hysteresis loops with strain (Fig. 6b) indicates that irreversible slip must be occurring at the interfaces [13]. The simplest geometric representation that encapsulates these features is shown in Fig. 10. The characteristics embodied in this figure have precedents [23,24], but conflict with proposals made by others [25–28]. In the model, the collagen fibrils stretch as the strain is imposed. If the system remained elastic, a highly concentrated stress would develop at the free edges of the much stiffer mineral platelets (at z = 0). Under such circumstances, the stresses must relax, locally, by shear yielding of the interlayer. That is, to accommodate this stretch,
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Fig. 10. A schematic of the representative unit cell used to analyze the tensile stress/strain response of cortical bone: it highlights the thin adhesive layer that undergoes inelastic shear deformation.
interface slip must occur. Using this representation, the mechanical response in tension can be explored (Appendix A), with the objective of establishing the roles of the constituent materials and the interfaces. Previous models have not brought out these roles in a manner consistent with the current measurements (Fig. 6). For modeling purposes, this inelastic shear process is considered to occur at a shear yield stress, sY (with no strain hardening). Accordingly, a slip zone, length ‘, extends from the platelet edges: with ‘ increasing as P increases [13,29]. This phenomenon results in a non-linear stress/strain response, even when the platelets and the collagen, otherwise, remain elastic. For analytic simplicity, a shear lag approximation has been used [13,29]. Namely, the stresses in each constituent are considered to vary only in the axial direction. The analysis is presented in Appendix A and predicted stress/strain curves are shown in Fig. 11. Comparison with the measurements is achieved (Fig. 11b) by regarding the modulus of the collagen fibrils, E2, and the shear yield strength of the adhesive layer, sY, as fitting parameters. Upon fitting the model to the measurements, the inferred fibril modulus, E2 = 1 GPa is comparable to values reported for mammalian tendons (1.5 GPa) [30– 32]. Based on this modulus, upon invoking a platelet aspect ratio, v = 10 [33], the shear yield stress needed to fit the measurements is, sY = 1.4 MPa: a reasonable magnitude for an organic solid. The reconstructed stress/strain curve plotted in Fig. 11b demonstrates that the model is capable of faithfully repro-
Fig. 11. The tensile stress/strain curve predicted using the model depicted in Fig. 10: (a) schematic; (b) superposition onto the measured stress/strain curve.
ducing the measurements. Moreover, it is consistent with a hysteresis loop width that approaches DeH/epl ! 0.5 at higher permanent strains (Fig. 6b). It is concluded that the model embodies the respective roles of the collagen, the mineral platelets and the interfaces. 5. Constitutive law The preceding measurements can be used to predict the bending response of cortical bone and, potentially, the buckling and bending responses of ligaments in trabecular bone, provided that they can be adequately represented by a constitutive law. The relevance of the mechanical response of cortical bone to the ligaments of trabecular bone has already been substantiated [34,35]. Various finite element studies have been conducted previously. Some studies have incorporated non-linear elastic effects pertinent to buckling, as well as damage [36]. Others have included the difference in behavior between tension and compression [37]. The rationale for another assessment is twofold. (i) Select an inelastic law already available in a commercial finite element code. Implementation in a commercial code has the distinct advantage that the results can be reproduced by many other groups and that the scope can be readily extended. (ii) Identify a representation that includes the dilatation as well as tension/compression asymmetry. The preceding assessment indicates that the most important features to be incorporated are as follows: (a) It must accommodate differing responses in tension and compression.
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(b) In tension the material must dilate (with mpl 0). (c) In compression, it must deform inelastically at (nearly) constant volume. The present choice is the linear Drucker–Prager model, available in the ABAQUS Analysis UserÕs Manual [38] (Appendix B), which satisfies (a–c). However, it is isotropic and thus limited to axial loads and bending moments. This approach complements other assessments based on damage models [11]. Its application to the current flexural tests is not included because it would merely duplicate characteristics presented by Fondrk et al. [11]. Namely, stress redistribution causes the compressive stress to attain its maximum value before the failure strain is attained in tension, confirming that the flexural load capacity is dictated by shear band formation in compression.
Fig. 13. Osteoporotic bone structure comprised almost entirely of struts (no faces).
6. Stress/strain responses of trabecular bone Based on the assumption that trabecular and cortical tissues have similar mechanical properties [34,35], trends in the stress/strain curves in uniaxial compression can be predicted using the foregoing constitutive law. The results are sensitive to the choice of the representative unit cell. For illustration purposes, the isotropic, ‘‘bending-dominated’’ configuration proposed by Gibson and Ashby [1]
Fig. 12. A representative unit cell for osteoporotic trabecular bone observed in the transverse orientation, based on the cell proposed by Gibson and Ashby [1]. The inset is a finite element simulation of the deformed shape upon uniaxial compression, indicating the bending and buckling of the ligaments.
(Fig. 12) is used. This choice has the attribute that it successfully correlates trends in the stiffness of trabecular bone with relative density [1]. The configuration might be representative of (osteoporotic) bone comprised almost entirely of struts (no faces) (Fig. 13). Anisotropy could be incorporated using alternative configurations applicable to healthy bone. The responses are obtained at three relative densities (f = 0.1, 0.15 and 0.2), incorporating the imperfection sensitivity of buckling [39]. The role of imperfections has not been explicitly addressed in prior assessments. Thus, before proceeding, an eigen-value analysis of the buckling modes is conducted [39] and small imperfections characteristic of the first mode are introduced. To demonstrate the importance of imperfections, a preliminary analysis conducted with and without imperfections (Fig. 14a), reveals the characteristic overestimate of the load capacity when imperfections are absent. However, provided that imperfections are present, the predicted response is insensitive to imperfection amplitude [39]. All ensuing results are obtained using first mode imperfections. Predicted stress/strain curves are presented in Fig. 14b. The curves have the same form as the measurements in the literature [1,4]. Recall that these results have been obtained with no adjustable parameters. The peak stresses are all coincident with the buckling of the vertical members, preceded by bending of the horizontal members (Fig. 15). Upon unloading, the results reveal that some of the strain is recovered, but most is retained as permanent strain. The distributions of inelastic strains in the ligaments at the peak stress (Fig. 15) illustrate that the cross-section has become fully inelastic: that is, the peak load is governed by inelastic buckling. While the representative unit cell (Fig. 12) appears to provide a meaningful representation of osteoporotic trabecular bone, it should not be construed that the stress/ strain response is governed solely by the relative density. Alternative topologies are expected along stress lines, which would elevate the load capacity and change the
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¼ 0:2) reveal that the Fig. 15. Displacements of the cell (relative density, q peak stress is coincident with the onset of inelastic buckling. The elastic zones are highlighted as well as the ‘‘plastic hinges’’ where the inelastic strains exceed 8%. An intermediate strain contour (epl = 4%) is also shown.
overestimated, initial imperfections have been incorporated, with shape dictated by an eigen-value analysis of the buckling modes. A model for the tensile deformation of cortical bone has been introduced. The inelastic deformation arises from shear yielding of the interfaces between the platelets and the collagen. The model predicts permanent strains and hysteresis loops consistent with the measurements. It appears to have the basic attributes needed to relate the stress/strain response to the properties of the constituents and the interfaces. Fig. 14. Compressive stress/strain curves for osteoarthritic trabecular bone calculated using the representative cell depicted in Fig. 12 and the Drucker–Prager constitutive law, calibrated using the tension and compression stress/strain curves for cortical bone. Results have been obtained for three relative densities.
Appendix A. Inelastic strain model Below the slip zone depicted in Fig. 10 (z > ‘), the strain in the platelets and the collagen is the same, whereupon r1 =E1 ¼ r2 =E2
relative roles of bending and buckling. Assessments of this type will be conducted in a future study. 7. Concluding remarks
with the stresses defined in the manner shown in Fig. 10. The subscript 1 refers to the mineral phase and 2 to the collagen. Force equilibrium requires that rd 2 ¼ r1 d 1 þ r2 d 2
Measurements of the inelastic deformation of cortical bone have been used to devise and calibrate an inelastic constitutive law. The law allows for different yield strengths in tension and compression, as well as the dilatation in tension. By assuming that comparable inelastic features occur in the ligaments of trabecular bone, stress/strain curves subject to uniaxial compression have been estimated. The trends are comparable to those in the literature, while also demonstrating how inelastic bending and buckling of the ligaments affect the stress evolution beyond yield. To assure that the loads at which buckling occurs are not
ðA:1Þ
ðA:2Þ
where d, d1, d2 are the thickness dimensions shown in the figure. Eliminating r1 gives r2 1 ¼ 1þk r
ðA:3Þ
where k¼
d 1 E1 d 2 E2
Within the slip region (z < ‘), force equilibrium in the collagen gives the strain gradient
C. Mercer et al. / Acta Biomaterialia 2 (2006) 59–68
dez2 sY ¼ dz E2 d 2
ðA:4Þ
Subject to the following boundary conditions: ez2 ¼ r2 =E2 ez2 ¼ r=E2
at z ¼ 0
ðA:5Þ
The corresponding slip length is ‘ r k ¼ d 2 sY 1 þ k The displacement is thus (‘ < L) Z L sY ‘2 uz E 2 E 2 e2z dz ¼ ðL ‘Þr2 þ 2d 2 0 Consequently, the overall strain is (‘ < L) eE2 r 1 k k 1 ð1 þ kÞ ¼ 1 sY v 1þk 2 r
ðA:6Þ
ðA:7Þ
ðA:8Þ
Thereafter, upon further straining, the extra load is supported solely by the collagen. Accordingly, if it remains elastic, the elevation in stress, Dr, and the extra strain, De, must be related by ðA:10Þ
Inspection of the preceding results reveals that, when e < e0, the strain can be re-expressed as ðA:11Þ
Here ecol is the strain that would exist at stress r if the system were made solely from collagen. The overall response is plotted in Fig. 11a. Upon unloading, interface slip occurs in the opposite direction, commencing at the edges of the platelets. When the load has been fully removed, reverse slip progress to: ‘rev = L/2 [29]. Consequently, there is a permanent strain epl ¼
r2max v k2 4E22 e0 ð1 þ kÞ2
ðB:1Þ
where S is the deviatoric stress. The linear yield criterion is ðB:2Þ
where 3
t ¼ 1=2q½1 þ 1=K ð1 1=KÞðr=qÞ
At larger stresses, the slip length reaches the center of the platelets (‘ = L). This happens at a critical strain, e0: sY v ðA:9Þ e0 E 2 ¼ 2
e2col ½1 k=2 2e0
1=3
F ¼ t p tan b d ¼ 0
r 1 k <1 sY v 1 þ k
e ¼ ecol
The Drucker–Prager model is expressed in terms of allthree stress invariants [38]. It provides for a non-circular yield surface in the deviatoric plane in order to match different yield values in tension and compression. It incorporates associated inelastic flow as well as separate dilation and friction angles. The yield surface employs two invariants, defined as the equivalent pressure, p, and the von Mises equivalent stress, q. In addition, the linear version of the model uses the third invariant r ¼ ð9=2S S : SÞ
where, v = d1/L, is the aspect ratio of the platelets. This result is subject to the stress satisfying the inequality
Dr ¼ E2 De
gen remains elastic) the permanent strain and loop width would remain constant. Appendix B. The constitutive law for cortical bone
at z ¼ ‘
The strain in the collagen becomes (z 6 ‘) r sY z e2z ¼ E2 E2 d 2
67
ðA:12Þ
where rmax is the maximum stress reached before unloading. Reloading results in hysteresis, with predicted loop width: DeH = epl/2. This behavior continues at strains up to e0. Thereafter, at larger strains (provided that the colla-
ðB:3Þ
with b the slope of the yield surface in t–q space (commonly referred to as the friction angle), while K and d are parameters used to fit the measurements in tension and compression. The law is calibrated by fitting the stress/strain curves from Fig. 4 and the Poisson ratios (elastic and inelastic) from Fig. 5. The fit gives: K = 1 (so that t = q), d = 171 MPa and b = 43.5°. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Oxford: Pergamon Press; 1998. [2] Currey JD. The mechanical properties of bone. Clin Orthopaed Relat Res 1970;73:209–31. [3] Currey JD. The mechanical adaptations of bones. Princeton, NJ: Princeton University Press; 1984. [4] Hayes WC, Carter DR. Post yield behavior of subchondral trabecular bone. J Biomed Mater Res 1976;10:537–44. [5] Gibson LJ. The mechanical behavior of cancellous bone. J Biomech 1985;18:317–28. [6] Reilly DT, Burstein AH. The elastic and ultimate properties of compact bone tissue. J Biomech 1975;8:393–405. [7] Currey JD, Brear K. Tensile yield in bone. Calc Tissue Res 1974;15:173–9. [8] Reilly GC, Currey JD. The development of microcracking and failure in bone depends on the load mode to which it is adapted. J Exp Biol 1999;202:543–52. [9] Reilly GC, Currey JD. The effects of damage and microcracking on the impact strength of bone. J Biomech 2000;33:337–43. [10] Fondrk MT, Bahniuk EH, Davy DT. Inelastic strain accumulation in cortical bone during rapid transient tensile loading. J Biomech Eng— Trans ASME 1999;121:616–21. [11] Fondrk MT, Bahniuk EH, Davy DT. A damage model for nonlinear tensile behavior of cortical bone. J Biomech Eng – Trans ASME 1999;121:533–41. [12] Wang RZ, Suo Z, Evans AG, Yao N, Aksay IA. Deformation mechanisms in nacre. J Mater Res 2001;16:2485–93. [13] Evans AG, Zok FW. Physics and mechanics of brittle matrix composites. Solid State Phys 1994;47:177–286.
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Mechanisms governing the inelastic deformation of cortical bone and application to trabecular bone C. Mercer
a,*
, M.Y. He a, R. Wang b, A.G. Evans
a
a
b
Materials Department, University of California Santa Barbara, Santa Barbara, CA 93106, USA Department of Materials Engineering, University of British Columbia, Vancouver, BC, Canada V6T 1Z4 Received 9 May 2005; received in revised form 28 July 2005; accepted 10 August 2005
Abstract To understand the inelastic response of bone, a three-part investigation has been conducted. In the first, unload/reload tests have been used to characterize the hysteresis and provide insight into the mechanisms causing the strain. The second part devises a model for the stress/strain response, based on understanding developed from the measurements. The model rationalizes the inelastic deformation in tension, as well as the permanent strain and hysteresis. In the third part, a constitutive law representative of the deformation is selected and used to illustrate the coupled buckling and bending of ligaments that arise when trabecular bone is loaded in compression. Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Bone; Cortical; Trabecular; Inelastic deformation; Hysteresis
1. Introduction Many studies of both cortical and trabecular bone have demonstrated their propensity to inelastic strain [1–4] (Fig. 1). Parallel to the long axis, cortical bone exhibits inelastic responses that differ in tension and compression [1–6] (Fig. 1a). In tension, it yields, followed by (linear) hardening up to a failure strain of order 2.5%. The inelastic strain has been attributed to the development of diffuse microcrack arrays [7–9]. Simultaneous measurements of the axial and transverse strains have revealed that such deformation involves dilatation [10], consistent with a role of microcracks. The strains have been characterized by means of a damage model with internal state variables [11]. In compression, cortical bone also yields, but at higher stress than in tension. It strain hardens rapidly to a peak, then softens and fails at strains of about 1.5%. The softening has been attributed to the formation of shear bands [7].
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Corresponding author. Tel.: +1 805893 5930. E-mail address: [email protected] (C. Mercer).
Such behavior is not unique, but akin to the deformation characteristics of nacre [12] and fibrous oxides [13]. When compressed, trabecular bone exhibits extensive inelastic deformation (Fig. 1b), often attaining strains exceeding 60% before failure [4]. A peak stress occurs at strains of 5–10%. The deformations preceding the peak dictate the overall load capacity. Yield surface determinations [14] suggest that inelastic mechanisms are involved. The characteristics of synthetic foams with topology similar to trabecular bone may provide benchmarks. The load capacity of elastomeric/polymer foams is dictated by the elastic buckling of its ligaments [15], while that for metal foams is controlled by yielding [16]. The consequence is that different scaling formulae relate the stress maximum, rmax, to the properties of the ligaments. For isotropic, open cell polymer foams [1]: rmax/E 0.05f2, where E is the YoungÕs modulus for the material in the ligaments and f is the volume concentration of the solid in the foam. For isotropic metal foams [16]: rmax/rY f3/2, where rY is the uniaxial yield strength of the material in the ligaments. The differences in these formulae simply underscore the need to confirm that the mechanism governing the load capacity of trabecular bone is inelastic.
1742-7061/$ - see front matter Ó 2005 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. doi:10.1016/j.actbio.2005.08.004
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Fig. 2. Schematic of the instrumented flexure test showing the locations of the strain gages.
Fig. 1. (a) Schematic of the tensile and compressive stress/strain curves for cortical bone along the axis of a long bone [1]. (b) Schematic of a compressive stress/strain curve for trabecular bone [1].
The present study has three parts. (i) A mechanical probe is used to ascertain the hysteresis associated with the inelastic strains, as well as the elastic modulus evolution. (ii) Motivated by these measurements, a model is presented that highlights the relative roles of the constituent phases (collagen and mineral platelets) and of the interfaces between them. (iii) A constitutive law is invoked that incorporates key aspects of the inelastic deformation. The criterion for the choice is that the law already exists within a commercial finite element code such that, upon calibration, it can be applied immediately to large-scale problems, such as the cell level response of trabecular bone [1]. 2. Materials and methods The cortical bone used in this study was obtained from bovine femurs. Cylindrical, mid-diaphyseal sections (40 mm long) were extracted from fresh femurs (exact age not known, presumably between 2 and 7 years old). Rectangular specimens were cut on a longitudinal/radial plane at the posterior sector of the femur, with a low-speed diamond blade saw under continuous irrigation. The four rectangu-
lar surfaces were mechanically ground using silicon carbide media, and then polished with diamond suspensions, followed by 0.05 lm alumina suspension on a precision machine (MULTIPOL 2 Precision Polishing Machine, Ultra Tec, Santa Ana, USA). The final specimens were 38 mm long, 3.85 mm wide (across the thickness of the bone), and 3.05 mm (four specimens) or 1.52 mm (six specimens) thick. The specimens were stored in water at 4 °C for 1–15 days prior to testing. Four-point bend tests (Fig. 2) were conducted using a servo-hydraulic testing machine at a cross-head speed of 0.5 mm/min. The inner loading span was 15 mm and the outer support span 30 mm (Fig. 2). The upper and lower loading surfaces were chosen to be perpendicular to the lamellar boundaries of the cortical bone to minimize the structural variations across the thickness of the specimens and therefore, to allow comparison between the compressive and tensile behavior. To measure the strains on both tensile and compressive surfaces, strain gages (from either Omega Engineering or Vishay Micro-Measurements) were applied to both surfaces (one on each side) using M-Bond 200 adhesive following degreasing and cleaning. In some cases, 0°/90° strain gages were used for measurement of the transverse strains in addition to axial strains. Specimens were either monotonically loaded to failure, or repeatedly unloaded/reloaded at incrementally higher plastic strains, in order to measure hysteresis. The specimens were kept hydrated during testing by using an agarose gel (Sigma–Aldrich, USA). The inelastic deformation has been reproducibly established using six specimens. After mechanical testing, the specimens were examined with a stereomicroscope for macro-scale damage. For this purpose, some specimens were stained with basic fuchsin [17]. The stained specimens were embedded in a mounting medium. Using a low-speed diamond saw, thin slices were cut along the longitudinal direction of the specimens, with each slice either parallel or perpendicular to the compressive (upper loading) surface. The slices were polished to
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80 lm thick and embedded in a polymer medium on glass slides. This procedure caused the damage to be stained pink, and easily discernable in transmitted light (Nikon 600, USA). 3. Flexural measurements A typical response relating the load to the axial strain upon monotonic testing to failure is summarized in Fig. 3a. The corresponding stress/strain curves obtained by deconvolution [12] are plotted in Fig. 4. All tests gave essentially the same inelastic strains and strain hardening. The YoungÕs modulus (E = 25 GPa) is the same in tension and compression, and consistent with the literature [2,3]. The yield strains in tension (0.6%) and compression (1.1%) are also consistent with those reported elsewhere [2,3]. This reproducibility encourages further analysis of the inelastic strains. The simultaneous measurements of the transverse and axial strains, typified by the results presented in Fig. 3b, enable determination of the volumetric strains. The inelastic Poisson ratio (the ratio of the transverse to the axial inelastic
Fig. 4. Tensile and compressive stress/strain curves for cortical bone ascertained from the results presented in Fig. 3. The curves are similar to those reported in the literature [2,3].
strains) has been determined from these measurements and plotted as a function of the inelastic strain (Fig. 5). The results highlight the differences in tension and compression, with implications discussed below. Note that, in tension (unlike plasticity) the strains are not volume conserving [10]. At strains beyond yield, the following features are most notable (Figs. 4–8). (i) In tension, the material behaves as an inelastic solid with linear strain hardening (Fig. 4): tangent modulus, EH 1 GPa. The transverse strain is negligible, signifying an inelastic Poisson ratio, mpl 0 (Fig. 5). That is, the material dilates [10].
Fig. 3. (a) The relationships between load and axial strain measured on cortical bone in flexure. The results obtained by simulation using the Drucker–Prager constitutive law are superposed. (b) Plots of the axial and transverse strains measured on the tension and compression faces, as a function of load.
Fig. 5. Trends in the inelastic Poisson ratio with inelastic strain ascertained from Fig. 3b.
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Fig. 7. Optical images of a specimen of cortical bone fractured in bending: (a) side face; (b) partial view of the compression face (arrows indicate direction of shear band formation).
mpl 0.30.4 (Fig. 5), suggesting that the shear mechanisms governing compressive deformation [7] are nearly volume-conserving.
Fig. 6. (a) Stress–strain curves in tension upon unloading and reloading, showing 12 hysteresis loops. (b) Variation in the width of the hysteresis loops with permanent strain. (c) Variation in initial loading modulus with permanent strain.
(ii) In compression, the peak stress is appreciably larger than in tension, but the strain at failure is smaller (ef 1.6% in compression and ef 2.4% in tension; Fig. 4). Consequently, the load maximum in flexure is compression-dominated, as described elsewhere [11]. The inelastic Poisson ratio is non-zero,
Periodic unloading/reloading tests (Fig. 6a) indicate appreciable stress/strain hysteresis in tension. The measurements, unprecedented on cortical bone, elucidate the underlying inelastic mechanisms. The hysteresis is a measure of the internal friction and its irreversibility [13]. It is quantified by plots of the maximum loop width (unloading to reloading) with permanent strain (Fig. 6b). The initial loading slope (shown in Fig. 6a) is a measure of the current YoungÕs modulus, E [13]. A plot against strain (Fig. 6c) indicates that E does not change significantly. The implication is that straining does not cause cracking of the mineral platelets nor does it cause separations between the mineral phase and the collagen. Taken together, the hysteresis and modulus measurements indicate that the deformation involves internal slip (with associated friction), with the mineral phase and interfaces remaining intact. These findings motivate the mechanism to be described in the following section. The cumulative strain ascertained both on the strain gages and from the final deformed shape (>4%), significantly
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Fig. 8. Shear bands that develop on the compression face: (a) overview; (b) bands nucleating at blood vessel.
exceeds the rupture strain upon monotonic loading. The extra strain appears to accumulate on a loop-by-loop basis, manifest as a strain bias per loop. Namely, higher strains are needed for the reloading curves to attain the load maximum achieved prior to unloading (Fig. 6a). One interpretation is that the sacrificial bonds [18–20] between the collagen and mineral platelets rupture on loading, but re-form upon unloading. Side and top views of fractured specimens (Fig. 7) confirm the differences in failure mode between tension and compression. In tension, a dominant crack develops normal to the surface, penetrating to the neutral plane. In compression, shear bands are evident [7]. The angle between the bands and the loading axis is / = 28 ± 5°. Staining and cross-sectioning establishes that the bands extend across the bone lamellae and are mirror symmetric with respect to the loading axis (Fig. 8a). Higher resolution images reveal that many of the bands nucleate at the blood vessels (Fig. 8b). The similarity of this failure mode with the kinking of fibrous oxides at high temperature (Fig. 9) [21], suggests a methodology for modeling the compressive strength. Similar bands have also been observed in the ligaments of trabecular bone [22].
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Fig. 9. Scanning electron microscope images of the kink bands found in columnar-grained oxides compressed at high temperature (where they deform plastically). Note the abrupt bending of the columns within the band and the fracturing that occurs at the boundaries [21]. It is suggested that the collagen fibrils in bone experience similar deformations when subject to failure in compression.
4. Inelastic deformation mechanism The results in Fig. 6 can only be rationalized if the collagen phase is continuous and the mineral phase discontinuous. Namely, if the brittle mineral phase were continuous it would crack at quite small strains causing the modulus to decrease with increasing inelastic strain [13], contrasting with the actual measurements (Fig. 6c). Moreover, the evolving nature of the hysteresis loops with strain (Fig. 6b) indicates that irreversible slip must be occurring at the interfaces [13]. The simplest geometric representation that encapsulates these features is shown in Fig. 10. The characteristics embodied in this figure have precedents [23,24], but conflict with proposals made by others [25–28]. In the model, the collagen fibrils stretch as the strain is imposed. If the system remained elastic, a highly concentrated stress would develop at the free edges of the much stiffer mineral platelets (at z = 0). Under such circumstances, the stresses must relax, locally, by shear yielding of the interlayer. That is, to accommodate this stretch,
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Fig. 10. A schematic of the representative unit cell used to analyze the tensile stress/strain response of cortical bone: it highlights the thin adhesive layer that undergoes inelastic shear deformation.
interface slip must occur. Using this representation, the mechanical response in tension can be explored (Appendix A), with the objective of establishing the roles of the constituent materials and the interfaces. Previous models have not brought out these roles in a manner consistent with the current measurements (Fig. 6). For modeling purposes, this inelastic shear process is considered to occur at a shear yield stress, sY (with no strain hardening). Accordingly, a slip zone, length ‘, extends from the platelet edges: with ‘ increasing as P increases [13,29]. This phenomenon results in a non-linear stress/strain response, even when the platelets and the collagen, otherwise, remain elastic. For analytic simplicity, a shear lag approximation has been used [13,29]. Namely, the stresses in each constituent are considered to vary only in the axial direction. The analysis is presented in Appendix A and predicted stress/strain curves are shown in Fig. 11. Comparison with the measurements is achieved (Fig. 11b) by regarding the modulus of the collagen fibrils, E2, and the shear yield strength of the adhesive layer, sY, as fitting parameters. Upon fitting the model to the measurements, the inferred fibril modulus, E2 = 1 GPa is comparable to values reported for mammalian tendons (1.5 GPa) [30– 32]. Based on this modulus, upon invoking a platelet aspect ratio, v = 10 [33], the shear yield stress needed to fit the measurements is, sY = 1.4 MPa: a reasonable magnitude for an organic solid. The reconstructed stress/strain curve plotted in Fig. 11b demonstrates that the model is capable of faithfully repro-
Fig. 11. The tensile stress/strain curve predicted using the model depicted in Fig. 10: (a) schematic; (b) superposition onto the measured stress/strain curve.
ducing the measurements. Moreover, it is consistent with a hysteresis loop width that approaches DeH/epl ! 0.5 at higher permanent strains (Fig. 6b). It is concluded that the model embodies the respective roles of the collagen, the mineral platelets and the interfaces. 5. Constitutive law The preceding measurements can be used to predict the bending response of cortical bone and, potentially, the buckling and bending responses of ligaments in trabecular bone, provided that they can be adequately represented by a constitutive law. The relevance of the mechanical response of cortical bone to the ligaments of trabecular bone has already been substantiated [34,35]. Various finite element studies have been conducted previously. Some studies have incorporated non-linear elastic effects pertinent to buckling, as well as damage [36]. Others have included the difference in behavior between tension and compression [37]. The rationale for another assessment is twofold. (i) Select an inelastic law already available in a commercial finite element code. Implementation in a commercial code has the distinct advantage that the results can be reproduced by many other groups and that the scope can be readily extended. (ii) Identify a representation that includes the dilatation as well as tension/compression asymmetry. The preceding assessment indicates that the most important features to be incorporated are as follows: (a) It must accommodate differing responses in tension and compression.
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(b) In tension the material must dilate (with mpl 0). (c) In compression, it must deform inelastically at (nearly) constant volume. The present choice is the linear Drucker–Prager model, available in the ABAQUS Analysis UserÕs Manual [38] (Appendix B), which satisfies (a–c). However, it is isotropic and thus limited to axial loads and bending moments. This approach complements other assessments based on damage models [11]. Its application to the current flexural tests is not included because it would merely duplicate characteristics presented by Fondrk et al. [11]. Namely, stress redistribution causes the compressive stress to attain its maximum value before the failure strain is attained in tension, confirming that the flexural load capacity is dictated by shear band formation in compression.
Fig. 13. Osteoporotic bone structure comprised almost entirely of struts (no faces).
6. Stress/strain responses of trabecular bone Based on the assumption that trabecular and cortical tissues have similar mechanical properties [34,35], trends in the stress/strain curves in uniaxial compression can be predicted using the foregoing constitutive law. The results are sensitive to the choice of the representative unit cell. For illustration purposes, the isotropic, ‘‘bending-dominated’’ configuration proposed by Gibson and Ashby [1]
Fig. 12. A representative unit cell for osteoporotic trabecular bone observed in the transverse orientation, based on the cell proposed by Gibson and Ashby [1]. The inset is a finite element simulation of the deformed shape upon uniaxial compression, indicating the bending and buckling of the ligaments.
(Fig. 12) is used. This choice has the attribute that it successfully correlates trends in the stiffness of trabecular bone with relative density [1]. The configuration might be representative of (osteoporotic) bone comprised almost entirely of struts (no faces) (Fig. 13). Anisotropy could be incorporated using alternative configurations applicable to healthy bone. The responses are obtained at three relative densities (f = 0.1, 0.15 and 0.2), incorporating the imperfection sensitivity of buckling [39]. The role of imperfections has not been explicitly addressed in prior assessments. Thus, before proceeding, an eigen-value analysis of the buckling modes is conducted [39] and small imperfections characteristic of the first mode are introduced. To demonstrate the importance of imperfections, a preliminary analysis conducted with and without imperfections (Fig. 14a), reveals the characteristic overestimate of the load capacity when imperfections are absent. However, provided that imperfections are present, the predicted response is insensitive to imperfection amplitude [39]. All ensuing results are obtained using first mode imperfections. Predicted stress/strain curves are presented in Fig. 14b. The curves have the same form as the measurements in the literature [1,4]. Recall that these results have been obtained with no adjustable parameters. The peak stresses are all coincident with the buckling of the vertical members, preceded by bending of the horizontal members (Fig. 15). Upon unloading, the results reveal that some of the strain is recovered, but most is retained as permanent strain. The distributions of inelastic strains in the ligaments at the peak stress (Fig. 15) illustrate that the cross-section has become fully inelastic: that is, the peak load is governed by inelastic buckling. While the representative unit cell (Fig. 12) appears to provide a meaningful representation of osteoporotic trabecular bone, it should not be construed that the stress/ strain response is governed solely by the relative density. Alternative topologies are expected along stress lines, which would elevate the load capacity and change the
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¼ 0:2) reveal that the Fig. 15. Displacements of the cell (relative density, q peak stress is coincident with the onset of inelastic buckling. The elastic zones are highlighted as well as the ‘‘plastic hinges’’ where the inelastic strains exceed 8%. An intermediate strain contour (epl = 4%) is also shown.
overestimated, initial imperfections have been incorporated, with shape dictated by an eigen-value analysis of the buckling modes. A model for the tensile deformation of cortical bone has been introduced. The inelastic deformation arises from shear yielding of the interfaces between the platelets and the collagen. The model predicts permanent strains and hysteresis loops consistent with the measurements. It appears to have the basic attributes needed to relate the stress/strain response to the properties of the constituents and the interfaces. Fig. 14. Compressive stress/strain curves for osteoarthritic trabecular bone calculated using the representative cell depicted in Fig. 12 and the Drucker–Prager constitutive law, calibrated using the tension and compression stress/strain curves for cortical bone. Results have been obtained for three relative densities.
Appendix A. Inelastic strain model Below the slip zone depicted in Fig. 10 (z > ‘), the strain in the platelets and the collagen is the same, whereupon r1 =E1 ¼ r2 =E2
relative roles of bending and buckling. Assessments of this type will be conducted in a future study. 7. Concluding remarks
with the stresses defined in the manner shown in Fig. 10. The subscript 1 refers to the mineral phase and 2 to the collagen. Force equilibrium requires that rd 2 ¼ r1 d 1 þ r2 d 2
Measurements of the inelastic deformation of cortical bone have been used to devise and calibrate an inelastic constitutive law. The law allows for different yield strengths in tension and compression, as well as the dilatation in tension. By assuming that comparable inelastic features occur in the ligaments of trabecular bone, stress/strain curves subject to uniaxial compression have been estimated. The trends are comparable to those in the literature, while also demonstrating how inelastic bending and buckling of the ligaments affect the stress evolution beyond yield. To assure that the loads at which buckling occurs are not
ðA:1Þ
ðA:2Þ
where d, d1, d2 are the thickness dimensions shown in the figure. Eliminating r1 gives r2 1 ¼ 1þk r
ðA:3Þ
where k¼
d 1 E1 d 2 E2
Within the slip region (z < ‘), force equilibrium in the collagen gives the strain gradient
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dez2 sY ¼ dz E2 d 2
ðA:4Þ
Subject to the following boundary conditions: ez2 ¼ r2 =E2 ez2 ¼ r=E2
at z ¼ 0
ðA:5Þ
The corresponding slip length is ‘ r k ¼ d 2 sY 1 þ k The displacement is thus (‘ < L) Z L sY ‘2 uz E 2 E 2 e2z dz ¼ ðL ‘Þr2 þ 2d 2 0 Consequently, the overall strain is (‘ < L) eE2 r 1 k k 1 ð1 þ kÞ ¼ 1 sY v 1þk 2 r
ðA:6Þ
ðA:7Þ
ðA:8Þ
Thereafter, upon further straining, the extra load is supported solely by the collagen. Accordingly, if it remains elastic, the elevation in stress, Dr, and the extra strain, De, must be related by ðA:10Þ
Inspection of the preceding results reveals that, when e < e0, the strain can be re-expressed as ðA:11Þ
Here ecol is the strain that would exist at stress r if the system were made solely from collagen. The overall response is plotted in Fig. 11a. Upon unloading, interface slip occurs in the opposite direction, commencing at the edges of the platelets. When the load has been fully removed, reverse slip progress to: ‘rev = L/2 [29]. Consequently, there is a permanent strain epl ¼
r2max v k2 4E22 e0 ð1 þ kÞ2
ðB:1Þ
where S is the deviatoric stress. The linear yield criterion is ðB:2Þ
where 3
t ¼ 1=2q½1 þ 1=K ð1 1=KÞðr=qÞ
At larger stresses, the slip length reaches the center of the platelets (‘ = L). This happens at a critical strain, e0: sY v ðA:9Þ e0 E 2 ¼ 2
e2col ½1 k=2 2e0
1=3
F ¼ t p tan b d ¼ 0
r 1 k <1 sY v 1 þ k
e ¼ ecol
The Drucker–Prager model is expressed in terms of allthree stress invariants [38]. It provides for a non-circular yield surface in the deviatoric plane in order to match different yield values in tension and compression. It incorporates associated inelastic flow as well as separate dilation and friction angles. The yield surface employs two invariants, defined as the equivalent pressure, p, and the von Mises equivalent stress, q. In addition, the linear version of the model uses the third invariant r ¼ ð9=2S S : SÞ
where, v = d1/L, is the aspect ratio of the platelets. This result is subject to the stress satisfying the inequality
Dr ¼ E2 De
gen remains elastic) the permanent strain and loop width would remain constant. Appendix B. The constitutive law for cortical bone
at z ¼ ‘
The strain in the collagen becomes (z 6 ‘) r sY z e2z ¼ E2 E2 d 2
67
ðA:12Þ
where rmax is the maximum stress reached before unloading. Reloading results in hysteresis, with predicted loop width: DeH = epl/2. This behavior continues at strains up to e0. Thereafter, at larger strains (provided that the colla-
ðB:3Þ
with b the slope of the yield surface in t–q space (commonly referred to as the friction angle), while K and d are parameters used to fit the measurements in tension and compression. The law is calibrated by fitting the stress/strain curves from Fig. 4 and the Poisson ratios (elastic and inelastic) from Fig. 5. The fit gives: K = 1 (so that t = q), d = 171 MPa and b = 43.5°. References [1] Gibson LJ, Ashby MF. Cellular solids: structure and properties. Oxford: Pergamon Press; 1998. [2] Currey JD. The mechanical properties of bone. Clin Orthopaed Relat Res 1970;73:209–31. [3] Currey JD. The mechanical adaptations of bones. Princeton, NJ: Princeton University Press; 1984. [4] Hayes WC, Carter DR. Post yield behavior of subchondral trabecular bone. J Biomed Mater Res 1976;10:537–44. [5] Gibson LJ. The mechanical behavior of cancellous bone. J Biomech 1985;18:317–28. [6] Reilly DT, Burstein AH. The elastic and ultimate properties of compact bone tissue. J Biomech 1975;8:393–405. [7] Currey JD, Brear K. Tensile yield in bone. Calc Tissue Res 1974;15:173–9. [8] Reilly GC, Currey JD. The development of microcracking and failure in bone depends on the load mode to which it is adapted. J Exp Biol 1999;202:543–52. [9] Reilly GC, Currey JD. The effects of damage and microcracking on the impact strength of bone. J Biomech 2000;33:337–43. [10] Fondrk MT, Bahniuk EH, Davy DT. Inelastic strain accumulation in cortical bone during rapid transient tensile loading. J Biomech Eng— Trans ASME 1999;121:616–21. [11] Fondrk MT, Bahniuk EH, Davy DT. A damage model for nonlinear tensile behavior of cortical bone. J Biomech Eng – Trans ASME 1999;121:533–41. [12] Wang RZ, Suo Z, Evans AG, Yao N, Aksay IA. Deformation mechanisms in nacre. J Mater Res 2001;16:2485–93. [13] Evans AG, Zok FW. Physics and mechanics of brittle matrix composites. Solid State Phys 1994;47:177–286.
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