Application of the Tsai-Wu quadratic multiaxial failure criterion to bovine trabecular bone

As a first step toward development of a multiaxial failure criterion for human trabecular bone, the Tsai-Wu quadratic failure criterion was modified as a function of apparent density and applied to bovine tibial trabecular bone. Previous data from uniaxial compressive, tensile, and torsion tests (n = 139 total) were combined with those from new triaxial tests (n = 17) to calibrate and then verify the criterion. Combinations of axial compression and radial pressure were used to produce the triaxial compressive stress states. All tests were performed with minimal end artifacts in the principal material coordinate system of the trabecular network. Results indicated that the stress interaction term F12 exhibited a strong nonlinear dependence on apparent density (r2 > 0.99), ranging from -0.126 MPa-2 at low densities (0.29 g/cm3) to 0.005 MPa-2 at high densities (0.63 g/cm3). After calibration and when used to predict behavior of new-specimens without any curve-fitting, the Tsai-Wu criterion had a mean (+/- SD) error of -32.6 +/- 10.6 percent. Except for the highest density triaxial specimens, most (15/17 specimens) failed at axial stresses close to their predicted uniaxial values, and some reinforcement for transverse loading was observed. We conclude that the Tsai-Wu quadratic criterion, as formulated here, is at best only a reasonable predictor of the multiaxial failure behavior of trabecular bone, and further work is required before it can be confidently applied to human bone.     

Homogenized stiffness matrices for mineralized collagen fibrils and lamellar bone using unit cell finite element models

Mineralized collagen fibrils have been usually analyzed like a two-phase composite material where crystals are considered as platelets that constitute the reinforcement phase. Different models have been used to describe the elastic behavior of the material. In this work, it is shown that when Halpin–Tsai equations are applied to estimate elastic constants from typical constituent properties, not all crystal dimensions yield a model that satisfy thermodynamic restrictions. We provide the ranges of platelet dimensions that lead to positive definite stiffness matrices. On the other hand, a finite element model of a mineralized collagen fibril unit cell under periodic boundary conditions is analyzed. By applying six canonical load cases, homogenized stiffness matrices are numerically calculated. Results show a monoclinic behavior of the mineralized collagen fibril. In addition, a 5-layer lamellar structure is also considered where crystals rotate in adjacent layers of a lamella. The stiffness matrix of each layer is calculated applying Lekhnitskii transformations, and a new finite element model under periodic boundary conditions is analyzed to calculate the homogenized 3D anisotropic stiffness matrix of a unit cell of lamellar bone. Results are compared with the rule-of-mixtures showing in general good agreement.     

A comparison of uniaxial and biaxial mechanical properties of the annulus fibrosus: a porcine model

The annulus fibrosus of the intervertebral disk experiences multidirectional tension in vivo, yet the majority of mechanical property testing has been uniaxial. Therefore, our understanding of how this complex multilayered tissue responds to loading may be deficient. This study aimed to determine the mechanical properties of porcine annular samples under uniaxial and biaxial tensile loading. Two-layer annulus samples were isolated from porcine disks from four locations: anterior superficial, anterior deep, posterior superficial, and posterior deep. These tissues were then subjected to three deformation conditions each to a maximal stretch ratio of 1.23: uniaxial, constrained uniaxial, and biaxial. Uniaxial deformation was applied in the circumferential direction, while biaxial deformation was applied simultaneously in the circumferential and compressive directions. Constrained uniaxial consisted of a stretch ratio of 1.23 in the circumferential direction while holding the tissue stationary in the axial direction. The maximal stress and stress-stretch ratio (S-S) moduli determined from the biaxial tests were significantly higher than those observed during both the uniaxial tests (maximal stress, 97.1% higher during biaxial; p=0.002; S-S moduli, 117.9% higher during biaxial; p=0.0004) and the constrained uniaxial tests (maximal stress, 46.8% higher during biaxial; S-S moduli, 82.9% higher during biaxial). These findings suggest that the annulus is subjected to higher stresses in vivo when under multidirectional tension.     

Influence of boundary conditions on computed apparent elastic properties of cancellous bone

High-resolution finite element models of trabecular bone can be used to study trabecular structure–function relationships, elasticity, multiaxial strength, and tissue remodelling in more detail than experiments. Beside effects of the model size, scan/analysis resolution, segmentation process, etc., the type of the applied boundary conditions (BCs) have a strong influence on the predicted elastic properties. Appropriate BCs have to be applied on hexahedral digital finite element models in order to obtain effective elastic properties. Homogeneous displacement BCs as proposed by Van Rietbergen et al. (J Biomech 29(12):1653–1657, 1996) lead to “apparent” rather than to “effective” elastic properties. This study provides some answers concerning such differences by comparing various BC types (uniform displacement, mixed BCs, periodic BCs), different volume element definitions (original and mirrored models), and several bone volume fractions (BVTV ranging from 6.5 to 37.6%). First, the mixed BCs formulated by Hazanov (Arch Appl Mech 68(6):385–394, 1998) are theoretically extended to shear loading of a porous media. Second, six human bone samples are analyzed, their orthotropic Young’s moduli, shear moduli, and Poisson’s ratios computed and compared. It is found that the proposed mixed BCs give exactly the same effective elastic properties as periodic BCs if a periodic and orthotropic micro-structured material is used and thus denoted as “periodicity compatible” mixed uniform BCs (PMUBCs). As bone samples were shown to be nearly orthotropic for volume element side lengths ≥5 mm the proposed mixed BCs turn out to be the best choice because they give again essentially the same overall elastic properties as periodic BCs. For bone samples of smaller dimensions ( < 5 mm) with a strong anisotropy (beyond orthotropy) uniform displacement BCs remain applicable but they can significantly overestimate the effective stiffness. In Memoriam, Prof. Christian Huet.     

Shear modulus of porcine coronary artery: contributions of media and adventitia

The epicardial coronary arteries experience significant torsion in the axial direction due to changes in the shape of the heart during the cardiac cycle. The objective of this study was to determine the torsional mechanical properties of the coronary arteries under various circumferential and longitudinal loadings. The coronary artery was treated as a two-layer composite vessel consisting of intima-medial and adventitial layers, and the shear modulus of each layer was determined. Eight porcine hearts were obtained at a local abattoir, and their right coronary and left anterior descending arteries were isolated and tested in vitro with a triaxial torsion machine (inflation, longitudinal stretch, and circumferential twist). After the intact vessel was tested, the adventitia was dissected away, leaving an intact media that was then tested under identical triaxial loading conditions. We proposed a biomechanical analysis to compute the shear modulus of the adventitia from the measured shear moduli of the intact vessel and the media. To validate our predictions, we used four additional hearts in which the shear modulus of the adventitia was measured after dissection of media. Our results show that the shear modulus does not depend on the shear stress or strain but varies linearly with circumferential and longitudinal stresses and in a nonlinear way with the corresponding strains. Furthermore, we found that the shear modulus of the adventitia is larger than that of the intact vessel, which is larger than the vessel media. These results may have important implications for baroreceptor sensitivity, circulation of the vasa vasorum, and coronary dissection.     

Determination of dynamically adapting anisotropic material properties of bone under cyclic loading

Because bone tissue adapts to loading conditions, finite element simulations of remodelling bone require a precise prediction of dynamically changing anisotropic elastic parameters. We present a phenomenological theory that refers to the tissue in terms of the tendency of the structure to align with principal stress directions. We describe the material parameters of remodelling bone. This work follows findings by the same research group and independently by Danilov (1971) in the field of plasticity, where the dependencies of the components of the stiffness tensor in terms of time are based on Hill's anisotropy. We modify such an approach in this novel theory that addresses bone tissue that can regenerate. The computational assumption of the theory is that bone trabeculae have the tendency to orient along one of the principal stress directions but during remodelling the principal stresses change continuously and the resulting orientation of the trabeculae can differ from the principal stress direction at any given time. The novelty of this work consists in the limited number of parameters needed to compute the twenty-one anisotropic material parameters at any given location in the bone tissue. In addition to the theory, we present here two cases of simplified geometry, loading and boundary conditions to show the effect of (1) time on the material properties; and (2) change of loading conditions on the anisotropic parameters. The long term goal is to experimentally verify that the predictions generated by theory provide a reliable simulation of cancellous bone properties.     

Validation of a voxel-based FE method for prediction of the uniaxial apparent modulus of human trabecular bone using macroscopic mechanical tests and nanoindentation

Assessment of the mechanical properties of trabecular bone is of major biological and clinical importance for the investigation of bone diseases, fractures and their treatments. Finite element (FE) methods are getting increasingly popular for quantifying the elastic and failure properties of trabecular bone. In particular, voxel-based FE methods have been previously used to calculate the effective elastic properties of trabecular microstructures. However, in most studies, bone tissue moduli were assumed or back-calculated to match the apparent elastic moduli from experiments, which often lead to surprisingly low values when compared to nanoindentation results. In this study, voxel-based FE analysis of trabecular bone is combined with physical measures of volume fraction, micro-CT (microCT) reconstructions, uniaxial mechanical tests and specimen-specific nanoindentation tests for proper validation of the method. Cylindrical specimens of cancellous bone were extracted from human femurs and their volume fraction determined with Archimede's method. Uniaxial apparent modulus of the specimens was measured with an improved tension-compression testing protocol that minimizes boundary artefacts. Their microCT reconstructions were segmented to match the measured bone volume fraction and used to create full-size voxel models with 30-45 microm element size. For each specimen, linear isotropic elastic material properties were defined based on specific nanoindentation measurements of its embedded bone tissue. Linear FE analyses were finally performed to simulate the uniaxial mechanical tests. Additional parametric analyses were performed to evaluate the potential errors on the predicted apparent modulus arising from variations in segmentation threshold, tissue modulus, and the use of 125-mm(3) cubic sub-regions. The results demonstrate an excellent correspondence between experimental measures and FE predictions of uniaxial apparent modulus. In conclusion, the adopted voxel-based FE approach is found to be a robust method to predict the linear elastic properties of human cancellous bone, provided segmentation of the microCT reconstructions is carefully calibrated, tissue modulus is known a priori and the entire region of interest is included in the analysis.     

Passive material properties of intact ventricular myocardium determined from a cylindrical model

The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.     

The open section effect in a long bone with a longitudinal defect - a theoretical modeling study

A longitudinal defect dramatically alters the stress distribution within a long bone. The altered stress distribution can influence the structural properties of the bone and the stimulus for repair and remodeling of the defect and the surrounding bone. For applied torsion, the defect interrupts the normal shear flow around the bone. Reversal of the shear flow along the inner cortex of the bone is the primary characteristic of the "open-section" effect. Stress concentration effects also produce large stresses at the defect corners. A finite element model of a femur mid-diaphysis with a rectangular defect in the posterior cortex was developed to quantify the femur stress distribution and torsional stiffness for defect widths ranging from one-tenth of the femur outer diameter (0.1 OD) to 0.3 OD, and defect lengths ranging from 0.5 to 5 OD. Defects with a length of 1 OD or shorter had little influence on the femur torsional stiffness or the femur shear-stress distribution. The torsional stiffness decreased most dramatically as the defect length increased from 2 to 3 OD, but began to approach an asymptote near 5 OD. Shear flow reversal peaked at the center of the defect for defects longer than 1 OD, and the magnitude of the reversal began to approach an asymptote near 5 OD. For each defect, the largest stresses within the bone, developed at the defect corners. The results indicate that the open-section effect decreases the torsional stiffness and stress concentration effects decrease the torsional strength of a long bone with a longitudinal defect.     

Heterogeneous linear elastic trabecular bone modelling using micro-CT attenuation data and experimentally measured heterogeneous tissue properties

High-resolution voxel-based finite element software, such as FEEBE developed at the NCBES, is widely used for studying trabecular bone at the micro-scale. A new approach to determine heterogeneous bone tissue material properties for computational models was proposed in this study. The specimen-specific range of tissue moduli across strut width was determined from nanoindentation testing. This range was mapped directly using linear interpolation to that specimen's micro-computed tomography (microCT) grey value range as input material properties for finite element analysis. The method was applied to cuboid trabecular bone samples taken from eight, 4-year-old (skeletally mature) ovine L5 vertebrae. Before undergoing experimental uniaxial compression tests, the samples were microCT scanned and 30 microm resolution finite element models were generated. The linear elastic finite element models were compressed to 1% strain. This material property assignment method for computational models accurately reproduced the experimentally determined apparent modulus and concentrations of stress at locations of failure.     

The twist-to-bend compliance of the Rheum rhabarbarum petiole: integrated computations and experiments

Plant petioles can be considered as hierarchical cellular structures, displaying geometric features defined at multiple length scales. Their macroscopic mechanical properties are the cumulative outcome of structural properties attained at each level of the structural hierarchy. This work appraises the compliance of a rhubarb stalk by determining the stalk’s bending and torsional stiffness both computationally and experimentally. In our model, the irregular cross-sectional shape of the petiole and the layers of the constituent tissues are considered to evaluate the stiffness properties at the structural level. The arbitrary shape contour of the petiole is generated with reasonable accuracy by the Gielis superformula. The stiffness and architecture of the constituent layered tissues are modeled by using the concept of shape transformers so as to obtain the computational twist-to-bend ratio for the petiole. The rhubarb stalk exhibits a ratio of flexural to torsional stiffness 4.04 (computational) and 3.83 (experimental) in comparison with 1.5 for isotropic, incompressible, circular cylinders, values that demonstrate the relative structural compliance to flexure and torsion.     

Twisting and bending of biological beams: distribution of biological beams in a stiffness mechanospace

Most biological beams bend and twist relatively easily compared to human-made structures. This paper investigates flexibility in 57 diverse biological beams in an effort to identify common patterns in the relationship between flexural stiffness and torsional stiffness. The patterns are investigated by mapping both ideal and biological beams into a mechanospace defined by flexural and torsional stiffness. The distribution of biological beams is not random, but is generally limited to particular regions of the mechanospace. Biological beams that are stiff in bending are stiff in torsion, while those that bend easily also twist easily. Unoccupied regions of the mechanospace represent rare combinations of mechanical properties, without proving that they are impossible. The mechanical properties of biological beams closely resemble theoretical expectations for ideal beams. Both distributions are potentially being driven by the interdependence of the material and structural properties determining stiffness. The mechanospace can be used as a broadly comparative tool to highlight systems that fall outside the general pattern observed in this study. These outlying beams may be of particular interest to both biologists and engineers due to either material or structural innovations.     

Evaluation of bone plate with low-stiffness material in terms of stress distribution

The aim of this study is to evaluate a newly developed bone plate with low-stiffness material in terms of stress distribution. In this numerical study, 3D finite element models of the bone plate with low-stiffness material and traditional bone plates made of stainless steel and Ti alloy have been developed by using the ANSYS software. Stress analyses have been carried out for all three models under the same loading and boundary conditions. Compressive stresses occurring in the intact portion of the bone (tibia) and at the fractured interface at different stages of bone healing have been investigated for all three types of bone-plate systems. The results obtained have been compared and presented in graphs. It has been seen that the bone plate with low-stiffness material offers less stress-shielding to the bone, providing a higher compressive stress at the fractured interface to induce accelerated healing in comparison with Ti alloy and stainless-steel bone plate. In addition, the effects of low-stiffness materials with different Young's modulus on stress distribution at the fractured interface have been investigated in the newly developed bone-plate system. The results showed that when a certain value of Young's modulus of low-stiffness material is exceeded, increase in stiffness of the bone plate does not occur to a large extent and stress distributions and micro-motions at the fractured interface do not change considerably.     

Anisotropic bone remodelling model based on a continuum damage-repair theory.

The purpose of this paper is to present and discuss a new model for internal bone remodelling. We state that the evolution of the internal variables of the bone microstructure and its incidence on the modification of the elastic constitutive parameters may be formulated following the exact principles of Continuum Damage Mechanics (CDM), although no actual damage is considered. A remodelling tensor, analogous to the standard damage tensor, is proposed which completely characterises the state of the homogenised bone microstructure and, therefore, its stiffness. This tensor is defined in terms of the apparent density and the "fabric tensor" associated with porosity and directionality of the trabeculae, respectively. Contrary to standard damage mechanics, its variation may be negative to allow for material "repair". The different elements that compose the theory are then established, such as the mechanical stimulus which, as is common in CDM, was chosen as the tensor that is thermodynamically associated to the remodelling one. The resorption and apposition criteria (similar to the damage criterion) were also expressed in terms of this mechanical stimulus and, finally, the evolution law, to define the change rate of the remodelling tensor. An associated flow rule is proposed that fulfils a principle of minimum mechanical dissipation for convex remodelling criteria (the case in question here). Some other important experimental features are deduced naturally from the model, such as the coincidence of the principal directions of the fabric tensor with those of the elasticity tensor, or the fact that the principal directions of the fabric tensor tend to align with the principal directions of stress. This model is applied to the study of the remodelling evolution of the proximal extremity of the femur, obtaining results very similar to experimental data.     

An anisotropic elastic-viscoplastic damage model for bone tissue

A new anisotropic elastic-viscoplastic damage constitutive model for bone is proposed using an eccentric elliptical yield criterion and nonlinear isotropic hardening. A micromechanics-based multiscale homogenization scheme proposed by Reisinger et al. is used to obtain the effective elastic properties of lamellar bone. The dissipative process in bone is modeled as viscoplastic deformation coupled to damage. The model is based on an orthotropic ecuntric elliptical criterion in stress space. In order to simplify material identification, an eccentric elliptical isotropic yield surface was defined in strain space, which is transformed to a stress-based criterion by means of the damaged compliance tensor. Viscoplasticity is implemented by means of the continuous Perzyna formulation. Damage is modeled by a scalar function of the accumulated plastic strain ${D(\kappa)}$ , reducing all element s of the stiffness matrix. A polynomial flow rule is proposed in order to capture the rate-dependent post-yield behavior of lamellar bone. A numerical algorithm to perform the back projection on the rate-dependent yield surface has been developed and implemented in the commercial finite element solver Abaqus/Standard as a user subroutine UMAT. A consistent tangent operator has been derived and implemented in order to ensure quadratic convergence. Correct implementation of the algorithm, convergence, and accuracy of the tangent operator was tested by means of strain- and stress-based single element tests. A finite element simulation of nano- indentation in lamellar bone was finally performed in order to show the abilities of the newly developed constitutive model.     

Finite element models predict cancellous apparent modulus when tissue modulus is scaled from specimen CT-attenuation

High-resolution architecture-based finite element models are commonly used for characterizing the mechanical behavior of cancellous bone. The vast majority of studies use homogeneous material properties to model trabecular tissue. The objectives of this study were to demonstrate that inhomogeneous finite element models that account for microcomputed tomography-measured tissue modulus variability more accurately predict the apparent stiffness of cancellous bone than homogeneous models, and to examine the sensitivity of an inhomogeneous model to the degree of tissue property variability. We tested five different material cases in finite element models of ten cancellous cubes in simulated uniaxial compression. Three of these cases were inhomogeneous and two were homogeneous. Four of these cases were unique to each specimen, and the remaining case had the same tissue modulus for all specimens. Results from all simulations were compared with measured elastic moduli from previous experiments. Tissue modulus variability for the most accurate of the three inhomogeneous models was then artificially increased to simulate the effects of non-linear CT-attenuation-modulus relationships. Uniqueness of individual models was more critical for model accuracy than level of inhomogeneity. Both homogeneous and inhomogeneous models that were unique to each specimen had at least 8% greater explanatory power for apparent modulus than models that applied the same material properties to all specimens. The explanatory power for apparent modulus of models with a tissue modulus coefficient of variation (COV) range of 21-31% was 13% greater than homogeneous models (COV=0). The results of this study indicate that inhomogenous finite element models that have tissue moduli unique to each specimen more accurately predict the elastic behavior of cancellous cubic specimens than models that have common tissue moduli between all specimens.     

Design and validation of a testing system to assess torsional cancellous bone failure in conjunction with time-lapsed micro-computed tomographic imaging

When compressed axially, cancellous bone often fails at an oblique angle along well-defined bands, highlighting the importance of cancellous bone shear properties. Torsion testing to determine shear properties of cancellous bone has often been conducted under conditions appropriate only for axis-symmetric specimens comprised of homogeneous and isotropic materials. However, most cancellous bone specimens do not meet these stringent test conditions. Therefore, the aim of this study was to design and validate a uniaxial, incremental torsional testing system for non-homogeneous orthotropic or non-axis-symmetric specimens.Precision and accuracy of the newly designed torsion system was validated by using Plexiglas rods and beams, where obtained material properties were compared to those supplied by the manufacturer. Additionally, the incremental step-wise application of angular displacement and simultaneous time-lapsed μCT imaging capability of the system was validated using whale cancellous bone specimens, with step-wise application of angular displacement yielding similar torsional mechanical properties to continuous application of angular displacement in a conventional torsion study.In conclusion, a novel torsion testing system for non-homogeneous, orthotropic materials using the incremental step-wise application of torsion and simultaneous time-lapsed μCT imaging was designed and validated.     

Consideration of multiple load cases is critical in modelling orthotropic bone adaptation in the femur

Functional adaptation of the femur has been investigated in several studies by embedding bone remodelling algorithms in finite element (FE) models, with simplifications often made to the representation of bone’s material symmetry and mechanical environment. An orthotropic strain-driven adaptation algorithm is proposed in order to predict the femur’s volumetric material property distribution and directionality of its internal structures within a continuum. The algorithm was applied to a FE model of the femur, with muscles, ligaments and joints included explicitly. Multiple load cases representing distinct frames of two activities of daily living (walking and stair climbing) were considered. It is hypothesised that low shear moduli occur in areas of bone that are simply loaded and high shear moduli in areas subjected to complex loading conditions. In addition, it is investigated whether material properties of different femoral regions are stimulated by different activities. The loading and boundary conditions were considered to provide a physiological mechanical environment. The resulting volumetric material property distribution and directionalities agreed with ex vivo imaging data for the whole femur. Regions where non-orthogonal trabecular crossing has been documented coincided with higher values of predicted shear moduli. The topological influence of the different activities modelled was analysed. The influence of stair climbing on the properties of the femoral neck region is highlighted. It is recommended that multiple load cases should be considered when modelling bone adaptation. The orthotropic model of the complete femur is released with this study.     

The role of the lamellar interface during torsional yielding of human cortical bone

Fragility fractures are a result of alterations in bone quantity, tissue properties, applied loads, or a combination of these factors. The current study addresses the contribution of cortical bone tissue properties to skeletal fragility by characterizing the shear damage accumulation processes which occur during torsional yielding in normal bone. Samples of human femoral cortical bone were loaded in torsion and damaged at a post-yield twist level. The number of microcracks within osteons, interstitial tissue, and along cement lines were assessed using basic fuchsin staining. Damage density measures (number of cracks/mm2) were correlated with stiffness degradation and changes in relaxation. Damaged samples exhibited a wide variation in total microcrack density, ranging from 1.1 to 43.3 cracks/mm2 with a mean density of 19.7 +/- 9.8 cracks/mm2. Lamellar interface cracks comprised more than 75% of the total damage, indicating that the lamellar interface is weak in shear and is a principal site of shear damage accumulation. Damage density was positively correlated with secant stiffness degradation, but only explained 22% of the variability in degradation. In contrast, damage density was uncorrelated with the changes in relaxation, indicating that a simple crack counting measure such as microcrack density was not an appropriate measure of relaxation degradation. Finally, a nonuniform microcrack density distribution was observed, suggesting that internal shear stresses were redistributed within the torsion samples during post-yield loading. The results suggested that the lamellar interface in human cortical bone plays an important role in torsional yielding by keeping cracks physically isolated from each other and delaying microcrack coalescence in order to postpone the inevitable formation of the fatal crack.     

Structural behaviour and strain distribution of the long bones of the human lower limbs

Although stiffness and strength of lower limb bones have been investigated in the past, information is not complete. While the femur has been extensively investigated, little information is available about the strain distribution in the tibia, and the fibula has not been tested in vitro. This study aimed at improving the understanding of the biomechanics of lower limb bones by: (i) measuring the stiffness and strain distributions of the different low limb bones; (ii) assessing the effect of viscoelasticity in whole bones within a physiological range of strain-rates; (iii) assessing the difference in the behaviour in relation to opposite directions of bending and torsion. The structural stiffness and strain distribution of paired femurs, tibias and fibulas from two donors were measured. Each region investigated of each bone was instrumented with 8–16 triaxial strain gauges (over 600 grids in total). Each bone was subjected to 6–12 different loading configurations. Tests were replicated at two different loading speeds covering the physiological range of strain-rates. Viscoelasticity did not have any pronounced effect on the structural stiffness and strain distribution, in the physiological range of loading rates explored in this study. The stiffness and strain distribution varied greatly between bone segments, but also between directions of loading. Different stiffness and strain distributions were observed when opposite directions of torque or opposite directions of bending (in the same plane) were applied. To our knowledge, this study represents the most extensive collection of whole-bone biomechanical properties of lower limb bones.