Some mechanical properties of goose femoral cortical bone

The ultimate compressive strength and modulus of elasticity of femoral cortical bone from adult geese (Anser anser), were determined by sex and by quadrant by compressing small right circular cylinders which were 2.4 mm in height and 0.8 mm in diameter. The average ultimate compressive strength was 183 +/- 29 MPa. The average modulus of elasticity was 13.2 +/- 3.4 GPa. The bending strength and bending modulus of elasticity were determined by a three point bend test on rectangular prisms which had the approximate dimensions 0.75 mm X 0.75 mm X 25 mm. The average bending strength was 263 +/- 44 MPa while the average bending modulus was 19.6 +/- 3.1 GPa. The calcium content was determined by atomic absorption spectrophotometry and no correlation was found with the mechanical properties. The histology of the cortical bone was examined both quantitatively and qualitatively. A unique type of Haversian bone is described. Goose bone was found to be morphologically similar to adolescent human bone and to have mechanical properties similar to those of adult human bone.     

Mechanical loading attenuates bone loss due to immobilization and calcium deficiency.

Our purpose was to determine the effects of a mechanical loading intervention on mass, geometry, and strength of rat cortical bone during a period of disuse concurrent with calcium deficiency (CD). Adult female rats were assigned to unilateral hindlimb immobilization, immobilized-loaded, or control (standard chow, 1.85% calcium) treatments. Both immobilized groups were fed a CD rat chow (0.01% calcium) to induce high bone turnover. Three times weekly, immobilized-loaded rats were subjected to 36 cycles of 4-point bending of the immobilized lower leg. After 6 wk, the immobilized rats exhibited decreased tibial shaft bone mineral density (-12%), ultimate load (-19%), and stiffness (-20%; tested in 3-point bending to failure) vs. control rats. Loading prevented this decline in bone density and attenuated decreases in ultimate load and stiffness. Elastic modulus was unaffected by disuse or loading. Bone cross-sectional area in the immobilized-loaded rats was equivalent to that of control animals, even though endocortical resorption continued unabated. On the medial periosteum, percent mineralizing surface doubled vs. that in immobilized rats. This loading regimen stimulated periosteal mineralization and maintained bone mineral density, thereby attenuating the loss in bone strength incurred with disuse and concurrent calcium deficiency.     

Mechanics of bone/PMMA composite structures: an in vitro study of human vertebrae

The goal of this study was to provide material property data for the cement/bone composite resulting from the introduction of PMMA bone cement into human vertebral bodies. A series of quasistatic tensile and compressive mechanical tests were conducted using cement/bone composite structures machined from cement-infiltrated vertebral bodies. Experiments were performed both at room temperature and at body temperature. We found that the modulus of the composite structures was lower than bulk cement (p<0.0001). For compression at 37( composite function)C: composite =2.3+/-0.5GPa, cement =3.1+/-0.2GPa; at 23( composite function)C: composite =3.0+/-0.3GPa, cement =3.4+/-0.2GPa. Specimens tested at room temperature were stiffer than those tested at body temperature (p=0.0004). Yield and ultimate strength factors for the composite were all diminished (55-87%) when compared to cement properties. In general, computational models have assumed that cement/bone composite had the same modulus as cement. The results of this study suggest that computational models of cement infiltrated vertebrae and cemented arthroplasties could be improved by specifying different material properties for cement and cement/bone composite.     

Mechanical behavior of a new biphasic calcium phosphate bone graft

The aim of this study was to create a new porous calcium phosphate implant for use as a synthetic bone graft substitute. Porous bioceramic was fabricated using a foam-casting method. By using polyurethane foam and a slurry containing hydroxyapatite-dicalcium phosphate powder, water, and additives, a highly porous structure (66 ± 5%) was created. The porous specimens possess an elastic modulus of 330 ± 32 MPa and a compressive strength of 10.3 ± 1.7 MPa. The X-ray diffraction patterns show hydroxyapatite and beta-pyrophosphate phases after sintering. A rabbit model was developed to evaluate the compressive strength and elastic modulus of cancellous bone defects treated with these porous synthetic implants. The compressive mechanical properties became weaker until the second month post implantation. After the second month, these properties increased slightly and remained higher than control values. New bone formed on the outside surface and on the macropore walls of the specimens, as osteoids and osteoclasts were evident two months postoperatively. Considering these properties, these synthetic porous calcium phosphate implants could be applicable as cancellous bone substitutes.     

High-resolution finite element models with tissue strength asymmetry accurately predict failure of trabecular bone

The ability to predict trabecular failure using microstructure-based computational models would greatly facilitate study of trabecular structure–function relations, multiaxial strength, and tissue remodeling. We hypothesized that high-resolution finite element models of trabecular bone that include cortical-like strength asymmetry at the tissue level, could predict apparent level failure of trabecular bone for multiple loading modes. A bilinear constitutive model with asymmetric tissue yield strains in tension and compression was applied to simulate failure in high-resolution finite element models of seven bovine tibial specimens. Tissue modulus was reduced by 95% when tissue principal strains exceeded the tissue yield strains. Linear models were first calibrated for effective tissue modulus against specimen-specific experimental measures of apparent modulus, producing effective tissue moduli of (mean±S.D.) 18.7±3.4 GPa. Next, a parameter study was performed on a single specimen to estimate the tissue level tensile and compressive yield strains. These values, 0.60% strain in tension and 1.01% strain in compression, were then used in non-linear analyses of all seven specimens to predict failure for apparent tensile, compressive, and shear loading. When compared to apparent yield properties previously measured for the same type of bone, the model predictions of both the stresses and strains at failure were not statistically different for any loading case (p>0.15). Use of symmetric tissue strengths could not match the experimental data. These findings establish that, once effective tissue modulus is calibrated and uniform but asymmetric tissue failure strains are used, the resulting models can capture the apparent strength behavior to an outstanding level of accuracy. As such, these computational models have reached a level of fidelity that qualifies them as surrogates for destructive mechanical testing of real specimens.     

3 种不同弹性模量钛合金股骨假体在羊股骨置换模型中应力分布的三维 有限元分析

目的:通过三维有限元分析方法来观察并比较3种不同弹性模量钛合金股骨假体在羊股骨置换模型中von-Mises应力分布的情况。方法:采用64排螺旋CT对一健康成年羊的下肢股骨进行全长的CT扫描,扫描层厚为0.5 mm,扫描所得的数据存储为DICOM文件。将得到的DICOM文件导入到CT图像分析软件Mimics 10.0,然后利用Mimics 10.0软件来生成股骨的骨质点云数据,再将生成的骨质点云数据导入到Simpleware分析软件,通过机械加工反求中的复杂曲面造型技术建立起精确的三维实体模型。对三维实体模型进行网格划分,确定了髓腔的形状,并根据羊下肢股骨髓腔的形状设计了作者实验用的羊股骨假体模型,然后在ANSYS 12.1软件中进行网格划分。给予加载缓慢行走载荷以及扭转载荷,分析并比较羊股骨以及3种不同弹性模量钛合金股骨假体在股骨置换模型中von-Mises应力分布的情况。结果:在缓慢行走载荷以及扭转载荷条件下,3种不同弹性模量钛合金股骨假体von-Mises应力分布变化趋势一致,假体的柄颈结合部以及假体柄上1/3为应力集中区域。3种不同弹性模量的最大应力集中点均位于柄颈结合部,60 GPa弹性模量的股骨假体植入后假体的最大应力最小(37.8 MPa、29.1 MPa),股骨的最大应力最大(12.6 MPa、24.5 MPa);80 GPa的次之,假体的最大应力(38.4 MPa、33.4 MPa),股骨的最大应力(12.5 MPa、24.5 MPa);110 GPa的股骨假体植入后假体的最大应力最大(38.9 MPa、38.1 MPa),股骨的最大应力最小(12.3 MPa、24.5 MPa)。60 GPa弹性模量的股骨假体植入后的假体最大位移和相对位移均最小(缓慢行走载荷下假体最大位移为0.551 mm、相对位移为0.008 mm,扭转载荷下假体最大位移为0.730 mm、相对位移为0.011 mm)。结论:较低弹性模量的钛合金股骨假体(60 GPa)由于其弹性模量更接近于骨组织的弹性模量,股骨假体与股骨间的"应力遮挡"效应较小,更有利于应力在股骨假体及股骨间的传递,增加了股骨假体的早期稳定性,延长了其临床寿命。     

Bending properties, porosity, and ash fraction of black bear (Ursus americanus) cortical bone are not compromised with aging despite annual periods of disuse

In many species, including humans, disuse causes an imbalance in bone remodeling that leads to increased bone porosity as a result of increased bone resorption and decreased bone formation. However, black bears (Ursus americanus) may not develop disuse osteopenia, to the extent that other animals do, during long periods of disuse (i.e. hibernation) because they maintain osteoblastic bone formation during hibernation, even though bone resorption is increased during hibernation. Black bears may also have a mechanism to rapidly and completely recover the bone lost (by increased resorption during hibernation) during their remobilization period. Our findings suggest that cortical bone bending strength (211-328 MPa), bending modulus (16.0-29.5 MPa), fracture energy (0.0118-0.0205 J mm(-2)), porosity (2.3-7.1%), and ash fraction (0.638-0.672) are not compromised with age in black bears, despite annual periods of disuse. In fact, the ultimate strength (p=0.01), modulus (p=0.04), and ash fraction (p=0.03) of cortical bone were shown to significantly increase with age (2-14 yrs). Female bears give birth and nurse during hibernation; however, we found no significant (p>0.16) differences between male and female bone properties. Other animals require remobilization periods 2-3 times longer than the immobilization period to recover the bone lost during disuse. Our findings support the idea that black bears, which hibernate 5-7 months annually, have evolved a biological mechanism to mitigate the adverse effects of disuse on bone porosity and mechanical behavior.     

The mechanical properties of human tibial trabecular bone as a function of metaphyseal location

Experimental determination of the elastic modulus and ultimate strength of human tibial trabecular bone as a function of metaphyseal location is presented.

A 1 cm cubic matrix with planes parallel to the subchondral plate was defined on five fresh frozen cadaver tibias. Approximately 400, 7 mm × 10 mm cylindrical bone plugs were cut from the locations defined by the matrix and tested in uniaxial compressive stress at a strain rate of 0.1%s⁻¹. Results of the study indicate that the trabecular bone properties vary as much as two orders of magnitude from one location to another. As might be predicted from Wolff's law, and noted by previous investigators, concentrations of strength arise from the medial and lateral metaphyseal cortices toward the major medial and lateral contact regions.

These results may be valuable for improved analytical modeling and optimal prosthetic design.     

Heterogeneity of the mechanical properties of demineralized bone

Knowledge of the mechanical properties of the collagenous component of bone is required for composite modeling of bone tissue and for understanding the age- and disease-related reductions in the ductility and strength of bone. The overall goal of this study was to investigate the heterogeneity of the mechanical properties of demineralized bone which remains unexplained and may be due to differences in the collagen structure or organization or in experimental protocols. Uniaxial tension tests were conducted to measure the elastic and failure properties of demineralized human femoral (n = 10) and tibial (n = 13) and bovine humeral (n = 8) and tibial (n = 8) cortical bone. Elastic modulus differed between groups (p = 0.02), varying from 275 +/- 94 MPa (mean +/- SD) to 450 + 50 MPa. Similarly, ultimate stress varied across groups from 15 + 4.2 to 26 + 4.7 MPa (p = 0.03). No significant differences in strain-to-failure were observed between any groups in this study (pooled mean of 8.4 +/- 1.6%; p = 0.42). However, Bowman et al. (1996) reported an average ultimate strain of 12.3 +/- 0.5% for demineralized bovine humeral bone, nearly 40% higher than our value. Taken together, it follows that all the monotonic mechanical properties of demineralized bone can display substantial heterogeneity. Future studies directed at explaining such differences may therefore provide insight into aging and disease of bone tissue.     

Micromechanical response of mineral and collagen phases in bone

We report the first simultaneous quantification of Young's modulus in the separate material phases of bone: collagen and carbonated hydroxyapatite. High-energy X-ray scattering and in situ loading revealed macroscopic, mineral, and collagen Young's moduli (90% confidence limit) for a canine fibula equaled 24.7(0.2) GPa, 38.2(0.5) GPa {for 00.4 and 43.6(1.4) GPa for 22.2}, and 18(1.2) GPa, respectively. The mineral contained compressive residual stresses on the order of -60 to -80 MPa before loading and had a stress enhancement (ratio of internal to applied stress) between 2.0 and 2.3. The diffraction peak width increased with increasing applied stress, mainly along the bone's longitudinal direction, and peak widths returned to pre-deformation values when load was removed. In a second fibula section from the same animal, the mineral's internal stress changed from -50 MPa (22.2 reflection) to -75 MPa (00.4) just after removal from formalin to -10 MPa after eight hours immersion in phosphate-buffered saline; the corresponding change in collagen D-spacing DeltaD/D equaled 4.2x10(-3).     

Intrinsic mechanical properties of trabecular calcaneus determined by finite-element models using 3D synchrotron microtomography

To determine intrinsic mechanical properties (elastic and failure) of trabecular calcaneus bone, chosen as a good predictor of hip fracture, we looked for the influence of image's size on a numerical simulation. One cubic sample of cancellous bone (9 x 9 x 9 mm(3)) was removed from the body of the calcaneus (6 females, 6 males, 79+/-9 yr). These samples were tested under compressive loading. Before compressive testing, these samples were imaged at 10.13 microm resolution using a 3D microcomputed tomography (muCT) (ESRF, France). The muCT images were converted to finite-element models. Depending on the bone density values (BV/TV), we compared two different finite element models: a linear hexahedral and a linear beam finite element models. Apparent experimental Young's modulus (E(app)(exp)) and maximum apparent experimental compressive stress (sigma(max)(exp)) were significantly correlated with bone density obtained by Archimedes's test (E(app)(exp)=236+/-231 MPa [19-742 MPa], sigma(max)(exp)=2.61+/-1.97 MPa [0.28-5.81 MPa], r>0.80, p<0.001). Under threshold at 40 microm, the size of the numerical samples (5.18(3) and 6.68(3)mm(3)) seems to be an important parameter on the accuracy of the results. The numerical trabecular Young's modulus was widely higher (E(trabecular)(num)=34,182+/-22,830 MPa [9700-87,211 MPa]) for the larger numerical samples and high BV/TV than those found classically by other techniques (4700-15,000 MPa). For rod-like bone samples (BV/TV<12%, n=7), Young's modulus, using linear beam element (E(trabecular)(num-skeleton): 10,305+/-5500 MPa), were closer to the Young's modulus found by other techniques. Those results show the limitation of hexahedral finite elements at 40 microm, mostly used, for thin trabecular structures.     

Nanoindentation and whole-bone bending estimates of material properties in bones from the senescence accelerated mouse SAMP6

The senescence accelerated mouse, strain P6 (SAMP6) has been described as a model of senile osteoporosis. Recent results from whole-bone bending tests indicate that, despite having increased moments of inertia, SAMP6 long bones are weak and brittle compared to SAMR1 controls. In the current study we determined material properties of cortical bone from SAMP6 and SAMR1 femora and tibiae by two methods-nanoindentation and whole-bone bending tests combined with simple beam theory. We hypothesized that: (1) SAMP6 mice have reduced cortical bone material properties compared to SAMR1 controls; and (2) modulus estimated from whole-bone bending tests correlates well with modulus determined by nanoindentation. Results from nanoindentation indicated that modulus and hardness are approximately 10% higher in SAMP6 mice compared to SAMR1 controls (p<0.001), a finding consistent with slightly higher mineralization in SAMP6 bones. Despite their superior elastic and hardness properties, the bending failure properties of SAMP6 bones were markedly inferior--ultimate stress and toughness were reduced by 40% and 60%, respectively (p<0.001). Comparisons between the two testing methods for determining modulus showed poor agreement. Modulus estimated from whole-bone bending tests was not correlated with modulus determined by nanoindentation (p=0.054; r2=0.03) and the absolute values differed by a factor of five between the two methods (bending [wet], 6GPa; nanoindentation [dry], 31GPa). Moreover, relative differences between groups were inconsistent between the two methods. We conclude: (1) cortical bone from the SAMP6 mouse has increased modulus and hardness but poor material strength and toughness, which underscores the relevance of the SAMP6 mouse for studies of skeletal fragility, and (2) values of elastic modulus of bone tissue estimated using simple beam theory and bending tests of mouse femora and tibiae are inaccurate and should be interpreted with caution.     

Mechanical properties of a biodegradable bone regeneration scaffold

Poly (Propylene Fumarate) (PPF), a novel, bulk erosion, biodegradable polymer, has been shown to have osteoconductive effects in vivo when used as a bone regeneration scaffold (Peter, S. J., Suggs, L. J., Yaszemski, M. J., Engel, P. S., and Mikos, A. J., 1999, J. Biomater. Sci. Polym. Ed., 10, pp. 363-373). The material properties of the polymer allow it to be injected into irregularly shaped voids in vivo and provide mechanical stability as well as function as a bone regeneration scaffold. We fabricated a series of biomaterial composites, comprised of varying quantities of PPF, NaCl and beta-tricalcium phosphate (beta-TCP), into the shape of right circular cylinders and tested the mechanical properties in four-point bending and compression. The mean modulus of elasticity in compression (Ec) was 1204.2 MPa (SD 32.2) and the mean modulus of elasticity in bending (Eb) was 1274.7 MPa (SD 125.7). All of the moduli were on the order of magnitude of trabecular bone. Changing the level of NaCl from 20 to 40 percent, by mass, did not decrease Ec and Eb significantly, but did decrease bending and compressive strength significantly. Increasing the beta-TCP from 0.25 g/g PPF to 0.5 g/g PPF increased all of the measured mechanical properties of PPF/NVP composites. These results indicate that this biodegradable polymer composite is an attractive candidate for use as a replacement scaffold for trabecular bone.     

The role of fabric in the large strain compressive behavior of human trabecular bone

Osteoporosis-related vertebral body fractures involve large compressive strains of trabecular bone. The small strain mechanical properties of the trabecular bone such as the elastic modulus or ultimate strength can be estimated using the volume fraction and a second order fabric tensor, but it remains unclear if similar estimations may be extended to large strain properties. Accordingly, the aim of this work is to identify the role of volume fraction and especially fabric in the large strain compressive behavior of human trabecular bone from various anatomical locations. Trabecular bone biopsies were extracted from human T12 vertebrae (n=31), distal radii (n=43), femoral head (n=44), and calcanei (n=30), scanned using microcomputed tomography to quantify bone volume fraction (BV/TV) and the fabric tensor (M), and tested either in unconfined or confined compression up to very large strains (～70%). The mechanical parameters of the resulting stress-strain curves were analyzed using regression models to examine the respective influence of BV/TV and fabric eigenvalues. The compressive stress-strain curves demonstrated linear elasticity, yielding with hardening up to an ultimate stress, softening toward a minimum stress, and a steady rehardening followed by a rapid densification. For the pooled experiments, the average minimum stress was 1.89 ± 1.77 MPa, while the corresponding mean strain was 7.15 ± 1.84%. The minimum stress showed a weaker dependence with fabric as the elastic modulus or ultimate strength. For the confined experiments, the stress at a logarithmic strain of 1.2 was 8.08 ± 7.91 MPa, and the dissipated energy density was 5.67 ± 4.42 MPa. The latter variable was strongly related to the volume fraction (R(2)=0.83) but the correlation improved only marginally with the inclusion of fabric (R(2)=0.84). The influence of fabric on the mechanical properties of human trabecular bone decreases with increasing strain, while the role of volume fraction remains important. In particular, the ratio of the minimum versus the maximum stress, i.e., the relative amount of softening, decreases strongly with fabric, while the dissipated energy density is dominated by the volume fraction. The collected results will prove to be useful for modeling the softening and densification of the trabecular bone using the finite element method.     

Habitual loading results in tendon hypertrophy and increased stiffness of the human patellar tendon.

The purpose of this study was to examine patellar tendon (PT) size and mechanical properties in subjects with a side-to-side strength difference of >/=15% due to sport-induced loading. Seven elite fencers and badminton players were included. Cross-sectional area (CSA) of the PT obtained from MRI and ultrasonography-based measurement of tibial and patellar movement together with PT force during isometric contractions were used to estimate mechanical properties of the PT bilaterally. We found that distal tendon and PT, but not mid-tendon, CSA were greater on the lead extremity compared with the nonlead extremity (distal: 139 +/- 11 vs. 116 +/- 7 mm(2); mid-tendon: 85 +/- 5 vs. 77 +/- 3 mm(2); proximal: 106 +/- 7 vs. 83 +/- 4 mm(2); P < 0.05). Distal tendon CSA was greater than proximal and mid-tendon CSA on both the lead and nonlead extremity (P < 0.05). For a given common force, stress was lower on the lead extremity (52.9 +/- 4.8 MPa) compared with the nonlead extremity (66.0 +/- 8.0 MPa; P < 0.05). PT stiffness was also higher in the lead extremity (4,766 +/- 716 N/mm) compared with the nonlead extremity (3,494 +/- 446 N/mm) (P < 0.05), whereas the modulus did not differ (lead 2.27 +/- 0.27 GPa vs. nonlead 2.16 +/- 0.28 GPa) at a common force. These data show that a habitual loading is associated with a significant increase in PT size and mechanical properties.     

Biomechanical properties of bones from rats treated with sevelamer

Sevelamer hydrochloride is used for ten years in patients on dialysis as a phosphate binder. We have previously shown that oral application of sevelamer prevents the bone loss and increases the bone volume in ovariectomized rats. In this study we further analysed the biomechanical properties of bones from rats treated with sevelamer utilizing a threepoint bending test to determine the mechanical properties of the cortical bone of the mid-shaft femur, while the indentation test was used to determine the mechanical properties of cancellous bone in the marrow cavity of the distal femoral metaphysis. Parameters analyzed included: maximum load (F(u)), stiffness (S), energy absorbed (W), toughness (T) and ultimate strength (sigma). The intrinsic properties, stress, elastic modulus and toughness were determined from measured maximum load, strains, stiffness, energy absorbed, outer and inner diameters, and calculated bone cross-sectional moment of inertia. Sevelamer was given to rats for 25 weeks with a content of 3% of sevelamer in a standard diet, starting immediately following ovariectomy (OVX). Animals were divided to the following groups: (1) Sham; (2) Sham + sevelamer 3%; (3) OVX; (4) OVX + sevelamer 3%. Our results showed that sevelamer particularly influenced the rat trabecular bone by increasing the maximum load for 26.2%, energy absorbed for 24.2% and the ultimate strength for 26.2% in sham animals treated with sevelamer 3%, as compared to sham rats. Sevelamer 3% in OVX rats also increased the maximum load for 71.4%, stiffness for 70.7%, energy absorbed for 55.9% and the ultimate strength for 71.3% as compared to OVX controls. In the three bending test sevelamer had a very little effect on preventing loss of bone strenght in the cortical bone. These results collectively suggest that sevelamer improves bone biomechanical properties, mainly affecting trabecular bone quality in both normal and ovariectomized rats.     

Tensile experiments and SEM fractography on bovine subchondral bone

Subchondral bone undecalcified samples, extracted from bovine femoral heads, are subjected to a direct tensile load. The Young's modulus of each sample is determined from repeated tests within the elastic limit. In a last test, the tensile load is increased up to the specimen failure, determining the ultimate tensile strength. The investigation is performed on both dry and wet specimens. The measured Young's modulus for dry samples is 10.3+/-2.5GPa, while that of wet samples is 3.5+/-1.2GPa. The ultimate tensile strengths are 36+/-10 and 30+/-7.5MPa for dry and wet specimens, respectively. SEM micrographs of failure surfaces show characteristic lamellar bone structures, with lamellae composed of calcified collagen fibers. Rudimentary osteon-like structures are also observed. Failure surfaces of wet samples show a marked fiber pull-out, while delamination predominates in dry samples. The obtained results are interpreted on the basis of the deformation mechanisms typical of fiber-reinforced laminated composite materials.     

A Three-Dimensional Finite Element Scheme to Investigate the Apparent Mechanical Properties of Trabecular Bone

A computational technique is described for investigating the apparent mechanical properties of trabecular bone based on tissue geometry obtained from the marching cubes volume rendering scheme. Using this scheme, a 3D representation of the trabecular bone was extracted from two-dimensional cross-sections of the tissue originating from a quantitative serial sectioning procedure. Surface information consists of node coordinates and polygon connectivity in a 3D space. A custom, adaptive mesh generation technique using a normal offset was used to prepare 3D finite element volume meshes (4-node tetrahedral elements) of variable mesh density from the extracted surface geometry. Nine target mesh resolutions (32 μm to 107 μm) were examined for a (1.5 mmx 1.5 mmx 2 mm) volume of trabecular bone. A mesh density of 50,000 elements/mm(3) of bone tissue was found to be adequate for convergence of apparent (bulk) modulus for 1% uniaxial compression. For this convergent case, the maximum local normal compressive tissue stress was 400 MPa which was six hundred-fold greater than the computed apparent stress. Variation in the apparent modulus was less than 5% when Poisson's ratio values were varied between 0.1 and 0.4. Poisson's ratio values greater than 0.4 had a more marked effect on the apparent modulus. Based upon these results, approximately 1 million, 4-node tetrahedral elements are required to analyze a continuum scale model of trabecular bone (5 mm cube).     

Effects of differences in mineralization on the mechanical properties of bone

There is a considerable variation in the mineralization of bone; normal, non-pathological compact bone has ash masses ranging from 45 to 85% by mass. This range of mineralization results in an even greater range of mechanical properties. The Young modulus of elasticity can range from 4 to 32 GPa, bending strength from 50 to 300 MPa, and the work of fracture from 200 to 7000 Jm-2. It is not possible for any one type of bone to have high values for all three properties. Very high values of mineralization produce high values of Young modulus but low values of work of fracture (which is a measure of fracture toughness). Rather low values of mineralization are associated with high values of work of fracture but low values of Young modulus and intermediate values of bending strength. The reason for the high value for the Young modulus associated with high mineralization is intuitively obvious, but has not yet been rigorously modelled. The low fracture toughness associated with high mineralization may be caused by the failure of various crack-stopping mechanisms that can act when the mineral crystals in bone have not coalesced, but which become ineffective when the volume fraction of mineral becomes too high. The adoption of different degrees of mineralization by different bones, leading to different sets of mechanical properties, is shown to be adaptive in most cases studied, but some puzzles still remain.     

Effect of specimen size on work-of-fracture measurements

It has been suggested that work-of-fracture, which quantifies the ability of a material to resist fracture, is dependent on specimen size. This experiment compared work-of-fracture, calculated as energy per unit area, for different specimen sizes of Plexiglas, bovine tibial bone and aluminum. Three different geometrically similar cross sections were tested for each material for a total of 54 specimens. Work-of-fracture was measured by loading a notched beam (triangular cross section) in three-point bending at a constant deformation rate. The energy necessary to cause fracture was measured from a load-deformation curve. Specimen fracture area was determined using macrophotography. Atomic absorption spectrophotometry was used to determine weight percent calcium of bone specimens and quantitative light microscopy was used to determine fractional void area. Analysis of variance showed no effect of specimen size on work-of-fracture for aluminum or Plexiglas specimens (p greater than 0.05). A significant difference was found, however, between the large (area = 11.7 +/- 1.9 mm2) and small (area = 3.48 +/- 0.68 mm2) bone specimens and between the medium (area = 5.89 +/- 0.69 mm2) and small (area = 3.48 +/- 0.68 mm2) bone specimens. No correlation was found between work-of-fracture and either calcium content (r2 = 0.128) or fractional void area (r2 = 0.0713). The mean work-of-fracture values found are as follows: aluminum, 59.8 +/- 13.7 kJ m-2; Plexiglas, 0.620 +/- 0.074 kJ m-2; bone (area 5.89 +/- 0.69 mm2-11.7 +/- 1.9 mm2), 9.72 +/- 1.93 kJ m-2 and bone (area 3.48 +/- 0.68 mm2), 5.48 +/- 1.79 kJ m-2.