Numerical errors and uncertainties in finite-element modeling of trabecular bone

Although micromechanical finite-element models are being increasingly used to help interpret the results of bio-mechanical tests, there has not yet been a systematic study of the numerical errors and uncertainties that occur with these methods. In this work, finite-element models of human L1 vertebra have been used to analyze the sensitivity of the calculated elastic moduli to resolution, boundary conditions, and variations in the Poisson's ratio of the tissue material. Our results indicate that discretization of the bone architecture, inherent in the tomography process, leads to an underestimate in the calculated elastic moduli of about 20% at 20 microm resolution; these errors vary roughly linearly with the size of the image voxels. However, it turns out that there is a cancellation of errors between the softening introduced by the discretization of the bone architecture and the excess bending resistance of eight-node hexahedral finite elements. Our empirical finding is that eight-node cubic elements of the same size as the image voxels lead to the most accurate calculation for a given number of elements, with errors of less than 5% at 20 microm resolution. Comparisons with mechanical testing are also hindered by uncertainties in the grip conditions: our results show that these uncertainties are of comparable magnitude to the systematic differences in mechanical testing methods. Both discretization errors and uncertainties in grip conditions have a smaller effect on relative moduli, used when comparing between different specimens or different load directions, than on an absolute modulus. The effects of variations in the Poisson's ratio of the bone tissue were found to be negligible.     

The ability of quantitative ultrasound to predict the mechanical properties of trabecular bone under different strain rates

The ability of quantitative ultrasound to predict the mechanical properties of trabecular bone under different strain rates was investigated. Ultrasound velocity (UV) and broadband attenuation (BUA) were measured for 60 specimens of human trabecular bone. Samples were divided into two equal groups and loaded in compression at the strain rates of 0.0004 and 0.08 s-1. The ultimate strength, elastic modulus, and energy absorption capacity were determined for each specimen. Specimens tested at 0.08 s-1 had a mean value of strength 63% higher than the specimens tested at 0.0004 s-1. The elastic modulus and energy absorption capacity were 82% and 42% higher, respectively, for the higher strain rate. UV and BUA were significantly associated with most mechanical properties at both strain rates. All mechanical properties were also correlated strongly with a linear combination of UV and BUA for both the low and high loading rates. The use of ultrasound parameters may provide good clinical means for assessing the resistance of trabecular bone to both low and high energy trauma.     

Steel Hexagonal Honeycomb Core Equivalent Elastic Moduli for Bridge Deck Sandwich Panels

Glass fiber-reinforced polymer (GFRP) materials possess inherently high strength-to-weight ratios, but their effective elastic moduli are low relative to civil engineering (CE) construction materials. While elastic modulus may be comparable to that of some CE materials, the lower shear modulus adversely affects stiffness. As a result, serviceability issues are what govern GFRP deck design in the CE bridge industry. An innovative solution to increase the stiffness of a commercial GFRP reinforced-sinusoidal honeycomb sandwich panel was proposed; this solution would completely replace the GFRP honeycomb core with a hexagonal honeycomb core constructed from commercial steel roof decking. The purpose of this study was to perform small-scale tests to characterize the steel hexagonal honeycomb core equivalent elastic moduli in an effort to simplify the modeling of the core. The steel core equivalent moduli experimental results were compared with theoretical hexagonal honeycomb elastic modulus equations from the literature, demonstrating the applicability of the theoretical equations to the steel honeycomb core. Core equivalent elastic modulus equations were then proposed to model and characterize the steel hexagonal honeycomb as applicable to sandwich panel design. The equivalent honeycomb core will enable an efficient sandwich panel stiffness design technique, both for structural analysis methods (i.e., hand calculations) and finite-element analysis procedures.     

Human vertebral body apparent and hard tissue stiffness

Cancellous bone apparent stiffness and strength are dependent upon material properties at the tissue level and trabecular architecture. Microstructurally accurate, large-scale finite element (LS-FE) models were used to predict the experimental apparent stiffness of human vertebral cancellous bone and to estimate the trabecular hard tissue stiffness. Twenty-eight LS-FE models of cylindrical human vertebral cancellous bone specimens (8 mm in diameter, 9.5 mm in height, one each from twenty-eight individuals) were generated directly from microcomputed tomography images and solved by a special purpose iterative finite element program. The experimental apparent stiffness and strength of the specimens were determined by mechanical testing to failure in the infero superior direction. Morphometric measurements including bone volume fraction (BV/TV), three eigenvalues of the fabric tensor and average P(L) were also calculated. The finite element estimate of apparent stiffness explained much of the variance in both experimental apparent stiffness (r2=0.89) and experimental apparent strength (r2=0.87). Stepwise linear regression analysis demonstrated that the LS-FE estimated apparent stiffness was the only significant predictor of experimental apparent stiffness and strength when it was included with all measured morphometric values. Hard tissue stiffness was quite variable between individuals (mean, 5.7 GPa; S.D. 1.6 GPa), but was not significantly related to age, sex, race, weight or morphometric measures for this sample.     

A 3D damage model for trabecular bone based on fabric tensors

Motivated by the role of damage in normal and pathological conditions of trabecular bone, a novel 3D constitutive law was developed that describes anisotropic elasticity and the rate-independent degradation in mechanical properties resulting from the growth of cracks or voids in the trabecular tissue. The theoretical model was formulated within the framework of continuum damage mechanics and based on two fabric tensors characterizing the local trabecular morphology. Experimental validation of the model was achieved by uniaxial and torsional testing of waisted bovine trabecular bone specimens. Strong correlations were found between cumulated permanent strain, reduction in elastic moduli and nonlinear postyield stress, which support the hypothesis that these variables reflect the same underlying damage process.     

Trabecular bone mineral and calculated structure of human bone specimens scanned by peripheral quantitative computed tomography: relation to biomechanical properties

The relationship of cortical bone mineral density (BMD), and geometry to bone strength has been well documented. In this study, we used peripheral quantitative computerized tomography (pQCT) to acquire trabecular BMD and high-resolution images of trabeculae from specimens to determine their relationship with biomechanical properties. Fifty-eight human cubic trabecular bone specimens, including 26 from the vertebral bodies, were scanned in water and air. Trabecular structure was quantitated using software developed with Advanced Visual Systems interfaced on a Sun/Sparc Workstation. BMD was also obtained using a whole-body computerized tomography scanner (QCT). Nondestructive testing of the specimens was performed to assess their elastic modulus. QCT and pQCT measurements of BMD of specimens in water were strongly correlated (r2 = 0.95, p < 0.0001), with a slope (0.96) statistically not significantly different from 1. Strong correlations were found between pQCT measurements of specimens in water and in air, for BMD (r2 = 0.96, p < 0.0001), and for apparent trabecular structural parameters (r2 = 0.89-0.93, p < 0.0001). Correlations were moderate between BMD and apparent trabecular structural parameters (r2 = 0.37-0.64, p < 0.0001). Precision as coefficient of variation (CV) and standardized coefficient of variation (SCV) for these measurements was < 5%. For the vertebral specimens, the correlation was higher between elastic modulus and BMD (r2 = 0.76,p < 0.0001) than between elastic modulus and apparent trabecular structural parameters (r2 = 0.58-0.72, p < 0.0001), while the addition of apparent trabecular nodes and branches to BMD in a multivariate regression model significantly increased the correlation with the elastic modulus (r2 = 0.86, p < 0.01). Thus, pQCT can comparably and reproducibly measure trabecular bone mineral in water or air, and trabecular structure can be quantitated from pQCT images. The combination of volumetric BMD with trabecular structural parameters rather than either alone improves the prediction of biomechanical properties. Such a noninvasive approach may be useful for the preclinical study of osteoporosis.     

Elastic properties of human cortical and trabecular lamellar bone measured by nanoindentation

An experimental investigation was undertaken to measure the intrinsic elastic properties of several of the microstructural components of human vertebral trabecular bone and tibial cortical bone by the nanoindentation method. Specimens from two thoracic vertebrae (T-12) and two tibiae were obtained from frozen, unembalmed human male cadavers aged 57 and 61 years. After drying and mounting in epoxy resin nanoindentation tests were conducted to measure Young's modulus and the hardness of individual trabeculae in the vertebrae and single osteons, and interstitial lamellae in the tibiae. Measurements on the vertebral trabeculae were made in the transverse direction, and the average Young's modulus was found to be 13.5 +/- 2.0 GPa. The tibial specimens were tested in the longitudinal direction, yielding moduli of 22.5 +/- 1.3 GPa for the osteons and 25.8 +/- 0.7 GPa for the interstitial lamellae. Analysis of variance showed that the differences in the measured moduli are statistically significant. Hardness differences among the various microstructural components were also observed.     

Fabrication and properties of graphite fiber reinforced magnesium

Graphite fiber reinforced magnesium composites have been fabricated by infiltration and liquid phase hot pressing techniques. Although pure magnesium does not wet graphite, weting and bonding were achieved by precoating the graphite fibers with titanium. Tensile strengths and elastic moduli of flat rectangular composite specimens were measured and compared with Rule of Mixtures strengths. Significant strengthening and stiffening were achieved. Tensile strength was increased by 170 pct and the elastic modulus by 100 pct. Measured elastic moduli were in good agreement with Rule of Mixtures values. The highest tensile strength was about 83 pct of the Rule of Mixtures strength. It is believed that specimen porosity and fiber misalignment account for the bulk of this difference and that an improved fabrication technique will yield specimens with higher strengths. Fracture modes are presented and discussed.     

Three-dimensional methods for quantification of cancellous bone architecture

Recent development in three-dimensional (3-D) imaging of cancellous bone has made possible true 3-D quantification of trabecular architecture. This provides a significant improvement of the tools available for studying and understanding the mechanical functions of cancellous bone. This article reviews the different techniques for 3-D imaging, which include serial sectioning, X-ray tomographic methods, and NMR scanning. Basic architectural features of cancellous bone are discussed, and it is argued that connectivity and architectural anisotropy (fabric) are of special interest in mechanics-architecture relations. A full characterization of elastic mechanical properties is, with traditional mechanical testing, virtually impossible, but 3-D reconstruction in combination with newly developed methods for large-scale finite element analysis allow calculations of all elastic properties at the cancellous bone continuum level. Connectivity has traditionally been approached by various 2-D methods, but none of these methods have any known relation to 3-D connectivity. A topological approach allows unbiased quantification of connectivity, and this further allows expressions of the mean size of individual trabeculae, which has previously also been approached by a number of uncertain 2-D methods. Anisotropy may be quantified by fundamentally different methods. The well-known mean intercept length method is an interface-based method, whereas the volume orientation method is representative of volume-based methods. Recent studies indicate that volume-based methods are at least as good as interface-based methods in predicting mechanical anisotropy. Any other architectural property may be quantified from 3-D reconstructions of cancellous bone specimens as long as an explicit definition of the property can be given. This challenges intuitive and vaguely defined architectural properties and forces bone scientists toward 3-D thinking.     

Plastic deformation in a multifunctional Ti-Nb-Ta-Zr-O alloy

Mechanisms for plastic deformation in the newly developed Ti-24 at. pct (Ta + Nb + V)-(Zr,Hf)-O alloys (Gum Metal) were investigated in relation to their unique properties. Transmission electron microscopy revealed that the microstructure after deformation was characterized by highly distorted crystal images, which are accompanied by numerous “giant faults.” Such plastic behavior implies that a large amount of elastic stain energy was stored discretely and hierarchically during cold working. Calculated elastic constants of the Ti-X (Nb,Ta,Mo,V) binary systems predicted that Young’s modulus in 〈001〉 and shear moduli along some directions including slip systems in a bcc crystal were extraordinary small. The low modulus not only well explains the highly distorted microstructure observed in the cold-worked specimens, but also signifies that ideal shear strength of the developed alloys is a very small value, which is close to the practical strength required for plastic deformation in the alloy. This implies that the giant faults observed in the deformed specimen were formed without the aid of dislocation glide.     

Mechanistic Corrections for Determining the Resilient Modulus of Base Course Materials Based on Elastic Wave Measurements

The mechanical performance of pavement systems depends on the stiffness of subsurface soil and aggregate materials. The moduli of base course, subbase, and subgrade soils included in pavement systems need to be characterized for their use in the new empirical-mechanistic design procedure (NCHRP 1-37A). Typically, the resilient modulus test is used in the design of base and subbase layers under repetitive loads. Unfortunately, resilient modulus tests are expensive and cannot be applied to materials that contain particles larger than 25 mm (for 125-mm diameter specimens) without scalping the large grains. This paper examines a new methodology for estimating resilient modulus based on the propagation of elastic waves. The method is based on using a mechanistic approach that relates the P-wave velocity-based modulus to the resilient modulus through corrections for stress, void ratio, strain, and Poisson’s ratio effects. Results of this study indicate that resilient moduli are approximately 30% of Young’s moduli based on seismic measurements. The technique is then applied to specimens with large-grain particles. Results show that the methodology can be applied to large-grained materials and their resilient modulus can be estimated with reasonable accuracy based on seismic techniques. An approach is proposed to apply the technique to field determinations of modulus.     

Three-Dimensional Discrete Element Models for Asphalt Mixtures

The main objective of this paper is to develop three-dimensional (3D) microstructure-based discrete element models of asphalt mixtures to study the dynamic modulus from the stress-strain response under compressive loads. The 3D microstructure of the asphalt mixture was obtained from a number of two-dimensional (2D) images. In the 2D discrete element model, the aggregate and mastic were simulated with the captured aggregate and mastic images. The 3D models were reconstructed with a number of 2D models. This stress-strain response of the 3D model was computed under the loading cycles. The stress-strain response was used to predict the asphalt mixture’s stiffness (modulus) by using the aggregate and mastic stiffness. The moduli of the 3D models were compared with the experimental measurements. It was found that the 3D discrete element models were able to predict the mixture moduli across a range of temperatures and loading frequencies. The 3D model prediction was found to be better than that of the 2D model. In addition, the effects of different air void percentages and aggregate moduli to the mixture moduli were investigated and discussed.     

Mechanical properties of titanium connectors

The tensile mechanical properties of welded titanium joints were studied, and intact titanium was used as controls. Welded joints were fabricated with either a stereographic laser-welding technique or a gas tungsten arc welding technique. The effect of heat treatment following a simulated porcelain application was also investigated. Heat-treated laser welds had significantly lower ultimate tensile strengths. Heat treatment had no effect on the modulus of elasticity or elongation, but generally significantly decreased the yield strength of the titanium specimens. The gas tungsten are welding specimens had significantly higher yield strengths and elastic moduli than the other two groups. The elongation of the control specimens was significantly greater than the elongation of the gas tungsten arc welding specimens, which was in turn significantly higher than that of the laser-welded specimens.     

Finite-Element Model for Failure Study of Two-Dimensional Triaxially Braided Composite

A new three-dimensional finite-element model of two-dimensional, triaxially braided composites is presented in this paper. This mesoscale modeling technique is used to examine and predict the deformation and damage observed in tests of straight-sided specimens. A unit cell-based approach is used to consider the braiding architecture and the mechanical properties of the fiber tows, the matrix, and the fiber tow-matrix interface. A 0°/±60° braiding configuration has been investigated by conducting static finite-element analyses. Failure initiation and progressive degradation has been simulated in the fiber tows by using the Hashin failure criteria and a damage evolution law. The fiber tow-matrix interface was modeled by using a cohesive zone approach to capture any fiber-matrix debonding. By comparing the analytical results with those obtained experimentally, the applicability of the developed model was assessed and the failure process was investigated.     

单压下节理密度及倾角对类岩石试件强度及变形的影响

通过预制张开节理类岩石试件,在单轴压缩条件下,研究节理密度及倾角的组合作用对试件强度和变形特征的影响.试验结果表明:(1)随着节理倾角的增大,应力-应变曲线由多峰值转变为单峰值,试件脆性增强,延性减弱;(2)节理密度对当量峰值强度的影响与节理倾角大小有关,对当量弹模的影响呈“V”形变化,即当量弹模随着节理密度的增大呈现先减小后增大的变化规律;(3)当量弹模随节理倾角的增大而增大,在节理倾角为90°的时候达到最大值,为完整试件弹性模量的70%~80%;(4)节理倾角对多节理类岩石试件当量峰值强度和当量弹性模量的影响大于节理密度的影响.对试验结果进一步分析发现:节理密度及节理倾角与应力-应变曲线、当量峰值强度及当量弹性模量之间的关系,其变化规律与试件的破坏过程息息相关,其破坏模式可分为张拉破坏、剪切破坏和复合破坏.      

Shear Modulus of Standard Pultruded Fiber Reinforced Plastic Material

Research to determine the shear modulus of standard pultruded fiber reinforced plastic (FRP) material is reviewed and appraised. It is found that different test methods have given shear moduli data in the range from 1.3 to 5.1 GPa, with varying degrees of scatter. Pultruded material is comprised of alternate layers of two distinct glass reinforcement types. By applying micromechanical modeling, it is shown that the in-plane shear modulus of the continuous unidirectional rovings layer is similar to that of the continuous filament (or strand) mat layer, and that these layer moduli, generally, lie in the range 3.5 to 4.8 GPa (depending on fiber volume fraction). This finding indicates that the significant difference (>1.3 times) between the in-plane (3 GPa or less) and the St. Venant torsion (always >4 GPa) shear moduli is likely to be due to the experimental test procedures and the physical interpretation of shearing, rather than the layer construction of the material. For structural profiles, it is seen that the shear modulus of 3 GPa in company design manuals is often less than measured. Researchers require correlated elastic constant data if elastic deflections and instability loads for structural members can be accurately predicted using elastic theory. Further work is, therefore, recommended to establish standard test and analytical methods for the determination of shear moduli of pultruded FRP material.     

Mesomechanical Model for Numerical Study of Two-Dimensional Triaxially Braided Composite

A mesoscale three-dimensional finite-element model is set up to model two-dimensional triaxially braided composites. Unit cell scheme is used to take into account braiding architecture as well as mechanical behavior of fiber tows, matrix, and fiber tow interface. A 0°/±60° braiding configuration has been studied. A failure criterion and progressive damage evolution model taking into account fiber tow and tow interface has been applied to theoretically predict interlaminar and intralaminar failure mode. Straight-sided specimen testing has been carried out in both axial and transverse direction. Results obtained in the tests as well as finite-element approaches are discussed. This paper also discusses the main feature of the model through an extensive parameter study. Overall, by comparison of experiment and model results, the applicability of the developed model is assessed and the failure process is investigated; furthermore, conducted parameter study enhances the strength of the model, which lies in the correlation of model parameters and identification of damage modes with experimental data on the overall stress strain curves.     

Possible role of decorin glycosaminoglycans in fibril to fibril force transfer in relative mature tendons--a computational study from molecular to microstructural level

Experimental studies on immature tendons have shown that the collagen fibril net is discontinuous. Manifold evidences, despite not being conclusive, indicate that mature tissue is discontinuous as well. According to composite theory, there is no requirement that the fibrils should extend from one end of the tissue to the other; indeed, an interfibrillar matrix with a low elastic modulus would be sufficient to guarantee the mechanical properties of the tendon. Possible mechanisms for the stress-transfer involve the interfibrillar proteoglycans and can be related to the matrix shear stress and to electrostatic non-covalent forces. Recent studies have shown that the glycosaminoglycans (GAGs) bound to decorin act like bridges between contiguous fibrils connecting adjacent fibril every 64-68 nm; this architecture would suggest their possible role in providing the mechanical integrity of the tendon structure. The present paper investigates the ability of decorin GAGs to transfer forces between adjacent fibrils. In order to test this hypothesis the stiffness of chondroitin-6-sulphate, a typical GAG associated to decorin, has been evaluated through the molecular mechanics approach. The obtained GAG stiffness is piecewise linear with an initial plateau at low strains (<800%) and a high stiffness region (3.1 x 10(-11)N/nm) afterwards. By introducing the calculated GAG stiffness in a multi-fibril model, miming the relative mature tendon architecture, the stress-strain behaviour of the collagen fibre was determined. The fibre incremental elastic modulus obtained ranges between 100 and 475 MPa for strains between 2% and 6%. The elastic modulus value depends directly on the fibril length, diameter and inversely on the interfibrillar distance. In particular, according to the obtained results, the length of the fibril is likely to play the major role in determining stiffness in mature tendons.     

EFFECT OF POROSITY AND PORE SIZE ON THE ELASTIC MODULI OF SINTERED IRON AND COPPER-TIN

Abstract

The Young’s modulus and shear modulus of iron and copper-tin sintered materials have been measured by the resonance method, with a view to determining the effects of porosity and pore size upon the values of the elastic moduli. The experimental values obtained were lower than those calculated on the basis of the Mackenzie theory. The discrepancy is attributed to a pore-size effect, since in the case of pores of sufficiently small dimensions the two results tend to coincide