Cosserat micromechanics of human bone: strain redistribution by a hydration sensitive constituent

Experimental determination of the strain distribution in prismatic, square cross-section bars of human compact bone in torsion disclosed nonclassical effects associated with the microstructure. Specifically, in wet bone at small strain, significant deviations from the classically predicted strain distribution were observed. The measured strain distribution in wet bone followed predictions based on Cosserat (micropolar) elasticity. In dry bone, the strain distribution was very close to the prediction of classical elasticity. The interaction between Haversian osteons and the cement substance between them was hypothesized to be the principal mechanism for the phenomena. To evaluate this hypothesis, additional specimens were subjected to prolonged torsional load and the cement lines were observed by reflected light microscopy. Localized deformation at the cement lines was observed, but it was less than values reported earlier for bovine plexiform bone.     

The effect of Haversian remodeling on the tensile properties of human cortical bone

The purpose of this study was to determine the effect of Haversian remodeling on the tensile properties of human cortical bone by testing specimens containing, as far a possible, a single type of bone tissue. Fifty-one specimens were prepared from sixteen fresh tibias, removed at autopsy. Age range was 19-35. Regions were selected so that the specimens would consist almost exclusively of either primary bone or Haversian bone. The ultimate tensile strength, ultimate strain and Young's modulus of elasticity were determined at a loading rate of 0.05 mm s-1. The primary bone specimens were found to have a significantly higher ultimate tensile strength and modulus of elasticity than those formed of Haversian bone.     

Can the theory of critical distances predict the failure of shape memory alloys?

Components made from shape memory alloys (SMAs) such as nitinol often fail from stress concentrations and defects such as notches and cracks. It is shown here for the first time that these failures can be predicted using the theory of critical distances (TCDs), a method which has previously been used to study fracture and fatigue in other materials. The TCD uses the stress at a certain distance ahead of the notch to predict the failure of the material due to the stress concentration. The critical distance is believed to be a material property which is related to the microstructure of the material. The TCD is simply applied to a linear model of the material without the need to model the complication of its non-linear behaviour. The non-linear behaviour of the material at fracture is represented in the critical stress. The effect of notches and short cracks on the fracture of SMA NiTi was studied by analysing experimental data from the literature. Using a finite element model with elastic material behaviour, it is shown that the TCD can predict the effect of crack length and notch geometry on the critical stress and stress intensity for fracture, with prediction errors of less than 5%. The value of the critical distance obtained for this material was L?=?90?μm; this may be related to its grain size. The effects of short cracks on stress intensity were studied. It was shown that the same value of the critical distance (L?=?90?μm) could estimate the experimental data for both notches and short cracks.     

A critical distance study of stress concentrations in bone

The Theory of Critical Distances (TCD) is a method used to study the failure of material in situations where stress concentrations, such as holes and notches, are present. This method uses two material constants: a critical length and a critical stress. The elastic stress field close to the stress concentration is examined, applying a fracture criterion. The TCD has been applied to predict brittle fracture in various different materials and various types of notches but it has not previously been applied to bone. Since bone fails by brittle fracture with limited plasticity, it is expected that the TCD will be applicable. Experimental data were obtained from the literature on the effects of sharp notches and holes loaded in various ways (tension, torsion and bending). These tests were modelled using finite element analysis. It was found that the TCD could be successfully applied to predict the load required for brittle fracture as a function of the type and size of the stress concentration feature. The critical distance was found to be almost constant, about 0.3-0.4mm, for all types of bone studied: the critical stress was found to be related to the material's ultimate tensile strength by a constant factor of T=1.33. The results of this study will be of practical value in the assessment of stress concentration features introduced during surgery and of naturally occurring bone defects.     

A phenomenological model for predicting fatigue life in bovine trabecular bone

Cyclic loading of bone during daily activities can lead to fatigue degradation and increased risk of fracture in both the young and elderly population. Damage processes under cyclic loading in trabecular bone result in the reduction of the elastic modulus and accumulation of residual strain. These effects increase with increasing stress levels, leading to a progressive reduction in fatigue life. The present work analyzes the effect of stress and strain variation on the above damage processes in bovine trabecular bone, and develops a phenomenological model relating fatigue life to the imposed stress level. The elastic modulus reduction of the bone specimens was observed to depend on the maximum compressive strain, while the rate of residual strain accumulation was a function of the stress level. A model was developed for the upper and lower bounds of bone elastic modulus reduction with increasing number of cycles, at each stress range. The experimental observations were described well by the model. The model predicted the bounds of the fatigue life with change in fatigue stress. The decrease in the fatigue life with increasing stress was related to corresponding increases in the residual strain accumulation rates at the elevated stress levels. The model shows the validity of fatigue predictions from relatively few cyclic experiments, by combining trends observed in the monotonic and the cyclic tests. The model also presents a relatively simple procedure for predicting the endurance limit for bovine trabecular bone specimens.     

Prediction of cortical bone elastic constants by a two-level micromechanical model using a generalized self-consistent method

A two-level micromechanical model of cortical bone based on a generalized self-consistent method was developed to take into consideration the transversely isotropic elasticity of many microstructural features in cortical bone, including Haversian canals, resorption cavities, and osteonal and interstitial lamellae. In the first level, a single osteon was modeled as a two-phase composite such that Haversian canals were represented by elongated pores while the surrounding osteonal lamellae were considered as matrix. In the second level, osteons and resorption cavities were modeled as multiple inclusions while interstitial lamellae were regarded as matrix. The predictions of cortical bone elasticity from this two-level micromechanical model were mostly in agreement with experimental data for the dependence of transversely isotropic elasticity of human femoral cortical bone on porosity. However, variation in cortical bone elastic constants was greater in experimental data than in model predictions. This could be attributed to variations in the elastic properties of microstructural features in cortical bone. The present micromechanical model of cortical bone will be useful in understanding the contribution of cortical bone porosity to femoral neck fractures.     

Micromechanics modeling of Haversian cortical bone properties.

A finite-element micromechanics model for Haversian cortical bone tissue has been developed and studied. The model is an extension of two-dimensional micromechanics techniques for fiber-reinforced composite materials. Haversian systems, or secondary osteons, are considered to be the fiber component, and interstitial lamellar bone the matrix material. The cement line is included as an 'interphase' component along the fiber/matrix interface. The model assumes a regular repeatable spacing of the longitudinally aligned continuous fibers and is, therefore, restricted to approximating Haversian cortical bone in its present form. Haversian porosity is modeled explicitly by incorporating a hollow fiber to represent the Haversian canal. Solutions have been obtained by applying uniform macroscopic stresses to the boundaries of the repeating unit cell model. Macroscopic mechanical property predictions correspond reasonably well with the experimental data for cortical bone, but are necessarily dependent on the input properties for each constituent, which are not well established. The predicted variation in the elastic modulus with porosity is not as sensitive as that observed experimentally. Stresses within the constituents can also be modeled with this method and are demonstrated to deviate from the macroscopic applied stress levels.     

Determining effective centroid position in biomechanical testing: a technique for simplifying whole bone analysis

BACKGROUND: Whole bone in vitro biomechanical compressive testing can be complicated by three factors: sample asymmetry, heterogeneous material properties, and unknown effective centroid location. METHOD OF APPROACH: The technique presented here facilitates the calculation of effective centroid position, modulus of elasticity and equivalent uniform strain magnitude for a cross section of bone from a simple whole bone compressive test. Simplification of section response to load is achieved through a combination of linear beam and simple planer geometry theory. The technique requires three longitudinal strain gauges be affixed around the test specimen cross section of interest, gauge position need not be determined. Sample loading is then accomplished using a simple four point loading jig. RESULTS: Results of the technique are presented on an object with known elasticity and geometry, an aluminium tube, and seven pairs of equine third metacarpal whole bones. CONCLUSIONS: Average cross section modulus of elasticity, equivalent uniform cross section strain, and effective centroid locations were all predicted to within the range of published values. Employing the testing setup and analysis technique presented in this paper resulted in a significant savings in both implementation complexity and cost over previously available techniques.     

Micromechanical modelling of cortical bone

Cortical bone is a heterogeneous material with a complex hierarchical microstructure. In this work, unit cell finite element models were developed to investigate the effect of microstructural morphology on the macroscopic properties of cortical bone. The effect of lacunar and vascular porosities, percentage of osteonal bone and orientation of the Haversian system on the macroscopic elastic moduli and Poisson's ratios was investigated. The results presented provide relationships for applying more locally accurate material properties to larger scale and whole bone models of varying porosity. Analysis of the effect of the orientation of the Haversian system showed that its effects should not be neglected in larger scale models. This study also provides insight into how microstructural features effect local distributions and cause a strain magnification effect. Limitations in applying the unit cell methodology approach to bone are also discussed.     

Incorporating tissue anisotropy and heterogeneity in finite element models of trabecular bone altered predicted local stress distributions

Trabecular bone is composed of organized mineralized collagen fibrils, which results in heterogeneous and anisotropic mechanical properties at the tissue level. Recently, biomechanical models computing stresses and strains in trabecular bone have indicated a significant effect of tissue heterogeneity on predicted stresses and strains. However, the effect of the tissue-level mechanical anisotropy on the trabecular bone biomechanical response is unknown. Here, a computational method was established to automatically impose physiologically relevant orientation inherent in trabecular bone tissue on a trabecular bone microscale finite element model. Spatially varying tissue-level anisotropic elastic properties were then applied according to the bone mineral density and the local tissue orientation. The model was used to test the hypothesis that anisotropy in both homogeneous and heterogeneous models alters the predicted distribution of stress invariants. Linear elastic finite element computations were performed on a 3 mm cube model isolated from a microcomputed tomography scan of human trabecular bone from the distal femur. Hydrostatic stress and von Mises equivalent stress were recorded at every element, and the distributions of these values were analyzed. Anisotropy reduced the range of hydrostatic stress in both tension and compression more strongly than the associated increase in von Mises equivalent stress. The effect of anisotropy was independent of the spatial redistribution high compressive stresses due to tissue elastic heterogeneity. Tissue anisotropy and heterogeneity are likely important mechanisms to protect bone from failure and should be included for stress analyses in trabecular bone.     

Strain rate dependence of the mechanical properties of reindeer antler and the cumulative damage model of bone fracture

Carter and Caler have produced a 'cumulative damage' model for the fracture of bone, based on creep experiments on human bone, which has been corroborated by monotonic tensile tests on bone, loaded at various strain rates. Monotonic tensile tests on reindeer's antler, which has a lower modulus of elasticity than human bone, produce very similar results. Unlike human bone, reindeer antler always shows a large post-yield strain, and it is possible to distinguish pre-yield and post-yield behaviour. The 'final stiffness' (ultimate stress/ultimate strain) is invariant with strain rate. This is confirmation that bone fractures when a certain amount of damage has accumulated. However, reindeer antler shows a considerable post-yield increase in stress. This is difficult to accommodate in a cumulative damage model.     

The effects of modeling simplifications on craniofacial finite element models: the alveoli (tooth sockets) and periodontal ligaments

Several finite element models of a primate cranium were used to investigate the biomechanical effects of the tooth sockets and the material behavior of the periodontal ligament (PDL) on stress and strain patterns associated with feeding. For examining the effect of tooth sockets, the unloaded sockets were modeled as devoid of teeth and PDL, filled with teeth and PDLs, or simply filled with cortical bone. The third premolar on the left side of the cranium was loaded and the PDL was treated as an isotropic, linear elastic material using published values for Young's modulus and Poisson's ratio. The remaining models, along with one of the socket models, were used to determine the effect of the PDL's material behavior on stress and strain distributions under static premolar biting and dynamic tooth loading conditions. Two models (one static and the other dynamic) treated the PDL as cortical bone. The other two models treated it as a ligament with isotropic, linear elastic material properties. Two models treated the PDL as a ligament with hyperelastic properties, and the other two as a ligament with viscoelastic properties. Both behaviors were defined using published stress-strain data obtained from in vitro experiments on porcine ligament specimens. Von Mises stress and strain contour plots indicate that the effects of the sockets and PDL material behavior are local. Results from this study suggest that modeling the sockets and the PDL in finite element analyses of skulls is project dependent and can be ignored if values of stress and strain within the alveolar region are not required.     

Fracture toughness and fatigue crack propagation rate of short fiber reinforced epoxy composites for analogue cortical bone

Third-generation mechanical analogue bone models and synthetic analogue cortical bone materials manufactured by Pacific Research Laboratories, Inc. (PRL) are popular tools for use in mechanical testing of various orthopedic implants and biomaterials. A major issue with these models is that the current third-generation epoxy-short fiberglass based composite used as the cortical bone substitute is prone to crack formation and failure in fatigue or repeated quasistatic loading of the model. The purpose of the present study was to compare the tensile and fracture mechanics properties of the current baseline (established PRL "third-generation" E-glass-fiber-epoxy) composite analogue for cortical bone to a new composite material formulation proposed for use as an enhanced fourth-generation cortical bone analogue material. Standard tensile, plane strain fracture toughness, and fatigue crack propagation rate tests were performed on both the third- and fourth-generation composite material formulations using standard ASTM test techniques. Injection molding techniques were used to create random fiber orientation in all test specimens. Standard dog-bone style tensile specimens were tested to obtain ultimate tensile strength and stiffness. Compact tension fracture toughness specimens were utilized to determine plane strain fracture toughness values. Reduced thickness compact tension specimens were also used to determine fatigue crack propagation rate behavior for the two material groups. Literature values for the same parameters for human cortical bone were compared to results from the third- and fourth-generation cortical analogue bone materials. Tensile properties of the fourth-generation material were closer to that of average human cortical bone than the third-generation material. Fracture toughness was significantly increased by 48% in the fourth-generation composite as compared to the third-generation analogue bone. The threshold stress intensity to propagate the crack was much higher for the fourth-generation material than for the third-generation composite. Even at the higher stress intensity threshold, the fatigue crack propagation rate was significantly decreased in the fourth-generation composite compared to the third-generation composite. These results indicate that the bone analogue models made from the fourth-generation analogue cortical bone material may exhibit better performance in fracture and longer fatigue lives than similar models made of third-generation analogue cortical bone material. Further fatigue testing of the new composite material in clinically relevant use of bone models is still required for verification of these results. Biomechanical test models using the superior fourth-generation cortical analogue material are currently in development.     

Tensile behavior of cortical bone: dependence of organic matrix material properties on bone mineral content

A porous composite model is developed to analyze the tensile mechanical properties of cortical bone. The effects of microporosity (volksman's canals, osteocyte lacunae) on the mechanical properties of bone tissue are taken into account. A simple shear lag theory, wherein tensile loads are transferred between overlapped mineral platelets by shearing of the organic matrix, is used to model the reinforcement provided by mineral platelets. It is assumed that the organic matrix is elastic in tension and elastic-perfectly plastic in shear until it fails. When organic matrix shear stresses at the ends of mineral platelets reach their yield values, the stress-strain curve of bone tissue starts to deviate from linear behavior. This is referred as the microscopic yield point. At the point where the stress-strain behavior of bone shows a sharp curvature, the organic phase reaches its shear yield stress value over the entire platelet. This is referred as the macroscopic yield point. It is assumed that after macroscopic yield, mineral platelets cannot contribute to the load bearing capacity of bone and that the mechanical behavior of cortical bone tissue is determined by the organic phase only. Bone fails when the principal stress of the organic matrix is reached. By assuming that mechanical properties of the organic matrix are dependent on bone mineral content below the macroscopic yield point, the model is used to predict the entire tensile mechanical behavior of cortical bone for different mineral contents. It is found that decreased shear yield stresses and organic matrix elastic moduli are required to explain the mechanical behavior of bones with lowered mineral contents. Under these conditions, the predicted values (elastic modulus, 0.002 yield stress and strain, and ultimate stress and strain) are within 15% of experimental data.     

The mechanical behaviour of intracondylar cancellous bone of the femur at different loading rates

The compressive properties of human cancellous bone of the distal intracondylar femur in its wet condition were determined. Specimens were obtained from six cadaveric femora and were tested at a strain rate of 0.002, 0.10 and 9.16 sec⁻¹. It was found that the compressive strength decreases with an increasing vertical distance from the joint. The highest compressive strength level was recorded in the posterior medial condyle. Correlations among the mechanical properties, the bulk specimen density and the bone mineral content yield (i) highly significant correlations between the compressive strength and the elastic modulus (ii) highly significant correlations between the compressive strength or the modulus of elasticity and the bulk specimen density (iii) a doubtful correlation between the compressive strength and the bone mineral content. All recorded graphs of the impact loaded specimens displayed several well defined stress peaks, unlike the graphs recorded at low loading rates. It can be concluded that upon impact loading the localized trabecular failure which is associated with each peak, does not affect the spongy bone's stress capacity in a detrimental way.     

Are plasticity models required to predict relative risk of lag screw cut-out in finite element models of trochanteric fracture fixation?

Using a finite element model of unstable trochanteric fracture stabilized with a sliding hip screw, the benefits of two plasticity-based formulations, Drucker–Prager and crushable foam, were evaluated and compared to the commonly used linear elastic model of trabecular bone in order to predict the relative risk of lag screw cut-out for five distinct load cases. The crushable foam plasticity formulation leads to a much greater strain localization, in comparison to the other two models, with large plastic strains in a localized region. The plastic zone predicted with Drucker–Prager is relatively more diffuse. Linear elasticity associated with a minimum principal strain criterion provides the smallest volume of elements susceptible to yielding for all loading modes. The region likely to undergo plastic deformation, as predicted by the linear elastic model, is similar to that obtained from plasticity-based formulations, which indicates that this simple criterion provides an adequate estimate of the risk of cut-out.     

Elastic moduli, yield stress, and ultimate stress of cancellous bone in the canine proximal femur

Elastic moduli, yield stress and ultimate compressive stress were determined for cancellous bone from the femoral head and neck regions of the canine femur. Unconfined compression tests were performed on 5 mm cubic samples which were cut from two femurs. Elastic moduli were measured in three orthogonal directions, and the yield stress and ultimate stress were measured along the proximal-distal axis. The results from this investigation support previous assumptions that the mechanical behavior of canine cancellous bone is qualitatively similar to human cancellous bone. The canine cancellous bone was observed to be anisotropic in elastic modulus. For two thirds of the cubic specimens tested, the elastic modulus was largest in the load-bearing, proximal-distal direction. A linear relationship between yield stress and elastic modulus was observed for canine bone, as is typical of human bone. A similar linear relationship between ultimate stress and elastic modulus was observed. Thus, for canine bone as well as for human bone, failure appears to be governed by a strain level which is position independent. The yield strain of 0.0259 and ultimate strain of 0.0288 for canine bone were both less than the yield strain of 0.0395 reported for human bone.     

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.     

Self-healing fish gelatin/sodium montmorillonite biohybrid coacervates: structural and rheological characterization

Complex coacervation driven by associative electrostatic interactions was studied in mixtures of exfoliated sodium-montmorillonite (Na(+)-MMT) nanoplatelets and fish gelatin, at a specific mixing ratio and room temperature. Structural and viscoelastic properties of the coacervate phase were investigated as a function of pH by means of different complementary techniques. Independent of the technique used, the results consistently showed that there is an optimum pH value at which the coacervate phase shows the tightest structure with highest elasticity. The solid-like coacervates showed an obvious shear-thinning behavior and network fracture but immediately recovered back into their original elastic character upon removal of the shear strain. The nonlinear mechanical response characterized by single step stress relaxation experiments revealed the same trend for the yield stress and isochronal shear modulus of the coacervates as a function of pH with a maximum at pH 3.0 and lower values at 2.5 and 3.5 pHs, followed by a very sharp drop at pH 4.0. Finally, small-angle X-ray scattering (SAXS) data confirmed that at pHs lower than 4.0 the coacervate phases were dense and structured with a characteristic length scale (ξ(SAXS)) of ~7-9 nm. Comparing the ξ(SAXS) with rheological characteristic length (ξ(rheol)) estimated from low-frequency linear viscoelastic data and network theory, it was concluded that both the strength of the electrostatic interactions and the conformation of the gelatin chains before and during of the coacervation process are responsible for the structure and rigidity of the coacervates.