Inelastic strain accumulation in cortical bone during rapid transient tensile loading

An experimental study examined the tensile stress-strain behavior of cortical bone during rapid load cycles to high strain amplitudes. Machined bovine and human cortical bone samples were subjected to loading cycles at a nominal load/unload rate of +/- 420 MPa/s. Loads were reversed at pre selected strain levels such that load cycles were typically completed in 0.5-0.7 seconds. Axial strain behavior demonstrated considerable nonlinearity in the first load cycle, while transverse strain behavior was essentially linear. For the human bone 29.1 percent (S.D. = 4.7 percent), and for the bovine bone 35.1 percent (S.D. = 10.8 percent) of the maximum nonlinear strain accumulated after load reversal, where nonlinear strain was defined as the difference between total strain and strain corresponding to linear elastic behavior. Average residual axial strain on unloading was 35.4 percent (S.D. = 1.2 percent) for human bone and 35.1 percent (S.D. = 2.9 percent) of maximum nonlinear strain. Corresponding significant volumetric strains and residual volumetric strains were found. The results support the conclusions that the nonlinear stress-strain behavior observed during creep loading also occurs during transient loading at physiological rates. The volume increases suggest that damage accumulation, i.e., new internal surfaces and voids, plays a major role in this behavior. The residual volume increases and associated disruptions in the internal structure of bone provide a potential stimulus for a biological repair response.     

Osteocyte lacunae tissue strain in cortical bone

Current theories suggest that bone modeling and remodeling are controlled at the cellular level through signals mediated by osteocytes. However, the specific signals to which bone cells respond are still unknown. Two primary theories are: (1) osteocytes are stimulated via the mechanical deformation of the perilacunar bone matrix and (2) osteocytes are stimulated via fluid flow generated shear stresses acting on osteocyte cell processes within canaliculi. Recently, much focus has been placed on fluid flow theories since in vitro experiments have shown that bone cells are more responsive to analytically estimated levels of fluid shear stress than to direct mechanical stretching using macroscopic strain levels measured on bone in vivo. However, due to the complex microstructural organization of bone, local perilacunar bone tissue strains potentially acting on osteocytes cannot be reliably estimated from macroscopic bone strain measurements. Thus, the objective of this study was to quantify local perilacunar bone matrix strains due to macroscopically applied bone strains similar in magnitude to those that occur in vivo. Using a digital image correlation strain measurement technique, experimentally measured bone matrix strains around osteocyte lacunae resulting from macroscopic strains of approximately 2000 microstrain are significantly greater than macroscopic strain on average and can reach peak levels of over 30,000 microstrain locally. Average strain concentration factors ranged from 1.1 to 3.8, which is consistent with analytical and numerical estimates. This information should lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Analysis of creep strain during tensile fatigue of cortical bone

During fatigue tests of cortical bone specimens, at the unload portion of the cycle (zero stress) non-zero strains occur and progressively accumulate as the test progresses. This non-zero strain is hypothesised to be mostly, if not entirely, describable as creep. This work examines the rate of accumulation of this strain and quantifies its stress dependency. A published relationship determined from creep tests of cortical bone (Journal of Biomechanics 21 (1988) 623) is combined with knowledge of the stress history during fatigue testing to derive an expression for the amount of creep strain in fatigue tests. Fatigue tests on 31 bone samples from four individuals showed strong correlations between creep strain rate and both stress and "normalised stress" (sigma/E) during tensile fatigue testing (0-T). Combined results were good (r(2)=0.78) and differences between the various individuals, in particular, vanished when effects were examined against normalised stress values. Constants of the regression showed equivalence to constants derived in creep tests. The universality of the results, with respect to four different individuals of both sexes, shows great promise for use in computational models of fatigue in bone structures.     

Understanding site-specific residual strain and architecture in bovine cortical bone

Living bone is considered as adaptive material to the mechanical functions, which continually undergoes change in its histological arrangement with respect to external prolonged loading. Such remodeling phenomena within bone depend on the degree of stimuli caused by the mechanical loading being experienced, and therefore, are specific to the sites. In the attempts of understanding strain adaptive phenomena within bones, different theoretical models have been proposed. Also, the existing literatures mostly follow the measurement of surface strains using strain gauges to experimentally quantify the strains experienced in the functional environment. In this work, we propose a novel idea of understanding site-specific functional adaptation to the prolonged load in bone on the basis of inherited residual strains and structural organization. We quantified the residual strains and amount of apatite crystals distribution, i.e., the degree of orientation, using X-ray diffraction procedures. The sites of naturally existing hole in bone, called foramen, are considered from bovine femur and metacarpal samples. Significant values of residual strains are found to exist in the specimens. Trends of residual strains noted in the specimens are mostly consistent with the degree of orientation of the crystallites. These features explain the response behavior of bone to the mechanical loading history near the foramen sites. Preferential orientation of crystals mapped around a femoral foramen specimen showed furnished tailored arrangement of the crystals around the hole. Effect of external loading at the femoral foramen site is also explained by the tensile loading experiment.     

Computational homogenisation based extraction of transverse tensile cohesive responses of cortical bone tissue

Biomechanics and Modeling in Mechanobiology - The numerical assessment of fracture properties of cortical bone is important in providing suggestions on patient-specific clinical treatments. We...     

Microcracking damage and the fracture process in relation to strain rate in human cortical bone tensile failure

It is difficult to define the 'physiological' mechanical properties of bone. Traumatic failures in-vivo are more likely to be orders of magnitude faster than the quasistatic tests usually employed in-vitro. We have reported recently [Hansen, U., Zioupos, P., Simpson, R., Currey, J.D., Hynd, D., 2008. The effect of strain rate on the mechanical properties of human cortical bone. Journal of Biomechanical Engineering/Transactions of the ASME 130, 011011-1-8] results from tests on specimens of human femoral cortical bone loaded in tension at strain rates (epsilon ) ranging from low (0.08s(-1)) to high (18s(-1)). Across this strain rate range the modulus of elasticity generally increased, stress at yield and failure and strain at failure decreased for rates higher than 1s(-1), while strain at yield was invariant for most strain rates and only decreased at rates higher than 10s(-1). The results showed that strain rate has a stronger effect on post-yield deformation than on initiation of macroscopic yielding. In general, specimens loaded at high strain rates were brittle, while those loaded at low strain rates were much tougher. Here, a post-test examination of the microcracking damage reveals that microcracking was inversely related to the strain rate. Specimens loaded at low strain rates showed considerable post-yield strain and also much more microcracking. Partial correlation and regression analysis suggested that the development of post-yield strain was a function of the amount of microcracking incurred (the cause), rather than being a direct result of the strain rate (the excitation). Presumably low strain rates allow time for microcracking to develop, which increases the compliance of the specimen, making them tougher. This behaviour confirms a more general rule that the degree to which bone is brittle or tough depends on the amount of microcracking damage it is able to sustain. More importantly, the key to bone toughness is its ability to avoid a ductile-to-brittle transition for as long as possible during the deformation. The key to bone's brittleness, on the other hand, is the strain and damage localisation early on in the process, which leads to low post-yield strains and low-energy absorption to failure.     

Measurement of microstructural strain in cortical bone

It is well known that mechanical factors affect bone remodeling such that increased mechanical demand results in net bone formation, whereas decreased demand results in net bone resorption. Current theories suggest that bone modeling and remodeling is controlled at the cellular level through signals mediated by osteocytes. The objective of this study was to investigate how macroscopically applied bone strains similar in magnitude to those that occur in vivo are manifest at the microscopic level in the bone matrix. Using a digital image correlation strain measurement technique, experimentally determined bone matrix strains around osteocyte lacuna resulting from macroscopic strains of approximately 2,000 microstrain (0.2%) reach levels of over 30,000 microstrain (3%) over fifteen times greater than the applied macroscopic strain. Strain patterns were highly heterogeneous and in some locations similar to observed microdamage around osteocyte lacuna indicating the resulting strains may represent the precursors to microdamage. This information may lead to a better understanding of how bone cells are affected by whole bone functional loading.     

Viscoelastic behaviour and failure of bovine cancellous bone under constant strain rate

A programme has been established to characterize the long-term behaviour of cancellous bone. Fresh bovine cancellous specimens of dimensions 10 x 10 x 10 mm3 and 10 x 40 x 3.6 mm3 were manufactured and used within the testing programme. Results published in the literature indicate that the long-term behaviour of cancellous bone is well described by a power law, which is a very similar response of typical polymers. So far, dynamic mechanical tests (DMA) in three-point bending, under frequencies between 0.01 and 100 Hz at room temperature, confirmed the published results in a qualitative way. Nevertheless, the measured dimensionless damping, tan delta, was slightly higher than the values reported in the literature for the compact bone. The relaxation curves were obtained from dynamic tests and confirmed that bone relaxation modulus can be described by a power law function of time. Tests under constant compression strain rate were performed at four different strain rates: 0.15/s, 0.015/s, 0.0015/s and 0.00015/s and strain rate dependent behaviour was observed. An average elastic bending modulus of 300 MPa was obtained.     

Relationship between streaming potential and compressive stress in bovine intervertebral tissue

The intervertebral disc is formed by the nucleus pulposus (NP) and annulus fibrosus (AF), and intervertebral tissue contains a large amount of negatively charged proteoglycan. When this tissue becomes deformed, a streaming potential is induced by liquid flow with positive ions. The anisotropic property of the AF tissue is caused by the structural anisotropy of the solid phase and the liquid phase flowing into the tissue with the streaming potential. This study investigated the relationship between the streaming potential and applied stress in bovine intervertebral tissue while focusing on the anisotropy and loading location. Column-shaped specimens, 5.5 mm in diameter and 3 mm thick, were prepared from the tissue of the AF, NP and the annulus–nucleus transition region (AN). The loading direction of each specimen was oriented in the spinal axial direction, as well as in the circumferential and radial directions of the spine considering the anisotropic properties of the AF tissue. The streaming potential changed linearly with stress in all specimens. The linear coefficients k_e of the relationship between stress and streaming potential depended on the extracted positions. These coefficients were not affected by the anisotropy of the AF tissue. In addition, these coefficients were lower in AF than in NP specimens. Except in the NP specimen, the k_e values were higher under faster compression rate conditions. In cyclic compression loading the streaming potential changed linearly with compressive stress, regardless of differences in the tissue and load frequency.     

On shear properties of trabecular bone under torsional loading: effects of bone marrow and strain rate

To further improve our understanding of trabecular bone mechanical behavior in torsion, our objective was to determine the effects of strain rate, apparent density, and presence of bone marrow on trabecular bone shear material properties. Torsion tests of cylindrical trabecular bone specimens from sheep lumbar vertebrae with and without bone marrow were conducted. The bones with marrow were divided into two groups and tested at shear strain rates of 0.002 and 0.05s(-1) measured at the specimen perimeter. The bones without marrow were divided into three groups and tested at shear strain rates of 0.002, 0.015, and 0.05s(-1). Comparing the results of bones with and without marrow tested at low (0.002s(-1)) and high (0.05s(-1)) strain rates, presence of bone marrow did not have any significant effect on trabecular bone shear modulus and strength. In specimens without marrow, power relationships were used to define shear strength and modulus as dependent variables in terms of strain rate and apparent density as independent variables. The shear strength was proportional to the apparent density raised to the 1.02 power and to the strain rate raised to the 0.13 power. The shear modulus was proportional to the apparent density raised to the 1.08 power and to the strain rate raised to the 0.07 power. This study provides further insight into the mechanism of bone failure in trauma as well as failure at the interface between bone and implants as it relates to prediction of trabecular bone shear properties.     

Damage rate is a predictor of fatigue life and creep strain rate in tensile fatigue of human cortical bone samples

We present results on the growth of damage in 29 fatigue tests of human femoral cortical bone from four individuals, aged 53-79. In these tests we examine the interdependency of stress, cycles to failure, rate of creep strain, and rate of modulus loss. The behavior of creep rates has been reported recently for the same donors as an effect of stress and cycles. In the present paper we first examine how the evolution of damage (drop in modulus per cycle) is associated with the stress level or the "normalized stress" level (stress divided by specimen modulus), and results show the rate of modulus loss fits better as a function of normalized stress. However, we find here that even better correlations can be established between either the cycles to failure or creep rates versus rates of damage than any of these three measures versus normalized stress. The data indicate that damage rates can be excellent predictors of fatigue life and creep strain rates in tensile fatigue of human cortical bone for use in practical problems and computer simulations.     

Two-dimensional strain fields on the cross-section of the bovine humeral head under contact loading

The objective of this study was to provide a detailed experimental assessment of the two-dimensional cartilage strain distribution on the cross-section of immature and mature bovine humeral heads subjected to contact loading at a relatively rapid physiological loading rate. Six immature and six mature humeral head specimens were loaded against glass and strains were measured at the end of a 5s loading ramp on the textured articular cross-section using digital image correlation analysis. The primary findings indicate that elevated tensile and compressive strains occur near the articular surface, around the center of the contact region. Few qualitative or quantitative differences were observed between mature and immature joints. Under an average contact stress of approximately 1.7 MPa, the peak compressive strains averaged -0.131+/-0.048, which was significantly less than the relative change in cartilage thickness, -0.104+/-0.032 (p<0.05). The peak tensile strains were significantly smaller in magnitude, at 0.0325+/-0.013. These experimental findings differ from a previous finite element analysis of articular contact, which predicted peak strains at the cartilage-bone interface even when accounting for the porous-hydrated nature of the tissue, its depth-dependent inhomogeneity, and the disparity between its tensile and compressive properties. These experimental results yield new insights into the local mechanical environment of the tissue and cells, and suggest that further refinements are needed in the modeling of contacting articular layers.     

Dependence of ultrasonic attenuation on bone mass and microstructure in bovine cortical bone

The development of the axial transmission technique now enables in vivo evaluation of cortical bone quality, which plays an important role in bone fragility. Cortical bone is a complex multiscale material, which may be made of different types of microstructure. The interaction between ultrasound and cortical bone remains unclear and most studies have been confined to wave speed analysis. The first aim of this study is to investigate the dependence of the frequency-dependent attenuation on the type of bone microstructure. The second goal is to determine whether broadband ultrasonic attenuation (BUA) is related to volumetric bone mineral density (vBMD) and mass density. Parallelepipedic samples of bovine cortical bone were cut from three specimens and tested in the axial, radial and tangential directions using an ultrasonic transmission device. BUA was evaluated over a 1-MHz wide bandwidth around 4MHz. In addition, the microstructure of each sample was determined using an optical microscope. BUA values measured in porotic microstructure are significantly higher than in Haversian microstructure. The lowest BUA values are obtained for plexiform microstructure. For all structures, BUA in the axial direction is significantly smaller than in the radial and tangential directions. Moreover, BUA is correlated with both vBMD and density (determination coefficient (R2) equal to 0.44 and 0.65, respectively, in the axial direction). BUA variations can be explained by scattering and viscoelastic mechanisms. This study suggests that BUA measurements have the potential to discriminate among different cortical bone microstructures in addition to providing material properties.     

Some viscoplastic characteristics of bovine and human cortical bone

Multiple cycle tensile creep tests were performed on human and bovine cortical bone specimens. The tests enabled total strain to be decomposed into elastic, linear viscoelastic, creep and permanent plastic components. The results indicate that a stress threshold exists; above which time dependent effects dominate material response and below which the behavior is primarily linear viscoelastic, with time effects playing only a secondary role. A constant stress above the threshold produces a constant steady state creep rate, with the magnitude of the creep rate being an exponential function of the stress magnitude. Additionally, it was found that a major portion of the inelastic strain is always recovered on unloading and that the accumulation of creep strain increases the material compliance on subsequent loadings below the threshold. These two factors suggest that a damage mechanism is responsible for the nonlinear behavior.     

Reorientation of cellulose nanowhiskers in agarose hydrogels under tensile loading

Agarose hydrogels filled with cellulose nanowhiskers were strained in uniaxial stretching under different humidity conditions. The orientation of the cellulose whiskers was examined before and after testing with an X-ray laboratory source and monitored in situ during loading by synchrotron X-ray diffraction. The aim of this approach was to determine the process parameters for reorienting the cellulose nanowhiskers toward a preferential direction. Results show that a controlled drying of the hydrogel is essential to establish interactions between the matrix and the cellulose nanowhiskers which allow for a stress transfer during stretching and thereby promote their alignment. Rewetting of the sample after reorientation of the cellulose nanowhiskers circumvents a critical increase of stress. This improves the extensibility of the hydrogel and is accompanied by a further moderate alignment of the cellulose nanowhiskers. Following this protocol, cellulose nanowhiskers with an initial random distribution can be reoriented toward a preferential direction, creating anisotropic nanocomposites.     

Experimental and finite element analysis of the rat ulnar loading model-correlations between strain and bone formation following fatigue loading

The rat forelimb compression model has been used widely to study bone response to mechanical loading. We used strain gages to assess load sharing between the ulna and radius in the forelimb of adult Fisher rats. We used histology and peripheral quantitative computed tomography (pQCT) to quantify ulnar bone formation 12 days after in vivo fatigue loading. Lastly, we developed a finite element model of the ulna to predict the pattern of surface strains during compression. Our findings indicate that at the mid-shaft the ulna carries 65% of the applied compressive force on the forelimb. We observed large variations in fatigue-induced bone formation over the circumference and length of the ulna. Bone formation was greatest 1-2 mm distal to the mid-shaft. At the mid-shaft, we observed woven bone formation that was greatest medially. Finite element analysis indicated a strain pattern consistent with a compression-bending loading mode, with the greatest strains occurring in compression on the medial surface and lesser tensile strains occurring laterally. A peak strain of -5190 microepsilon (for 13.3N forelimb compression) occurred 1-2 mm distal to the mid-shaft. The pattern of bone formation in the longitudinal direction was highly correlated to the predicted peak compressive axial strains at seven cross-sections (r2 = 0.89, p = 0.014). The in-plane pattern of bone formation was poorly correlated to the predicted magnitude of axial strain at 51 periosteal locations (r2 = 0.21, p < 0.001), because the least bone formation was observed where tensile strains were highest. These findings indicate that the magnitude of bone formation after fatigue loading is greatest in regions of high compressive strain.     

Deformation behaviour and damage accumulation of cortical bone specimens from the equine tibia under cyclic loading

Despite its clinical importance, the fatigue behaviour of cortical bone has not been examined as widely as its static behaviour. In the present study, specimens from the tibiae of horses have been subjected to load-controlled single step tests. The cyclic deformation behaviour was described by the development of stress-strain hysteresis parameters over the lifetime. The fatigue behaviour of bone is characterised by cyclic softening which is most distinctive towards the end of the lifetime. The microstructural damage accumulated during cyclic loading results in a loss of stiffness, asymmetrical deformation of the bone in tension and compression in cyclic creep. As shown by light and scanning electron microscopy, microcrack formation and growth is the main damage mechanism. The crack growth behaviour is strongly influenced by the microstructure, the stress components and the absolute value of the local stresses. Lower local stresses and/or compressive mean stresses lead to a dominant influence of the shear stress components with shear failure at inner interfaces. With increasing crack length, that is, higher local stress amplitudes, or tensile mean stresses, the microstructure is more and more ignored and failure occurs primarily under the influence of the normal stress components. This can be clearly seen on the fracture and specimen surfaces.