Anisotropic constitutive equations and experimental tensile behavior of brain tissue

The present study deals with the experimental analysis and mechanical modeling of tensile behavior of brain soft tissue. A transversely isotropic hyperelastic model recently proposed by Meaney (2003) is adopted and mathematically studied under uniaxial loading conditions. Material parameter estimates are obtained through tensile tests on porcine brain materials accounting for regional and directional differences. Attention is focused on the short-term response. An extrapolation of tensile test data to the compression range is performed theoretically, to study the effect of the heterogeneity in the tensile/compressive response on the material parameters. Experimental and numerical results highlight the sensitivity of the adopted model to the test direction.     

An experimental and numerical study on the transverse deformations in tensile test of tendons

The transverse deformations of tendons assessed in tensile tests seems to constitute a controversial issue in literature. On the one hand, large positive variations of the Poisson’s ratio have been reported, indicating volume reduction under tensile states. On the other hand, negative values were also observed, pointing out an auxetic material response. Based on these experimental observations, the following question is raised: Are these large and discrepant transverse deformations intrinsically related to the constitutive response of tendons or they result from artifacts of the mechanical test setup? In order to provide further insights to this question, an experimental and numerical study on the transverse kinematics of tendons was carried out. Tensile experiments were performed in branches of deep digital flexor tendons of domestic porcine, where the transverse displacements were measured by two high-speed, high-accuracy optical digital micrometers placed transversely to one another. Aiming at a better understanding of the effects of the mechanical test setup in the transverse measurements, a three-dimensional finite element model is proposed to resemble the tensile experiment. The main achieved results strongly support the following hypotheses regarding tensile tests of tendons: the clamping region considerably affects the kinematics of the specimen even at a large distance from the clamps; the transverse deformations are mainly ruled by stiff fibers embedded in a soft matrix; the generalization of the Poisson’s ratio to draw conclusions about changes in volume of tendons may lead to misinterpretations.     

Anisotropy of fibrous tissues in relation to the distribution of tensed and buckled fibers

Fibrous tissues are characterized by a much higher stiffness in tension than compression. This study uses microstructural modeling to analyze the material symmetry of fibrous tissues undergoing tension and compression, to better understand how material symmetry relates to the distribution of tensed and buckled fibers. The analysis is also used to determine whether the behavior predicted from a microstructural model can be identically described by phenomenological continuum models. The analysis confirms that in the case when all the fibers are in tension in the current configuration, the material symmetry of a fibrous tissue in the corresponding reference configuration is dictated by the symmetry of its fiber angular distribution in that configuration. However, if the strain field exhibits a mix of tensile and compressive principal normal strains, the fibrous tissue is represented by a material body which consists only of those fibers which are in tension; the material symmetry of this body may be deduced from the superposition of the planes of symmetry of the strain and the planes of symmetry of the angular fiber distribution. Thus the material symmetry is dictated by the symmetry of the angular distribution of only those fibers which are in tension. Examples are provided for various fiber angular distribution symmetries. In particular, it is found that a fibrous tissue with isotropic fiber angular distribution exhibits orthotropic symmetry when subjected to a mix of tensile and compressive principal normal strains, with the planes of symmetry normal to the principal directions of the strain. This anisotropy occurs even under infinitesimal strains and is distinct from the anisotropy induced from the finite rotation of fibers. It is also noted that fibrous materials are not stable under all strain states due to the inability of fibers to sustain compression along their axis; this instability can be overcome by the incorporation of a ground matrix. It is concluded that the material response predicted using a microstructural model of the fibers cannot be described exactly by phenomenological continuum models. These results are also applicable to nonbiological fiber-composite materials.     

Continuum description of the Poisson׳s ratio of ligament and tendon under finite deformation

Ligaments and tendons undergo volume loss when stretched along the primary fiber axis, which is evident by the large, strain-dependent Poisson?s ratios measured during quasi-static tensile tests. Continuum constitutive models that have been used to describe ligament material behavior generally assume incompressibility, which does not reflect the volumetric material behavior seen experimentally. We developed a strain energy equation that describes large, strain dependent Poisson?s ratios and nonlinear, transversely isotropic behavior using a novel method to numerically enforce the desired volumetric behavior. The Cauchy stress and spatial elasticity tensors for this strain energy equation were derived and implemented in the FEBio finite element software (www.febio.org). As part of this objective, we derived the Cauchy stress and spatial elasticity tensors for a compressible transversely isotropic material, which to our knowledge have not appeared previously in the literature. Elastic simulations demonstrated that the model predicted the nonlinear, upwardly concave uniaxial stress–strain behavior while also predicting a strain-dependent Poisson?s ratio. Biphasic simulations of stress relaxation predicted a large outward fluid flux and substantial relaxation of the peak stress. Thus, the results of this study demonstrate that the viscoelastic behavior of ligaments and tendons can be predicted by modeling fluid movement when combined with a large Poisson?s ratio. Further, the constitutive framework provides the means for accurate simulations of ligament volumetric material behavior without the need to resort to micromechanical or homogenization methods, thus facilitating its use in large scale, whole joint models.     

Modelling and simulation of porcine liver tissue indentation using finite element method and uniaxial stress–strain data

We hypothesize that both compression and elongation stress–strain data should be considered for modeling and simulation of soft tissue indentation. Uniaxial stress–strain data were obtained from in vitro loading experiments of porcine liver tissue. An axisymmetric finite element model was used to simulate liver tissue indentation with tissue material represented by hyperelastic models. The material parameters were derived from uniaxial stress–strain data of compressions, elongations, and combined compression and elongation of porcine liver samples. in vitro indentation tests were used to validate the finite element simulation. Stress–strain data from the simulation with material parameters derived from the combined compression and elongation data match the experimental data best. This is due to its better ability in modeling 3D deformation since the behavior of biological soft tissue under indentation is affected by both its compressive and tensile characteristics. The combined logarithmic and polynomial model is somewhat better than the 5-constant Mooney–Rivlin model as the constitutive model for this indentation simulation.     

Finite element implementation of a new model of slight compressibility for transversely isotropic materials

Modelling transversely isotropic materials in finite strain problems is a complex task in biomechanics, and is usually addressed by using finite element (FE) simulations. The standard method developed to account for the quasi-incompressible nature of soft tissues is to decompose the strain energy function (SEF) into volumetric and deviatoric parts. However, this decomposition is only valid for fully incompressible materials, and its use for slightly compressible materials yields an unphysical response during the simulation of hydrostatic tension/compression of a transversely isotropic material. This paper presents the FE implementation as subroutines of a new volumetric model solving this deficiency in two FE codes: Abaqus and FEBio. This model also has the specificity of restoring the compatibility with small strain theory. The stress and elasticity tensors are first derived for a general SEF. This is followed by a successful convergence check using a particular SEF and a suite of single-element tests showing that this new model does not only correct the hydrostatic deficiency but may also affect stresses during shear tests (Poynting effect) and lateral stretches during uniaxial tests (Poisson's effect). These FE subroutines have numerous applications including the modelling of tendons, ligaments, heart tissue, etc. The biomechanics community should be aware of specificities of the standard model, and the new model should be used when accurate FE results are desired in the case of compressible materials.     

Nonlinear analysis of intervertebral disk under dynamic load

This study pertains to the response of intervertebral joint under dynamic axial load. The numerical model represents two vertebral bodies with an interposed disk and uses three-dimensional elements. A transversely isotropic material law is adopted for cortical bone and an isotropic law for cancellous bone. Annulus collagen fibers are modelled using truss elements with no compressive resistance. The disk material is assumed hyperelastic, using a mixed finite element approach, allowing a representation of the disk involving the incompressibility characteristics for the material. The analysis considers finite displacement and strain fields under dynamic load. Intensity, trend and distribution of loads on the vertebral body are deduced from the literature. The problem is investigated with reference to different compressibility levels of disk material related to disk degenerationn phenomena.     

Time and depth dependent poisson’s ratio of cartilage explained by an inhomogeneous orthotropic fiber embedded biphasic model

A time- and depth-dependent Poisson’s ratio has been observed during unconfined compression experiments on articular cartilage, but existing cartilage models have not fully addressed these phenomena. The goal of this study was to develop a model which is able to predict and explain these phenomena, while also being able to fit other experimental scenarios on full depth cartilage specimens such as confined and unconfined compressions. A biphasic (poroelastic), fiber-embedded cartilage model was developed. The heterogeneous material properties of the cartilage (aggregate modulus, void ratio tensile modulus) were extracted from reported experiments on individual layers of bovine articular cartilage. The nonlinear permeability material constants were found by fitting the overall response to published experimental data from confined compression. The matrix of the cartilage was modelled as an inhomogeneous isotropic biphasic material with nonlinear strain dependent permeability. Orthotropic layers were added as embedded elements to represent collagen fibers. Material parameters for these layers were derived from tensile tests of different layers of cartilage. With these predefined tensile parameters, the model showed a good fit with multi-step confined and unconfined compression experiments (R²=0.984 and 0.977, respectively) and could also predict the depth-dependent Poisson’s ratio (R²=0.981). The highlight of the model is the ability to explain the time-depth dependent Poisson's ratio and, by association, the strong effect of material inhomogeneity on local stress and strain patterns within the cartilage layer. This material model’s response may provide valuable new insight into potential initiation of cartilage fibrillation or delamination in whole-joint simulations.     

Material characterization of the pig kidney in relation with the biomechanical analysis of renal trauma

The objective of this study was an investigation of the material properties of the fresh pig kidney and parametric characterization of its elastic and inelastic material behavior. The material investigation included density measurements, uniaxial as well as three-dimensional compression tests, tensile tests. and shear tests on the samples extracted from the fresh pig kidney. For comparison, density measurements on a number of soft synthetic materials were also performed. Compression tests on the radial and the tangential specimens from the cortex tissue were performed at various loading rates. Three-axial compression tests were performed on the cortex tissues placed in a compression chamber. Shear tests were performed by punching a cylinder into a slice of the cortex. Tensile tests were carried out on the outer capsule. For characterization of the material behavior, a non-linear theoretical simulation based on a two parameter Blatz model was used. For characterization of the time-dependent behavior of the pig kidney cortex, a four-parameter linear viscoelastic model was employed. From the present experimental and theoretical studies, a number of conclusions were derived: (1) The general behavior of the pig kidney cortex samples under compression showed the general non-linear features typical of the soft tissues; the stress strain diagram was composed of a very flat part at very low stress level to about 30% relative deformation which was followed by a steeply rising stiffening leading to the radial rupture of samples marked by a maximum nominal rupture strain of about 50%. (2) The uniaxial compression tests on the radial and the tangential samples from the cortex tissue showed an increase of the rupture stress with the increase in the loading rate, but a decrease in the related rupture strain. (3) The long-term uniaxial compression tests on the cortex specimens under sustained constant load showed an instantaneous deformation followed by a creep response which eventually approached an asymptote. (4) Simulation of the non-linear material behavior of the cortex tissue under uniaxial compression by the Blatz model gave two pairs of material parameters for the cortex in the radial and the tangential directions. Furthermore, fitting of the assumed four-parameter linear viscoelastic model with the experimental data resulted in the viscoelastic material parameters.     

The effect of a lathyritic diet on the sensitivity of tendon to strain rate

While the tensile failure properties of rat-tail tendon depend on strain rate, the sensitivity to strain rate decreases with age, especially during sexual maturation. The object of this study was to determine the effect of an experimental model of chronic lathyrism on age-dependent changes in the sensitivity of developing tendon strength to strain rate. Tensile failure experiments were conducted at high and low strain rate on tendons excised from test and control animals aged 1 to 6 mo. The tensile "yield" response of tendon was significantly affected by the diet resulting in a reduced tensile strength and failure strain. While the sensitivity of tendon failure to strain rate was slightly elevated by the experimental diet, age-dependent changes compared with controls. Since the diet supplement is thought to inhibit covalent crosslinking of collagen in the developing tendon, other factors are likely responsible for decrease in the sensitivity of tendon strength to strain rate during maturation.     

The role of tendon elasticity in hopping in a wallaby (Macropus rufogriseus)

The elastic properties of the major tendons of wallaby feet have been investigated in dynamic tensile tests at a frequency close to normal hopping frequencies. In addition, bending tests have been performed on the intertarsal and tarso-metatarsal joints. The data are used to calculate the strain energy stored temporarily in the tendons and ligaments of the legs during a typical hopping stride. The contribution made by stored strain energy to the work required for hopping is assessed.     

Dynamic response of immature bovine articular cartilage in tension and compression, and nonlinear viscoelastic modeling of the tensile response

Very limited information is currently available on the constitutive modeling of the tensile response of articular cartilage and its dynamic modulus at various loading frequencies. The objectives of this study were to (1) formulate and experimentally validate a constitutive model for the intrinsic viscoelasticity of cartilage in tension, (2) confirm the hypothesis that energy dissipation in tension is less than in compression at various loading frequencies, and (3) test the hypothesis that the dynamic modulus of cartilage in unconfined compression is dependent upon the dynamic tensile modulus. Experiment 1: Immature bovine articular cartilage samples were tested in tensile stress relaxation and cyclical loading. A proposed reduced relaxation function was fitted to the stress-relaxation response and the resulting material coefficients were used to predict the response to cyclical loading. Adjoining tissue samples were tested in unconfined compression stress relaxation and cyclical loading. Experiment 2: Tensile stress relaxation experiments were performed at varying strains to explore the strain-dependence of the viscoelastic response. The proposed relaxation function successfully fit the experimental tensile stress-relaxation response, with R2 = 0.970+/-0.019 at 1% strain and R2 = 0.992+/-0.007 at 2% strain. The predicted cyclical response agreed well with experimental measurements, particularly for the dynamic modulus at various frequencies. The relaxation function, measured from 2% to 10% strain, was found to be strain dependent, indicating that cartilage is nonlinearly viscoelastic in tension. Under dynamic loading, the tensile modulus at 10 Hz was approximately 2.3 times the value of the equilibrium modulus. In contrast, the dynamic stiffening ratio in unconfined compression was approximately 24. The energy dissipation in tension was found to be significantly smaller than in compression (dynamic phase angle of 16.7+/-7.4 deg versus 53.5+/-12.8 deg at 10(-3) Hz). A very strong linear correlation was observed between the dynamic tensile and dynamic compressive moduli at various frequencies (R2 = 0.908+/-0.100). The tensile response of cartilage is nonlinearly viscoelastic, with the relaxation response varying with strain. A proposed constitutive relation for the tensile response was successfully validated. The frequency response of the tensile modulus of cartilage was reported for the first time. Results emphasize that fluid-flow dependent viscoelasticity dominates the compressive response of cartilage, whereas intrinsic solid matrix viscoelasticity dominates the tensile response. Yet the dynamic compressive modulus of cartilage is critically dependent upon elevated values of the dynamic tensile modulus.     

A transversely isotropic constitutive model of excised guinea pig spinal cord white matter

Narrowing of the spinal canal generates an amalgamation of stresses within the spinal cord parenchyma. The tissue’s stress state cannot be quantified experimentally; it must be described using computational methods, such as finite element analysis. The objective of this research was to propose a compressible, transversely isotropic constitutive model, an augmentation of the isotropic Mooney–Rivlin hyperelastic strain energy function, to describe the guinea pig spinal cord white matter. Model parameters were derived from a combination of inverse finite element analysis on transverse compression experiments and least squared error analysis applied to quasi-static longitudinal tensile tests. A comparison of the residual errors between the predicted response and the experimental measurements indicated that the transversely isotropic constitutive law that incorporates an offset stretch reduced the error by a factor of four when compared to other commonly used models.     

Ligament material behavior is nonlinear,viscoelastic and rate-independent under shear loading

The material behavior of ligament is determined by its constituents, their organization and their interaction with each other. To elucidate the origins of the multiaxial material behavior of ligaments, we investigated ligament response to shear loading under both quasi-static and rate-dependent loading conditions. Stress relaxation tests demonstrated that the tissue was highly viscoelastic in shear, with peak loads dropping over 40% during 30 min of stress relaxation. The stress relaxation response was unaffected by three decades of change in shear strain rate (1.3, 13 and 130%/s). A novel parameter estimation technique was developed to determine material coefficients that best described the experimental response of each test specimen to shear. The experimentally measured clamp displacements and reaction forces from the simple shear tests were used with a nonlinear optimization strategy based around function evaluations from a finite element program. A transversely isotropic material with an exponential matrix strain energy provided an excellent fit to experimental load-displacement curves. The shear modulus of human MCL showed a significant increase with increasing shear strain (p<0.001), reaching a maximum of 1.72+/-0.4871 MPa. The results obtained from this study suggest that viscoelasticity in shear does not likely result from fluid flow. Gradual loading of transversely oriented microstructural features such as intermolecular collagen crosslinks or collagen-proteoglycan crosslinking may be responsible for the stiffening response under shear loading.     

Mechanics of cranial sutures using the finite element method

To investigate how cranial suture morphology and the arrangement of sutural collagen fibres respond to compressive and tensile loads, an idealised bone–suture–bone complex was analysed using a two-dimensional finite element model. Three suture morphologies were simulated with an increasing interdigitation index (I.I.): butt-ended, moderate interdigitated, and complex interdigitated. The collagen matrix within all sutures was modelled as an isotropic material, and as an orthotropic material in the interdigitated sutures with fibre alignment as reported in studies of miniature pigs. Static uniform compressive or tensile loading was applied to the complex. In interdigitated sutures with isotropic material properties, the orientation of the maximum (tensile) principal stresses within the suture matched the collagen fibre orientation observed in compressed and tensed sutures of miniature pigs. This suggests that randomly arranged sutural collagen fibres could optimise to an orientation most appropriate to withstand the predominant type of loading. A compression-resistant fibre arrangement imparted the highest suture strain energy relative to the isotropic and tension-resistant arrangements, indicating that this configuration maximises energy storage. A comparison across the different suture morphologies indicated that bone strain energy generally decreased with a decrease in I.I., irrespective of the sutural fibre arrangement. However, high bone stress at the interdigitation apices shifted to the limbs of the suture with an increase in I.I. These combined findings highlight the importance of suture morphology and anisotropy as properties having a significant influence on sutural mechanics.     

A Coupled Experiment-finite Element Modeling Methodology for Assessing High Strain Rate Mechanical Response of Soft Biomaterials

This study offers a combined experimental and finite element (FE) simulation approach for examining the mechanical behavior of soft biomaterials (e.g. brain, liver, tendon, fat, etc.) when exposed to high strain rates. This study utilized a Split-Hopkinson Pressure Bar (SHPB) to generate strain rates of 100-1,500 sec^-1. The SHPB employed a striker bar consisting of a viscoelastic material (polycarbonate). A sample of the biomaterial was obtained shortly postmortem and prepared for SHPB testing. The specimen was interposed between the incident and transmitted bars, and the pneumatic components of the SHPB were activated to drive the striker bar toward the incident bar. The resulting impact generated a compressive stress wave (i.e. incident wave) that traveled through the incident bar. When the compressive stress wave reached the end of the incident bar, a portion continued forward through the sample and transmitted bar (i.e. transmitted wave) while another portion reversed through the incident bar as a tensile wave (i.e. reflected wave). These waves were measured using strain gages mounted on the incident and transmitted bars. The true stress-strain behavior of the sample was determined from equations based on wave propagation and dynamic force equilibrium. The experimental stress-strain response was three dimensional in nature because the specimen bulged. As such, the hydrostatic stress (first invariant) was used to generate the stress-strain response. In order to extract the uniaxial (one-dimensional) mechanical response of the tissue, an iterative coupled optimization was performed using experimental results and Finite Element Analysis (FEA), which contained an Internal State Variable (ISV) material model used for the tissue. The ISV material model used in the FE simulations of the experimental setup was iteratively calibrated (i.e. optimized) to the experimental data such that the experiment and FEA strain gage values and first invariant of stresses were in good agreement.     

Tensile properties of calcified and uncalcified avian tendons

The mechanical properties of turkey and heron leg tendons have been investigated in dynamic tensile tests. Heron tendons have properties similar to those found for various mammalian tendons. The Young's modulus and the density of turkey tendons increase with increasing calcification. Ultimate tensile stresses are similar to those found for uncalcified tendon, but Young's modulus may reach about 16 GPa, a value normally associated with bone. Calcification lowers the amount of strain energy that can be stored temporarily in the tendons of the legs. The contribution made by elastic strain energy storage to lowering the cost of running is reduced.     

Effects of stress shielding on the transverse mechanical properties of rabbit patellar tendons

With the aim of studying mechanisms of the remodeling of tendons and ligaments, the effects of stress shielding on the rabbit patellar tendon were studied by performing tensile and stress relaxation tests in the transverse direction. The tangent modulus, tensile strength, and strain at failure of non-treated, control patellar tendons in the transverse direction were 1272 kPa, 370 kPa, and 40.5 percent, respectively, whereas those of the tendons stress-shielded for 1 week were 299 kPa, 108 kPa, and 40.4 percent, respectively. Stress shielding markedly decreased tangent modulus and tensile strength in the transverse direction, and the decreases were larger than those in the longitudinal direction, which were determined in our previous study. For example, tensile strength in the transverse and longitudinal direction decreased to 29 and 50 percent of each control value, respectively, after 1 week stress shielding. In addition, the stress relaxation in the transverse direction of stress-shielded patellar tendons was much larger than that of nontreated, control ones. In contrast to longitudinal tensile tests for the behavior of collagen, transverse tests reflect the contributions of ground substances such as proteoglycans and mechanical interactions between collagen fibers. Ground substances provide lubrication and spacing between fibers, and also confer viscoelastic properties. Therefore, the results obtained from the present study suggest that ground substance matrix, and interfiber and fiber-matrix interactions have important roles in the remodeling response of tendons to stress.     

Mechanical characteristics of biological soft tissue]

In a detailed study mechanical properties of tendons, muscles, nerves, blood-vessels and skin of just slaughtered pigs have been investigated in nearly stationary stress tests. Tensile tests have produced tensile strength, ultimate stress and their appropriate strains, Young's modulus and the work up to fatigue of samples. In hysteresis tests the deformation work has been determined as a function of numbers of stress cycles. The hysteresis decrease with the number of stress cycles and approaches asymptotically to cero. By preconditioning of tendons, nerves and blood-vessels to steady state significant differences of strain at tensile strength and of Young's modulus have been established. Moreover for nerves the tests have revealed significant deviations of tensile strength. Bruise tests have been carried out with muscle tissue. For the described setup the limit force can be specified, at which pathological changes appear. Subsequently conducted histological investigations have demonstrated this. In dynamical bruise tests there appeared no pathological changes in muscle tissue in spite of higher transmitted energy.