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.     

Viscoelastic material model for the temporomandibular joint disc derived from dynamic shear tests or strain-relaxation tests

Viscoelastic material models for the temporomandibular joint disc, based upon strain relaxation, were considered to underestimate energy absorption for loads with time constants beyond the relaxation time. Therefore, the applicability of a material model that takes the viscous behavior at a wide range of frequencies into account was assessed. To that purpose a non-linear multi-mode Maxwell model was tested in cyclic large-strain compression tests. Its material constants were approximated from dynamic small-strain shear deformation tests. The storage and loss moduli as obtained from a disc sample could be approximated with a four-mode Maxwell model. In simulated large-strain compression tests it behaved similarly as observed from the experimental tests. The underestimation of energy dissipation, as obtained from a single-mode Maxwell model was considerably reduced, especially for deformations with a higher strain rate. Furthermore, in contrast to the latter it was able to predict the increase of the stress amplitude with the compression frequency much better. In conclusion, the applied four-mode Maxwell model, based upon dynamic shear tests, was considered more suitable to predict higher frequency viscoelastic response, for instance during shock absorption, than a model based upon strain-relaxation.     

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.     

Mechanical properties of anterior malleolar ligament from experimental measurement and material modeling analysis

In this paper, mechanical properties of the anterior malleolar ligament (AML) of human middle ear were studied through the uniaxial tensile, stress relaxation and failure tests. The digital image correlation (DIC) method was used to assess the boundary effect in experiments and calculate the strain on specimens. The constitutive behavior of the AML was described by a transversely isotropic hyperelastic model which consists of a first-order Ogden model augmented by a I (4)-type reinforcing term. The material parameters of the model were estimated and the viscoelasticity of the AML was illustrated by hysteresis phenomena and stress relaxation function. The mechanical strength of the AML was obtained through the failure test and the mean ultimate stress and stretch ratio were measured as 1.05 MPa and 1.51, respectively. Finally, a linear Young's modulus-stress relationship of the AML was derived based on constitutive equation of the AML within a stress range of 0-0.5 MPa.     

Design and numerical implementation of a 3-D non-linear viscoelastic constitutive model for brain tissue during impact

Finite Element (FE) head models are often used to understand mechanical response of the head and its contents during impact loading in the head. Current FE models do not account for non-linear viscoelastic material behavior of brain tissue. We developed a new non-linear viscoelastic material model for brain tissue and implemented it in an explicit FE code. To obtain sufficient numerical accuracy for modeling the nearly incompressible brain tissue, deviatoric and volumetric stress contributions are separated. Deviatoric stress is modeled in a non-linear viscoelastic differential form. Volumetric behavior is assumed linearly elastic. Linear viscoelastic material parameters were derived from published data on oscillatory experiments, and from ultrasonic experiments. Additionally, non-linear parameters were derived from stress relaxation (SR) experiments at shear strains up to 20%. The model was tested by simulating the transient phase in the SR experiments not used in parameter determination (strains up to 20%, strain rates up to 8s(-1)). Both time- and strain-dependent behavior were predicted accurately (R2>0.96) for strain and strain rates applied. However, the stress was overestimated systematically by approximately 31% independent of strain(rate) applied. This is probably caused by limitations of the experimental data at hand.     

Fully non-linear hyper-viscoelastic modeling of skeletal muscle in compression

Understanding the behavior of skeletal muscle is critical to implementing computational methods to study how the body responds to compressive loading. This work presents a novel approach to studying the fully nonlinear response of skeletal muscle in compression. Porcine muscle was compressed in both the longitudinal and transverse directions under five stress relaxation steps. Each step consisted of 5% engineering strain over 1 s followed by a relaxation period until equilibrium was reached at an observed change of 1 g/min. The resulting data were analyzed to identify the peak and equilibrium stresses as well as relaxation time for all samples. Additionally, a fully nonlinear strain energy density–based Prony series constitutive model was implemented and validated with independent constant rate compressive data. A nonlinear least squares optimization approach utilizing the Levenberg–Marquardt algorithm was implemented to fit model behavior to experimental data. The results suggested the time-dependent material response plays a key role in the anisotropy of skeletal muscle as increasing strain showed differences in peak stress and relaxation time (p < 0.05), but changes in equilibrium stress disappeared (p > 0.05). The optimizing procedure produced a single set of hyper-viscoelastic parameters which characterized compressive muscle behavior under stress relaxation conditions. The utilized constitutive model was the first orthotropic, fully nonlinear hyper-viscoelastic model of skeletal muscle in compression while maintaining agreement with constitutive physical boundaries. The model provided an excellent fit to experimental data and agreed well with the independent validation in the transverse direction.     

Confined and unconfined stress relaxation of cartilage: appropriateness of a transversely isotropic analysis.

Previous studies have shown that stress relaxation behavior of calf ulnar growth plate and chondroepiphysis cartilage can be described by a linear transverse isotropic biphasic model. The model provides a good fit to the observed unconfined compression transients when the out-of-plane Poisson's ratio is set to zero. This assumption is based on the observation that the equilibrium stress in the axial direction (deltaz) is the same in confined and unconfined compression, which implies that the radial stress deltar = 0 in confined compression. In our study, we further investigated the ability of the transversely isotropic model to describe confined and unconfined stress relaxation behavior of calf cartilage. A series of confined and unconfined stress relaxation tests were performed on calf articular cartilage (4.5 mm diameter, approximately 3.3 mm height) in a displacement-controlled compression apparatus capable of measuring delta(z) and delta(r). In equilibrium, delta(r) > 0 and delta(z) in confined compression was greater than in unconfined compression. Transient data at each strain were fitted by the linear transversely isotropic biphasic model and the material parameters were estimated. Although the model could provide good fits to the unconfined transients, the estimated parameters overpredicted the measured delta(r). Conversely, if the model was constrained to match equilibrium delta(r), the fits were poor. These findings suggest that the linear transversely isotropic biphasic model could not simultaneously describe the observed stress relaxation and equilibrium behavior of calf cartilage.     

Viscoelastic properties of bovine orbital connective tissue and fat: constitutive models

Reported mechanical properties of orbital connective tissue and fat have been too sparse to model strain–stress relationships underlying biomechanical interactions in strabismus. We performed rheological tests to develop a multi-mode upper convected Maxwell (UCM) model of these tissues under shear loading. From 20 fresh bovine orbits, 30 samples of connective tissue were taken from rectus pulley regions and 30 samples of fatty tissues from the posterior orbit. Additional samples were defatted to determine connective tissue weight proportion, which was verified histologically. Mechanical testing in shear employed a triborheometer to perform: strain sweeps at 0.5–2.0 Hz; shear stress relaxation with 1% strain; viscometry at 0.01−0.5 s⁻¹ strain rate; and shear oscillation at 1% strain. Average connective tissue weight proportion was 98% for predominantly connective tissue and 76% for fatty tissue. Connective tissue specimens reached a long-term relaxation modulus of 668 Pa after 1,500 s, while corresponding values for fatty tissue specimens were 290 Pa and 1,100 s. Shear stress magnitude for connective tissue exceeded that of fatty tissue by five-fold. Based on these data, we developed a multi-mode UCM model with variable viscosities and time constants, and a damped hyperelastic response that accurately described measured properties of both connective and fatty tissues. Model parameters differed significantly between the two tissues. Viscoelastic properties of predominantly connective orbital tissues under shear loading differ markedly from properties of orbital fat, but both are accurately reflected using UCM models. These viscoelastic models will facilitate realistic global modeling of EOM behavior in binocular alignment and strabismus.     

Simple shear testing of parallel-fibered planar soft tissues

The simple shear test may provide unique information regarding the material response of parallel-fibered soft tissues because it allows the elimination of the dominant fiber material response from the overall stresses. However, inhomogeneities in the strain field due to clamping and free edge effects have not been documented. The finite element method was used to study finite simple shear of simulated ligament material parallel to the fiber direction. The effects of aspect ratio, clamping prestrain, and bulk modulus were assessed using a transversely isotropic, hyperelastic material model. For certain geometries, there was a central area of uniform strain. An aspect ratio of 1:2 for the fiber to cross-fiber directions provided the largest region of uniform strain. The deformation was nearly isochoric for all bulk moduli indicating this test may be useful for isolating solid viscoelasticity from interstitial flow effects. Results suggest this test can be used to characterize the matrix properties for the type of materials examined in this study, and that planar measurements will suffice to characterize the strain. The test configuration may be useful for the study of matrix, fiber-matrix, and fiber-fiber material response in other types of parallel-fibered transversely isotropic soft tissues.     

Does subcutaneous adipose tissue behave as an (anti-)thixotropic material?

Although subcutaneous adipose tissue undergoes large deformations on a daily basis, there is no adequate mechanical model to describe the transfer of mechanical load from the skin throughout the tissue to deeper layers. In order to develop such a non-linear model, a set of experimental data is required. Accordingly, this study examines the long term behavior of adipose tissue under small strain and its response to various large strain profiles. The results show that the shear modulus dramatically increases to about an order of magnitude after a loading period between 250 and 1250 s, but returns to its initial value within 3 h of recovery from loading. In addition, it was observed that the stress–strain responses for various large strain history sequences are reproducible up to a strain of 0.15. For increasing strains, the stress decreases for subsequent loading cycles and, above 0.3 strain, tissue structure changes such that the stress becomes independent of the applied strain. From the results, it can be concluded that adipose tissue likely behaves as an (anti-) thixotropic material and that a Mooney–Rivlin model might be appropriate to simulate behavior at physiologically relevant high strains. However, before the model is developed more fully, further experimental research is needed to ratify that the material is (anti-)thixotropic.     

Linear viscoelastic properties of bovine brain tissue in shear

We report the results from a series of rheological tests of fresh bovine brain tissue. Using a standard Bohlin VOR shear rheometer, shear relaxation and oscillating strain sweep experiments were performed on disks of brain tissue 30 mm in diameter, with a thickness of 1.5–2 mm. The strain sweep experiment showed that the viscoelastic strain limit is of the order of 0.1% strain. Shear relaxation data do not indicate the presence of a long-term elastic modulus, indicating fluid-like behavior. A relaxation spectrum was calculated by inverting the experimental data and used to predict oscillatory response, which agreed well with measured data.     

The prediction of stress-relaxation of ligaments and tendons using the quasi-linear viscoelastic model

Recent studies have questioned the ability of the quasi-linear viscoelastic (QLV) model to predict stresses and strains in response to loading conditions other than those used to fit the model. The objective of this study was to evaluate the ability of several models in the literature to predict the elastic stress response of ligament and tendon at strain levels higher than the levels used to fit the model. The constitutive models were then used to evaluate the ability of the QLV model to predict the overall stress response during stress relaxation. The models expressing stress as an exponential function of strain significantly overestimated stress when used at higher strain levels. The polynomial formulation of the Mooney–Rivlin model more accurately predicted the stress–strain behavior of ligament and tendon. The results demonstrate that the ability of the QLV model to accurately predict the stress-relaxation response is dependent in part on the accuracy of the function used to model the elastic response of the soft tissue.     

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.     

Three-dimensional finite element modelling of the human ACL: simulation of passive knee flexion with a stressed and stress-free ACL

In this study, a three-dimensional finite element model of the human anterior cruciate ligament (ACL) was developed and simulations of passive knee flexion were performed. The geometrical model of the ACL was built from experimental measurements performed on a cadaveric knee specimen which was also subjected to kinematics tests. These experiments were used to enforce the particular boundary conditions used in the numerical model. A previously developed transversely isotropic hyperelastic material model was implemented and the ability to pre-stress the ligament was also included. The model exhibited the key characteristics of connective soft tissues: anisotropy, nonlinear behaviour, large strains, very high compliance for compressive or bending loading along the collagen fibres and incompressibility. Simulations of passive knee flexion were performed, with and without pre-stressing the ACL. The resultant force generated by the ACL was monitored and the results compared to existing experimental data. The stress distribution within the ligament was also assessed. When the ACL was pre-stressed, there was a good correlation between the predicted and experimental resultant forces reported in the literature over the entire flexion-extension range. The stress distribution in the pre-stressed and stress-free ACL were similar, although the magnitudes in the pre-stressed ACL were higher, particularly at low flexion angles.     

A nonlinear viscoelastic finite element model of polyethylene

A nonlinear viscoelastic finite element model of ultra-high molecular weight polyethylene (UHMWPE) was developed in this study. Eight cylindrical specimens were machined from ram extruded UHMWPE bar stock (GUR 1020) and tested under constant compression at 7% strain for 100 sec. The stress strain data during the initial ramp up to 7% strain was utilized to model the "instantaneous" stress-strain response using a Mooney-Rivlin material model. The viscoelastic behavior was modeled using the time-dependent relaxation in stress seen after the initial maximum stress was achieved using a stored energy formulation. A cylindrical model of similar dimensions was created using a finite element analysis software program. The cylinder was made up of hexahedral elements, which were given the material properties utilizing the "instantaneous" stress-strain curve and the energy-relaxation curve obtained from the experimental data. The cylinder was compressed between two flat rigid bodies that simulated the fixtures of the testing machine. Experimental stress-relaxation, creep and dynamic testing data were then used to validate the model. The mean error for predicted versus experimental data for stress relaxation at different strain levels was 4.2%. The mean error for the creep test was 7% and for dynamic test was 5.4%. Finally, dynamic loading in a hip arthroplasty was modeled and validated experimentally with an error of 8%. This study establishes a working finite element material model of UHMWPE that can be utilized to simulate a variety of postoperative arthroplasty conditions.     

Stress relaxation behaviour of trabecular bone specimens

The present study defines several conditions under which stress relaxation tests can be performed and investigates the viscoelastic behaviour of trabecular bone in compression through a series of stress relaxation tests at three strain levels and in three loading directions of each cubic specimen. A visoelastic model is proposed to characterize the behaviour of trabecular bone and a spectrum of relaxation times is determined. Trabecular bone from the femoral head is non-linearly viscoelastic and displays anisotropic behaviour, which cannot be more symmetric elastically than orthotropic.