Stress-stretch Curves Research Articles

Compressive properties of passive skeletal muscle—The impact of precise sample geometry on parameter identification in inverse finite element analysis

Due to the increasing developments in modelling of biological material, adequate parameter identification techniques are urgently needed. The majority of recent contributions on passive muscle tissue identify material parameters solely by comparing characteristic, compressive stress–stretch curves from experiments and simulation. In doing so, different assumptions concerning e.g. the sample geometry or the degree of friction between the sample and the platens are required. In most cases these assumptions are grossly simplified leading to incorrect material parameters. In order to overcome such oversimplifications, in this paper a more reliable parameter identification technique is presented: we use the inverse finite element method (iFEM) to identify the optimal parameter set by comparison of the compressive stress–stretch response including the realistic geometries of the samples and the presence of friction at the compressed sample faces. Moreover, we judge the quality of the parameter identification by comparing the simulated and experimental deformed shapes of the samples. Besides this, the study includes a comprehensive set of compressive stress–stretch data on rabbit soleus muscle and the determination of static friction coefficients between muscle and PTFE.

Influence of a freeze–thaw cycle on the stress–stretch curves of tissues of porcine abdominal organs

The paper investigates both fresh porcine spleen and liver and the possible decomposition of these organs under a freeze–thaw cycle. The effect of tissue preservation condition is an important factor which should be taken into account for protracted biomechanical tests. In this work, tension tests were conducted for a large number of tissue specimens from twenty pigs divided into two groups of 10. Concretely, the first group was tested in fresh state; the other one was tested after a freeze-thaw cycle which simulates the conservation conditions before biomechanical experiments. A modified Fung model for isotropic behavior was adopted for the curve fitting of each kind of tissues. Experimental results show strong effects of the realistic freeze–thaw cycle on the capsule of elastin-rich spleen but negligible effects on the liver which virtually contains no elastin. This different behavior could be explained by the autolysis of elastin by elastolytic enzymes during the warmer period after thawing. Realistic biomechanical properties of elastin-rich organs can only be expected if really fresh tissue is tested. The observations are supported by tests of intestines.

Swellable elastomers under constraint

Swellable elastomers are widely used in the oilfield to seal the flow of downhole fluids. For example, when a crack appears in self-healing cement, the liquid in the surroundings flows into the crack and permeates into the cement, causing small particles of elastomers in the cement to swell, resulting in the blocking of the flow. Elastomers are also used as large components in swellable packers, which can swell and seal zones in the borehole. In these applications, the elastomers swell against the constraint of stiff materials, such as cement, metal, and rock. The pressure generated by the elastomer against the confinement is a key factor that affects the quality of the sealing. This work develops a systematic approach to predict the magnitude of the pressure in such components. Experiments are carried out to determine the stress-stretch curve, free swelling ratio, and confining pressure. The data are interpreted in terms of a modified Flory-Rehner model.

Extra-fibrillar matrix mechanics of annulus fibrosus in tension and compression

The annulus fibrosus (AF) of the disk is a highly nonlinear and anisotropic material that undergoes a complex combination of loads in multiple orientations. The tensile mechanical behavior of AF in the lamellar plane is dominated by collagen fibers and has been accurately modeled using exponential functions. On the other hand, AF mechanics perpendicular to the lamella, in the radial direction, depend on the properties of the ground matrix with little to no fiber contribution. The ground matrix is mainly composed of proteoglycans (PG), which are negatively charged macromolecules that maintain the tissue hydration via osmotic pressure. The mechanical response of the ground matrix can be divided in the contribution of osmotic pressure and an elastic solid part known as extra-fibrillar matrix (EFM). Mechanical properties of the ground matrix have been measured using tensile and confined compression tests. However, EFM mechanics have not been measured directly. The objective of this study was to measure AF nonlinear mechanics of the EFM in tension and compression. To accomplish this, a combination of osmotic swelling and confined compression in disk radial direction, perpendicular to the lamella, was used. For this type of analysis, it was necessary to define a stress-free reference configuration. Thus, a brief analysis on residual stress in the disk and a procedure to estimate the reference configuration are presented. The proposed method was able to predict similar swelling deformations when using different loading protocols and models for the EFM, demonstrating its robustness. The stress-stretch curve of the EFM was linear in the range 0.9 < λ₃ < 1.3 with an aggregate modulus of 10.18±3.32 kPa; however, a significant nonlinearity was observed for compression below 0.8. The contribution of the EFM to the total aggregate modulus of the AF decreased from 70 to 30% for an applied compression of 50% of the initial thickness. The properties obtained in this study are essential for constitutive and finite element models of the AF and disk and can be applied to differentiate between functional degeneration effects such as PG loss and stiffening due to cross-linking.

Circumferential variations of mechanical behavior of the porcine thoracic aorta during the inflation test

We developed an extension-inflation experimental apparatus with a stereo vision system and a stress–strain analysis method to determine the regional mechanical properties of a blood vessel. Seven proximal descending thoracic aortas were investigated during the inflation test at a fixed longitudinal stretch ratio of 1.35 over a transmural pressure range from 1.33 to 21.33 kPa. Four circumferential regions of each aorta were designated as the anterior (A), left lateral (L), posterior (P), and right lateral (R) regions, and the inflation test was repeated for each region of the aortas. We used continuous functions to approximate the surfaces of the regional aortic wall in the reference configuration and the deformed configuration. Circumferential stretch and stress at the four circumferential regions of the aorta were computed. Circumferential stiffness, defined as the tangent of the stress–stretch curve, and physiological aortic stiffness, named pressure–strain elastic modulus, were also computed for each region. In the low pressure range, the stress increased linearly with increased stretch, but the mechanical response became progressively stiffer in the high-pressure range above a transition point. At a transmural pressure of 12.00 kPa, mean values of stiffness were 416±104 kPa (A), 523±99 kPa (L), 634±91 kPa (P), and 489±82 kPa (R). The stiffness of the posterior region was significantly higher than that of the anterior region, but no significant difference was found in pressure–strain elastic modulus.

Optical measurements of vocal fold tensile properties: Implications for phonatory mechanics

In voice research, in vitro tensile stretch experiments of vocal fold tissues are commonly employed to determine the tissue biomechanical properties. In the standard stretch-release protocol, tissue deformation is computed from displacements applied to sutures inserted through the thyroid and arytenoid cartilages, with the cartilages assumed to be rigid. Here, a non-contact optical method was employed to determine the actual tissue deformation of vocal fold lamina propria specimens from three excised human larynges in uniaxial tensile tests. Specimen deformation was found to consist not only of deformation of the tissue itself, but also deformation of the cartilages, as well as suture alignment and tightening. Stress–stretch curves of a representative load cycle were characterized by an incompressible Ogden model. The initial longitudinal elastic modulus was found to be considerably higher if determined based on optical displacement measurements than typical values reported in the literature. The present findings could change the understanding of the mechanics underlying vocal fold vibration. Given the high longitudinal elastic modulus the lamina propria appeared to demonstrate a substantial level of anisotropy. Consequently, transverse shear could play a significant role in vocal fold vibration, and fundamental frequencies of phonation should be predicted by beam theories accounting for such effects.

Lonsdaleite Model of Open-Cell Elastic Foams: Theory and Calibration

We formulate a unit-cell model of open-cell elastic foams. In this model, a foam consists of four-bar tetrahedra arranged in the hexagonal diamond structure known as Lonsdaleite. The parameters of the model are the Young’s modulus of the bars and a few geometric parameters, the values of which may be roughly estimated for any given foam. We use the model to simulate a set of experiments in which elastic polyether polyurethane foams in a broad range of densities were tested under five loading conditions, namely tension along the rise direction; compression along the rise direction; compression along a transverse direction; simple shear combined with compression along the rise direction; and hydrostatic pressure combined with compression along the rise direction. With a suitable choice of values of the parameters of the model, the stress–stretch curves that we compute using the model compare favorably with the stress–stretch curves that were measured in the experiments. In some of the experiments a stress plateau in the stress–stretch curve was accompanied by heterogeneous stretch fields, even though the attendant stress fields were homogeneous. For these experiments we show that the model can be used to predict the occurrence of a second-order phase transition, so that the plateau stress can be interpreted as a Maxwell stress and the attendant heterogeneous stretch fields as two-phase fields, consistent with the experimental evidence. In other experiments the stress–stretch curve evinced a sudden and pronounced loss of stiffness, but no genuine stress plateau, and the attendant stretch fields remained homogeneous. For these experiments we show that the model can be used to predict the occurrence of a bifurcation of equilibrium in which the stress keeps rising as the deformation continues to increase in the post-buckling stage, so that the stretch fields remain homogeneous throughout, consistent with the experimental evidence. In general, to appraise the goodness of our model we put emphasis on the relation between the stress–stretch curve measured in an experiment and the nature of the attendant stretch fields. We submit that this emphasis should remain a guiding methodological trait in the appraisal of constitutive models of open-cell elastic foams.

Experimental study and constitutive modelling of the passive mechanical properties of the porcine carotid artery and its relation to histological analysis: Implications in animal cardiovascular device trials

The present study focusses on the determination, comparison and constitutive modelling of the passive mechanical properties of the swine carotid artery over very long stretches in both proximal and distal regions. Special attention is paid to the histological and mechanical variations of these properties depending on the proximity to the heart. The results can have clinical relevance, especially in the research field of intravascular device design. Before the final clinical trials on humans, research in the vascular area is conducted on animal models, swine being the most common due to the similarities between the human and swine cardiovascular systems as well as the fact that the swine size is suitable for testing devices, in this case endovascular carotid systems. The design of devices usually involves numerical techniques, and an important feature is the appropriate modelling of the mechanical properties of the vessel. Fourteen carotid swine arteries were harvested just after sacrifice and cyclic uniaxial tension tests in longitudinal and circumferential directions were performed for distal and proximal samples. The stress–stretch curves obtained were fitted with a hyperelastic anisotropic model. Stress-free configuration states were also analyzed. Finally, human and swine samples were processed in a histological laboratory and images were used to quantify their microconstituents. The statistical analysis revealed significant differences between the mechanical behavior of proximal and distal locations in the circumferential but not in the longitudinal direction. Circumferential direction samples show clear differences both in residual stretches and tensile curves between the two locations, while the features of longitudinal specimens are independent of the axial position. The statistical analysis provides significant evidence of changes depending on the position of the sample, mainly in elastin and SMC quantification.

Analysis of Mechanical Behavior of Red Blood Cell Membrane with Malaria Infection

Human red blood cells (RBCs) are responsible to transport oxygen and carbon dioxide for human bodies. The physiological functions of RBCs are greatly influenced by their mechanical properties. When RBC is infected by Malaria parasite called Plasmodium falciparum, it shows progressive changes in mechanical properties and loses its deformability. The infected red blood cells (IRBCs) develop properties of cytoadherence (stickiness) and rosetting (the binding of non-infected RBCs to parasitized RBCs). In this paper to analyze the mechanical properties and deformability of the IRBC, we applied stress-stretch ratio relation of its biomembrane .To express this constitutive relation, we proposed a mathematical model (Neo-Hookean model) based on membrane theory. On this model, we present continuous stress-stretch ratio curves for the relation derived from the model for different intracellular developmental stages of the parasite, to determine the mechanical properties of IRBC. The analytical results obtained from the mathematical model are more closed with the experimental data [1] which demonstrates the validity of the model. By restricting our attention to spherically symmetric deformation in the final schizont stage of parasite development, the pressure-extension ratio relation curve also adapted from the proposed strain energy function. The change in osmotic pressure versus volumetric ratio has been also considered for IRBC before hemolysis.

Hyperelastic properties of human meniscal attachments

Meniscal attachments are ligamentous tissues anchoring the menisci to the underlying subchondral bone. Currently little is known about the behavior of meniscal attachments, with only a few studies quantitatively documenting their properties. The objective of this study was to quantify and compare the tensile mechanical properties of human meniscal attachments in the transverse direction, curve fit experimental Cauchy stress–stretch data to evaluate the hyperelastic behavior, and couple these results with previously obtained longitudinal data to generate a more complete constitutive model. Meniscal attachment specimens were tested using a uniaxial tension test with the collagen fibers oriented perpendicular to the loading axis. Tests were run until failure and load-optical displacement data was recorded for each test. The medial posterior attachment was shown to have a significantly greater elastic modulus (6.42±0.78 MPa) and ultimate stress (1.73±0.32 MPa) when compared to the other three attachments. The Mooney–Rivlin material model was selected as the best fit for the transverse data and used in conjunction with the longitudinal data. A novel computational approach to determining the transition point between the toe and linear regions is presented for the hyperelastic stress–stretch curves. Results from piece-wise non-linear longitudinal curve fitting correlate well with previous linear elastic and SEM findings. These data can be used to advance the design of meniscal replacements and improve knee joint finite element models.

A formulation to model the nonlinear viscoelastic properties of the vascular tissue

Nearly all soft tissues, including the vascular tissue, present a certain degree of viscoelastic material response, which becomes apparent performing multiple relaxation tests over a wide range of strain levels and plotting the resulting stress relaxation curves, nonlinear viscoelasticity of the tissue. Changes in relaxation rate at each strain may occur at multiple strain levels. A constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these nonlinear viscoelastic phenomena is here presented and used to fit stress–stretch curves from experimental relaxation tests. This constitutive model was used to fit several data set of in vitro experimental stress relaxation tests performed on ovine and porcine aorta. The good fitting of the experimental data shows the capability of the model to reproduce the viscoelastic response of the vascular tissue.

The biomechanical effects of limb lengthening and botulinum toxin type A on rabbit tendon

Numerous studies have examined the effects of distraction osteogenesis (DO) on bone, but relatively fewer have explored muscle adaptation, and even less have addressed the concomitant alterations that occur in the tendon. The purpose herein was to characterize the biomechanical properties of normal and elongated rabbit ( N=20) tendons with and without prophylactic botulinum toxin type A (BTX-A) treatment. Elastic and viscoelastic properties of Achilles and Tibialis anterior (TA) tendons were evaluated through pull to failure and stress relaxation tests. All TA tendons displayed nonlinear viscoelastic responses that were strain dependent. A power law formulation was used to model tendon viscoelastic responses and tendon elastic responses were fit with a microstructural model. Distraction-elongated tendons displayed increases in compliance and stress relaxation rates over undistracted tendons; BTX-A administration offset this result. The elastic moduli of distraction-lengthened TA tendons were diminished ( p=0.010) when distraction was combined with gastrocnemius (GA) BTX-A administration, elastic moduli were further decreased ( p=0.004) and distraction following TA BTX-A administration resulted in TA tendons with moduli not different from contralateral control ( p>0.05). Compared to contralateral control, distraction and GA BTX-A administration displayed shortened toe regions, ( p=0.031 and 0.038, respectively), while tendons receiving BTX-A in the TA had no differences in the toe region ( p>0.05). Ultimate tensile stress was unaltered by DO, but stress at the transition from the toe to the linear region of the stress–stretch curve was diminished in all distraction-elongated TA tendons ( p<0.05). The data suggest that prophylactic BTX-A treatment to the TA protects some tendon biomechanical properties.

B-4 Direct and Indirect Biomechanical Changes during Gait Following Total Knee Arthroplasty

Nearly all soft tissues, among which the vascular tissue is included, present a certain degree of viscoelastic response. This behaviour may be attributed in part to fluid transport within the solid matrix, and to the friction between its fluid and solid constituents. After being preconditioned, the tissue displays highly repetitive behaviour, so that it can be considered pseudo-elastic, that is, elastic but behaving differently in loading and unloading. Because of this reason, very few constitutive laws accounting for the viscoelastic behaviour of the tissue have been developed. Nevertheless, the consideration of this inelastic effect is of crucial importance in surgeries—like vascular angioplasty—where the mentioned preconditioning cannot be considered since non-physiological deformation is applied on the vessel which, in addition, can cause damage to the tissue. A new constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these two inelastic phenomena is presented here and used to fit experimental stress–stretch curves from simple tension loading–unloading tests and relaxation test on porcine and ovine vascular samples.

Theory of Dielectric Elastomers Capable of Giant Deformation of Actuation

The deformation of a dielectric induced by voltage is limited by electrical breakdown if the dielectric is stiff, and by electromechanical instability if the dielectric is compliant. The interplay of the two modes of instability is analyzed for a dielectric elastomer, which is compliant at a small stretch, but stiffens steeply. The theory is illustrated with recent experiments of interpenetrating networks, and with a model of swollen elastomers. The theory predicts that, for an elastomer with a stress-stretch curve of a desirable form, the voltage can induce giant deformation.

Effects of region and sex on the mechanical properties of the glenohumeral capsule during uniaxial extension

Surgical repair of the glenohumeral capsule after dislocation ignores regional boundaries of the capsule and is not sex specific. However, each region of the capsule functions to stabilize the joint in different positions, and differences in joint laxity between men and women have been found. The objectives of this research were to determine the effects of region (axillary pouch and posterior capsule) and sex on the material properties of the glenohumeral capsule. Boundary conditions derived from experiments were used to create finite-element models that applied tensile deformations to tissue samples from the capsule. The material coefficients of a hyperelastic constitutive model were determined via inverse finite-element optimization, which minimized the difference between the experimental and finite-element model-predicted load-elongation curve. These coefficients were then used to create stress-stretch curves representing the material properties of the capsule regions for each sex in response to uniaxial extension. For the axillary pouch, the C1 (men: 0.28+/-0.39 MPa and women: 0.23+/-0.12 MPa) and C2 (men: 8.2+/-4.1 and women: 7.7+/-3.0) material coefficients differed between men and women by only 0.05 MPa and 0.5, respectively. Similarly, the posterior capsule coefficients differed by 0.15 MPa (male: 0.49+/-0.26 MPa and female: 0.34+/-0.20 MPa) and 0.6 (male: 7.8+/-2.9 and female: 7.2+/-3.0), respectively. No differences could be detected in the material coefficients between regions or sexes. As a result, surgeons may not need to consider region- and sex-specific surgical repair techniques. Furthermore, finite-element models of the glenohumeral joint may not need region- or sex-specific material coefficients when using this constitutive model.

A constitutive formulation of vascular tissue mechanics including viscoelasticity and softening behaviour

Nearly all soft tissues, among which the vascular tissue is included, present a certain degree of viscoelastic response. This behaviour may be attributed in part to fluid transport within the solid matrix, and to the friction between its fluid and solid constituents. After being preconditioned, the tissue displays highly repetitive behaviour, so that it can be considered pseudo-elastic, that is, elastic but behaving differently in loading and unloading. Because of this reason, very few constitutive laws accounting for the viscoelastic behaviour of the tissue have been developed. Nevertheless, the consideration of this inelastic effect is of crucial importance in surgeries—like vascular angioplasty—where the mentioned preconditioning cannot be considered since non-physiological deformation is applied on the vessel which, in addition, can cause damage to the tissue. A new constitutive formulation considering the particular features of the vascular tissue, such as anisotropy, together with these two inelastic phenomena is presented here and used to fit experimental stress–stretch curves from simple tension loading–unloading tests and relaxation test on porcine and ovine vascular samples.

Estimating zero-strain states of very soft tissue under gravity loading using digital image correlation.

This paper presents several experimental techniques and concepts in the process of measuring mechanical properties of very soft tissue in an ex vivo tensile test. Gravitational body force on very soft tissue causes pre-compression and results in a non-uniform initial deformation. The global digital image correlation technique is used to measure the full-field deformation behavior of liver tissue in uniaxial tension testing. A maximum stretching band is observed in the incremental strain field when a region of tissue passes from compression and enters a state of tension. A new method for estimating the zero-strain state is proposed: the zero strain position is close to, but ahead of the position of the maximum stretching band, or in other words, the tangent of a nominal stress-stretch curve reaches minimum at lambda greater or similar 1. The approach, to identify zero strain by using maximum incremental strain, can be implemented in other types of image-based soft tissue analysis. The experimental results of 10 samples from seven porcine livers are presented and material parameters for the Ogden model fit are obtained. The finite element simulation based on the fitted model confirms the effect of gravity on the deformation of very soft tissue and validates our approach.

Development of a three-dimensional unit cell to model the micromechanical response of a collagen-based extracellular matrix

The three-dimensional microstructure and mechanical properties of the collagen fibrils within the extracellular matrix (ECM) is now being recognized as a primary factor in regulating cell proliferation and differentiation. Therefore, an appreciation of the mechanical aspects by which a cell interacts with its ECM is required for the development of engineered tissues. Ultimately, using these interactions to design tissue equivalents requires mathematical models with three-dimensional architecture. In this study, a three-dimensional model of a collagen fibril matrix undergoing uniaxial tensile stress was developed by making use of cellular solids. A structure consisting of thin struts was chosen to represent the arrangement of collagen fibrils within an engineered ECM. To account for the large deformation of tissues, the collagen fibrils were modeled as hyperelastic neo-Hookean or Mooney–Rivlin materials. The use of cellular solids allowed the fibril properties to be related to the ECM properties in closed form, which, in turn, allowed the estimation of fibril properties using ECM experimental data. A set of previously obtained experimental data consisting of simultaneous measures of the fibril microstructure and mechanical tests was used to evaluate the model’s capability to estimate collagen fibril mechanical property when given tissue-scale data and to predict the tissue-scale mechanical properties when given estimated fibril stiffness. The fibril tangent modulus was found to be 1.26 ± 0.70 and 1.62 ± 0.88 MPa when the fibril was modeled as neo-Hookean and Mooney–Rivlin material, respectively. There was no statistical significance of the estimated fibril tangent modulus among the different groups. Sensitivity analysis showed that the fibril mechanical properties and volume fraction were the two input parameters which required accurate values. While the volume fraction was easily obtained from the initial image of the gel, the fibril mechanical properties were not readily available. Therefore the fibril mechanical properties were estimated in the leave-one-out cross-validation (LOOCV) analysis. The LOOCV analysis showed that the model was able to predict the ECM stress–stretch curve with an average mean squared error of 9.71 kPa 2. The three-dimensional architecture expands on previous continuum models and two-dimensional representations to provide a useful model for studying the hierarchical effects of ECM microstructure on cell function. This model can be used as a design tool to engineer the optimum microstructure for cells to function.

ゴムのシミュレーション技術 補強材充てんゴムの粘弾性変形応答シミュレーション

A brief review of constitutive equations based on the molecular chain network theory for elastic deformation behavior and the reptation theory for viscoelastic deformation behavior of rubbers were presented. The typical strain-rate-dependent deformation behaviors under monotonic and cyclic straining with different strain rates were reproduced and they corresponded well to the experimentally obtained results, particularly softening and hysteresis loss, that is, the Mullins effect, occurring in stress-stretch curves under cyclic deformation processes. The deformation behaviors of carbon-black (CB) filled rubber under monotonic and cyclic straining are modeled by using the proposed constitutive equation and 3D homogenization method. The results reveal the mechanism in substantial enhancement of the resistance of CB-filled rubber to macroscopic deformation, hysteresis loss, and the effects of the distribution morphology and the volume fraction of CB on the deformation behavior.

Computational evaluation of strain-rate-dependent deformation behavior of rubber and carbon-black-filled rubber under monotonic and cyclic straining

The molecular chain network model for elastic deformation behavior and the reptation theory for viscoelastic deformation behavior are used to derive a constitutive equation for rubber. The new eight-chain-like model contains eight standard models consisting of Langevin springs and dashpot to account for the interaction of chains with their surroundings. Monotonic and cyclic deformation behavior of rubber with relaxation under different strain rates have been examined. The results reveal the roles of the individual springs and dashpot, and the strain rate dependence of materials in the monotonic and cyclic deformation behaviors, particularly softening and hysteresis loss, that is, the Mullins effect, occurring in stress–stretch curves under cyclic deformation processes. The validity of the results is checked through comparison with experimental results. The deformation behaviors of a plane strain rubber unit cell containing carbon-black (CB) under monotonic and cyclic straining are investigated by computational simulation using the proposed constitutive equation and homogenization method. The results reveal the substantial enhancement of the resistance of CB-filled rubber to macroscopic deformation, which is caused by the marked orientation hardening due to the highly localized deformation of rubber. The role of strain rate sensitivity on such characteristic deformation behaviors as increases in the resistance to deformation, hysteresis loss, and the effects of the distribution morphology and the volume fraction of CB on the deformation behavior is clarified. The increases in the volume fraction and in the aggregation of the distribution of CB substantially raise the resistance to deformation and hysteresis loss.