Exponential Strain Energy Function Research Articles

3D biomechanical properties of the layered esophagus: Fung-type SEF and new constitutive model.

Most current studies on the passive biomechanical properties of esophageal tissues directly use the exponential strain energy function (SEF) to fit and calculate the constants of the constitutive equation. In the context of the extensive application of exponential SEF, in-depth research on the exponential SEF is still lacking. The purpose of this study is to combine the exponential function with the polynomial SEF to obtain the most suitable constitutive equation to describe the three-dimensional passive behavior of the esophagus. fresh pig esophagus with a length of 13cm in the middle position was selected as esophageal samples. The esophageal sample was separated into muscular layer and mucosal layer with surgical scissors. Stretch-inflation mechanical tests of the intact esophagus, esophageal muscular, and esophageal mucosa were carried out on a triaxial test machine. The external radius, axial force, and internal pressure were recorded simultaneously. The seven-parameter Fung-type SEF and several new SEFs combining polynomials and exponents were used to fit the experimental data curves. The stretch-inflation test data and the morphometric parameters at the zero-stress state of the layered esophagus were obtained. The new SEF with polynomial and exponential combination is more suitable to describe describing the three-dimensional passive biomechanical properties of esophageal tissue. Among them, New-Fung13 SEF is more suitable for describing the passive biomechanical properties of intact esophageal tissue, Sokolis-Fung13 SEF is more suitable for the esophageal muscle layer, and New-Fung10 SEF is more suitable for the esophageal mucosa. The constitutive parameters of the optimal constitutive model for each layer of the esophagus were obtained.

Relative Contributions of Matrix and Myocytes to Biaxial Mechanics of the Right Ventricle in Pulmonary Arterial Hypertension.

Pulmonary arterial hypertension (PAH) commonly leads to right ventricular (RV) hypertrophy and fibrosis that affect the mechanical properties of the RV myocardium (MYO). To investigate the effects of PAH on the mechanics of the RV MYO and extracellular matrix (ECM), we compared RV wall samples, isolated from rats in which PAH was induced using the SuHx protocol, with samples from control animals before and after the tissues were decellularized. Planar biaxial mechanical testing, a technique first adapted to living soft biological tissues by Fung, was performed on intact and decellularized samples. Fung's anisotropic exponential strain energy function fitted the full range of biaxial test results with high fidelity in control and PAH samples both before and after they were decellularized. Mean RV myocardial apex-to-outflow tract and circumferential stresses during equibiaxial strain were significantly greater in PAH than control samples. Mean RV ECM circumferential but not apex-to-outflow tract stresses during equibiaxial strain were significantly greater in the PAH than control group. The ratio of ECM to myocardial stresses at matched strains did not change significantly between groups. Circumferential stresses were significantly higher than apex-to-outflow tract stresses for all groups. These findings confirm the predictions of a mathematical model based on changes in RV hemodynamics and morphology in rat PAH, and may provide a foundation for a new constitutive analysis of the contributions of ECM remodeling to changes in RV filling properties during PAH.

Importance of material parameters and strain energy function on the wall stresses in the left ventricle

Patient-specific estimates of the stress distribution in the left ventricles (LV) may have important applications for therapy planning, but computing the stress generally requires knowledge of the material behaviour. The passive stress-strain relation of myocardial tissue has been characterized by a number of models, but material parameters (MPs) remain difficult to estimate. The aim of this study is to implement a zero-pressure algorithm to reconstruct numerically the stress distribution in the LV without precise knowledge of MPs. We investigate the sensitivity of the stress distribution to variations in the different sets of constitutive parameters. We show that the sensitivity of the LV stresses to MPs can be marginal for an isotropic constitutive model. However, when using a transversely isotropic exponential strain energy function, the LV stresses become sensitive to MPs, especially to the linear elastic coefficient before the exponential function. This indicates that in-vivo identification efforts should focus mostly on this MP for the development of patient-specific finite-element analysis.

A finite strain integral-type anisotropic damage model for fiber-reinforced materials: Application in soft biological tissues

A large deformation finite element method is adopted to study anisotropic damage behavior of geometrically nonlinear problems with specific application in soft biological tissues. Mesh dependency, a general drawback of local damage models, is circumvented using an integral type nonlocal damage model. Three different algorithms for updating the integration search zone and the distance between the Gaussian Points are investigated to demonstrate the best solution for a large deformation based integral approach. Since there exists no record for the effect of the characteristic length of the soft biological tissues, which plays an important role in an integral type nonlocal method, this effect is also investigated. Moreover, the anisotropic highly nonlinear mechanical response of the soft fiber-reinforced tissue is captured by an exponential type strain energy function, combined with the Neo-Hookean type function, to account for distributed collagen fibers and the surrounding ground matrix. In addition, soft tissues are considered to be incompressible materials; hence, a nearly incompressible numerical model is applied to preserve the volume. Several numerical examples are presented to illustrate the accuracy and good agreement with available experimental data. The major role of fibers’ orientation during the loading process is also investigated in numerical examples. It is expected to help more accurate analysis of practical cases such as aortic dissection.

Noninvasive in vivo determination of residual strains and stresses.

Vascular growth and remodeling during embryonic development are associated with blood flow and pressure induced stress distribution, in which residual strains and stresses play a central role. Residual strains are typically measured by performing in vitro tests on the excised vascular tissue. In this paper, we investigated the possibility of estimating residual strains and stresses using physiological pressure-radius data obtained through in vivo noninvasive measurement techniques, such as optical coherence tomography or ultrasound modalities. This analytical approach first tested with in vitro results using experimental data sets for three different arteries such as rabbit carotid artery, rabbit thoracic artery, and human carotid artery based on Fung's pseudostrain energy function and Delfino's exponential strain energy function (SEF). We also examined residual strains and stresses in the human swine iliac artery using the in vivo experimental ultrasound data sets corresponding to the systolic-to-diastolic region only. This allowed computation of the in vivo residual stress information for loading and unloading states separately. Residual strain parameters as well as the material parameters were successfully computed with high accuracy, where the relative errors are introduced in the range of 0-7.5%. Corresponding residual stress distributions demonstrated global errors all in acceptable ranges. A slight discrepancy was observed in the computed reduced axial force. Results of computations performed based on in vivo experimental data obtained from loading and unloading states of the artery exhibited alterations in material properties and residual strain parameters as well. Emerging noninvasive measurement techniques combined with the present analytical approach can be used to estimate residual strains and stresses in vascular tissues as a precursor for growth estimates. This approach is also validated with a finite element model of a general two-layered artery, where the material remodeling states and residual strain generation are investigated.

Reality based modeling and simulation of gallbladder shape deformation using variational methods

Accurate soft tissue deformation modeling is important for realistic surgical simulation. The aim of this study is to develop a reality-based gallbladder model and to determine material constants that represent gallbladder wall mechanical properties. Mechanical experiments on porcine gallbladder were performed to investigate tissue deformation, and an exponential strain energy function was used to describe the nonlinear stress-strain behavior of the gallbladder wall. A new volumetric function based upon the exponential strain energy function was proposed to model the gallbladder organ. A genetic algorithm was used to identify the material parameters of the proposed biomechanical model from the experimental data. The material constants of the exponential strain energy model were determined based on the experimental data. Deformation simulation and haptic rendering using the proposed gallbladder model were presented. Comparison between deformation predicted by the proposed model and that of the experimental data on gallbladder wall and gallbladder organ tissues demonstrates the applicability of this reality-based variational method for deformation simulation. An accurate soft tissue deformation model was developed using material constants identified for gallbladder. The model is suitable for interactive haptic rendering and deformation simulation. This model has potential applications for simulation of other hollow organs.

A Nonlinear Thin-Wall Model for Vein Buckling

Tortuous or twisted veins are often seen in the retina, cerebrum, and legs (varicose veins) of one-third of the aged population, but the underlying mechanisms are poorly understood. While the collapse of veins under external pressure has been well documented, the bent buckling of long vein segments has not been studied. The objectives of this study were to develop a biomechanical model of vein buckling under internal pressure and to predict the critical pressure. Veins were modeled as thin-walled nonlinear elastic tubes with the Fung exponential strain energy function. Our results demonstrated that veins buckle due to high blood pressure or low axial tension. High axial tension stabilized veins under internal pressure. Our buckling model estimated the critical pressure accurately compared to the experimental measurements. The buckling equation provides a useful tool for studying the development of tortuous veins.

A New Three-Dimensional Exponential Material Model of the Coronary Arterial Wall to Include Shear Stress Due to Torsion

The biomechanical milieu of the coronary arteries is unique in that they experience mechanical deformations of twisting, bending, and stretching due to their tethering to the epicardial surface. Spatial variations in stresses caused by these deformations could account for the heterogeneity of atherosclerotic plaques within the coronary tree. The goal of this work was to utilize previously reported shear moduli to calculate a shear strain parameter for a Fung-type exponential model of the arterial wall and determine if this single constant can account for the observed behavior of arterial segments under torsion. A Fung-type exponential strain-energy function was adapted to include a torsional shear strain term. The material parameter for this term was determined from previously published data describing the relationship between shear modulus and circumferential stress and longitudinal stretch ratio. Values for the shear strain parameter were determined for three geometries representing the mean porcine left anterior descending coronary artery dimensions plus or minus one standard deviation. Finite element simulation of triaxial biomechanical testing was then used to validate the model. The mean value calculated for the shear strain parameter was 0.0759+/-0.0009 (N=3 geometries). In silico triaxial experiments demonstrated that the shear modulus is directly proportional to the applied pressure at a constant longitudinal stretch ratio and to the stretch ratio at a constant pressure. Shear moduli determined from these simulations showed excellent agreement to shear moduli reported in literature. Previously published models describing the torsional shear behavior of porcine coronary arteries require a total of six independent constants. We have reduced that description into a single parameter in a Fung-type exponential strain-energy model. This model will aid in the estimation of wall stress distributions of vascular segments undergoing torsion, as such information could provide insight into the role of mechanical stimuli in the localization of atherosclerotic plaque formation.

Three-dimensional mechanical properties of porcine coronary arteries: a validated two-layer model

The normal coronary artery consists of two mechanically distinct layers: intima-media and adventitia. The objective of this study is to establish a two-layer three-dimensional (3-D) stress-strain relation of porcine coronary arteries. Experimental measurements were made by a series of biaxial tests (inflation and axial extension) of intact coronary arteries and, subsequently, their corresponding intima-media or adventitia layer. The Fung-type exponential strain energy function was used to describe the 3-D strain-stress relation for each layer and the intact wall. A genetic algorithm was used to determine the material constants in the Fung-type constitutive equation by curve fitting the experimental data. Because one layer must be sacrificed before the other layer can be tested, the material property of the missing layer was computed from the material constants of the intact vessel and the tested layer. A total of 20 porcine hearts were used: one group of 10 hearts for the left anterior descending artery and another group of 10 hearts for the right coronary artery. Each group was further divided into two subgroups of five specimens tested for the intact wall and the intima-media layer and for the intact wall and the adventitia layer. Our results show statistically significant differences in the material properties of the two layers. The mathematical model was validated by experimental stress-strain data for individual layers. The validated 3-D constitutive model will serve as a foundation for formulation of layer-specific boundary value problems in coronary physiology and cardiology.

Remodeling of strain energy function of common bile duct post obstruction

Biliary duct obstruction is an important clinical condition that affects millions of people worldwide. We have previously shown that the common bile duct (CBD) undergoes significant growth and remodelling post obstruction. The mechanical stress-strain relation is expected to change due to growth and remodeling in response to obstruction and hence pressure-overload. The objective of the present study was to characterize the material properties of the CBD of the sham group and at 3 hours, 12 hours, 2 days, 8 days and 32 days (n=5 in each group) after obstruction. The Fung's exponential strain energy function was used to relate stress to strain of the normal and remodeled ducts. The Marquardt-Levenberg and genetic algorithm methods were used for the determination of material constants for the exponential strain energy function. The elastic constants obtained by the two methods did not differ (p>0.254). We obtained stable and convergent material constants at every time point. The material constant that dictates the nonlinear rate of change in the circumferential stress with respect to strain increased significantly after obstruction acutely in the first 12 hrs., decreased at 2 days and finally increased montonically during the remaining 30 days. The acute initial increase in stiffness was simply due to the elastic response. The chronic remodeling process results in an initial circumferential softening followed by stiffening of the CBD. This study has important clinical significance for patients with CBD obstruction and for endoscopists and surgeons who treat the condition.

The effects of aneurysm on the biaxial mechanical behavior of human abdominal aorta

The biomechanical response of normal and pathologic human abdominal aortic tissue to uniaxial loading conditions is insufficient for the characterization of its three-dimensional (3D) mechanical behavior. Planar biaxial mechanical evaluation allows for 3D constitutive modeling of nearly incompressible tissues, as well as the investigation of the nature of mechanical anisotropy. In the current study, 26 abdominal aortic aneurysm (AAA) tissue samples and 8 age-matched (>60 years of age) nonaneurysmal abdominal aortic (AA) tissue samples were obtained and tested using a tension-controlled biaxial testing protocol. Graphical response functions (Sun et al., 2003. J. Biomech. Eng. 125, 372–380) were used as a guide to describe the pseudo-elastic response of AA and AAA. Based on the observed pseudo-elastic response, a four-parameter exponential strain energy function developed by Vito (1990. J. Biomech. Eng. 112, 153–159) was used from which both an individual specimen and group material parameter sets were determined for both AA and AAA. Peak Green strain values in the circumferential ( E θ θ , max ) and longitudinal ( E LL , max ) directions under an equibiaxial tension of 120 N/m were also compared. The strain energy function fit all of the individual specimens well with an average R 2 of 0.95±0.02 and 0.90±0.02 (mean±SEM) for the AA and AAA groups, respectively. The average E θ θ , max at 200 N/m equibiaxial tension was found to be significantly smaller for AAAs as compared to AAs (0.07±0.01 versus 0.13±0.03, respectively; p < 0.01 ). There was also a pronounced increase in the circumferential stiffness for AAA tissue as compared to AA tissue, indicating a larger degree of anisotropy for this tissue as compared to age-matched AA tissue. We also observed that the four-parameter Fung-elastic model was not able to fit the AAA tissue mechanical response using physically realistic material parameter values. It was concluded that aneurysmal degeneration of the abdominal aorta is associated with an increase in mechanical anisotropy, with preferential stiffening in the circumferential direction.

Biaxial elastic material properties of porcine coronary media and adventitia

The importance of mechanical stresses and strains has become well recognized in vascular physiology and pathology. To compute the stress and strain on the various components of the vessel wall, we must know the constitutive equations for the different layers of the vessel wall. The objective of the present study is to determine the constitutive equation of the coronary artery treated as a two-layer composite: intima-media and adventitial layers. Twelve hearts were obtained from a local slaughterhouse, and the right coronary artery and left anterior descending artery were dissected free from the myocardium. The vessel wall was initially mechanically tested biaxially (inflation and axial extension) as a whole (intact wall) and subsequently as intima-media or adventitial layer. A Fung-type exponential strain energy function was used to curve fit the experimental data for the intact wall and individual layers for the right coronary artery and left anterior descending artery. Two methods were used for the determination of material constants, including the Marquardt-Levenberg nonlinear least squares method and the genetic algorithm method. Our results show that there were no statistically significant differences in the material constants obtained from the two methods and that either set of elastic constants results in good fit of the data. Furthermore, at an in vivo value of axial stretch ratio, we find that the stiffness is as follows: intima-media > intact > adventitia. These results underscore the composite nature of coronary arteries with different material properties in each layer. The present results are necessary for analysis of coronary artery mechanics and to provide a fundamental understanding of vessel physiology.

A Two-Point Boundary-Value Problem for the Axial Shear of Hardening Isotropic Incompressible Nonlinearly Elastic Materials

The purpose of this research is to investigate a pure axial shear problem for the annular region between two concentric rigid cylinders occupied by an incompressible isotropic nonlinearly elastic material. Our main concern is with the subclass of these materials that exhibits hardening at large deformations. Such hardening at large strains is observed experimentally but is often not predicted by commonly used strain-energy densities. The particular axial shear problem that we investigate arises when the elastic material is perfectly bonded to an inner rigid circular core and to an outer rigid circular container. The deformation is driven by an axial pressure gradient. The problem is formulated as a two-point boundary-value problem for a second-order nonlinear ODE. For a broad class of incompressible isotropic hyperelastic materials, existence of smooth solutions is established on conversion to an initial value problem. We then specialize the material class to be of generalized neo-Hookean type, that is, where the strain-energy density depends only on the first invariant of the strain tensor. Two classes of such materials that exhibit hardening at large deformations are considered. The first class models limiting chain extensibility at the molecular level, and the second class is of power-law type. An interesting limiting case of the latter leads to an exponential strain-energy function commonly used in modeling biological materials. Some specific strain-energy densities from each class are examined in detail. Numerical boundary-value methods with quasilinearization for nonlinear second-order ODEs are used to illustrate a special feature of the solutions. It was previously shown by other authors that, for softening power-law materials, a boundary layer behavior is exhibited near the bonded surfaces: the axial displacement undergoes a sharp increase from zero at the endpoints and is slowly varying in the interior. We provide further numerical results regarding this phenomenon here. Our main concern is, however, with hardening materials. For such materials, the numerical results exhibit an interior localization at a location which is almost the midpoint between the boundaries. The axial displacement has a sharp change of slope which, in the limit of infinitely large pressure gradients, gives rise to a cusp in the displacement profile. The results highlight the contrasting behavior between softening and hardening rubber-like or biological materials.

Orthotropic mechanical properties of chemically treated bovine pericardium.

To facilitate bioprosthetic heart valve design, especially in the use of novel antimineralization chemical technologies, a thorough understanding of the multiaxial mechanical properties of chemically treated bovine pericardium (BP) is needed. In this study, we utilized a small angle light scattering based tissue pre-sorting procedure to select BP specimens with a high degree of structural uniformity. Both conventional glutaraldehyde (GL) and photo-oxidation (PO) chemical treatment groups were studied, with untreated tissue used as the control group. A second set of GL and PO groups was prepared by prestretching them along the preferred fiber direction during the chemical treatment. An extensive biaxial test protocol was used and the resulting stress-strain data fitted to an exponential strain energy function. The high structural uniformity resulted in both a consistent mechanical response and low variability in the material constants. For free fixed tissues, the strain energy per unit volume for GL treated BP was approximately 2.8 times that of PO treated BP at an equibiaxial Green's strain level of 0.16. Pre-stretched tissues exhibited a profound increase in both stiffness and the degree of anisotropy, with the GL treatment demonstrating a greater effect. Thus, structural control leads to an improved understanding of chemically treated BP mechanical properties. Judicious use of this knowledge can facilitate the design and enhanced long-term performance of bioprosthetic heart valves.

Myocardial material property determination in the in vivo heart using magnetic resonance imaging.

To determine nonlinear material properties of passive, diastolic myocardium using magnetic resonance imaging (MRI) tissue-tagging, finite element analysis (FEA) and nonlinear optimization. Alterations in the diastolic material properties of myocardium may pre-date the onset of or exist exclusive of systolic ventricular dysfunction in disease states such as hypertrophy and heart failure. Accordingly, significant effort has been expended recently to characterize the material properties of myocardium in diastole. The present study defines a new technique for determining material properties of passive myocardium using finite element (FE) models of the heart, MRI tissue-tagging and nonlinear optimization. This material parameter estimation algorithm is employed to estimate nonlinear material parameter sin the in vivo canine heart and provides the necessary framework to study the full complexities of myocardial material behavior in health and disease. Material parameters for a proposed exponential strain energy function were determined by minimizing the least squares difference between FE model-predicted and MRI-measured diastolic strains. Six mongrel dogs underwent MRI imaging with radiofrequency (RF) tissue-tagging. Two-dimensional diastolic strains were measured from the deformations of the MRI tag lines. Finite element models were constructed from early diastolic images and were loaded with the mean early to late left ventricular and right ventricular diastolic change in pressure measured at the time of imaging. A nonlinear optimization algorithm was employed to solve the least squares objective function for hte material parameters. Average material parameters for the six dogs were E = 28,722 +/- 15984 dynes/cm2 and c = 0.00182 +/- 0.00232 cm2/dyne. This parameter estimation algorithm provides the necessary framework for estimating the nonlinear, anisotropic and non-homogeneous material properties of passive myocardium in health and disease in the in vivo beating heart.

Physiological modeling of speech production: Methods for modeling soft-tissue articulators

As the initial components of a comprehensive physiological model of speech production, methods have been developed for modeling soft tissue structures, such as the tongue and lips, as continua. This approach provides a foundation for applying finite element methods to simulate these structures in a biomechanical model of speech production. Movements and deformations of the structures can then be computed as the solutions of a nonlinear second-order constraint system of ordinary differential equations, which is obtained from a finite element approximation of an energy rate equation. The muscle fibers in the soft tissue structures are represented as fields that specify the directions in which active and passive tensile stress is produced. The elastic behavior of the passive components is modeled using an isotropic exponential constitutive strain energy function, and the viscous stress components, by using linear viscosity. Incompressibility is maintained elementwise in the simulation using a simultaneously solved system for the computation of Lagrange multipliers. As first step towards modeling a vocal tract, a preliminary model of a tongue with eight muscles has been completed. Simulation results demonstrate the validity of the method and they support the feasibility of a physiologically based model of speech production.

An experimental method for evaluating constitutive models of myocardium in in vivo hearts

A new experimental method for the evaluation of myocardial constitutive models combines magnetic resonance (MR) radiofrequency (RF) tissue-tagging techniques with iterative two-dimensional (2-D) nonlinear finite element (FE) analysis. For demonstration, a nonlinear isotropic constitutive model for passive diastolic expansion in the in vivo canine heart is evaluated. A 2-D early diastolic FE mesh was constructed with loading parameters for the ventricular chambers taken from mean early diastolic-to-late diastolic pressure changes measured during MR imaging. FE solution was performed for regional, intramyocardial ventricular wall strains using small-strain, small-displacement theory. Corresponding regional ventricular wall strains were computed independently using MR images that incorporated RF tissue tagging. Two unknown parameters were determined for an exponential strain energy function that maximized agreement between observed (from MR) and predicted (from FE analysis) regional wall strains. Extension of this methodology will provide a framework in which to evaluate the quality of myocardial constitutive models of arbitrary complexity on a regional basis.

Measurement of strain and analysis of stress in resting rat left ventricular myocardium

A technique has been developed for measuring two-dimensional strains in the left ventricle of the isolated arrested rat heart subjected to passive ventricular loading. The pressure-volume relationship was found in eight hearts during inflation of a left ventricular balloon. With the zero-pressure state as reference, in-plane strain components were determined using a triangle of ultrasonic dimension transducers (0.6–0.8 mm diameter) placed 3–6 mm apart in the midwall of the left ventricle. Mean circumferential (fiber) strain was larger than longitudinal (cross-fiber) strain (0.108±0.045, 0.055±0.045, respectively, at 11 mmHg), and shear strain (−0.048±0.029) was negative, consistent with left-handed torsion. The in-plane angle of greatest stretch was uniform with inflation (range= −26.5° to −34.5°). The equatorial region of the left ventricle was modeled with finite element analysis of a transversely isotropic thick-walled cylindrical shell subjected to internal loading and axial forces. The material parameters of an exponential strain energy function were optimized so that the least-squares difference between the predicted and the measured midwall strains was minimized. Material properties, stress and strain in the rat heart were compared to values predicted for the dog. In both species the tissue was stiffer in the fiber direction than in the cross-fiber direction. The ratio of fiber to cross-fiber stiffness was lower in the rat (2.50) than in the dog (5.24) at low loads and approximately equal at higher loads (1.63 and 1.39, respectively). The computational and experimental analyses showed that the larger shear strain and more nonuniform in-plane extension in the rat may be an indication of significantly different anisotropic material properties in these two species, and implies differences in the collagen ultrastructure.

An approach to quantification of biaxial tissue stress-strain data

Delineation of the mechanical properties of biologic tissues is one of the cornerstones of biomechanics. Abundant data from uniaxial tests exist but these cannot be extrapolated to describe three-dimensional properties of tissue. Biaxial stress-strain studies have been performed using skin, blood vessels and pericardium. Quantitative description of tissue properties in these studies has employed either polynomial or exponential strain-energy functions. Interpretation of these data, however, is difficult because of wide variability of the estimated coefficients of these functions. This variability has been attributed to experimental noise, numerical instabilities in the algorithms, or to strain-history dependence. No systematic method has been proposed to evaluate the variability. This paper describes a statistically based approach to assessing the sources of and accounting for variability of coefficients in describing biaxial stress-strain data. Our data are from canine pericardium subjected to various combinations of simultaneous biaxial stretching. We first determine a suitable strain-energy function with the least number of free parameters that will fit the data reasonably. We then perform residual analysis to see if standard statistical methods can be used to assess the variability. If not, we use a nonparametric method called bootstrapping that is suitable for assessing the uncertainty in the coefficients. Using a five-parameter exponential strain-energy function, pericardial tissue is found to be strain-history dependent and anisotropic. These findings cannot be attributed to either experimental noise or instability in the numerical algorithms.

Large deformation mechanics of the enucleated eyeball.

Large deformation of enucleated pig eyeballs under rigid cylindrical indenters was studied analytically and experimentally. The analytic model for the eyeball consists of a fluid-filled spherical membrane composed of an incompressible, elastic material with an exponential strain energy function. The Rayleigh-Ritz technique provided an approximate solution via a potential energy formulation. Comparison with results from tests on eyeballs and a water-filled rubber (Mooney-Rivlin) shell shows good agreement at large deflection, where membrane action dominates. Due to the highly nonlinear stress-strain relations for the sclera, the load remains relatively small until the indenter displacement approaches 40-60 percent of the eyeball radius, and then the load increases rapidly. Depending on the indenter size, either a perforation or a rupture type of failure occurs.