Constitutive model for a class of hyperelastic materials with embedded rheological properties

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.

Logarithmic rate based elasto-viscoplastic cyclic constitutive model for soft biological tissues

This paper presents a three-dimensional finite element model of skeletal muscle and its validation incorporating inital tissue strains. A constitutive relation was determined by using a convex free strain energy function (SEF) where active and passive response contributions were obtained fitting experimental data from the rat tibialis anterior (TA) muscle. The passive and active finite strains response was modelled within the framework of continuum mechanics by a quasi-incompressible transversely isotropic material formulation. Magnetic resonance images (MRI) were obtained to reconstruct the external geometry of the TA. This geometry includes initial strains also taken into account in the numerical model. The numerical results show excellent agreement with the experimental results when comparing reaction force-extension curves both in passive and active tests. The proposed constitutive model for the muscle is implemented in a subroutine in the commercial finite element software package ABAQUS.

In general, materials with linear memory constitutive relations are characterized by a relaxation function. This leads to a situation where the free energy for most materials with memory is not unique. There is a convex set of free energy functionals with a minimum and a maximum element. An alternative procedure is proposed which characterizes a material by the kernel of the rate of dissipation functional. Using some recent results, we find that a unique free energy and relaxation function may then be deduced. An example is given for discrete spectrum materials. Also, the new results are used to show that a previously derived general representation of rate of dissipation and free energy functionals is not complete, in the sense that there are valid functionals which cannot be described by this general formula.

Hencky's strain-energy function for finite isotropic elasticity is obtained by the replacement of the infinitesimal strain measure occurring in the classical strain-energy function of infinitesimal isotropic elasticity with the Hencky or logarithmic strain measure. It has been shown recently by Anand that this simple strain-energy function, with two classical Lameacute; elastic constants, is in good agreement with a wide class of materials for moderately large deformations. Very recently, it has been shown by these authors that the hyperelastic relation with the foregoing Hencky strain–energy function may enter as a basic constituent into the Eulerian rate formulation of finite elastoplasticity for metals, etc. Now it is commonly used in finite–element method (FEM) computations and in commercial packets of FEM codes, etc. For this useful strain–energy function, there is a need to study the restrictions and consequences resulting from certain well–founded constitutive inequality conditions. Here, we consider the well–known Legendre–Hadamard, or the ellipticity,condition. We first derive simple, explicit necessary and sufficient conditions for ellipticity in terms of the principal stretches. Then we determine the largest common region for ellipticity in the principal stretch space, which applies to all Hencky strain–energy functions with non–negative Lame constants. In particular, we find out the largest cube contained in the common region just mentioned. We prove that the Hencky strain–energy function fulfils the Legendre–Hadamard condition whenever every principal stretch falls within the range[α,3√e], where the lower bound α= 0.211 62 .. . is the unique root of a certain transcendental equation involving the natural logarithmic function, and e = 2.718 28 ... , in the upper bound, is the base of the natural logarithm. The range mentioned above, i.e. [0.211 62,1.395 61], covers the range [0.7, 1.3] set by Anand for moderately large deformations. Moreover, it is shown that Hencky's strain–energy function obeys the well–known Baker-Ericksen inequality and Hill's inequality over the whole range of deformations.

Acoustoelastic analysis has usually been applied to compressible engineering materials. Many materials (e.g., rubber and biologic materials) are “nearly” incompressible and often assumed incompressible in their constitutive equations. These material models do not admit dilatational waves for acoustoelastic analysis. Other constitutive models (for these materials) admit compressibility but still do not model dilatational waves with fidelity (shown herein). In this article a new strain energy function is formulated to model dilatational wave propagation in nearly incompressible, isotropic materials. This strain energy function requires four material constants and is a function of Cauchy–Green deformation tensor invariants. This function and existing (compressible) strain energy functions are compared based upon their ability to predict dilatational wave propagation in uniaxially prestressed rubber. Results demonstrate deficiencies in existing functions and the usefulness of our new function for acoustoelastic applications. Our results also indicate that acoustoelastic analysis has great potential for the accurate prediction of active or residual stresses in nearly incompressible materials.

A strain energy function which depends on five independent variables that have immediate physical interpretation is proposed for finite strain deformations of transversely isotropic elastic solids. Three of the five variables (invariants) are the principal stretch ratios and the other two are squares of the dot product between the preferred direction and two principal directions of the right stretch tensor. The set of these five invariants is a minimal integrity basis. A strain energy function, expressed in terms of these invariants, has a symmetry property similar to that of an isotropic elastic solid written in terms of principal stretches. Ground state and stress-strain relations are given. The formulation is applied to several types of deformations, and in these applications, a mathematical simplicity is highlighted. The proposed model is attractive if principal axes techniques are used in solving boundary-value problems. Experimental advantage is demonstrated by showing that a simple triaxial test can vary a single invariant while keeping the remaining invariants fixed. A specific form of strain energy function can be easily obtained from the general form via a triaxial test. Using series expansions and symmetry, the proposed general strain energy function is refined to some particular forms. Since the principal stretches are the invariants of the strain energy function, the Valanis-Landel form can be easily incorporated into the constitutive equation. The sensitivity of response functions to Cauchy stress data is discussed for both isotropic and transversely isotropic materials. Explicit expressions for the weighted Cauchy response functions are easily obtained since the response function basis is almost mutually orthogonal.

A new 3-D finite element for nonlinear elasticity using the theory of a Cosserat point

The Hencky medium with softening under isothermal and quasistatic deformation is considered. It is believed that volume deformation is not positive. In this case softening is characterized by the part of union curve with negative slope. For aforementioned conditions function of free energy is presented. For all stages of deformation in the space “volume deformation - intensity of shear’s deformation” level lines of the free energy are constructed. It is established that level lines are ellipses in hardening and function of free energy increases with distance from their centers, while in softening hyperbolas are level lines and function of free energy decreases with distance from their centers. Obtained results indirectly confirm the change in type of boundary value problem from elliptic to hyperbolic under material transition to the softening.

Within the theory of isothermal, homogeneous, isotropic, nonlinear elasticity, the selection of the appropriate form for the strain energy function is still an issue. Over the years, a number of proposals have been made for specific strain energy functions, some based on experimental data, others based on mathematical elegance or specific theoretical constraints. Because of differing formulations, it is difficult to compare strain energy functions. In this paper, methods first introduced by Haines and Wilson are used to compare incompressible strain energy functions, and these methods are applied to a number of models from the literature, including six which have been correlated with the 1944 data of Treloar. Comparison is made easier by the fact that the value of the strain energy function is relevant only over a limited region of invariant space, known as the attainable region. The comparison shows that four of the models predict similar strain energy functions up to moderate strains, but that they do not reproduce all of the characteristics of the original experimental data.

The vascular wall exhibits nonlinear anisotropic mechanical properties. The identification of a strain energy function (SEF) is the preferred method to describe its complex nonlinear elastic properties. Earlier constituent-based SEF models, where elastin is modeled as an isotropic material, failed in describing accurately the tissue response to inflation-extension loading. We hypothesized that these shortcomings are partly due to unaccounted anisotropic properties of elastin. We performed inflation-extension tests on common carotid of rabbits before and after enzymatic degradation of elastin and applied constituent-based SEFs, with both an isotropic and an anisotropic elastin part, on the experimental data. We used transmission electron microscopy (TEM) and serial block-face scanning electron microscopy (SBFSEM) to provide direct structural evidence of the assumed anisotropy. In intact arteries, the SEF including anisotropic elastin with one family of fibers in the circumferential direction fitted better the inflation-extension data than the isotropic SEF. This was supported by TEM and SBFSEM imaging, which showed interlamellar elastin fibers in the circumferential direction. In elastin-degraded arteries, both SEFs succeeded equally well in predicting anisotropic wall behavior. In elastase-treated arteries fitted with the anisotropic SEF for elastin, collagen engaged later than in intact arteries. We conclude that constituent-based models with an anisotropic elastin part characterize more accurately the mechanical properties of the arterial wall when compared to models with simply an isotropic elastin. Microstructural imaging based on electron microscopy techniques provided evidence for elastin anisotropy. Finally, the model suggests a later and less abrupt collagen engagement after elastase treatment.

There is a scarcity of data regarding the mechanical properties of the ureter, although this would facilitate our understanding of its physiology and pathophysiology, and the development of suitable biomaterials for replacement. There is hence an urgent need for multiaxial experimental data and methodical constitutive formulations, which we aim at presenting through this report. The zero-stress state of wall tissue, serving as the starting geometry for biomechanical analyses, was accordingly determined and the 3D passive behaviour of ureteral specimens, isolated from healthy rabbits, was studied under a physiologic range of finite inflation and longitudinal extension. Two most-commonly employed descriptors of soft tissue behaviour were chosen to fit the material response: the Fung-type strain–energy function (SEF) and its combination with a quadratic function. Both SEFs were tested in the thick-walled setting, with incompressibility enforced explicitly or via a Lagrange multiplier. The deformational response of the ureter exhibited an exponential and not the sigmoidal dependency on pressure that requests implementation of two-term SEFs. Indeed, the four-parameter Fung-type SEF resulted in reasonable fit of both the external radius and longitudinal force vs. lumen pressure data, and fitting accuracy was not improved when attempting the seven-parameter Fung-type or biphasic SEFs. There were also serious over-parameterisation problems with those models, favouring the implementation of the SEF with the smallest number of parameters. The material parameters optimised revealed significant mechanical anisotropy, with longitudinal properties being stiffer than circumferential ones under equibiaxial stress states. We conclude that ureter displays a nonlinear anisotropic mechanical response that is well-characterised by the four-parameter Fung-type SEF.

In Part I, methods for determining the strain energy function and the associated constitutive stress-deformation law for rubber-like materials is undertaken and the mechanics of data reduction needed to determine some parameters of the theory are displayed. Experiments were performed in four different stress fields on a foamed polyurethane rubber (dilatable rubber) and on several kinds of continuum rubbers. A new strain energy function and the associated stress-deformation law for a foamed rubber are generated which correlate most of the data to a high degree of accuracy. A parameter appearing in the functional expression for a foam rubber has the same significance as Poisson's ratio in infinitesimal elastic theory. For continuum rubbers, the isotropic Neo-Hookean representations of quasi-static behavior is found to be sufficient over most of the whole range of extension. In Part II, geometrical representations of an isotropic failure surface based on various criteria are depicted both in principal stress and principal stretch spaces for elastic materials. The experimental data are compared with all criteria and the results are discussed. In Part III, finite elastic theory is used to determine the stress and deformation fields around the base of a radial crack in an infinitely long rubber log opened by a facially bonded rigid wedge-shaped bellow. In the last Part, the topology of interstices idealized as closest packed spherical holes (idealized foam structure) is investigated. Equivalent elastic constants are calculated for rubbery interstices of both hexagonal and face-centered cubic closest packings under small displacement.

Constitutive properties of concrete under dynamic loading are necessary for the rational analysis of concrete structures subject to impact and impulsive loads. The constitutive model presented herein models microcracking through the use of a continuous damage parameter for which a vectorial representation is adopted. The rate of increase of the damage is dependent on the state of strain as well as on the time rate of strain. The constitutive equations are derived from the strain energy function which is influenced by the accumulated damage. The constitutive model is calibrated using uniaxial tension (or flexural) and uniaxial compression test data. The calibrated model is then used to predict certain other load responses of concrete.

The material modeling of hyper-elastic properties in rubber is generally characterized by the strain energy function. The strain energy functions have been represented either in term of the strain in variants that are functions of the stretch ratios, or directly in terms of the principal stretch. Successful modeling and design of rubber components relies on both the selection of an appropriate strain energy function and an accurate determination of material constants in the function. Material constants in the strain energy functions can be determined from the curve fitting of experimental stress-strain data. The uniaxial tension, equi-biaxial tension and pure shear test were performed to acquire the constants of the strain energy functions which were Mooney-Rivlin and Ogden model. Nonlinear finite element analysis was executed to evaluate the behavior of deformation and strain distribute by using the commercial finite element code. Also, the fatigue tests were carried out to obtain the fatigue failure. Fatigue failure was initiated at the critical location was observed during the fatigue test of rubber component, which was the same result predicted by the finite element analysis.