Isochoric Part Research Articles

A robust anisotropic hyperelastic formulation for the modelling of soft tissue

The Holzapfel–Gasser–Ogden (HGO) model for anisotropic hyperelastic behaviour of collagen fibre reinforced materials was initially developed to describe the elastic properties of arterial tissue, but is now used extensively for modelling a variety of soft biological tissues. Such materials can be regarded as incompressible, and when the incompressibility condition is adopted the strain energy Ψ of the HGO model is a function of one isotropic and two anisotropic deformation invariants. A compressible form (HGO-C model) is widely used in finite element simulations whereby the isotropic part of Ψ is decoupled into volumetric and isochoric parts and the anisotropic part of Ψ is expressed in terms of isochoric invariants. Here, by using three simple deformations (pure dilatation, pure shear and uniaxial stretch), we demonstrate that the compressible HGO-C formulation does not correctly model compressible anisotropic material behaviour, because the anisotropic component of the model is insensitive to volumetric deformation due to the use of isochoric anisotropic invariants. In order to correctly model compressible anisotropic behaviour we present a modified anisotropic (MA) model, whereby the full anisotropic invariants are used, so that a volumetric anisotropic contribution is represented. The MA model correctly predicts an anisotropic response to hydrostatic tensile loading, whereby a sphere deforms into an ellipsoid. It also computes the correct anisotropic stress state for pure shear and uniaxial deformations. To look at more practical applications, we developed a finite element user-defined material subroutine for the simulation of stent deployment in a slightly compressible artery. Significantly higher stress triaxiality and arterial compliance are computed when the full anisotropic invariants are used (MA model) instead of the isochoric form (HGO-C model).

Non-Debye heat capacity formula refined and applied to GaP, GaAs, GaSb, InP, InAs, and InSb

Characteristic non-Debye behaviors of low-temperature heat capacities of GaP, GaAs, GaSb, InP, InAs, and InSb, which are manifested above all in form of non-monotonic behaviors (local maxima) of the respective Cp(T)/T3 curves in the cryogenic region, are described by means of a refined version of a recently proposed low-to-high-temperature interpolation formula of non-Debye type. Least-mean-square fittings of representative Cp(T) data sets available for these materials from several sources show excellent agreements, from the liquid-helium region up to room temperature. The results of detailed calculations of the respective material-specific Debye temperature curves, ΘD(T), are represented in graphical form. The strong, non-monotonic variations of ΘD(T) values confirm that it is impossible to provide reasonable numerical simulations of measured Cp(T) dependences in terms of fixed Debye temperatures. We show that it is possible to describe in good approximation the complete Debye temperature curves, from the cryogenic region up to their definitive disappearance (dropping to 0) in the high temperature region, by a couple of unprecedented algebraic formulas. The task of constructing physically adequate prolongations of the low-temperature Cp(T) curves up to melting points was strongly impeded by partly rather large differences (up to an order of 10 J/(K·mol)) between the high-temperature data sets presented in different research papers and/or data reviews. Physically plausible criteria are invoked, which enabled an a priori rejection of a series of obviously unrealistic high-temperature data sets. Residual uncertainties for GaAs and InAs could be overcome by re-evaluations of former enthalpy data on the basis of a novel set of properly specified four-parameter polynomial expressions applying to large regions, from moderately low temperatures up to melting points. Detailed analytical and numerical descriptions are given for the anharmonicity-related differences of isobaric vs. isochoric (harmonic) parts of heat capacities. Relevant sets of empirical parameters and representative collections of heat capacity and Debye temperature values for all materials under study are presented in tabulated form.

On a new model for inhomogeneous volume growth of elastic bodies

In general, growth characterises the process by which a material increases in size by the addition of mass. In dependence on the prevailing boundary conditions growth occurs in different, often complex ways. However, in this paper we aim to develop a model for biological systems growing in an inhomogeneous manner thereby generating residual stresses even when growth rates and material properties are homogeneous. Consequently, a descriptive example could be a body featuring homogeneous, isotropic material characteristics that grows against a barrier. At the moment when it contacts the barrier inhomogeneous growth takes place. If thereupon the barrier is removed, some types of bodies keep the new shape mainly fixed. As a key idea of the proposed phenomenological approach, we effort the theory of finite plasticity applied to the isochoric part of the Kirchhoff stress tensor as well as an additional condition allowing for plastic changes in the new grown material, only. This allows us to describe elastic bodies with a fluid-like growth characteristic. Prominent examples are tumours where the characteristic macro mechanical growth behaviour can be explained based on cellular arguments. Finally, the proposed framework is embedded into the finite element context which allows us to close this study with representative numerical examples.

Constitutive model for brain tissue under finite compression

While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10s–1 ramp rate and 20s hold time has been developed. The assumption of quasi-linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C10, C01 and C11) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06s and 14s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0s−1) and the results in compression and their extrapolation to tension were compared against previous models.

A finite strain Eulerian formulation for compressible and nearly incompressible hyperelasticity using high‐order B‐spline finite elements

SUMMARYWe present a numerical formulation aimed at modeling the nonlinear response of elastic materials using large deformation continuum mechanics in three dimensions. This finite element formulation is based on the Eulerian description of motion and the transport of the deformation gradient. When modeling a nearly incompressible solid, the transport of the deformation gradient is decomposed into its isochoric part and the Jacobian determinant as independent fields. A homogeneous isotropic hyperelastic solid is assumed and B‐splines‐based finite elements are used for the spatial discretization. A variational multiscale residual‐based approach is employed to stabilize the transport equations. The performance of the scheme is explored for both compressible and nearly incompressible applications. The numerical results are in good agreement with theory illustrating the viability of the computational scheme. Copyright © 2011 John Wiley & Sons, Ltd.

Computational modeling of passive myocardium

AbstractThis work deals with the computational modeling of passive myocardial tissue within the framework of mixed, non‐linear finite element methods. We consider a recently proposed, convex, anisotropic hyperelastic model that accounts for the locally orthotropic micro‐structure of cardiac muscle. A coordinate‐free representation of anisotropy is incorporated through physically relevant invariants of the Cauchy–Green deformation tensors and structural tensors of the corresponding material symmetry group. This model, which has originally been designed for exactly incompressible deformations, is extended towards entirely three‐dimensional inhomogeneous deformations by additively decoupling the strain energy function into volumetric and isochoric parts along with the multiplicative split of the deformation gradient. This decoupled constitutive structure is then embedded in a mixed finite element formulation through a three‐field Hu–Washizu functional whose simultaneous variation with respect to the independent pressure, dilatation, and placement fields results in the associated Euler–Lagrange equations, thereby minimizing the potential energy. This weak form is then consistently linearized for uniform‐pressure elements within the framework of an implicit finite element method. To demonstrate the performance of the proposed approach, we present a three‐dimensional finite element analysis of a generic biventricular heart model, subjected to physiological ventricular pressure. The parameters employed in the numerical analysis are identified by solving an optimization problem based on six simple shear experiments on explanted cardiac tissue. Copyright © 2010 John Wiley & Sons, Ltd.

A visco-hyperelastic model for skeletal muscle tissue under high strain rates

In this paper, a visco-hyperelastic skeletal muscle model is developed. The constitutive relation is based on the definition of a Helmholtz free energy function. It is assumed that the Helmholtz energy can be decomposed into volumetric and isochoric parts; furthermore, the isochoric energy can be decoupled into hyperelastic and viscous parts. The model developed involves 14 material parameters and its performance is evaluated by comparing the finite element simulation results with the published experimental studies on the New Zealand white rabbit tibialis anterior muscle. Results show that this model is able to describe the visco-hyperelastic behaviour of both passive and active skeletal muscle tissues under high strain rates (10/s and 25/s).

Geometrically nonlinear analysis of hyperelastic solids by high-order tetrahedral finite elements

The purpose of this study is to develop a computer code for accurate prediction of the mechanical behavior of hyperelastic solids under large deformation and finite strains. It is used, in the present work, the Lagrangian positional version of the Finite Element Method, in which the degrees of freedom are current positions instead of displacements. The main mechanical variables are the Green-Lagrange strain and the second Piola-Kirchhoff stress tensors. Isoparametric tetrahedral solid finite element of any approximation degree is employed with full integration in order to obtain accuracy. In order to be general, it is developed a numerical strategy to determine the shape functions coefficients of any order, following tetrahedral basis. The solid equilibrium is achieved at the current position by means of the Minimal Potential Energy Principle regarding positions and is solved by the Newton-Raphson iterative scheme. It is important to mention that the adopted constitutive laws are nonlinear, isotropic and homogeneous hyperelastic, normally used in mechanical analysis of elastomers, with the near compressibility assumption. It is applied the Flory decomposition, i.e., the multiplicative split of the deformation gradient into a volumetric and an isochoric part. A parallel processing strategy is used to increase the simulation speed and memory capacity, justifying the use of high order elements for solid analysis. Results confirm that the developed computer code is capable of predicting the static behavior of complex problems, such as Cook's membrane and partially loaded block. The shape functions generator code exhibits simplicity and, thus, it may be easily implemented in other finite elements. The increase in the order of approximation and the mesh refinement improve accuracy and, therefore, the proposed methodology together with parallel computing indicate a safe way for further developments in plasticity and damage mechanics for large strain and deformations.

A novel constitutive model of skeletal muscle taking into account anisotropic damage

The purpose of this study is to develop a constitutive model of skeletal muscle that describes material anisotropy, viscoelasticity and damage of muscle tissue. A free energy function is described as the sum of volumetric elastic, isochoric elastic and isochoric viscoelastic parts. The isochoric elastic part is divided into two types of shear response and the response in the fiber direction. To represent the dependence of the mechanical properties on muscle activity, we incorporate a contractile element into the model. The viscoelasticity of muscle is modeled as a three-dimensional model constructed by extending the one-dimensional generalized Maxwell model. Based on the framework of continuum damage mechanics, the anisotropic damage of muscle tissue is expressed by a second-order damage tensor. The evolution of the damage is assumed to depend on the current strain and damage. The evolution equation is formulated using the representation theorem of tensor functions. The proposed model is applied to the experimental data on tensile mechanical properties in the fiber direction and the compression properties in the fiber and cross-fiber directions in literature. The model can predict non-linear mechanical properties and breaking points.

Anisotropic stress-softening model for compressible solids

A general constitutive theory for anisotropic stress softening in compressible solids is presented. The constitutive equation describes anisotropic strain induced behaviour of an initially “isotropic” virgin material. Parameters which characterise damage are proposed together with a concept of damage function. In order to develop an anisotropic stress-softening theory for compressible materials in close parallel to a recent incompressible anisotropic theory, the right stretch tensor is decomposed into its isochoric and dilatational parts. The ’free’ energy is expressed as a function of the dilatation, modified principal stretches, a volume change parameter and invariants of the dyadic products of the principal directions of the right stretch tensor and two structural tensors. A class of free energy functions is discussed and a special form of this class which satisfies the Clausius–Duhem inequality is proposed. Results of the theory applied to uniaxial tension, bulk compression and simple shear deformations are given. A sequence of deformations involving shear, hydrostatic-compression and hydrostatic-tension deformations is also investigated. In the case of hydrostatic-tension deformation, the stress softening is due to cavitation damage. The theoretical results obtained are consistent with expected behaviour and compare well with experimental data.

On the Mullins effect and hysteresis of fibered biological materials: A comparison between continuous and discontinuous damage models

Deformation induced softening is an inelastic phenomenon frequently accompanying mechanical behavior of soft biological tissues. This paper presents and compare continuous and discontinuous damage approaches to model softening effects in fibered materials such as soft biological tissues. The structural model is formulated using the concept of internal variables that provides a very general description of materials involving irreversible effects. We consider the internal variables associated to damage to correspond to separated contributions of the matrix and the fibers. Local damage accumulation is related to two phenomenological variables, the maximum value and the arclength of the effective free energy attained during the loading process, respectively. A local multiplicative decomposition of the deformation gradient into volume-preserving and dilatational parts is used that permits to model the incompressibility property of most types of soft biological tissues. In this context, damage is related only to the isochoric part of the deformation. Finally, simulations of biaxial and uniaxial tests in two directions are used to compare the performance of both models. Numerical simulations indicate that only a mixed model that consider both, continuous and discontinuous, damage models is able to capture the softening phenomenon of soft biological tissues.

Quantization of two-dimensional affine bodies with stabilized dilatations

Discussed are some quantization problems of two-dimensional affine bodies. Quantum dilatational motion is stabilized by some appropriately chosen model potentials. Isochoric part of the dynamics is geodetic, i.e. potential-free. Surprisingly enough, this is compatible with the existence of discrete spectrum (bounded quantum motion). The Sommerfeld polynomial method is used to perform the quantization of such problems.

Computational aspects of a pseudo-elastic constitutive model for muscle properties in a soft-bodied arthropod

In this paper we discuss the computational implementation of a new constitutive model that describes the muscle properties in a soft-bodied arthropod. Qualitatively, the muscle tissues behave similar to particle-reinforced rubber and are capable of large non-linear elastic deformations, show a hysteretic behavior, and display stress softening during the first few cycles of repeated loading. Such behavior can be described by the framework of pseudo-elastic transversely isotropic hyperelasticity. The computational model assumes compressible overall response, and is based upon a multiplicative split of the deformation gradient tensor into volumetric and isochoric parts. Details regarding the implementation of the computational model in the context of an implicit finite element solution procedure are presented. In particular, an explicit expression is provided for the material tangent stiffness tensor. Results obtained utilizing the new implementation are also presented.

On the physical assumptions underlying the volumetric-isochoric split and the case of anisotropy

This paper discusses the multiplicative decomposition of the deformation gradient into its volumetric and isochoric parts and its implications in the case of anisotropy. An analysis is carried out showing that the volumetric-isochoric split of the stored energy function can be justified and systematically derived on the basis of the physical assumption that the spherical part of the stress depends on the determinant of the deformation gradient without ad hoc introduction of the multiplicative split. The analysis shows that care must be exercised in the case of anisotropic material description in order not to violate certain physical requirements. Additive splits of the energy can be justified on the basis of certain physical observations and independent of the multiplicative decomposition of the deformation gradient. Specifically, it is shown that a spherical state of stress will cause even in the incompressible case, a change of shape. In fibre reinforced materials, the split of the stored energy function into a part related to the matrix and a part related to the fibre is considered, showing that the volumetric-isochoric split should be applied to the matrix part only.

On the development of compressible pseudo-strain energy density for elastomers: Part 2 application to finite element

This paper presents the results of a finite element procedure and analysis for pure gum rubber, based on a compressible strain energy function proposed by the authors. The development of the proposed separable, one-term compressible constitutive equations is explained. An explanation of the isochoric part of the strain energy function based on constitutive relation proposed by the authors is also discussed. A brief review of material properties and the basic equations of the finite element procedure are given. Fortran 77 was used to code the user-subroutine that calculates the element's isochoric tangent stiffness and isochoric first Piola-Kirchhoff stress from the proposed one-term strain energy density function. The subroutine was then implemented into a commercial finite element package. The Updated Lagrangian (UL) procedure was used, based on the principal stretch ratios in which the material moduli are initially determined in the principal direction and then transformed to the local coordinate. The results of the finite element analysis based on the Ogden and the proposed models were compared against the experimental data. It is shown that although the Ogden model produced an accurate result, the proposed model improves the results on two fronts. Firstly, the accuracy of the results is slightly increased in the uniaxial case and more noticeably so in the equibiaxial analysis. Secondly, the proposed model improved the analysis run-time by approximately 50%. In short, it was shown that the proposed one-term model exhibited a better representation of the material behavior.

A structural model for the viscoelastic behavior of arterial walls: Continuum formulation and finite element analysis

In this paper we present a two-layer structural model suitable for predicting reliably the passive (unstimulated) time-dependent three-dimensional stress and deformation states of healthy young arterial walls under various loading conditions. It extends to the viscoelastic regime a recently developed constitutive framework for the elastic strain response of arterial walls (see Holzapfel et al. (2001)). The structural model is formulated within the framework of nonlinear continuum mechanics and is well-suited for a finite element implementation. It has the special merit that it is based partly on histological information, thus allowing the material parameters to be associated with the constituents of each mechanically-relevant arterial layer. As one essential ingredient from the histological information the constitutive model requires details of the directional organization of collagen fibers as commonly observed under a microscope. We postulate a fully automatic technique for identifying the orientations of cellular nuclei, these coinciding with the preferred orientations in the tissue. The biological material is assumed to behave incompressibly so that the constitutive function is decomposed locally into volumetric and isochoric parts. This separation turns out to be advantageous in avoiding numerical complications within the finite element analysis of incompressible materials. For the description of the viscoelastic behavior of arterial walls we employ the concept of internal variables. The proposed viscoelastic model admits hysteresis loops that are known to be relatively insensitive to strain rate, an essential mechanical feature of arteries of the muscular type. To enforce incompressibility without numerical difficulties, the finite element treatment adopted is based on a three-field Hu-Washizu variational approach in conjunction with an augmented Lagrangian optimization technique. Two numerical examples are used to demonstrate the reliability and efficiency of the proposed structural model for arterial wall mechanics as a basis for large scale numerical simulations.

Algorithms for computation of stresses and elasticity moduli in terms of Seth–Hill's family of generalized strain tensors

AbstractThe paper discusses algorithms for the computation of stresses and elasticity moduli for stored energy functions which are given in terms of a Seth–Hill‐type generalized strain measure. The key contribution is distinct computational representations of a chain rule representation for the stresses and moduli which may serve as a basic tool for an effective numerical implementation of complex elasticity models. The representations are formulated in the Lagrangian geometric setting based on a spectral decomposition of the right Cauchy–Green tensor and contain particular representations for the case of equal eigenvalues. Two algorithmic boxes are pointed out for three‐ and two‐dimensional problems, respectively. Furthermore, a distinct algorithmic setting is developed for applications of the Seth–Hill‐type measures to the isochoric part of the deformation based on a spectral decomposition of the unimodular part of the right Cauchy–Green tensor. The computation of the stress response in connection with generalized Hookean laws is considered as a simple application. Copyright © 2001 John Wiley & Sons, Ltd.

Easy-to-Compute Tensors With Symmetric Inverse Approximating Hencky Finite Strain and Its Rate

It is shown that there exist approximations of the Hencky (logarithmic) finite strain tensor of various degrees of accuracy, having the following characteristics: (1) The tensors are close enough to the Hencky strain tensor for most practical purposes and coincide with it up to the quadratic term of the Taylor series expansion; (2) are easy to compute (the spectral representation being unnecessary); and (3) exhibit tension-compression symmetry (i.e., the strain tensor of the inverse transformation is minus the original strain tensor). Furthermore, an additive decomposition of the proposed strain tensor into volumetric and deviatoric (isochoric) parts is given. The deviatoric part depends on the volume change, but this dependence is negligible for materials that are incapable of large volume changes. A general relationship between the rate of the approximate Hencky strain tensor and the deformation rate tensor can be easily established.

The simple tension problem at large volumetric strains computed from finite hyperelastic material laws

In this paper, different formulations of finite isotropic hyperelastic material laws for compressible solids are considered. Material laws with an additive split of the hyperelastic strain energy function into isochoric parts and volumetric parts are often used in the numerical treatment of nearly incompressible solids. It will be shown that this formulation leads to unphysical results in the simple tension problem when we do not restrict ourselves to nearly incompressible materials.

ON LARGE STRAIN VISCOELASTICITY: CONTINUUM FORMULATION AND FINITE ELEMENT APPLICATIONS TO ELASTOMERIC STRUCTURES

This article deals with the continuum formulation of finite strain viscoelasticity and provides its numerical simulation with the finite element method. In particular, elastomeric solids which are of essential engineering interest are discussed. In order to simulate the significant different bulk/shear-response of polymeric media the deformation is decomposed into volumetric elastic and isochoric viscoelastic parts. The constitutive equations are presented within the context of internal variable models and a Lagrangian kinematical description is adopted throughout. For sufficiently slow processes the material responds in a rubbery elastic manner which is assumed to be modelled with an Ogden-type strain energy function well-known from rubber elasticity. The stresses and the symmetric consistent tangent moduli are briefly discussed along with a second-order approximation of the constitutive rate equation. The main thrust of this paper on the computational side is to show the meaningful time-dependent behaviour and the general applicability of the three-dimensional constitutive model. By applying assumed enhanced strain elements which are well-suited for (nearly) incompressible problems three representative numerical examples illustrate relaxation and creeping phenomena at large strains and the equilibrium finite elastic response, which is asymptotically obtained.