Hyperelastic Body Research Articles

Restoration of the symmetries broken by reversible growth in hyperelastic bodies

In this paper, we interpret the development of material in homogeneities in continuum, hyper elastic bodies in the presence of reversible growth in terms of broken symmetries [1]. By applying Noether's Theorem [1, 2, 3, 4], we find a set of equations yielding the fields necessary to compensate for the broken symmetry. As growth occurs, these fields provide for an instantaneously updated reference configuration of the body, and are responsible for the dynamical restoring of the body symmetries. In addition, we propose to use these compensating fields in order to generalize the definition of the transplant operator given in [5,6]. This work has been motivated by the current theoretical investigations on the biomechanical aspects of growth in particular cartilage.

CONSTITUTIVE MODEL OF HIGH DAMPING RUBBER MATERIALS

A mathematical model of high-damping rubber materials is developed. First the material experiments necessary for modeling are systematically conducted. Then, on the basis of the results of the material experiments, a constitutive model for rubber materials is proposed. The model is decomposed into two parts. The first part consists of an elastoplastic body with a strain-dependent isotropic hardening law and it represents the energy dissipation of the material, while the second part consists of a hyperelastic body with a damage model and it expresses the evolutional direction of the stress tensor. By comparing the experimental results with the simulation by the model, the model is found to well approximate the behaviors of high-damping rubber materials. Finally, a hybrid analysis method is proposed. In this method, the strain field of laminated rubber bearings measured by image processing is combined with the numerical analysis to confirm the applicability of the proposed model to the bearing. In addition, by this hybrid analysis method, the bulk modulus of rubber material is also computed.

Updated Lagrangian Approach of Stress Rate and Finite Element Formulations of the Hyperelastic Body.

In this paper, we propose updated Lagrangian formulation of hyperelastic body, which has been formulated by total Lagrangian approach, and confirm the accuracy and stability of developed finite element analysis code. The referenced configuration of proposed hyperelastic constitutive equation consists with the configuration of elasto-plasticity for finite deformation by applying updated Lagrangian approach. So, both of constitutive equations can be treated simultaneously in analysis. Numerical results by updated Lagrangian formulation show higher stability under large deformation.

Unilateral contact with adhesion and friction between two hyperelastic bodies

In the present paper, we consider a thermodynamic model using the contact kinematics developed by A. Curnier, Q.C. He and J.J. Téléga [C. R. Acad. Sci. Paris Sér. II 314 (1992) 1] involving unilateral contact, adhesion and Coulomb friction between two homogeneous, isotropic and hyperelastic bodies. Adhesion is described by an internal state variable β ϕ introduced by M. Frémond [C. R. Acad. Sci. Paris Sér. II 295 (1982) 913; J. Theor. Appl. Mech. 6 (1987) 383]. Taking the case of contact between a hyperelastic solid and a plane support, we formulate the associated boundary value problem as a minimization problem when no friction is involved. When the intensity of the adhesion obeys a `static' law, we obtain an existence result for this problem.

Stability of deformation of isotropic hyperelastic bodies

The equations relating stress rates to strain rates are formulated and conditions of stable deformation of isotropic hyperelastic bodies are described. Stress–strain relations are presented for pure shear and uniaxial and axisymmetric loading of a material with a constitutive function obtained by generalization of the constitutive function of Hooke's law. In the case of small strains, the diagrams virtually coincide with the linear diagrams following from Hooke's law. Ramification of solutions and transition to declining diagrams begin at the same time, irrespective of values of the constants of the material, when large stresses of the order of the shear modulus are reached.

Formulation du contact avec adhérence en élasticité non linéaire entre deux solides déformables

Within the framework of finite deformations and using an approach to the kinematics of contact due to A. Curnier, Q.C. He and J.J. Téléga, we propose a spatial thermodynamic formulation for the problem coupling unilateral condition, friction and adhesion. The adhesion is characterized by its intensity introduced by M. Frémond. In the case of frictionless contact between an hyperelastic body and a plane rigid support, with a particular `static' law for the evolution of the intensity of adhesion, the problem can be reduced to a minimization one for which we can show the existence of a solution.

Constitutive Equations of an Isotropic Hyperelastic Body

Stress—strain equations for an isotropic hyperelastic body are formulated. It is shown that the strain energy density whose gradient determines stresses can be defined as a function of two rather than three arguments, namely, strain–tensor invariants. In the case of small strains, the equations become relations of Hooke's law with two material constants, namely, shear modulus and bulk modulus.

Finite element analysis and constitutive modelling of anisotropic nonlinear hyperelastic bodies with convected frames

The constitutive equation for anisotropic nonlinear hyperelastic materials is developed with general curvilinear material coordinates. Also, adequate finite element formulation with the same coordinate systems is developed. The formulation is based on hyperelastic theory of the material and mixed finite element method. To demonstrate the applicability of the developed constitutive equation and the method, several numerical examples are solved and presented. The present scheme could be effectively used in the analysis of the bodies of composites such as biological tissues and reinforced rubber-like composite materials.

Rotations which Make Strain and Stress Coaxial

At a material point of a hyperelastic anisotropic body the strain and the stress tensors do not share, in general, a common basis of proper vectors. It is shown that there are at least three rotations which, when applied to the reference configuration before a given deformation, yield a pair of coaxial strain and stress tensors. This is an improvement on a previous result, which showed such rotations to be no fewer than two. The proof is based on the application of Milnor's theorem from differential topology.

Directions of Coaxiality between Pure Strain and Stress in Linear Elasticity

For a linear anisotropic hyperelastic body subjected to a state of pure strain it is shown that there are at least three directions such that the stress and strain tensors are coaxial.

混合モード荷重下にある異方弾性体内のき裂のエネルギ解放率の数値解析

For a crack in an anisotropic elastic body subjected to remote inclined load under a plane stress condition, the energy release rate at the onset of crack kinking is analyzed by numerical analysis. The analysis is based on the path-independent E-integral using the finite element method. The E-integral gives the energy release rate at the onset of crack kinking for hyperelastic bodies. The eight-noded and six-noded isoparametric finite elements are used and the integral paths lie along the sides of the finite element. The numerical integration of the E-integral formula can be evaluated directly by using the nodal forces and nodal displacements. After the path independency is examined in an isotropic elastic body subjected to the remote constant tension stress, it is shown that the results for inclined loads agree very well with the results by Wu (1978). For the anisotropic bodies, in the cases of different planes of symmetry, the energy release rate is computed first under the perpendicular loading to the carck surface and then under the inclined loading. For the former, the results agree well with the perturbation solutions by Gao and Chiu (1992). For the inclined load, the direction of the maximum energy release rate is more sensitive to the direction of loading than the direction of the plane of symmetry.

Virtual Configurations and Constitutive Equations for Residually Stressed Bodies with Material Symmetry

The purpose of this paper is to construct the general forms for constitutive equations for residually stressed hyperelastic bodies that are composed of material with a specified symmetry. The effective mechanical properties of a material comprising a residually stressed body are dependent on both the intrinsic material properties of the underlying natural material and on the residual stress field itself. Thus, prediction of the mechanical behavior of a residually stressed body typically requires a constitutive model that explicitly includes the influence of the residual stress for deformations out of the residually stressed configuration. In this paper, constitutive equations are derived for the cases where the underlying natural material is transversely isotropic or orthotropic, two material symmetries of special interest in the description of biological tissues. In addition, it is established that the method used in the derivation can be applied to materials with any symmetry for which an integrity basis is known. The derivation requires that the constitutive equation for the underlying natural material be known and that the restriction of the response function to the set of positive definite symmetric tensors be locally invertible. The resulting constitutive equations are expressed as functions of the residual stress and the deformation gradient out of the residually stressed configuration. A major advantage of the method is that the only material parameters in the constitutive equation are those of the underlying stress free material.

A note on inhomogeneous deformations of nonlinear elastic layers

A class of inhomogeneous deformations is studied in the framework of the nonlinear theory of hyperelastic incompressible bodies whose material properties can depend on the material point in a particular way. Applications of the results obtained to finite thermoelasticity are given.

Fracture of an infinite incompressible hyperelastic material under compression along a cylindrical crack

For problems concerning fracture of materials under compression along crack, we will use the fracture criteria proposed earlier in [2], based on a mechanism for loss of stability near defects of the crack type in the initial stage of fracture. A quantitative expression for the indicated criterion is the critical values of the loading parameters (contractions along the axes, compressive stresses), character~ing the local instability near cracks. In this case, the critical loads are determined by using the relations of the three-dimensional linearized stability theory in [4]. In [5], a general approach is suggested for such problems. This approach to investigation of the initial stage of fracture within the linearized theory of deformable bodies currently seems to be most acceptable both because of its sufficient rigor and because of its generality for different models of materials. In [6, 7], a review is given of papers in which problems have been solved in the linearized formulation for different geometric schemes for the location of cracks, and values have been obtained for the breaking loads for different models of the materials. However, we should note that within the indicated approach, up to now only those problems have been investigated in which cracks in the material are located exclusively in planes along which compression of the material occurs [8]. In this paper, we investigate the question of fracture of an infinite material containing a crack of finite dimensions, located on a circular cylindrical surf.ace, under axial compression conditions. As an example, we consider such a crack in an incompressible hyperelastic body, i.e., in a body for which there exists an elastic potential which is specified as a function of the invariants 9 (11, 12) = ~[ll(Ai, Az), I2(A,, A2)] =" W(At, Az) [3]. We will model the crack, as in fracture mechanics, as a mathematical discontinuity. Furthermore, we will consider only the initial stage of fracture, and contact of the crack edges is not considered here. We also note that relatively few investigations have been carried out for the considered crack geometry even in the linear formulation [10-14]. 1. We will use a cylindrical coordinate system (r, 0, x3). In this coordinate system, let a circular cylindrical crack of radius b and length 2a occupy the region {r = b, 0 < 0 < 2x, a ~ x3 < a}. In the case of compression of the material along the directrixes of the cylindrical crack, a homogeneous subcritical stress-strain state arises in the material [4]

A quasi-variational problem in nonlinear elasticity

ω⊂R3 and Γ ⊂R2 are bounded, connected, Lipschitz open sets. v: Γ→R is the vertical displacement of an elastic membrane stretched on Γ and fixed at the boundary. The condition\(\int\limits_\Omega {det\nabla \psi } dx = vol\psi (\Omega )\) is imposed on the admissible deformations ψ:ω→R3 of a hyperelastic body whose reference configuration is θ. The additional constraint ψ3(x)≥v(ψ1,2(x)), forcing the body to stay above the membrane, is relaxed in order to show the existence of a minimizer of total energy of the mechanical sistem.

On continuum theories involving quasilinear first order non-conservative systems with involutions and a supplementary inequality

In this note we give a result of hyperbolicity for linearized quasilinear first ordernon-conservative systems with involutions and a supplementary inequality of conservative form. We also indicate some of the circumstances in which, for studying the evolution systems for a class of continua near their equilibria, such result can be applied. Some remarkable examples are the cases of a certain generalization of Landau's two-fluid scheme for superftuid Helium II, of a hyperelastic body acted upon by suitable live body forces, and of various other models appeared in the literature.

Variational principles of nonlinear fracture mechanics

A variational principle of total energy is formulated for finite strain statics of a hyperelastic body whose initial configuration contains a gap. From this principle statical equations and boundary conditions for the gapped body are derived. The equilibrium condition at a gap tip is associated with the well-knownJ-integrals. By including the reaction of inertia and the flux of kinetic energy the principle of total energy is transformed to the variational inequality of evolution for dynamics of a hyperelastic body with a propagating crack. A closed system of dynamical equations, boundary conditions and additional conditions on the unknown contact crack surfaces and crack tip is obtained. As example the antiplane shear of an infinite gapped body is considered.

Homogenization of fissured elastic solids in the presence of unilateral conditions and friction

The objective of this contribution is to find effective properties of hyperelastic bodies weakened by periodically distributed microfissures. Problems investigated are internal Signorini's problems with friction. Two such problems have been studied. The first problem is purely static and the friction law is of the deformational plasticity type. To find the overall properties an implicit variational inequality has been homogenized. The second problem concerns homogenization in the quasi-static case and the sliding rule of the flow law type. The variational formulation is obtained in the form of an implicit variational inequality coupled with a variational inequality. In both cases the macroscopic behaviour is elastic-plastic of nonstandard type.

A penalty approximation for a unilateral contact problem in non‐linear elasticity

AbstractUsing Ball's approach to non‐linear elasticity, and in particular his concept of polyconvexity, we treat a unilateral three‐dimensional contact problem for a hyperelastic body under volume and surface forces. Here the unilateral constraint is described by a sublinear function which can model the contact with a rigid convex cone. We obtain a solution to this generally non‐convex, semicoercive Signorinin problem as a limit of solutions of related energy minimization problems involving friction normal to the contact surface where the friction coefficient goes to infinity. Thus we extend an approximation result of Duvaut and Lions for linear‐elastic unilateral contact problems to finite deformations and to a class of non‐linear elastic materials including the material models of Ogden and of Mooney‐Rivlin for rubberlike materials.Moreover, the underlying penalty method is shown to be exact, that is a sufficiently large friction coefficient in the auxiliary energy minimization problems suffices to produce a solution of the original unilateral problem, provided a Lagrange multiplier to the unilateral constraint exists.

Injectivity and self-contact in nonlinear elasticity

Let Ω be a bounded open connected subset of ℝ3 with a sufficiently smooth boundary. The additional condition ∫ det ▽ψ dx ≦ vol ψ(Ω) is imposed on the admissible deformations ψ: ¯Ω → ℝ of a hyperelastic body whose reference configuration is ¯Ω. We show that the associated minimization problem provides a mathematical model for matter to come into frictionless contact with itself but not interpenetrate. We also extend J. Ball's theorems on existence to this case by establishing the existence of a minimizer of the energy in the space W1,p(Ω;ℝ3), p > 3, that is injective almost everywhere.