Topics in Finite Elasticity: Hyperelasticity of Rubber, Elastomers, and Biological Tissues—With Examples

1987 ◽  
Vol 40 (12) ◽  
pp. 1699-1734 ◽  
Author(s):  
Millard F. Beatty
Keyword(s):  
Constitutive Equations ◽  
Mechanical Energy ◽  
Finite Elasticity ◽  
Physical Nature ◽  
Decomposition Theorem ◽  
Elastic Stability ◽  
Biological Tissues ◽  
Field Equations ◽  
Energy Principle ◽  
Hyperelastic Materials

This is an introductory survey of some selected topics in finite elasticity. Virtually no previous experience with the subject is assumed. The kinematics of finite deformation is characterized by the polar decomposition theorem. Euler’s laws of balance and the local field equations of continuum mechanics are described. The general constitutive equation of hyperelasticity theory is deduced from a mechanical energy principle; and the implications of frame invariance and of material symmetry are presented. This leads to constitutive equations for compressible and incompressible, isotropic hyperelastic materials. Constitutive equations studied in experiments by Rivlin and Saunders (1951) for incompressible rubber materials and by Blatz and Ko (1962) for certain compressible elastomers are derived; and an equation characteristic of a class of biological tissues studied in primary experiments by Fung (1967) is discussed. Sample applications are presented for these materials. A balloon inflation experiment is described, and the physical nature of the inflation phenomenon is examined analytically in detail. Results for the different materials are compared. Two major problems of finite elasticity theory are discussed. Some results concerning Ericksen’s problem on controllable deformations possible in every isotropic hyperelastic material are outlined; and examples are presented in illustration of Truesdell’s problem concerning analytical restrictions imposed on constitutive equations. Universal relations valid for all compressible and incompressible, isotropic materials are discussed. Some examples of non-uniqueness, including that of a neo-Hookean cube subject to uniform loads over its faces, are described. Elastic stability criteria and their connection with uniqueness in the theory of small deformations superimposed on large deformations are introduced, and a few applications are mentioned. Some previously unpublished results are presented throughout.

Lateral Stress Effect on Fracture Quantities of a Crack in a Piezoelectric Medium

2007 ◽  
Vol 348-349 ◽  
pp. 957-960
Author(s):  
Erasmo Viola ◽  
Claudia Belmonte ◽  
Giuseppe Viola
Keyword(s):  
Analytical Approach ◽  
Decomposition Theorem ◽  
Piezoelectric Medium ◽  
Stress Effect ◽  
Lateral Stress ◽  
Intensity Factors ◽  
Field Equations ◽  
Piezoelectric Body ◽  
Electroelastic Fields ◽  
Electromechanical Field

The aim of this paper is to seek the solution to the electromechanical field equations for a cracked linear piezoelectric body using an analytical approach which is based on the decomposition theorem of linear algebra. The electroelastic fields around the crack tip are given. The energy release rate is written in terms of those fields intensity factors.

scholarly journals Meshless Radial Point Interpolation Method for Hyperelastic Materials

10.29007/r7sp ◽  
2020 ◽  
Author(s):  
Trong Khiem Bui ◽  
Vu Tuong Nguyen ◽  
Thanh Nha Nguyen ◽  
Tich Thien Truong
Keyword(s):  
Large Deformation ◽  
Interpolation Method ◽  
Biological Tissues ◽  
Material Behavior ◽  
Nonlinear Material ◽  
Radial Point Interpolation Method ◽  
Hyperelastic Materials ◽  
Radial Point Interpolation ◽  
Point Interpolation Method ◽  
Point Interpolation

Hyperelastic materials are special types of material that tends to behavior elastically when they are subjected to very large strains. These materials show not only the nonlinear material behavior but also the large deformation and stress-strain relationship is derived from a strain energy density function. Hyperelastic materials are widely used in many applications such as biological tissues, polymeric foams, and moreover. Neo - Hookean is a material model for hyperelastic solid which contains only two material parameters: bulk modulus and shear modulus. In the field of numerical analysis, the radial point interpolation method (RPIM) is a well-known meshfree method based on Garlekin's weak form. With the property of “free of mesh”, the RPIM approach shows its advantage for large deformation problems. In this study, a meshless radial point interpolation method is applied to demonstrate the elastic response of rubber-like materials based on the Mooney- Rivlin model. The obtained results are compared with the reference solutions given by other methods to verify the accuracy of the proposed method.

Constitutive Equations for Hyperelastic Materials Based on the Upper Triangular Decomposition of the Deformation Gradient

2018 ◽  
Vol 24 (6) ◽  
pp. 1785-1799 ◽  
Author(s):  
Y. Q. Li ◽  
X.-L. Gao
Keyword(s):  
Energy Density ◽  
Constitutive Equations ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Deformation Gradient ◽  
Density Functions ◽  
Triangular Decomposition ◽  
Deformation Gradient Tensor ◽  
Hyperelastic Materials ◽  
Distortion Tensor

The upper triangular decomposition has recently been proposed to multiplicatively decompose the deformation gradient tensor into a product of a rotation tensor and an upper triangular tensor called the distortion tensor, whose six components can be directly related to pure stretch and simple shear deformations, which are physically measurable. In the current paper, constitutive equations for hyperelastic materials are derived using strain energy density functions in terms of the distortion tensor, which satisfy the principle of material frame indifference and the first and second laws of thermodynamics. Being expressed directly as derivatives of the strain energy density function with respect to the components of the distortion tensor, the Cauchy stress components have simpler expressions than those based on the invariants of the right Cauchy-Green deformation tensor. To illustrate the new constitutive equations, strain energy density functions in terms of the distortion tensor are provided for unconstrained and incompressible isotropic materials, incompressible transversely isotropic composite materials, and incompressible orthotropic composite materials with two families of fibers. For each type of material, example problems are solved using the newly proposed constitutive equations and strain energy density functions, both in terms of the distortion tensor. The solutions of these problems are found to be the same as those obtained by applying the polar decomposition-based invariants approach, thereby validating and supporting the newly developed, alternative method based on the upper triangular decomposition of the deformation gradient tensor.

On a class of non-dissipative materials that are not hyperelastic

2008 ◽  
Vol 465 (2102) ◽  
pp. 493-500 ◽  
Author(s):  
K.R Rajagopal ◽  
A.R Srinivasa
Keyword(s):  
Physical Meaning ◽  
Constitutive Equations ◽  
Mechanical Response ◽  
Mass Density ◽  
Cauchy Stress ◽  
Response Functions ◽  
State Variables ◽  
Hyperelastic Materials ◽  
Current Mass ◽  
Helmholtz Potential

The purpose of this brief note is to develop fully Eulerian, implicit constitutive equations for the mechanical response of a class of materials that do not dissipate mechanical work in any process. We show that such materials can be modelled by obtaining a form for the Helmholtz potential as a function of the current mass density, the Cauchy stress and certain other parameters that capture anisotropic response. The resulting constitutive equations are of the form , where and are functions of the state variables of the system. The class of materials that can be obtained from such a constitutive relation is considerably more general than conventional Green-elastic hyperelastic materials. Such response functions may be suitable for the modelling of biological tissue where, due to the constant remodelling that takes place, there may be no physical meaning to a ‘reference configuration’.

The Application of the Principle of Stationary Potential Energy to Some Problems in Finite Elasticity

1965 ◽  
Vol 32 (3) ◽  
pp. 656-660 ◽  
Author(s):  
Mark Levinson
Keyword(s):  
Potential Energy ◽  
Finite Strain ◽  
Finite Elasticity ◽  
Energy Methods ◽  
Stationary Potential ◽  
Major Purpose ◽  
Energy Principle ◽  
Finite Strain Theory ◽  
Rubberlike Material ◽  
The Subject

Two applications of the principle of stationary potential energy to the finite straining of a neo-Hookean (rubberlike) material are given in this paper. The major purpose of the work presented is to illustrate the suitability of energy methods for the solution of problems in finite strain theory since the literature of the subject does not contain mention of such solutions. One problem not amenable to the usual inverse methods of finite elasticity is studied approximately. The other problem, involving a stability question of an unusual sort, is handled with ease by means of the energy principle.

Strain-Rate Sensitive Constitutive Modeling of Anisotropic Visco-Hyperelastic Materials

2012 ◽  
Author(s):  
Sahand Ahsanizadeh ◽  
LePing Li
Keyword(s):  
Strain Rate ◽  
Evolution Equations ◽  
Mechanical Response ◽  
Viscoelastic Behavior ◽  
Biological Tissues ◽  
Internal Variables ◽  
Current Configuration ◽  
Hyperelastic Materials ◽  
History Of ◽  
Viscoelastic Constitutive Model

Integral-based formulations of viscoelasticity have been widely used to describe the mechanical behavior of soft biological tissues and polymers. However, it is suggested that they are not suitable to be used under high strain rates. On the other hand, strain-rate sensitive models with an explicit dependence on the strain-rate have been developed for a certain class of materials. They predict the viscoelastic behavior during ramp loading more accurately while fail to account for the relaxation response. In order to overcome these drawbacks, a viscoelastic constitutive model has been proposed in this study based on the concept of internal variables. While the behavior of elastic materials is uniquely determined by the current state of deformation or external variables, the mechanical response of inelastic materials are regulated also by internal variables. The internal variables are associated with the dissipative mechanisms in the material and along with the evolution equations introduce the effect of history of the deformation to the current configuration. The current study employs short-term and long-term internal variables to account for the viscoelastic response during loading and relaxation respectively.

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