Comparison of results from four linear constitutive relations in isotropic finite elasticity (original) (raw)

Linear constitutive relations in isotropic finite elasticity

1998

Abstract For simple shearing and simple extension deformations of a homogeneous and isotropic elastic body, it is shown that a linear relation between the second Piola-Kirchhoff stress tensor and the Green-St. Venant strain tensor does not predict a physically reasonable response of the body. This constitutive relation implies that the slope of the curve between an appropriate component of the first Piola-Kirchhoff stress tensor and a deformation measure is an increasing functions of the deformation measure.

Universal relations for transversely isotropic elastic materials

2002

Abstract We adopt Hayes and Knops's approach and derive universal relations for finite deformations of a transversely isotropic elastic material. Explicit universal relations are obtained for homogeneous deformations corresponding to triaxial stretches, simple shear, and simultaneous shear and extension. Universal relations are also derived for five families of nonhomogeneous deformations.

Generalization of strain-gradient theory to finite elastic deformation for isotropic materials

Continuum Mechanics and Thermodynamics, 2016

This paper concerns finite deformation in the strain-gradient continuum. In order to take account of the geometric nonlinearity, the original strain-gradient theory which is based on the infinitesimal strain tensor is rewritten given the Green-Lagrange strain tensor. Following introducing the generalized isotropic Saint Venant-Kirchhoff material model for the strain-gradient elasticity, the boundary value problem is investigated in not only the material configuration but also the spatial configuration building upon the principle of virtual work for a three-dimensional solid. By presenting one example, the convergence of the strain-gradient and classical theories is studied.

Implicit constitutive relations for visco-elastic solids: Part II. Non-homogeneous deformations

International Journal of Non-linear Mechanics, 2020

A constitutive relation was developed in Part I for describing the response of a class of visco-elastic bodies, wherein the left Cauchy-Green tensor, the symmetric part of the velocity gradient, and the Cauchy stress tensor are related through an implicit constitutive relation. Here, we study a boundary value problem within the context of the model namely the inhomogeneous deformation of the body, corresponding to the response of an infinitely long slab due to the influence of gravity.

Deformation produced by a simple tensile load in an isotropic elastic body

1976

Page 1. Journal of Elasticity, Vol. 6, No. 1, January 1976 Noordhoff International Publishing - Leyden Printed in The Netherlands Deformation produced by a simple tensile load in an isotropic elastic body RC BATRA Engineerin 9 Mechanics Department, University of Missouri, Rolla, Missouri 65401, USA (Received April, 1975) ABSTRACT It is shown that a simple tensile load produces a simple extension provided the empirical inequalities (Truesdell and Noll [1], eqn.

A Primer in Elasticity

Journal of Elasticity - J ELAST, 2000

Let E be a 3-dimensional Euclidean space, and let V be the vector space associated with E. We distinguish a point p ∈ E both from its position vector p(p) := (p − o) ∈ V with respect to a chosen origin o ∈ E and from any triplet (ξ 1 , ξ 2 , ξ 3 ) ∈ IR 3 of coordinates that we may use to label p. Moreover, we endow V with the usual inner product structure, and orient it in one of the two possible manners. It then makes sense to consider the inner product a · b and the cross product a × b of two elements a, b ∈ V; in particular, we define the length of a vector a to be |a| = (a · a) 1/2 , and denote by U := {v ∈ V | |v| = 1} the sphere of all vectors having unit length. When needed or simply convenient, we think of E as equipped with a Cartesian frame {o; c 1 , c 2 , c 3 } with orthogonal basis vectors c i ∈ U (i = 1, 2, 3); the Cartesian components of a vector v ∈ V are then v i := v · c i and, in particular, the triplet (p 1 , p 2 , p 3 ) ∈ IR 3 , p i := p(p) · c i , of components of the position vector are the Cartesian coordinates of a point p ∈ E.

Nonlinear transversely isotropic elastic solids: an alternative representation

The Quarterly Journal of Mechanics and Applied Mathematics, 2008

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.

Finite homogeneous deformations of symmetrically loaded compressible membranes

Zeitschrift für angewandte Mathematik und Physik, 2007

The equilibrium problem of nonlinear, isotropic and hyperelastic square membranes, stretched by a double symmetric system of dead loads, is investigated. Depending on the form of the stored energy function, the problem considered may admit asymmetric solutions in addition to the expected symmetric solutions. For compressible materials, the mathematical condition allowing the computation of these asymmetric solutions is given. Moreover, explicit expressions for evaluating critical loads and bifurcation points are derived. Results and basic relations obtained for general isotropic materials are then specialized for a compressible Mooney-Rivlin material and a broad numerical analysis is performed. The qualitatively more interesting branches of asymmetric equilibrium are shown and the influence of the material parameters is discussed. Finally, using the energy criterion, some stability considerations are made. . 73G05, 73G10, 73H05.

Large Deformation Constitutive Laws for Isotropic Thermoelastic Materials

2012

We examine the approximations made in using Hooke's law as a constitutive relation for an isotropic thermoelastic material subjected to large deformation. For a general thermoelastic material, we employ the volume-preserving part of the deformation gradient to facilitate volumetric/shear strain decompositions of the free energy, its first derivatives (the Cauchy stress and entropy), and its second derivatives (the specific heat, Grüneisen tensor, and elasticity tensor). Specializing to isotropic materials, we calculate these constitutive quantities more explicitly. For deformations with limited shear strain, but possibly large changes in volume, we show that the differential equations for the stress involve new terms in addition to the traditional Hooke's law terms. These new terms are of the same order in the shear strain as the objective derivative terms needed for frame indifference; unless the latter terms are negligible, the former cannot be neglected. We also demonstrate that accounting for the new terms requires that the deformation gradient be included as a field variable.

A Generalized Strain Approach to Anisotropic Elasticity

2021

This work proposes a generalized Lagrangian strain function fα (that depends on modified stretches) and a volumetric strain function gα (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions fα and gα to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which...

Homogeneity conditions for elastic membranes

Acta Mechanica, 1983

This paper deals with the concept of homogeneity within the framework of hyperelastic anisotropic membranes. A frame field, i.e. an orthonormal set of vectors lying in the tangent plahe at each point of the membrane, is used to represent observers regarded as equivalent for comparing material response. The notions of homogeneity and unidirectional homogeneity are formulated in this setting and conditions required for a given strain energy function to define a homogeneous material are derived. The paper concludes with a discussion of certain additional features which illustrate the concepts and which arise out of special choices of the strain energy function.

ON THE THERMOPLASTICITY CONSTITUTIVE RELATIONS FOR ISOPTOPIC AND TRANSVERSELY ISOTROPIC MATERIALS

Transstellar Journals, 2019

The widely used engineering construction materials such as fiber and laminated composite materials are usually under the thermomechanical forces and undergoes thermoplastic deformations. These composites may be considered as a transversely isotropic or orthotropic materials. In this paper, the plasticity constitutive relations for isotropic and transversely isotropy materials proposed in [33]are developed taking into account the temperature and written up the strain and stress space thermoplasticity constitutive relations for aforementioned materials. For simplicity, thermoplasticity theories are restricted to a small deformations. The usefulness and privileges of the strain space thermoplasticity constitutive relations for the formulation the coupled thermomechanical boundary value problems are discussed. It is found that the strain space thermoplasticity constitutive relations are more convenient for numerical solution of the coupled thermoplasticity boundary value problems as compared to stress space theory.

Two-dimensional static deformation of an anisotropic medium

Sadhana, 2005

The problem of two-dimensional static deformation of a monoclinic elastic medium has been studied using the eigenvalue method, following a Fourier transform. We have obtained expressions for displacements and stresses for the medium in the transformed domain. As an application of the above theory, the particular case of a normal line-load acting inside an orthotropic elastic half-space has been considered in detail and closed form expressions for the displacements and stresses are obtained. Further, the results for the displacements for a transversely isotropic as well as for an isotropic medium have also been derived in the closed form. The use of matrix notation is straightforward and avoids unwieldy mathematical expressions. To examine the effect of anisotropy, variations of dimensionless displacements for an orthotropic, transversely isotropic and isotropic elastic medium have been compared numerically and it is found that anisotropy affects the deformation significantly.

On anisotropic elasticity and questions concerning its Finite Element implementation

Computational Mechanics, 2013

We give conditions on the strain-energy function of nonlinear anisotropic hyperelastic materials that ensure compatibility with the classical linear theories of anisotropic elasticity. We uncover the limitations associated with the volumetric-deviatoric separation of the strain-energy used, for example, in many Finite Element (FE) codes in that it does not fully represent the behavior of anisotropic materials in the linear regime. This limitation has important consequences. We show that, in the small deformation regime, a FE code based on the volumetric-deviatoric separation assumption predicts that a sphere made of a compressible anisotropic material deforms into another sphere under hydrostatic pressure loading, instead of the expected ellipsoid. For finite deformations, the commonly adopted assumption that fibres cannot support compression is incorrectly implemented in current FE codes and leads to the unphysical result that under hydrostatic tension a sphere of compressible anisotropic material deforms into a larger sphere.

On the coincidence of the principal axes of stress and strain in isotropic elastic bodies

1975

Page 1. LETTERS IN APPLIED AND ENGINEERING SCIENCE, Vol. 3, pp. 435-439. Pergamon Press, Inc. Printed in the United States. ON THE COINCIDENCE OF THE PRINCIPAL AXES OF STRESS AND STRAIN IN ISOTROPIC ELASTIC BODIES RC Batra Engineering Mechanics Department University of Missouri Rolla, Missouri 65401 ABSTRACT It is shown that for isotropic elastic materials the principal axes of stress are also the principal axes of strain provided the empirical inequalities hold. 1.