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The first constitutive equation (constitutive law) was developed by Robert Hooke and is known as Hooke's law.It deals with the case of linear elastic materials.Following this discovery, this type of equation, often called a "stress-strain relation" in this example, but also called a "constitutive assumption" or an "equation of state" was commonly used.
The hyperelastic material is a special case of a Cauchy elastic material. For many materials, linear elastic models do not accurately describe the observed material behaviour. The most common example of this kind of material is rubber, whose stress-strain relationship can be defined as non-linearly elastic, isotropic and incompressible.
Expressed in terms of components with respect to a rectangular Cartesian coordinate system, the governing equations of linear elasticity are: [1]. Equation of motion: , + = where the (), subscript is a shorthand for () / and indicates /, = is the Cauchy stress tensor, is the body force density, is the mass density, and is the displacement.
For an isotropic material the Cauchy stress tensor can be expressed as a function of the left Cauchy-Green tensor =.The constitutive equation may then be written: = (). In order to find the restriction on which will ensure the principle of material frame-indifference, one can write:
This does not mean all materials with twist effect fall in the bi-isotropic class. The twist effect of the class of bi-isotropic materials is caused by the chirality and non-reciprocity of the structure of the media, in which the electric and magnetic field of an electromagnetic wave (or simply, light) interact in an unusual way.
A biaxial tensile state can be derived starting from the most general constitutive law for isotropic materials in large strains regime: = (+ +) where S is the second Piola-Kirchhoff stress tensor, I the identity matrix, C the right Cauchy-Green tensor, and =, = and = the derivatives of the strain energy function per unit of volume in the ...
Structural analysis is a branch ... the formulation is based on the same three fundamental relations: equilibrium, constitutive, ... materials that are isotropic ...
Typical constitutive relations for rocks assume that the deformation process is isothermal, the material is isotropic, quasi-linear, and homogenous and material properties do not depend upon position at the start of the deformation process, that there is no viscous effect and therefore no intrinsic time scale, that the failure criterion is rate ...