Equilibrium equations linear elastic solid

Note that these equations are a special case of the equilibrium equations revealed in eqn (2.53) in the constitutive context of an isotropic linear elastic solid. 

To construct the elastic Green function for an isotropic linear elastic solid, we make a special choice of the body force field, namely, f(r) = fo5(r), where fo is a constant vector. In this case, we will denote the displacement field as Gik r) with the interpretation that this is the component of displacement in the case in which there is a unit force in the k direction, fo = e. To be more precise about the problem of interest, we now seek the solution to this problem in an infinite body in which it is presumed that the elastic constants are uniform. In light of these definitions and comments, the equilibrium equation for the Green function may be written as... 

Isotropic Elasticity and Nervier Equations Use the constitutive equation for an isotropic linear elastic solid given in eqn (2.54) in conjunction with the equilibrium equation of eqn (2.84), derive the Navier equations in both direct and indicial notation. Fourier transform these equations and verify eqn (2.88). 

Equation (3) is the equation of equilibrium of the porous medium. In this equation, it is assumed that the medium is non-linearly elastic, and G (Pa) and A (Pa) are Lame s constants of elasticity and P is the coefficient of volumetric thermal expansion of the solid matrix. G and A, and also A d the bulk modulus can also be expressed as functions of the... 

For small displacements from the equilibrium position, the force between the atoms is proportional to the displacement (see equation (2.12)). This is true not only for a single bond, but also for larger atomic compounds and thus for macroscopic solids. This linear-elastic behaviour is described mathematically by Hooke s law. It is valid only for small strains. In metals and ceramics, this is not an important constraint because the elastic part of any deformation is small. 

It is clear that all the specimens used to determine properties such as the tensile bar, torsion bar and a beam in pure bending are special solid mechanics boundary value problems (BVP) for which it is possible to determine a closed form solution of the stress distribution using only the loading, the geometry, equilibrium equations and an assumption of a linear relation between stress and strain. It is to be noted that the same solutions of these BVP s from a first course in solid mechanics can be obtained using a more rigorous approach based on the Theory of Elasticity. 

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