A special case of the principle of virtual displacements that deals with linear as well as nonlinear elastic bodies is known as the principle of minimum total potential energy. For elastic bodies (in...

A special case of the principle of virtual displacements that deals with linear as well as nonlinear elastic bodies is known as the principle of minimum total potential energy. For elastic bodies (in the absence of temperature variations), there exists a strain energy density function Ψ (measured per unit volume) such that

Equation (1) represents the constitutive equation of a hyper elastic material. The strain energy density Ψ is a single-valued function of strains at a point, and is assumed to be positive-definite. The statement of the principle of virtual displacements can be expressed in terms of the strain energy density Ψ:

The sum U + V = Π is called the total potential energy of the elastic body. The statement in Eq. (7) is known as the principle of minimum total potential energy. Derive the Euler equations for an isotropic Hookean solid using the principle of minimum total potential energy, Eq. (7). Assume small strains (i.e. linearized elasticity).