Consider a solid body surrounded by a boundary . In general, volume forces are applied on while surface forces and/or displacements are imposed on . Let denote the displacement vector defined on . which results from the forces and displacements imposed on . From . the deformation tensor can be calculated as the symmetric part of the gradient of :

(16–1)

In the present context of linear isothermal and isotropic elasticity, the constitutive equation of the stresses is given by

(16–2)

where is the unit tensor. This equation involves two physical parameters: the Young's modulus and the Poisson's ratio .

When introducing this expression for the stresses into the momentum equation, and after substituting by its definition, a second-order elliptic equation is eventually obtained for the displacement . After substitution, the momentum equation can be written as:

(16–3)

where is the density of the solid and is the acceleration. This equation requires that boundary conditions be given in terms of displacement or in terms of force. So, on a boundary where the solid body is fixed, a zero displacement will be assigned, whereas nodal force or continuous forces can be applied on freestanding walls. In an FSI simulation, on a boundary where the solid is in contact with a surrounding fluid, continuous forces originating from the fluid will be assigned as boundary conditions.

The von Mises stress is invoked in the von Mises criterion used for anticipating the occurrence of yielding of ductile materials, such as metals. It is evaluated as:

(16–4)

The equation is algebraic, so that the von Mises stress can directly be evaluated from the previously calculated stress components.

For 2D planar stress state, the relationship becomes:

(16–5)

The von Mises stress can be invoked to linear and non-linear elasticity cases. It is originally developed for ductile materials, such as metals. The application towards other solid materials is subject to validation and common sense.