Let us first consider only infinitesimal strains and linear elasticity. If we assume that the RVE is small enough, we can neglect inertial and body forces. In addition, if the RVE is in equilibrium, conservation of mass automatically holds. Then the equations that govern the motion of the RVE can be written as:
In the above equations, is the strain tensor (small strain), is the displacement vector, and is the fourth-order tensor of elastic moduli at the point .
To get a unique solution of the governing equations, we need boundary conditions on . These boundary conditions may be in the form of applied tractions:
where is the outward normal vector to the surface and is the applied traction.
Alternatively, the boundary conditions may be in the form of applied displacements:
where is the applied displacement.
We usually assume that the portions of the boundary on which tractions and displacements are applied are nonoverlapping, i.e., and .
If we need to solve the energy equation, we also have to specify heat flux or specified temperature boundary conditions on the RVE.