Basic Equations of Continuum Mechanics

The modeling of the behavior of pressure-sensitive materials is embedded in the general continuum mechanics. The basic equations of continuum mechanics can be split into the material-independent and the material-dependent equations. The starting point is the introduction of the kinematics based on pure mathematical considerations. In addition, the velocities and the accelerations of the relevant kinematical variables are presented. The next section is devoted to the introduction of the action on the continuum and the inner reaction. Starting with such properties like forces and stresses finally the static equilibrium is stated. The last part of the material-independent equations is the introduction of the balances. Limiting our discussions by thermo-mechanical actions only, the balance of mass, momentum, moment of momentum, energy and entropy are deduced. The specific properties and features of the pressure-sensitive materials are presented in the next sections. Within this chapter the general ideas of material modeling (deductive approach) are given. Finally, some examples of special constitutive equations for incompressible and compressible materials are presented. These examples are mostly related to rubber-like materials.