Forces, Torques, Balances

In this chapter, we discuss the representation of material interactions in continuous bodies with shape described just by their location in space. We consider the velocity of a material point and the power needed to move it as primitive quantities, so that a force appears as the entity furnishing that power when evaluated over the assigned velocity. We define the external power over a generic part of the body with nonvanishing volume, i.e., the power exerted by bulk and contact actions external to the part considered, and we derive integral balances of actions from the sole requirement of invariance of the external power under rigid-body changes in observers. Then, as a consequence of the integral balance of forces, we prove the action–reaction principle and, using it, the existence of a stress depending just on places and time (the Cauchy stress theorem). Local balances of forces and couples in Eulerian (actual) and Lagrangian (referential) representations follow from the pertinent integral balances by localization. Then we derive the expressions of the inertial terms and the inner power. We discuss also balances at discontinuity surfaces and conclude the chapter by proposing and discussing various exercises.