Overview of Forces in Continuum Mechanics
Cauchy’s equation of continuum mechanics reads
where ρ is the density, r denotes the coordinates of a material point, T is the stress tensor, and fext is an external volume force such as gravity (fext = ρg). This is the equation solved in the structural mechanics interfaces for the special case of a linear elastic material, neglecting the electromagnetic contributions.
In the stationary case there is no acceleration, and the equation representing the force balance is
The stress tensor must be continuous across a stationary boundary between two materials. This corresponds to the equation
where T1 and T2 represent the stress tensor in Materials 1 and 2, respectively, and n1 is the normal pointing out from the domain containing Material 1. This relation gives rise to a surface force acting on the boundary between Material 1 and 2.
In certain cases, the stress tensor T can be divided into one part that depends on the electromagnetic field quantities and one part that is the mechanical stress tensor,
For the special case of an elastic body, the mechanical stress tensor is proportional only to the strain and the temperature gradient. The exact nature of this split of the stress tensor into an electromagnetic and a mechanical part depends on the material model, if it can be made at all.
For more information on the mechanical stress tensor for elastic materials, see the documentation for the interfaces. For example, The Solid Mechanics Interface in the COMSOL Multiphysics Reference Manual.
It is sometimes convenient to use a volume force instead of the stress tensor. This force is obtained from the relation
This changes the force balance equation to
or, as stated in the structural mechanics interfaces,