Equation of Motion
The first Piola–Kirchhoff stress P is calculated from the second Piola–Kirchhoff stress as P = FS. The first Piola–Kirchhoff stress relates forces in the present configuration with areas in the reference configuration, and it is sometimes called the nominal stress.
Using the first Piola–Kirchhoff stress tensor, the equation of motion can be written in the following form:
(3-10)
where the density corresponds to the material density in the initial undeformed state, the volume force vector FV has components in the actual configuration but given with respect to the undeformed volume, and the tensor divergence operator is computed with respect to the coordinates on the material frame. Equation 3-10 is the strong form that corresponds to the weak form equations within the Solid Mechanics interface (and many related multiphysics interfaces) in COMSOL Multiphysics. Using vector and tensor components, the equation can be written as
The components of the first Piola–Kirchhoff stress tensor are nonsymmetric in the general case, thus
because the component indices correspond to different frames. Such tensors are called two-point tensors.
The boundary load vector FA in case of geometric nonlinearity can be related to the first Piola–Kirchhoff stress tensor via the following formula:
where the normal n0 corresponds to the undeformed surface element. Such a force vector is often referred to as the nominal traction. In components, it can be written as
The Cauchy stress, σ, can be calculated as
The Cauchy stress is a true stress that relates forces in the present configuration (spatial frame) to areas in the present configuration, and it is a symmetric tensor. Equation 3-10 can be rewritten in terms of the Cauchy stress as
where the density corresponds to the density in the actual deformed state, the volume force vector fV has components in the actual configuration (spatial frame) given with respect to the deformed volume, and the divergence operator is computed with respect to the spatial coordinates.
The pressure is computed as
which corresponds to the volumetric part of the Cauchy stress. The deviatoric part is defined as
The second invariant of the deviatoric stress
is used for the computation of von Mises (equivalent) stress