PiolaKirchhoffstresstensorer

Piola-Kirchhoff stress tensors are mathematical tools used in continuum mechanics to describe the internal forces within a deformed material. They are essential in the analysis of nonlinear elasticity, where large deformations occur, and serve to relate forces to the deformation state of a material body.

There are two primary types of Piola-Kirchhoff stress tensors: the first and second Piola-Kirchhoff stress tensors. The first Piola-Kirchhoff stress tensor (often denoted as P) is a two-point tensor that maps areas in the reference (undeformed) configuration to forces in the current (deformed) configuration. It is not symmetric and provides a convenient description of the forces in the original configuration, which simplifies the formulation of constitutive laws and boundary conditions.

The second Piola-Kirchhoff stress tensor (denoted as S) is derived from the first and is symmetric. It accounts for the material’s stress state in the reference configuration itself, making it useful in formulations where energy derivatives with respect to strain measures are required. S is often used in material modeling because of its symmetric property and direct relation to the strain energy density function.

Both tensors are frame-indifferent and work within the context of finite strain theories, where the key quantity is the deformation gradient. The Piola-Kirchhoff stresses are contrasted with the Cauchy stress tensor, which describes forces per unit area in the current configuration and is more physically intuitive but less convenient for large strains.

Overall, Piola-Kirchhoff stress tensors are foundational in the computational modeling of complex elastic and elastoplastic materials, facilitating the analysis of their behavior under large and nonlinear deformations.