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The traction on a plane bisecting the body is given by, where is the Cauchy stress tensor and is the outward unit normal vector to the plane. It can be shown that by the conservation of angular momentum that this tensor is symmetric ( . The Cauchy stress tensor, a spatial tensor, is the actual physical stress, that is, the force per unit deformed area. To simplify the equations of continuum mechanics, especially when working in the material configuration, several other stress measures are often used. The Kirchhoff stress tensor is defined as The first Piola-Kirchhoff stress tensor is given as Note that , like , is not symmetric. Also, like , is known as a two-point tensor, meaning it is neither a material nor a spatial tensor. Since we have two strain tensors, one spatial and one material tensor, it would be useful to have similar stress measures. The Cauchy stress is a spatial tensor and the second Piola-Kirchhoff (2 PK) stress tensor, defined as is a material tensor. The inverse relations are: In many practical applications it is physically relevant to separate the hydrostatic stress and the deviatoric stress of the Cauchy stress tensor: Here, the pressure is defined as . Note that the deviatoric Cauchy stress tensor satisfies .
The directional derivative of the 2 PK stress tensor needs to be calculated for the linearization of the finite element equations. For a hyperelastic material, a linear relationship between the directional derivative of and the linearized strain can be obtained: Here, is a fourth-order tensor known as the material elasticity tensor. Its components are given by, where is the strain-energy density function for the hyperelastic material. The spatial equivalent – the spatial elasticity tensor – can be obtained by,