The '''nominal stress''' is a mathematical description of the [[stress (mechanics)|mechanical stress state]] within a material, used in [[continuum mechanics]], that takes into account the displacement and deformation of the material. Like the [[Cauchy stress tensor]], it is a linear function {$P$} that maps a direction {$n$} in space to the stress vector {$P(n)$} across a surface perpendicular to {$n$}. Unlike the Cauchy tensor however, it relates forces in the current configuration of the material to areas in some fixed reference configuration. The nominal stress tensor is useful in [[stress analysis]] when the object under load is significantly displaced or deformed relative to the original "unloaded" state. For example, the Cauchy Stress tensor is variant to a pure rotation, while the deformation strain tensor is invariant. For [[infinitesimal strain theory|infinitesimal displacements]] the Cauchy and the nominal stress tensor are identical. The transpose of the nominal stress tensor is also known as the '''first Piola–Kirchhoff stress tensor'''. There is a [[second Piola–Kirchhoff stress tensor]] that expresses the stress force, as well as the area, in the original configuration. ==Definition== [[File:StressMeasures.png|thumb|400px|Quantities used in the definition of the nominal stress]] The nominal stress tensor {$\boldsymbol{P}$} relates forces in the ''present'' configuration with areas in the ''reference'' ("material") configuration. :{$ \boldsymbol{P} = J~\boldsymbol{\sigma}~\boldsymbol{F}^{-T} ~$} where {$\boldsymbol{F}$} is the [[deformation gradient]] and {$J= \det\boldsymbol{F}$} is the [[Jacobian matrix and determinant|Jacobian]] [[determinant]]. The nominal stress {$\boldsymbol{N}=\boldsymbol{P}^T$} is the transpose of the first Piola-Kirchhoff stress (PK1 stress) {$\boldsymbol{P}$} and is defined via :{$ d\mathbf{f} = \mathbf{t}_0~d\Gamma_0 = \boldsymbol{N}^T\cdot\mathbf{n}_0~d\Gamma_0 = \boldsymbol{P}\cdot\mathbf{n}_0~d\Gamma_0 $} or :{$ \mathbf{t}_0 = \boldsymbol{N}^T\cdot\mathbf{n}_0 = \boldsymbol{P}\cdot\mathbf{n}_0 $} This stress is unsymmetric and is a two point tensor like the deformation gradient. This is because it ''relates the force in the deformed configuration to an oriented area vector in the reference configuration.'' In terms of components with respect to an [[orthonormal basis]], the first Piola–Kirchhoff stress is given by :{$P_{iL} = J~\sigma_{ik}~F^{-1}_{Lk} = J~\sigma_{ik}~\cfrac{\partial X_L}{\partial x_k}~\,\!$} Because it relates different coordinate systems, the 1st Piola–Kirchhoff stress is a [[two-point tensor]]. In general, it is not symmetric. The 1st Piola–Kirchhoff stress is the 3D generalization of the 1D concept of [[engineering stress]]. If the material rotates without a change in stress state (rigid rotation), the components of the 1st Piola–Kirchhoff stress tensor will vary with material orientation. The 1st Piola–Kirchhoff stress is energy conjugate to the deformation gradient.