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Table 1 Kinematics, constitutive quantities and stresses

From: An extension of assumed stress finite elements to a general hyperelastic framework

Symbol Continuum mechanical description
\({{\varvec{u}}}\) Displacement vector
\(\varvec{F}={{\varvec{I}}}+ {\mathrm{Grad}}{{\varvec{u}}}\) Deformation gradient
\(\varvec{C}={{\varvec{F}}}^T{{\varvec{F}}}\) Right Cauchy-Green tensor
\(\varvec{E} = \frac{1}{2}({{\varvec{C}}}-{{\varvec{I}}})\) Green-St. Venant strain tensor
\(\chi \) Complementary stored energy
\(\psi \) Helmholtz free energy
\(\varvec{S}\) Second Piola-Kirchhoff stress tensor
\(\varvec{P} = {{\varvec{F}}}{{\varvec{S}}}\) First Piola-Kirchhoff stress tensor
\(\varvec{\tau } = {{\varvec{P}}}{{\varvec{F}}}^T \) Kirchhoff stress tensor
\(\varvec{\sigma } = (\det {{\varvec{F}}})^{-1}{\varvec{\tau }}\) Cauchy stress tensor