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 |