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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