Tag: FEM

FEM: Material tangent of neoHookean material
Let’s calculate material tangent of hyperelastic material, taking neoHookean for instance. The elastic energy of neoHookean material is written as \psi^e=\frac{\mu}2tr(\bold{C})\mu\ln J A penalty method is applied to constrain volume unchanged \psi^b=\frac{\kappa}2(J^22\ln J1) The total free energy density is therefore written into \psi=\psi^e+\psi^b The stress is obtained by derivative P_{iA}=\frac{\partial\psi}{\partial F_{iA}}=\frac{\partial\psi^e}{\partial F_{iA}}+\frac{\partial\psi^b}{\partial F_{iA}} The elastic…

FEM: Fbar in total lagrangian shceme
Fbar method is a element technique used in linear elements to alleviate volumetric locking. This method replaces F of each integration point with Fbar. Index notation Displacement interpolation u_i=N_{iK}d_k Deformation gradient is calculated as F_{iA}=\delta_{iA}+N_{iK,A}d_K=\delta_{iA}+B_{iKA}d_K where d_K is nodal displacement Total lagrangian shceme without traction and body force is \delta U=\int P_{iA}\delta F_{iA}dV and nodal…