Energetically Consistent Invertible Elasticity

文章来源:
Energetically Consistent Invertible Elasticity

2 Isotropic hyperelasticity
证明 P(F)=UP^(σ)VT

其实有更简单的证明方法:

1.Invertible Finite Elements For Robust Simulation of Large
Deformation

其中第5节 Diagonalization

if stress P is invariant under world rotation(rigid body rotation), then:
P(UF)=UP(F)
if stress P is invariant under material rotation(isotropic), then:
P(FVT)=P(F)VT

其中
2.Robust Quasistatic Finite Elements and Flesh Simulation
这篇也引用了文献1中这个概念

第5节 Finite Element Forces

As noted in [1], the first Piola-Kirchhoff stress is invariant under rotations of either material or world space for isotropic materials.

Combining the rotational invariance of the first Piola-Kirchoff stress with the diagonalization of the deformation gradient yields
P(F)=UUTP(F)VVT=UP(UTFV)VT=UP(F^)VT

看上去我们的证明没有用到isotropic materials这一特点,其实也利用了
∂Fij∂σk求导时,我们默认U,V对σ的导数为0.