Material frame indifference

A constitutive equation is frame indifferent when a superposed rigid spatial rotation changes its outputs by their tensor transformation rules. For an elastic Cauchy stress tensor, this requires $\sigma (QF)=Q\sigma (F)Q^T$. It differs from material isotropy, which rotates reference material directions.

For an objective smooth strain energy density $W(QF)=W(F)$, differentiation along $Q=e^{tK}$ shows $DW(F)F^T$ is symmetric. Hence the associated Cauchy stress tensor is symmetric. At a stress-free reference $DW(I)=0$, differentiation of this identity shows $D^{2}W(I)[H]$ is symmetric for every $H$. In components this gives output minor symmetry. The Hessian has pair-interchange symmetry, which then gives the input minor symmetry and the displayed annihilation of skew increments. The stress-free condition matters: at a prestressed reference the derivative includes the extra term $DW(I)H^T$.

For a continuously differentiable energy on the connected set of positive-determinant matrices, material frame indifference differentiates to the displayed covariance of the first Piola-Kirchhoff stress tensor. Conversely that covariance makes $W(QF)-W(F)$ independent of $F$. Call the constant $c(Q)$. Composition gives $c(Q_1{Q}_2)=c(Q_1)+c(Q_2)$. Continuity and compactness of the rotation group make $c$ bounded, while $c(Q^n)=nc(Q)$ forces every $c(Q)$ to vanish. Thus no additive rotation-dependent energy ambiguity remains. Homogeneity means the energy has no explicit dependence on the material point; it is separate from this observer-independence property.

An objective spatial second-rank tensor transforms by the displayed rule under every rigid observer change $x^{\ast }=c(t)+Q(t)x$. This permits time-dependent rotations, not merely constant rotations. Cauchy stress tensor is objective because contact traction and surface normal both rotate by Q. The symmetric rate-of-strain tensor is objective because the additional observer spin in the velocity gradient is antisymmetric.