= Antiplane energy does not determine the full stress
On an <antiplane shear> $F=I+e_3\otimes(p,q,0)$, the <first strain invariant> satisfy $I_1=I_2=3+p^2+q^2$, $I_3=1$. Restricting an isotropic objective <strain energy density> to this family determines only derivatives tangent to the family. It fixes $S_{31}=2w'(p^2+q^2)p$ and $S_{32}=2w'(p^2+q^2)q$, but does not specify the whole <Cauchy stress tensor>. For example the <neo-Hookean solid> energy $W=\mu(|F|^2-3)/2-\mu\log\det F$ gives $S=\mu(B-I)$ on this family. Even though the reference state is stress free, this is not generally a scalar multiple of $B$.
Back to article page
