5.3 Principle of Material Objectivity

Consider an elastic body deformed from its reference configuration by fields F, ?, G to the present configuration. Assume that a rigid rotation represented by the orthogonal matrix O is superimposed on the present configuration so that ( X, t) = O _ij x _j ( X, t) gives the coordinates of the material point X in the rotated configuration. Then the deformation gradient of the rotated configuration is related to F _iL = ? x _i / ?X _L by . It is reasonable to assume that the Helmholtz free energy is unaltered by the superimposed rotation. Furthermore, G and Q are unaffected as they are defined with respect to coordinates in the reference configuration. The requirement that

for every proper orthogonal matrix O is called material objectivity. In Continuum Mechanics, we require that the constitutive equations must be objective. Equation (5.19) ₁ implies that when F is replaced by F* the value of the Helmholtz free energy stays unchanged. Similarly, Eq. (5.19) ₂ requires that values of the function Q evaluated at ( F, ?, G) and ( F*, ?, G) are the same. Recalling the polar decomposition F _iL = R _iM U _ML of F, Eq. (5.19) must hold when O = R ^T, i.e., O _ij