By Erwin Schrödinger
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10) at an arbitrary t, we obtain d d . 12) 0, because OCt) QT(t) is skew symmetric since Q(t)E Y(f). 10). 9). We shall now assume that a is a frame-indifferent function on the constraint surface re. 13) Hyperelastic Materials with Internal Constraints 23 for all FE Cfl and Q E Y'(9. 9). 14) where we have suppressed the argument t. 14) is equivalent to (dQFaf Q - (dFaf E (f/(F)lf. 7), we can rewrite this condition as F(dQFaf Q - F(dFaf E %(F). 8) imply that QF(dQFaf E QY'(F) QT. 12) for the reaction spaces, we then obtain (3_13).
Stored Energy Function For a hyperelastic material point with internal constraint the stored energy function is a (scalar-valued) smooth function a defined on the constraint surface C(j relative to the local reference configuration x. 1) where (l denotes the mass density of the point in the process. 2) where (l" denotes the mass density of the material point in the local configuration x. In a purely mechanical theory, energy is present in the forms of kinetic energy, potential energy, and stored energy.
R(F) and such that . (dFa, F) . 5) 22 H. -C. 2). 3) as tr ([e(t) (dp(t) af - F-l(t) T(t)] F(t)) = O. K(F). K(F). 8) This is the constitutive equation of a hyperelastic material point relative to the local reference configuration. H. 9) for all FE re and Q E Y(f). 10) such that Q(t) E Y(f) with Q(O) = I is contained in re at each FE re. 5), we obtain tr [F(dpaf 0(0)] = o. 11) implies that F(dpaf must be a symmetric tensor. Conversely, suppose that F(dpaf is symmetric at all points FE re. 10) at an arbitrary t, we obtain d d .