By A. Cemal Eringen

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This vector after deformation becomes dx =x dX . 24) dx =g R C dX = k K L I L M M K K ^ R g dX l X m m K K f For a bibliography on this theorem see Eringen (1962, Section 10). 24) may be decomposed as follows (Fig. 2): (a) The vector dX ax ( . 24) , thus proving the theorem. 2 FIG. 2. Decomposition of deformation. Note that if and only if dX is a proper vector of C will the stretching not involve a further rotation of the vector dx\ . The order of operations translation, rotation, and stretching is clearly not important.

3) X = X(S) as S varies. 5) X = X(U,V) or F(X) = 0 where U and F a r e parameters. 6) x = x(X(U, V), i), F[X(x, t)] = 0 A material volume is a region of material particles. Theorem 1 (Lagrange's criterion). 7) =0 Proof. Suppose that f(x, t) = 0 is a material surface on which a typical point is moving with a velocity v which is not necessarily the velocity x of the material point instantaneously occupying that point on the surface. 7). 10) i/2 I=(x -v )(g %f^i k {n) 2 in) This asserts that at a point x on the surface f— 0 , / is proportional to the normal speed, relative to the surface, of the material point situated instantaneously at x.

When these conditions are violated, the corresponding displacement field in the body is not unique. The body may then possess dislocations. 1), by partial differentiation. For the nonlinear theory this is tedious. 2) a a a + a ([ln, s][km, r] — [Im, s][kn, r]) rs where [kl, m] = Wkm,i + a - a ) _! -i a a =a a = b Both C and c are nonsingular, symmetric, positive definite tensors, and when they are known we can calculate the arc length in B and b, which is part of the Euclidean space. 1 DEFORMATION AND MOTION 43 In the former C is used as the metric tensor a and partial differentiation is with respect to X and in the latter c is used as the metric tensor and partial differentiation is with respect to x .