By Frederick Bloom (auth.)
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Extra resources for Modern Differential Geometric Techniques in the Theory of Continuous Distributions of Dislocations
Example text
For s o l i d bodies, Riemannian o t h e r t h a n the i s o t r o p i c and, connection; in fact, ones, the do not u s u a l l y give rise to a none of the i n d u c e d R i e m a n n i a n connection on an iso- are the only type of solid b o d i e s intrinsic Riemannian metrics unique connection. connectior by T h e o r e m 11-4, connections a torsion-free therefore, can be a m a t e r i a l material connection does not exist on B. We w i l l h a v e m o r e to say about this s i t u a t i o n in the next section.
There is a vector-valued all points stress on CoB is given function t(x,n) x in St(C) and all unit vectors defined for n such that the acting on C is given by t(x; C) = t(x,n) where n is the exterior boundary unit normal of C in the configuration stress vector at x and the above stress principle at the point x on the ~t" We call t(x,n) concept the is known as the of Cauchy. , of moment 25 of momentum. If the momentum M(C) of C in the configuration ~t is given by / x dm then the principle of balance of ~t(C ) ~ momentum requires that f(C) = M(C) and implies, under suitable continuity conditions, the existence of a tensor field ~S such that t(~,n) = Ts(x)n for all xs~t(~C); the superposed dot above indicates, respect to time.
Also, and the above remark, a s s o c i a t e d w i t h the as is e v i d e n t the e x i s t e n c e from T h e o r e m 11-7 of a flat "G" c o n n e c t i o n on E(B,~) is r e l a t e d to the local h o m o g e n e i t y we may use the t o r s i o n material connections g e n e i t y of B. and c u r v a t u r e tensors on B to c h a r a e t e r i z e Actually, 11-18, as a " m e a s u r e " In fact, tions connection, whose of the in m o s t of those of d i s l o c a t i o n s was p r e s c r i b e d theories in c r y s t a l a priori, symbols w i t h no r e f e r e n c e the p r e s c r i b e d c o n n e c t i o n was integrable and the a s s o c i a t e d the d e n s i t y we may integrable density.