Download PDF by Arnold Sommerfeld: Partial Differential Equations in Physics

By Arnold Sommerfeld

ISBN-10: 0126546568

ISBN-13: 9780126546569

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The choice of C is customarily restricted by three symmetry requirements that we now motivate. (i) We know from our local analysis of deformation in Chapter I that a deformation of small gradient H may be regarded as consisting of a simple deformation measured by E = sym H and a rotation measured by W = skw H; the linearity of C implies that qH] = qE] + qW]. 2) Now, translations induce no stress, since the gradient of a translation is the null tensor and C[O] = O. If we wish to guarantee that rotations induce no stress as well, we are led to stipulate that C[skwH] = 0, HELin.

To fulfill this requirement it is sufficient to assume that skw C[H] = 0, HELin. 4) (iii) The stored energy of a linearly elastic material described by C is a smooth mapping a: Sym ---+ JR, a = a (E), P. 5) 48 P. 6) and that the following normalization condition is satisfied: 0'(0) = O. 7) The value of a at E is interpreted as the elastic energy stored per unit volume at a point of the reference shape when the infinitesimal strain tensor takes the value E at that point. 7). It can be shown (cf.

One may ask which gradient the displacement field u+ has at p. Let H and H+ denote, respectively, the gradients of the mappings y ~ u(y) and y+ ~ u+(y+). 1), V(u+ 0 y+) = H+Q, V(Qu(y)) = QH. 5) (Exercise 1). 7). 10) 54 P. 11) (here and henceforth the dependence on p is again left tacit). 11) is a subgroup of Rot that is called the material symmetry group of C. * EXERCISES 1. Show that the algebraic operations of orthogonal conjugation and symmetrization commute. 2. 11). 3. 11) (this is the definition usually found in textbooks, cf.

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Partial Differential Equations in Physics by Arnold Sommerfeld

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