By Arnold Sommerfeld
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Additional resources for Partial Differential Equations in Physics
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.
Partial Differential Equations in Physics by Arnold Sommerfeld