By P. Podio-Guidugli
I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her unique feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for invaluable comments caused through their interpreting of 1 or one other of the various earlier drafts, from 1988 to this point. because it has taken me goodbye to convey this writing to its current shape, many different colleagues and scholars have episodically provided precious reviews and stuck errors: an inventory could chance to be incomplete, yet i'm heartily thankful to all of them. eventually, I thank V. Nicotra for skillfully remodeling my hand sketches into book-quality figures. P. PODIO-GUIDUGLI Roma, April 2000 magazine of Elasticity fifty eight: 1-104,2000. 1 P. Podio-Guidugli, A Primer in Elasticity. © 2000 Kluwer educational Publishers. bankruptcy I pressure 1. Deformation. Displacement enable eight be a third-dimensional Euclidean house, and allow V be the vector house linked to eight. We distinguish some degree p E eight either from its place vector p(p):= (p-o) E V with appreciate to a selected beginning zero E eight and from any triplet (~1, ~2, ~3) E R3 of coordinates that we may perhaps use to label p. furthermore, we endow V with the standard internal product constitution, and orient it in a single of the 2 attainable manners. It then is sensible to think about the internal product a .
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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.
A Primer in Elasticity by P. Podio-Guidugli