By P. Podio-Guidugli
I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her specific feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for beneficial comments triggered by way of their analyzing of 1 or one other of the numerous prior drafts, from 1988 up to now. because it has taken me see you later to deliver this writing to its current shape, many different colleagues and scholars have episodically provided priceless reviews and stuck error: a listing may threat to be incomplete, yet i'm heartily thankful to all of them. ultimately, 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 enable V be the vector house linked to eight. We distinguish some extent p E eight either from its place vector p(p):= (p-o) E V with admire to a selected foundation zero E eight and from any triplet (~1, ~2, ~3) E R3 of coordinates that we may well 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 smart to think about the interior product a .
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4. 26) HELin, and, granted the necessary smoothness, show that aCe) = a (0) a(O) + ea'(O) + ~e2a"(0) + 0(e 3), = 0'(1), a"(O) a'(O) = oFa(l) . 22), show that a(F) = ~[F - I] . oj2)a(I)[F - I] + O(IF - 11 3). 28) 5. 25). 14. Material Symmetry It is common experience that certain rigid transfonnations of the reference shape are mechanically undetectable, in that two otherwise identical defonnational experiments, the one perfonned before and the other after anyone of those rigid 52 P. PODIO-GUIDUGLI transfonnations, give identical results in tenns of induced stresses and stored energies.
Lin. 6) 1~1=1l. 8) of the identity of Sym. 9) Q:=Q~Q. Lin. It is not difficult to see that, for each Q E Rot, Q commutes with sph and dey: Qsph = sphQ = sph, Qdev = devQ. 11) 56 P. PODIO-GUIDUGLI EXERCISES 1. 12) lI= sym+skw. 13) 8ik 8jh (here 8i) is the Kronecker symbol, whose value is 1 if i = j, 0 if i :j:. j). 2. 5). (ii) Show that (A ® B)(C ~ D) = A ® (CT ~ DT[B1) , (A ~ B)(C ® D) = (A ~ B[C]) ® D. 14) 3. 11). 4. Prove that, if p E 'U and P = P ® p, then P~P=P®P. 15) 16. c of C if Q and C commute (cf.
31 STRESS in terms of TO, a tensor-valued field over f(Q) having the smoothness stipulated for s(-, n). 4h is T E Sym in f(Q). 1); similarly, the applied traction at a typical point of af(Q) is given by the action of T on the exterior normal to af(Q) at that point. REMARK. 2)2 remains the same. Let a motion m = Ut I t ETC R} be given, and let 7 be the corresponding trajectory. A pair consisting of m and a symmetric-valued tensor field T defined over 7, with T (', t) smooth for each t E T, is called a dynamical process.