By P. Podio-Guidugli

ISBN-10: 9048155924

ISBN-13: 9789048155927

ISBN-10: 9401705941

ISBN-13: 9789401705943

I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her targeted feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for worthwhile comments caused by way of their examining of 1 or one other of the various prior drafts, from 1988 to this point. because it has taken me goodbye to carry this writing to its current shape, many different colleagues and scholars have episodically provided valuable reviews and stuck error: an inventory could probability 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 allow eight be a three-d Euclidean area, and allow 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 recognize to a selected foundation zero E eight and from any triplet (~1, ~2, ~3) E R3 of coordinates that we might use to label p. in addition, 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 contemplate the interior product a .

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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. [19]). For linearly elastic materials, such commutativity is the algebraic substance of the physical property that certain rotations of the reference shape are not detectable by means of experiments measuring stress or energy (Exercise 2). c is a group. c for all C. c. c. cc itself. cc = {I}. In between these extreme cases, a fistful of discrete groups and one continuous group suffice to describe all symmetries of interest in the constitutive theory of anisotropic linearly elastic materials.

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. [11]). Show that, just as Orth can be represented as the direct product of Rot and the two-element group {-I, I}, 9:'c is the direct product of 9-c and {-I, I}. 15. Fourth-Order Tensors Various fourth-order tensors are encountered in the study of the linearly elastic response; in this section we collect some related algebraic material to be used in the sequel.

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.

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