By P. Podio-Guidugli
I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her exact feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for helpful comments caused through their examining of 1 or one other of the numerous past 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 necessary reviews and stuck error: an inventory might 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 permit eight be a three-dimensional Euclidean area, 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 interior product a .
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3) suggest other pairs of dual stress and strain measures. The following forms involving Piola and Cosserat stresses are important in the mechanics of solids: p(s)(n) = f TR. F" 1n = f TR . D =! 2). REMARK. We give two examples that illustrate the mechanical significance of the stress power. 5) with 1l' a pressure field that is not constitutively specified, and JL the viscosity. 7) u(F) is interpreted as the stored energy per unit referential volume in a defor- mation of gradient F. Clearly, in this case, TR .
J 9. must be such to satisfy 9. 11) for each index k, here unsummed. , is left invariant under the action of each Q associated with a rotation belonging to the given group 9.. 12) for 9. = Rot are 1 = Devand e2 = Sph (cf. 11)). 4). L (with Y1 of multiplicity 5, and Y2 simple): e C iso = Y1 dev + Y2 sph. l, its orthogonal complement in Sym, obeys it as well (Exercise 4). 59 CONSTITUTIVE ASSUMPTIONS In the transversely isotropic case, when 9. ::J Rot(a), the solution of the representation problem is slightly more complicated.
Note that the operator Div (div) involves differentiation with respect to the space variables in the reference (defonned) shape (Exercise 3). 2)z are equivalent. ** The Cosserat stress is called the second Piola-Kirchhoff stress by those who call first PiolaKirchhoff stress the Piola stress; the use of the denomination Piola-Cosserat stress for the Cosserat stress is also documented. Resting on the historical notes found in [35, Section 210], we find unnecessary to take away from either Piola or the Cosserat brothers to give to Kirchhoff, who deserves his fame for other achievements; and we find that the qualifier Piola-Cosserat is an arbitrary artifact.