By P. Podio-Guidugli
I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her specified feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for priceless comments caused via their studying of 1 or one other of the various past drafts, from 1988 to this point. because it has taken me see you later to deliver this writing to its current shape, many different colleagues and scholars have episodically provided invaluable reviews and stuck blunders: a listing might chance to be incomplete, yet i'm heartily thankful to all of them. eventually, I thank V. Nicotra for skillfully reworking 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 enable V be the vector area 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 foundation zero E eight and from any triplet (~1, ~2, ~3) E R3 of coordinates that we could 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 think about the interior product a .
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5), both an exact and a linearized equilibrium problem can be fonnulated with those data. 17) a relation that allows for qualitative estimates of the solution that are independent of the constitutive law (Exercise 2). 18) of the reference shape can always be found such that Mo(Q, r) E Sym. 19) * The general question of consistency between exact and linearized 3-dimensional elastostatics was posed by A. Signorini in the thirties. By means of formal asymptotic expansions of the type already introduced by the Cosserat brothers, Signorini found consistency conditions, as well as instances of inconsistency, that were later variously generalized, and given both mechanical and geometrical interpretation.
14) and specifies the pull-back to the reference shape of the surface force: SR dA = F-1sda. 16) EXERCISES 1. 3), respectively, show that mo is the axial vector of -2 Skw Mo (cf. 2). * 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.
8h. 42 P. 3). 21) that TR = TF* = T(O + C\ trH)I - c\HT + ... ) = T + O(C\). 9) of a smallness parameter is performed, then the Piola and Cauchy stress measures coalesce. For this reason, when we deal with the linear theory, we shall need not distinguish between these two stress measures, * and indicate stress by the letter S. 6) helps us to evaluate further the position of the linearized versus the exact formulation of eqUilibrium problems in solid mechanics. 6) can be written as Mo(Q, J) = Mo(Q, i) +( JaQ u® So + 0) 1 + u(p), u ® bo E Sym.