By P. Podio-Guidugli
I are looking to thank R. L. Fosdick, M. E. Gurtin and W. O. Williams for his or her distinct feedback of the manuscript. I additionally thank F. Davi, M. Lembo, P. Nardinocchi and M. Vianello for worthwhile feedback caused by way of their analyzing of 1 or one other of the various prior drafts, from 1988 up to now. because it has taken me goodbye to deliver this writing to its current shape, many different colleagues and scholars have episodically provided necessary reviews and stuck error: an inventory may chance to be incomplete, yet i'm heartily thankful to all of them. ultimately, 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 house, and permit 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 may well use to label p. additionally, 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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24) (Exercises 2, 6, and 7). Notice that, for both Mo(Q, f) and Mo(Q, i) to be symmetric, the information needed is, roughly speaking, that the reference and the deformed shape have the same overall symmetries; since an assumption on the deformed shape is involved, that information has an implicit, but indubitable, constitutive nature. EXERCISES 1. Let the Kirchhoff stress measure be defined by T := (detF)T. 25) 45 STRESS Show that, to within O(IVul)-terms, T = T = TR = 2. In Figure 10, let a = a e, = 1.
5. 20) fR, then (Y'v)v = -v x Curl v. 21) 6. Let v be a vector field, and let A be a symmetric-valued tensor field over a region fR. 'R An· v = f (Div A) . v + LA. sym(Y'v). 15) holds for a smooth, divergenceless constraint field V(p) overQ. 7. Let 'Un be the n-dimensional unit sphere in the vector space V n. 23) where Ivn is the identity of v(n). 8. Let Eo E Sym be given, and let u(p) = Eo(p - 0), p E afR. 1), show that IE = Eo. 25 CHAPTER II Stress 7. Forces. Balances Once a body Q and a deformation f of Q have been fixed, we confront the problem of modelling the accompanying mechanical interaction, both between parts of Q and between Q and its environment.
The problem of finding all pairs (y, e), with Y e R and e a subspace of Sym, such that C[C]=yC, ICI=l, Cee. , ) if (Yi, C i ) and (Yj, C j ) are proper pairs with Yi #- Yj, then C i . Cj = O. 6) C= LYiCi ®Ci, i=l where it is understood that proper numbers are repeated as many times as their algebraic multiplicity. H For example, if Y has multiplicity 2, there are solutions * Cf. [9, 10; 11, Section 21]. , multiple of 11:/2 about a fixed axis. ** Hence, transversely isotropic materials have the symmetry group of a circle in the plane.