By Jerrold E. Marsden;Thomas J. R. Hughes
This graduate-level learn ways mathematical foundations of third-dimensional elasticity utilizing sleek differential geometry and practical research. it really is directed to mathematicians, engineers and physicists who desire to see this classical topic in a contemporary surroundings with examples of more moderen mathematical contributions. proper difficulties look through the textual content. 1983 variation.
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Example text
If c(t) is a curve in 1R 3 , the tangent to c(t) is defined by C/(t) = limith~o (c(t + h) - c(t»/h. If the standard Euclidean coordinates of c(t) are (cl(t), c 2 (t), c 3(t», then C/(t) l 2 = (dc , dc , dC dt dt 3 ). dt To avoid confusion with other coordinate systems we shall write t/>~, and so on, for the Euclidean components of t/>. Since IR 3 is the set of all real triples, denoted z = (Zl, Z2, Z3), and for fixed X E (X, t) is a curve in IR\ we get VeX, t) = (V~(X, t), V;(X, t), V;(X, t» at/>; -_ (at/>~ ~(X,t), at/>; ~(X, t), ~(X, t)) .
Tr b)2 - (tr b2)] = 3 + K2, and det b = 1. 2 + 1) o 0 0 0 Po ] . + Pl + P2 The columns of a give the forces acting on planes with normals in the three coordinate directions. Notice that the normal stresses, 0' 11'0' 22' and 0' 33 need not vanish, so that simple shear cannot be maintained by a shear stress alone. However, note that if the reference state is unstressed, then a = 0 when K = 0; that is, Po + PI + P2 = 0 when K = O. In this case, 0'33' 0'11' and 0'22 are 0(K 2), that is, are second order in K.
CN) For hyperelastic materials, (CN) may be translated to a convexity-type condition on the stored energy function. For homogeneous isotropic hyperelastic materials, (CN) implies that is strictly convex and is subject therefore to the same criticisms as above. That being strictly convex is unreasonable has been pointed out by Hill [1968], [1970], Rivlin [1973], and Sidoroff [1974]. That (eN) implies (SC