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Mathematical Foundations of Elasticity by Jerrold E. Marsden;Thomas J. R. Hughes

By Jerrold E. Marsden;Thomas J. R. Hughes

This graduate-level research methods mathematical foundations of three-d elasticity utilizing glossy differential geometry and sensible research. it truly 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. correct difficulties seem during the textual content. 1983 variation.

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Anyone who wishes to follow the mainstream of current research in bifurcation theory and its applications to elasticity must be literate, or better fluent, in both geometry and functional analysis. Geometry and functional analysis go together very well. " It is our conviction that in future years these subjects will become basic tools for many researchers in elasticity. This is why we have used them in this book. CHAPTER GEOMETRY AND KINEMATICS OF BODIES Normally a solid continuum body occupies an open subset of three-dimensional Euclidean space (or the closure of an open subset if the boundary is counted).

In Chapter 6 we shall see that strong ellipticity is just what is needed for a good existence and uniqueness theory for linear elastostatics and elastodynamics. The violation of strong ellipticity has little or nothing to do with the loss of stability and buckling, as we shall see in Chapter 7 (compare Truesdell and Noll [1965], p. 275). Antman [1973a] has shown t\1at strong ellipticity is compatible with the necking of bars in tension. Ericksen [1975] has suggested that strong ellipticity may fail when "phase transitions" occur.

To distinguish between the two types of vectors, the terminologies covariant and contravariant are applied to ea and ea, respectively. ) Tensor Analysis on Manifolds Let {xa} denote a coordinate system defined on an open set in a manifold S. The coordinate basis corresponding to {xa} is denoted a/axa, and the dual basis is denoted dx a. The terminology "vector" is reserved for a/axa and the terminology "one-form" (or covector) is used for dxa. The transformation rules for a/axa and dx a are suggested by the differential operator notations; that is, if :Xa denotes another coordinate system on S, then a ax b a d d -a a:x ad b a:xa = a:xa ax b an x = axb x.

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