By Andrei Constantinescu
This ebook is meant for researchers, engineers and scholars in strong mechanics, fabrics technological know-how and physics who're attracted to utilizing the ability of recent computing to resolve a wide selection of difficulties of either sensible and basic importance in elasticity. large use of Mathematica within the ebook makes on hand to the reader a variety of recipes that may be with no trouble adjusted to check specific tastes or specifications, to imagine ideas, and to hold out symbolic and numerical research and optimization.
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Extra resources for Elasticity with Mathematica: An introduction to continuum mechanics and linear elasticity
M = n · M · n q = n · 2divM The resultant body force f per unit area, resultant bending moment m per unit length, and resultant shear force q per unit length are related to the body force f and surface tractions t by the formulas f (xˆ ) = h/2 −h/2 e 3 · f (xˆ , x3 ) dx3 m(xˆ ) = h/2 −h/2 x3t (xˆ , x3 ) dx3 q(xˆ ) = h/2 −h/2 e 3 · t (xˆ , x3 ) dx3 . PAB CUFX161-Constantinescu August 13, 2007 17:14 46 Dynamics and statics: stresses and equilibrium The above expressions demonstrate that force fields acting within thin plates are given by integral moments of different orders of the three-dimensional body force and surface tractions.
4 Potential representations of self-equilibrated stress tensors 49 The Beltrami stress potential If is bounded by a simple closed surface, then the following statements are equivalent: (i) symmetric stress tensor field σ is divergence-free, div σ = 0 (ii) symmetric stress tensor field σ is self-equilibrated, σ · n ds = 0 x × σ · n ds = 0 (iii) there exists a second-order symmetric tensor field B such that σ = inc B. The equivalence of the first two statements for a simply bounded body has already been established.
The above equations are obtained by integrating the local form of the balance equations and applying the formula for the time derivative of a moving domain. As a particular case of the above expressions, the principle of virtual work can be applied to the virtual velocity field corresponding to general rigid body movement, giving the balance of linear and angular momentum for the entire body . Bending of thin plates A plate can be defined as a solid occupying a domain of the form = ω × [−h/2, h/2].