ISBN-10: 0817642722

ISBN-13: 9780817642723

* complete textbook/reference applies mathematical equipment and sleek symbolic computational tools to anisotropic elasticity * Presents unified method of an enormous range of structural types * state of the art recommendations are supplied for quite a lot of composite fabric configurations, together with: 3D anisotropic our bodies, 2-D anisotropic plates, laminated and thin-walled buildings

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Extra info for Analytical Methods in Elasticity

Example text

I j are usually referred to as the“generalized stresses” and are not stresses in the strict sense. 75), these quantities are based on the element volume before deformation. However, they have important symmetry characteristics that are missing in σi j , namely, σi j = σ j i (while σi j = σ j i ). Therefore, under a coordinate system transformation the tensor σ = {σi j } acts like a secondorder symmetric tensor. Subsequently, this tensor should be regarded as the stress tensor when a fully nonlinear analysis is employed.

For example, the x coordinates of the A and B vertices (see Fig. 3) are, respectively, f1, α1 dα1 , f1, α1 dα1 + f1, α2 dα2 . When the same material element after deformation is examined, it may be no longer described as a cubic even in its coordinate space. The material element in the deformed state in both the coordinate space and the Euclidean space is generally called an “oblique angle parallelepiped”, and may be generally viewed as a cubic, the corners of which have been displaced differently, so its six faces are now different quadrangles (in essence, as previously discussed, this general description holds for the undeformed case in Euclidean space as well).

The corresponding Lagrange functional has the form JL (y) = x1 x0 [F(x, y, y ) + ∑ k=1 λk Fk (x, y, y )] dx. 5 Euler’s Equations 35 In this case we solve the variation problem JL (y) → min, by considering Lagrange multipliers, λk , as constants. 3 Elastica. The analysis described in this example deals with large deformations of an elastic rod. This kind of problems are traditionally termed “Elastica”. For further reading see (Frisch-Fay, 1962), (Stronge and Yu, 1993). 14. e. 2. Here s ∈ [0, l] is the natural length parameter of the curve.