By Omri Rand
This paintings specializes in mathematical equipment and sleek symbolic computational instruments required to unravel basic and complex difficulties in anisotropic elasticity. particular purposes are provided to the category of difficulties which are encountered within the conception.
Key beneficial properties: specific emphasis is put on the choice of analytic technique for a selected challenge and the opportunity of symbolic computational suggestions to aid and improve the analytic method of problem-solving · the actual interpretation of actual and approximate mathematical suggestions is punctiliously tested and gives new insights into the concerned phenomena · cutting-edge options are supplied for a variety of composite fabric configurations constructed via the authors, together with nonlinear difficulties and complicated research of laminated and thin-walled constructions · plentiful photo examples, together with animations, extra facilitate an knowing of the most steps within the answer procedure.
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Extra resources for Analytical methods in anisotropic elasticity: with symbolic computational tools
62), one may employ a standard Taylor series expansion, and therefore, in some cases the above truncated form represents an approximation. We subsequently expect the displacements and rotations to be expressed in levels as well, namely m u(k) (x, y)zk , u = ∑ k=0 m v = ∑ k=0 v(k) (x, y)zk , K Km ωi = ∑ ωi (x, y)zk , (k) m w = ∑ k=0 w(k) (x, y)zk , K K i = 1, 2, 3. 63b) k=0 In what follows, Km stands for the maximal expansion degree of all analysis components (clearly, the expansion degree of the strain components will be less than those of the displacement components).
2. 2 Displacement by Strain Integration 13 In general, we shall expand all spatial functions as truncated polynomials of degree Km ≥ 0 of the axial variable, z, and as continuous functions of the cross-section variables x and y. 60) we shall refer to Km as the expansion degree, and to G(k) as the kth (level) component of G. Clearly, G may be integrated and differentiated with respect to z as z 0 m G(k) G dz = ∑ k=0 K zk+1 , k+1 K −1 m G, z = ∑ k=0 (k + 1)G(k+1) zk . 61) To derive the present approach, we also assume that the strain components are truncated as m εi = ∑ k=0 εi (x, y)zk .
124), which is set for a small virtual perturbation of the displacements and not their magnitude in the equilibrium state. 3 Theorem of Reciprocity Consider two observations of the response of a given elastic body to two different systems of (1) (1) loads. First, the loading system Fs , Fb has been applied and created a deformation u(1) and strain energy U (1) . Then, (after the first loading system has been removed) a loading system (2) (2) Fs , Fb has created the deformation u(2) and the associated strain energy U (2) .
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