Download e-book for iPad: Analytical methods in anisotropic elasticity: with symbolic by Omri Rand
By Omri Rand
This paintings makes a speciality of mathematical tools and smooth symbolic computational instruments required to unravel basic and complicated difficulties in anisotropic elasticity. particular functions are offered to the category of difficulties which are encountered within the concept.
Key gains: specific emphasis is put on the choice of analytic technique for a selected challenge and the possibility of symbolic computational ideas to aid and boost the analytic method of problem-solving · the actual interpretation of tangible and approximate mathematical ideas is punctiliously tested and offers new insights into the concerned phenomena · state of the art options are supplied for quite a lot of composite fabric configurations built via the authors, together with nonlinear difficulties and complicated research of laminated and thin-walled constructions · plentiful photo examples, together with animations, additional facilitate an knowing of the most steps within the answer strategy.
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Additional resources for Analytical methods in anisotropic elasticity: with symbolic computational tools
However, we preserve the same notation logic in which σi j is defined as the stress component in the deformed jth direction of a given coordinate system that acts on a plane, which before deformation was perpendicular to the kˆ i -direction. On each of the deformed faces one may define stress vectors. We will denote by σ 1 , σ 2 and σ 3 the stress vectors over faces #1,2 and 3, see Fig. 3. 215)), and their respective areas are Ai = H j Hk dα j dαk (apply cyc-i jk). The areas of the corresponding faces after deformation are denoted A∗i .
152) where gk are constants and Fk are continuous functions of three arguments. 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).
168) ∂y More generally, if the functional F(x1 , . . , xn , u, u x1 , . . , u xn ) depends on 2n + 1 (n > 1) variables including the function u(x1 , . . , xn ) of n variables and its first derivatives, then the resulting Euler’s equation takes the form F, u − ∑ i=1 n ∂ (F, u xi ) = 0. 170) where the admissible functions u(x, y) belong to the C3 class on the domain Ω, and take specified continuous values on the boundary. These calculations lead to the following Euler’s equation F, u − ∂ ∂ ∂2 ∂2 ∂2 (F, u x ) − (F, u y ) + 2 (F, u xx ) + (F, u xy ) + 2 (F, u yy ) = 0.
Analytical methods in anisotropic elasticity: with symbolic computational tools by Omri Rand