By Omri Rand;Vladimir Rovenski
* accomplished textbook/reference applies mathematical equipment and glossy symbolic computational tools to anisotropic elasticity * Presents unified method of an enormous range of structural versions * state of the art strategies are supplied for quite a lot of composite fabric configurations, together with: 3-D anisotropic our bodies, 2-D anisotropic plates, laminated and thin-walled constructions
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Additional info for Analytical Methods in Elasticity
The result is a function of the specific point under discussion, and may be written as a function of the curvilinear coordinates α1 , α2 , α3 . 208). 11 executes stress tensor transformations between curvilinear and Cartesian coordinates. 20 1. 8 2 4 1 x 2 3 x_new 4 column (a) Axis orientation. 5 row 6 6 (b) The matrix Mσ . 4: Example of axis orientation and the resulting Mσ for ψ = 30◦ , θ = φ = 0. 1 Principal Stresses We shall now seek expressions for the principal stresses at a point. In essence, we are looking for a set of rotation angles that will define a new system orientation, in which only the normal stress components σα (α = x, y, z) are nonzero at a point.
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) . Applying now the first system loading and then the second one (without removing the first one), we get the deformation u(1) + u(2) and strain energy U (1) +U (2) +U (12) .
The converse theorem is true as well, see (Sokolnikoff, 1983). g. 125) where T stands for the total kinetic energy stored in the system, and t1 and t2 are times where the deformation is known. This principle may be invoked to derive the equations of motion of an elastic body, or in other words, the resulting Euler’s equations of such functional are time-dependent. 8. 2 The Theorem of Minimum Complementary Energy In contrast with the previous case where displacement components variations were dealt with, we shall now employ small variations of the stress distribution.
Analytical Methods in Elasticity by Omri Rand;Vladimir Rovenski