Analytical Methods in Elasticity by Omri Rand;Vladimir Rovenski

By Omri Rand;Vladimir Rovenski

* entire textbook/reference applies mathematical tools and smooth symbolic computational tools to anisotropic elasticity * Presents unified method of an enormous range of structural versions * state of the art suggestions are supplied for a variety of composite fabric configurations, together with: 3-D anisotropic our bodies, 2-D anisotropic plates, laminated and thin-walled constructions

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3 Stress Measures (a) Geometrical relations between the Θ4 , Θ5 invariants and the principal σDP i eigenvalue. 23 (b) An infinitesimal cube at an arbitrary orientation. 6: Stress invariants and transformation notation. 112b) 2 T13 = σ2T + σN − σP1 σN − σP2 . s. 113b) ≥ 0. 113c) σN − σP3 σN − σP1 We may now plot the above conditions in the σN − σT plane, in order to find a region where all inequalities are satisfied, which will yield the valid combinations of normal and shear stress at the point under discussion.

110b). An immediate and clear result of Mohr’s diagram is that the maximal shear stress is given by σP −σP σP +σP m 1 3 1 3 σm T = 2 , which occurs at σN = 2 . 12 has produced Mohr’s diagram shown in Fig. 7(b). 095 (a) General notation. 1. 7: Mohr’s diagram. For each combination of σN and σT in Mohr’s diagram corresponds a pair of angles (ψ, θ). Note that the value of φ has no importance in this case, since it represents a rotation about the x-direction (as it will change neither σN nor σT ). 208:a,b).

2. 2: Normal and tangential loads over a small area in an elastic domain. (in-plane) force component will be denoted “T”. Both N and T are functions of the location of the circle over the plane (or essentially the location of its center point, P in the plane). By narrowing the area A, the point P and the area A collide, and the normal and tangential stress components at P are defined as N , A→0 A σN (P) = lim T . 72) Hence, the dimension of the stress components is force per unit area. At each point, one may examine an infinitesimally small material element, as shown in Fig.

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