2-Stress+strain+isotropic+elasticity

# 2-Stress+strain+isotropic+elasticity - 15 Stress Strain and...

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Unformatted text preview: 15 Stress, Strain and Isotropic Elasticity We give a brief review of the principles of stress, strain and isotropic elasticity that will be needed in the study of defects. Probably students will have seen this elementary material in other courses. It is given here as a reference that will be used from time to time in the remainder of these notes. Stress The basic definition of stress is force/unit area. An axial stress is: ¡ = P A And a shear stress is: Dimensions: f / ¡ 2 Units: N / m 2 = Pascal = Pa Stress – A second rank tensor Nine quantities are needed to completely define a state of stress: ¡ ij ( i , j = x , y , z ) . Here i = directionof force , j = directionof planenormal . Three directions and three plane normals makes for nine components, giving a second rank tensor. Number of components of a tensor = N; Rank of a Tensor = R: N = 3 R so nine components for a second rank tensor. 16 Stress at a point (infinitesimal volume). The positive notation is shown. Positive components of stress involve positive forces acting on faces with positive normals (or equivalently, negative forces on faces with normals pointing in the negative directions (as shown for the component ¡ xx ) It is convenient to write the second rank stress tensor as a 3x3 matrix: ¡ ij = ¡ xx ¡ xy ¡ xz ¡ yx ¡ yy ¡ yz ¡ zx ¡ zy ¡ zz ¢ £ ¤ ¤ ¤ ¥ ¦ § § § . A similar notation is used for other orthogonal coordinate systems. Cylindrical coordinates ¡ ij = ¡ rr ¡ r ¢ ¡ rz ¡ ¢ r ¡ ¢¢ ¡ ¢ z ¡ zr ¡ z ¢ ¡ zz £ ¤ ¥ ¥ ¥ ¦ § ¨ ¨ ¨ 17 Spherical coordinates ¡ ij = ¡ rr ¡ r ¢ ¡ r £ ¡ ¢ r ¡ ¢¢ ¡ ¢£ ¡ £ r ¡ £¢ ¡ ££ ¤ ¥ ¦ ¦ ¦ § ¨ © © © The stress tensor is a symmetric tensor, wherein ¡ ij = ¡ ji . A simple moment equilibrium analysis shows this. The sum of the moments about the z axis should be zero if no inertial moments exist (no dynamics). Thus M z ¡ = ¢ xy £ x ( ) £ y ¤ ¢ yx £ y ( ) £ x = ¢ xy = ¢ yx and, generally, ¡ ij = ¡ ji . So there are only six independent stress components. It is sometimes convenient to rename these components as follows ¡ ij 18 ¡ ij = ¡ xx ¡ xy ¡ xz ¡ yx ¡ yy ¡ yz ¡ zx ¡ zy ¡ zz ¢ £ ¤ ¤ ¤ ¥ ¦ § § § = ¡ 1 ¡ 6 ¡ 5 • ¡ 2 ¡ 4 • • ¡ 3 ¢ £ ¤ ¤ ¥ ¦ § § This is called the matrix notation, where the stress components are ¡ i ( i = 1,2... 6) ¡ 1 = ¡ xx , ¡ 2 = ¡ yy , ¡ 3 = ¡ zz ¡ 4 = ¡ yz = ¡ zy ¡ 5 = ¡ xz = ¡ zx ¡ 6 = ¡ xy = ¡ yx sometimes written as ¡ i = ¡ 1 ¡ 2 ¡ 3 ¡ 4 ¡ 5 ¡ 6 ¢ £ ¤ ¤ ¤ ¤ ¤ ¤ ¤ ¥ ¦ § § § § § § § For cylindrical coordinates: ¡ 1 = ¡ rr , ¡ 2 = ¡ ¢¢ , ¡ 3 = ¡ zz ¡ 4 = ¡ ¢ z = ¡ z ¢ ¡ 5 = ¡ rz = ¡ zr ¡ 6 = ¡ r ¢ = ¡ ¢ r and for spherical coordinates ¡ 1...
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## This note was uploaded on 12/01/2011 for the course MS&E 206 taught by Professor Nix during the Spring '08 term at Stanford.

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2-Stress+strain+isotropic+elasticity - 15 Stress Strain and...

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