StressStrain_08_Analysis of Strain

StressStrain_08_Analysis of Strain - Section 3.8 3.8...

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Section 3.8 Solid Mechanics Part I Kelly 85 3.8 Properties of the Strain Stress transformation formulae, principal stresses, stress invariants and formulae for maximum shear stress were presented in §3.4-§3.5. The strain is very similar to the stress. They are both tensors, having nine components, and all the formulae for stress hold also for the strain. Formulae for a two-dimensional state of strain are given in what follows. 3.8.1 Strain Transformation Formula Consider two perpendicular line-elements lying in the coordinate directions x and y , and suppose that it is known that the strains are xy yy xx ε , , , Fig. 3.8.1. Consider now a second coordinate system, with axes y x , , oriented at angle θ to the first system, and consider line-elements lying along these axes. It can be shown that the line-elements in the second system undergo strains according to the following (two dimensional) strain transformation equations : xy xx yy xy xy yy xx yy xy yy xx xx θε 2 cos ) ( cos sin 2 sin cos sin 2 sin sin cos 2 2 2 2 + = + = + + = Strain Transformation Formulae (3.8.1) Figure 3.8.1: A rotated coordinate system Note the similarity between these equations and the stress transformation formulae, Eqns. 3.4.7.
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StressStrain_08_Analysis of Strain - Section 3.8 3.8...

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