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Kinematics_of_CM_07_Small_Strain_Theory

Kinematics_of_CM_07_Small_Strain_Theory - Section 2.7 2.7...

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Section 2.7 Solid Mechanics Part III Kelly 258 2.7 Small Strain Theory When the deformation is small, from 2.2.43-4, ( ) u I F u I U I F grad grad Grad + + = + = (2.7.1) neglecting the product of u grad with U Grad , since these are small quantities. Thus one can take u U grad Grad = and there is no distinction to be made between the undeformed and deformed configurations. The deformation gradient is of the form α I F + = , where α is small. 2.7.1 Decomposition of Strain Any second order tensor can be decomposed into its symmetric and antisymmetric part according to 1.10.28, so that ij ij i j j i i j j i j i x u x u x u x u x u Ω + = + + = + = + + = ε 2 1 2 1 2 1 2 1 T T ε x u x u x u x u x u (2.7.2) where ε is the small strain tensor 2.2.48 and , the anti-symmetric part of the displacement gradient, is the small rotation tensor , so that F can be written as ε I F + + = Small Strain Decomposition of the Deformation Gradient (2.7.3) It follows that (for the calculation of e , one can use the relation ( ) δ I δ I + 1 for small δ ) ε e E ε I b C = = + = = 2 (2.7.4) Rotation Since is antisymmetric, it can be written in terms of an axial vector ω , cf . §1.10.11, so that for any vector a , 3 12 1 13 1 23 , e e e ω a ω a Ω Ω + Ω = × = (2.7.5) The relative displacement can now be written as
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