Section 2.7 Solid Mechanics Part III Kelly 2582.7 Small Strain Theory When the deformation is small, from 2.2.43-4, ()uIFuIUIFgradgradGrad+≈+=+=(2.7.1) neglecting the product of ugradwith UGrad, since these are small quantities. Thus one can take uUgradGrad=and there is no distinction to be made between the undeformed and deformed configurations. The deformation gradient is of the form αIF+=, where αis 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 ijijijjiijjijixuxuxuxuxuΩ+=⎟⎟⎠⎞⎜⎜⎝⎛∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛∂∂+∂∂=∂∂+=⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛∂∂−∂∂+⎟⎟⎠⎞⎜⎜⎝⎛⎟⎠⎞⎜⎝⎛∂∂+∂∂=∂∂ε21212121TTΩεxuxuxuxuxu(2.7.2) where εis the small strain tensor 2.2.48 and Ω, the anti-symmetric part of the displacement gradient, is the small rotation tensor, so that Fcan be written as ΩεIF++=Small Strain Decomposition of the Deformation Gradient(2.7.3) It follows that (for the calculation of e, one can use the relation ()δIδI−≈+−1for small δ) εeEεIbC==+==2(2.7.4) Rotation Since Ωis antisymmetric, it can be written in terms of an axial vectorω, cf. §1.10.11, so that for any vector a, 312113123,eeeωaωΩaΩ−Ω+Ω−=×=(2.7.5) The relative displacement can now be written as
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