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Kinematics_of_CM_06_Deformation_Rates_Further

Kinematics_of_CM_06_Deformation_Rates_Further - Section 2.6...

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Section 2.6 Solid Mechanics Part III Kelly 253 2.6 Deformation Rates: Further Topics 2.6.1 Relationship between l, d, w and the rate of change of R and U Consider the polar decomposition RU F = . Since R is orthogonal, I RR = T , and a differentiation of this equation leads to T T R R R R R & & = (2.6.1) with R skew-symmetric (see Eqn. 1.14.2). Using this relation, the expression 1 = F F l & , and the definitions of d and w , Eqn. 2.5.7, one finds that { Problem 1} ( ) [ ] ( ) [ ] T 1 T 1 1 T 1 T 1 1 T 1 sym 2 1 skew 2 1 R U U R R U U U U R d R U U R R U U U U R w R U U R l R R R = + = + = + = + = & & & & & & & (2.6.2) Note that R being skew-symmetric is consistent with w being skew-symmetric, and that both w and d involve R , and the rate of change of U . When the motion is a rigid body rotation, then 0 U = & , and T R R w R & = = (2.6.3) 2.6.2 Deformation Rate Tensors and the Principal Material and Spatial Bases The rate of change of the stretch tensor in terms of the principal material base vectors is { } = + + = 3 1 ˆ ˆ ˆ ˆ ˆ ˆ i i i i i i i i i i N N N N N N U & & & & & λ λ λ (2.6.4) Consider the case when the principal material axes stay constant, as can happen in some simple deformations. In that case, U & and 1 U are coaxial (see §1.11.5): = = 3 1 ˆ ˆ i i i i N N U λ & & and = = 3 1 1 ˆ ˆ 1 i i i i N N U λ (2.6.5)
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