Kinematics_of_CM_05_Deformation_Rates

# Kinematics_of_CM_05_Deformation_Rates - Section 2.5 2.5...

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Section 2.5 Solid Mechanics Part III Kelly 244 2.5.2 Material Derivatives of the Deformation Gradient The spatial velocity gradient may be written as x X X x x X x X x X X v x v = = = t t or 1 = F F l & so that the material derivative of F can be expressed as F l F = & Material Time Derivative of the Deformation Gradient (2.5.4) Also, it can be shown that { Problem 1} T T . T 1 . 1 T . T = = = F l F l F F F F & (2.5.5) 2.5.3 The Rate of Deformation and Spin Tensors The velocity gradient can be decomposed into a symmetric tensor and a skew-symmetric tensor as follows (see §1.10.10): w d l + = (2.5.6) where d is the rate of deformation tensor (or rate of stretching tensor ) and w is the spin tensor (or rate of rotation , or vorticity tensor ), defined by () = = + = + = i j j i ij i j j i ij x v x v w x v x v d 2 1 , 2 1 2 1 , 2 1 T T l l w l l d Rate of Deformation and Spin Tensors (2.5.7) The physical meaning of these tensors is next examined. The Rate of Deformation Consider first the rate of deformation tensor d and note that x v x l d dt d d d = = (2.5.8)
Section 2.5 Solid Mechanics Part III Kelly 245 The rate at which the square of the length of x d is changing is then ( ) () x d x x l x x x x x x x x x d d d d d dt d d d d dt d d dt d d dt d d d dt d 2 2 2 , 2 2 2 = = = = = (2.5.9) the last equality following from 2.5.6 and 1.10.31e. Dividing across by 2 2 x d , then leads to n d n ˆ ˆ = λ & Rate of stretching per unit stretch in the direction n ˆ (2.5.10) where X x d d / = is the stretch and x

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Kinematics_of_CM_05_Deformation_Rates - Section 2.5 2.5...

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