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 243 2.5 Deformation Rates In this section, rates of change of the deformation tensors introduced earlier, F , C , E , etc., are evaluated, and special tensors used to measure deformation rates are discussed, for example the velocity gradient l , the rate of deformation d and the spin tensor w . 2.5.1 The Velocity Gradient The velocity gradient is used as a measure of the rate at which a material is deforming. Consider two fixed neighbouring points, x and x x d + , Fig. 2.5.1. The velocities of the material particles at these points at any given time instant are ) ( x v and ) ( x x v d + , and x x v x v x x v d d + = + ) ( ) ( , The relative velocity between the points is x l x x v v d d d = ( 2 . 5 . 1 ) with l defined to be the (spatial) velocity gradient, j i ij x v l = = = , grad v x v l Spatial Velocity Gradient (2.5.2) Figure 2.5.1: velocity gradient The spatial velocity gradient is commonly used in both solid and fluid mechanics. Less commonly used is the material velocity gradient, which is related to the rate of change of the deformation gradient: F X X x X x X X X V V & = = = = ) , ( ) , ( ) , ( Grad t t t t t (2.5.3) and use has been made of the fact that, since X and t are independent variables, material time derivatives and material gradients commute. x x x d + () x v ( ) x x v d + v d x d
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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)
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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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