Kinematics_of_CM_11_Convected_Coordinates_Rates

Kinematics_of_CM_11_ - j i j i ij j i m i mj j i j i j i j i i j j i j i ij l g l l l g g lg g g g g g lg g g g lg g g g lg g&&&&&

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Section 2.11 Solid Mechanics Part III Kelly 296 Convected Coordinates: Time Rates of Change In this section, the time derivatives of kinematic tensors described in §2.4-2.6 are now described using convected coordinates. 2.11.1 Deformation Rates Time Derivatives of the Base Vectors and the Deformation Gradient First, the material time derivatives of the deformed base vectors are, from 2.10.8, i i i i i i i i g F F g F F G F g g F F g F F G F g T T T T T 1 1 = = = = = = (2.11.1) with, again from 2.10.8, i i i i i i i i g G F G g F g G F G g F = = = = T T 1 (2.11.2) The Velocity Gradient The velocity gradient is defined by 2.5.2, v l grad = , so that, using 1.16.5, j j i i j j i i x x g v e v e v x v l Θ = Θ Θ = = = (2.11.3) Also, from 1.16.3, i i i Θ = Θ = v x g (2.11.4) so that, as an alternative to 2.11.3, i i g g l = (2.11.5) This is consistent with Eqn. 2.5.4, lF F = , which gives, with 1.11.2a and 2.10.8b, ( )( ) i i j j i i g g g G G g F F l = = = 1 (2.11.6)
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Section 2.11 Solid Mechanics Part III Kelly 297 The components of the spatial velocity gradient are
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Unformatted text preview: j i j i ij j i m i mj j i j i j i j i i j j i j i ij l g l l l g g lg g g g g g lg g g g lg g g g lg g & & & & & ⋅ = = ⋅ = ⋅ = = ⋅ = = ⋅ = = ⋅ ⋅ (2.11.7) Further, from 2.11.1, 2.11.2 and 2.11.5, l g l g g l g lg g i i i i i i − = = − = = T T & & (2.11.8) Contracting the first of these with i d Θ leads to i i i i d d Θ = Θ lg g & (2.11.9) which is equivalent to 2.5.1, x l v d d = . The Rate of Deformation and Spin Tensors From 2.5.6, w d l + = . The covariant components of the rate of deformation and spin are ( ) ( ) ( ) ( ) ( ) ( ) j i j i j m m m m i j i ij j i j i j i j m m m m i j i ij w d g g g g g g g g g g g l l g g g g g g g g g g g g g g l l g ⋅ − ⋅ = ⊗ − ⊗ = − = ⋅ = ⋅ + ⋅ = ⊗ + ⊗ = + = ⋅ & & & & & & & & 2 1 2 1 2 1 2 1 2 1 2 1 2 1 T T (2.11.10)...
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This note was uploaded on 01/20/2012 for the course ENGINEERIN 3 taught by Professor Staff during the Fall '11 term at Auckland.

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Kinematics_of_CM_11_ - j i j i ij j i m i mj j i j i j i j i i j j i j i ij l g l l l g g lg g g g g g lg g g g lg g g g lg g&&&&&

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