Kinematics_of_CM_10_Convected_Coordinates

Kinematics_of_CM_10_Convected_Coordinates - Section 2.10...

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Section 2.10 Solid Mechanics Part III Kelly 279 2.10 Convected Coordinates In this section, the deformation and strain tensors described in §2.2-3 are now described using convected coordinates (see §1.16). Note that all the tensor relations expressed in symbolic notation already discussed, such as C U = , i i i n N F λ = ˆ , lF F = & , are independent of coordinate system, and hold also for convected coordinates. 2.10.1 Convected Coordinates Introduce the curvilinear coordinates i Θ . The material coordinates can then be written as ) , , ( 3 2 1 Θ Θ Θ = X X (2.10.1) so i i X E X = and i i i i d dX d G E X Θ = = , (2.10.2) where i G are the covariant base vectors in the reference configuration, with corresponding contravariant base vectors i G , Fig. 2.10.1, with i j j i δ = G G (2.10.3) Figure 2.10.1: Curvilinear Coordinates The coordinate curves, curves of constant i Θ , form a net in the undeformed configuration. One says that the curvilinear coordinates are convected or embedded , that is, the coordinate curves are attached to material particles and deform with the body, so that each material 1 1 , x X 2 2 , x X 3 3 , x X X 1 1 , e E 2 2 , e E 1 g 2 g current configuration reference configuration 1 G 2 G x
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Section 2.10 Solid Mechanics Part III Kelly 280 particle has the same values of the coordinates i Θ in both the reference and current configurations. In the current configuration, the spatial coordinates can be expressed in terms of a new, “current”, set of curvilinear coordinates ) , , , ( 3 2 1 t Θ Θ Θ = x x , (2.10.4) with corresponding covariant base vectors i g and contravariant base vectors i g , with i i i i d dx d g e x Θ = = , (2.10.5) Example Consider a motion whereby a cube of material, with sides of length 0 L , is transformed into a cylinder of radius R and height H , Fig. 2.10.2. Figure 2.10.2: a cube deformed into a cylinder A plane view of one quarter of the cube and cylinder are shown in Fig. 2.10.3. Figure 2.10.3: a cube deformed into a cylinder 0 L R 0 L H 1 X 2 X 1 x 2 x 0 L R X x P p
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Section 2.10 Solid Mechanics Part III Kelly 281 The motion and inverse motion are given by ) ( X χ x = , ( ) ()() 3 0 3 2 2 2 1 2 1 0 2 2 2 2 1 2 1 0 1 2 2 X L H x X X X X L R x X X X L R x = + = + = (basis: i e ) and ) ( 1 x χ X = , () () 3 0 3 2 2 2 1 1 2 0 2 2 2 2 1 0 1 2 2 x H L X x x x x R L X x x R L X = + = + = (basis: i E ) Introducing a set of convected coordinates, Fig. 2.10.4, the material and spatial coordinates are ) , , ( 3 2 1 Θ Θ Θ = X X , 3 0 3 2 1 0 2 1 0 1 tan 2 2 Θ = Θ Θ = Θ = H L X R L X R L X and (these are simply cylindrical coordinates) ) , , ( 3 2 1 Θ Θ Θ = x x , 3 3 2 1 2 2 1 1 sin cos Θ = Θ Θ = Θ Θ = x x x A typical material particle (denoted by p ) is shown in Fig. 2.10.4. Note that the position vectors for p have the same i Θ values, since they represent the same material particle.
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Section 2.10 Solid Mechanics Part III Kelly 282 Figure 2.10.4: curvilinear coordinate curves 2.10.2 The Deformation Gradient With convected curvilinear coordinates, the deformation gradient is i i G g F = , (2.10.6) which is consistent with ( ) X F G G g g x d d d d j i i j j j = Θ = Θ = (2.10.7) The deformation gradient F , the transpose T F and the inverses T 1 ,
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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_10_Convected_Coordinates - Section 2.10...

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