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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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