Section 2.10 Solid Mechanics Part III Kelly 2792.10 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 CU=, iiinNFλ=ˆ, lFF=&, are independent of coordinate system, and hold also for convected coordinates. 2.10.1 Convected Coordinates Introduce the curvilinear coordinates iΘ. The material coordinates can then be written as ),,(321ΘΘΘ=XX(2.10.1) so iiXEX=and iiiiddXdGEXΘ==, (2.10.2) where iGare the covariant base vectors in the reference configuration, with corresponding contravariant base vectors iG, Fig. 2.10.1, with ijjiδ=⋅GG(2.10.3) Figure 2.10.1: Curvilinear Coordinates The coordinate curves, curves of constant iΘ, form a net in the undeformed configuration. One says that the curvilinear coordinates are convectedor embedded, that is, the coordinate curves are attached to material particles and deform with the body, so that each material 11,xX22,xX33,xXX11,eE22,eE1g2gcurrent configuration reference configuration 1G2Gx
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