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Tham khảo tài liệu 'nonlinear continua part 4', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | 2.13 Strain rates 51 It is important to remember that the functional dependence is lxaA aA oXB f . 2.109b Using the chain rule in Eq. 2.109a toXaA va i toXlA . 2.109c We define in the spatial configuration the velocity gradient tensor ỉ tga tg 2.110a we can write the above as Ị v V 2.110b tlT Vtv . 2.110c Hence we can write Eq. 2.109c as t V a __ tia t vl fci 1 1 1 OA A I I A a 2.111a and therefore X Ị oX 2.111b It is important to realize that the above is the material time derivate of D t X the deformation gradient tensor X t . 2.13.2 The Eulerian strain rate tensor and the spin vorticity tensor We can decompose the velocity gradient tensor into its symmetric and skew-symmetric components l d 2.112a where d tdT 2 tl 2.112b is the Eulerian strain rate tensor defined in the spatial configuration and - t T 2 _ 2.112c is the spin or vorticity tensor also defined in the spatial configuration. Let us assume a deformation process referred to a fixed Cartesian system. The principal directions of U form in the reference configuration a Cartesian system known as Lagrangian system. The principal directions of ỊV form in the spatial configuration a Cartesian system known as a Eulerian system Hill 1978 . 52 Nonlinear continua Lagrangean system time t 0 see Eq. 2.47.b R Eulerian system time t Fig. 2.5. Rotations We can go from one of the above-defined coordinate systems to another one using the rotation tensors sketched in Fig. 2.5. From Fig. 2.5 we get tiE R R 2.113 For two consecutive rotations Í ÍR t tR R 2.114a and therefore ịR _ lim At 0 1 4tR - g At oR 2.114b We can define a rotation rate . _ H 4tR - g _ lim At 0 ----- -- R At 2.114c and using it in Eq. 2.114b Hill 1978 we get in the same way oR _ Rr ỒR 2.115a ĨRl _ Rl Rl 2.115b iu _ Re Re 2.115c 2.13 Strain rates 53 Since the rotation tensors are orthogonal we can write IRT oR g 2.116a taking the time derivative of the above equation and using Eq. 2.115a we have tữR t R 0 2.116b in the same way ỒOl ỒOĨ 0 2.116c tữE t E 0
