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Tham khảo tài liệu 'mechanics of saouma part 5', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | omposition 25 Given X1 X1 x2 3X3 x3 2X2 find the deformation gradient F the right stretch tensor U the rotation tensor R and the left stretch tensor V. Solution From Eq. 4.25 From Eq. 4.126 F dX1 dxi dxi dXi XXX iX2 dX2 iX3 dX2 dXi dX3 dX ỜX3 dX3 dX3 dX1 dX2 ỔX3 3 0 U2 1 0 0 1 0 0 1 0 0 ft f 0 0 2 0 0 3 0 4 0 0 3 0 0 2 0 0 0 9 1 0 0 thus From Eq. 4.127 R Fl Finally from Eq. 4.128 V frt 0 0 3 0 0 3 0 10 1 1 0 0 0 3 0 0 0 2 4.130 4.131 4.132 4.133 4.134 1 0 0 0 0 2 0 U 1 0 0 0 2 0 1 0 0 0 2 0 0 2 0 0 3 1 0 0 1 0 0 0 0 1 2 0 0 1 3 0 0 0 1 0 1 0 0 0 0 1 Example 4-11 Polar Decomposition II For the following deformation X1 A1X1 x2 A3X3 and x3 A2X2 find the rotation tensor. Solution F U 2 A1 0 0 0 0 A3 0 A2 0 Ff F A1 0 0 A1 0 0 A21 0 0 0 0 A2 0 0 A3 0 A22 0 0 A3 0 0 A2 0 0 0 A2 A3 U A1 0 0 0 A2 0 0 0 A3 R F U -1 A1 0 0 0 0 A 0 A2 0 3 1 A1 0 0 0 X Ă2 0 0 0 1 A3 1 4.135 4.136 4.137 4.138 4.139 1 0 0 0 0 1 0 0 Thus we note that R corresponds to a 90 rotation about the e1 axis. Victor Saouma Mechanics of Materials II KINEMATIC Example 4-12 Polar Decomposition III Victor Saouma Mechanics of Materials II 4.3 Strain Decomposition -ỹrait 27 m-polar.nb m- In 4 v1 v2 v3 N Eigenvectors CST 4 Determine U and U-1 with respect to the e basis In. 0 0 1. y Pola . ._ . ._. U_e N vnormalized . Ueigen . vnormalized 3 Given X1 matrix U Out 50 In 11 In 1 Out 6 Out 11 Out l In 7 0.707 0.707 0. y 0.707 2.12 0. 0. 0. 1. z vnormalized GramSchmidt v3 -v2 v1 u_einverse N Inverse 3 0.382683 0.92388 0 y 2.12 -0.707 0. y -0.707 0.707 0 0. 0. 1. z CSTeigen Chop N vnormalized . CST . vnormalized 4 gofôe In 12 In 2 -5.82 . ine Rwith7r1eisp0ect to the R N F . 3 0.707 0.707 0. y -707 0 07 0. 0. 0. 1. In 8 seterr e basis Ueigen N Sqrt CSTeigen 4 2.414 0 0 y . 0 0.4142 0 0 0 1. In 3 N Eigenvalues CST In 9 Ueigenminus1 Inverse Ueigen Out 3 Out 9 11. 0.171573 5.82843 0.414214 0. 0. y 0. 2.41421 0. __0 0. 1. Victor Saouma Mechanics of Materials .