MIT2_094S11_lec12

MIT2_094S11_lec12 - 2.094 Finite Element Analysis of Solids...

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Unformatted text preview: 2.094 Finite Element Analysis of Solids and Fluids Fall 08 Lecture 12- Total Lagrangian formulation Prof. K.J. Bathe MIT OpenCourseWare We discussed: t t X = x x i j d t x = t X d x , d x = t X 1 d t x (12.1) t C = t X T t X (12.2) d x = t X d t x where t X = t X 1 = t x i (12.3) x j The Green-Lagrange strain: 1 1 t = t X T t X I = t C I (12.4) 2 2 Polar decomposition: 1 2 t X = t R t U t = 2 t U I (12.5) We see, physically that: where d t + t x and d t x are the same lengths the components of the G-L strain do not change. Note in FEA k x i = h k x i k for an element (12.6) t t k u i = h k u i k t t x i = x i + u i for any particle (12.7) Hence for the element t k t k x i = h k x i + h k u i (12.8) k k = h k x i k + t u i k (12.9) k = h k t x k k (12.10) k 49 MIT 2.094 12. Total Lagrangian formulation E.g., k = 4 2nd Piola-Kirchhoff stress t S = t t X t t X T components also independent...
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MIT2_094S11_lec12 - 2.094 Finite Element Analysis of Solids...

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