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MIT2_094S11_lec11 - 2.094 Finite Element Analysis of Solids...

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2.094 Finite Element Analysis of Solids and Fluids Fall ‘08 Lecture 11 - Deformation, strain and stress tensors Prof. K.J. Bathe MIT OpenCourseWare We stated that we use Reading: Ch. 6 t V t τ ij δ e ij d V t = 0 V t S t 0 V = t R (11.1) t 0 ij δ 0 ij d The deformation gradient We use t x i = 0 x i + t u i t x 1 t x 1 t x 1 0 x 1 0 x 2 0 x 3 t x 2 t x 2 t x 2 t X 0 (11.2) = 0 0 0 x x x 1 2 3 x t 3 x t 3 x t 3 x 0 1 x 0 2 x 0 3 t d x 1 d t x (11.3) t d x = 2 t d x 3 0 d x 1 d 0 x 0 (11.4) (11.5) d x 2 0 d x 3 Implies that d t x = 0 t X d 0 x = ( 0 t X is frequently denoted by 0 t F or simply F , but we use F for force vector) We will also use the right Cauchy-Green deformation tensor t C t X T t X = (11.6) 0 0 0 Some applications 45
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MIT 2.094 11. Deformation, strain and stress tensors The stretch of a fiber ( t λ ): 2 t t t 2 d x T d x d s t λ = = (11.7) d 0 x T d 0 x d 0 s The length of a fiber is 0 0 0
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