Midterm Exam (Continuum Mechanics, 2016 Fall) Instruction: By default, the graduate students are required to answer all the question below. Question 1 part a) and Question 5 are not required for undergraduate students but are bonus problem for undergraduate students 1. Write both the expanded form and matrix form of a)   ij ijkl kl  C with     ij ji kl lk ijkl klij    (stress), (strain), (Stiffness tensor) C C (hint: for Cijkl it has 81 components, (this symmetry will reduce from 81 to 54), (this symmetry will further reduce from 54 to 36), (this symmetry will further reduce from 54 to 21) ij ji kl kl C C ijkl klij        ) Question a) is not required for undergraduate students but is a bonus problem for undergraduates. b) T S ij ij where i j k l , , , 1,2,3   2. In the Cartesian coordinate, the stress-strain relationship can be given as below for isotropic and linear elastic materials, please use the Einstein summation index notation to rewrite the long form of the following equations into one compact form equation.       11 11 22 33 12 12 22 22 11 33 23 23 33 33 11 22 31 31 1 1 , 1 1 , 1 1 , v v E E v v E E v v E E                                              3. Write the components of b T S a b TS a   ( ) or  in both index notation and matrix form, where a and b are vectors, T and S are 2nd order tensors. 4. Prove that (a) u v w u w v u v w           (b) u v w w u v v w u            (c)    ij jk ki  3where u, v, and w are vectors, " " and " "  are scalar products (or inner products or dot products) and cross product, respectively. ij is the Kronecker Delta function. (d) Is u v w     equal to u v w    , why? 5. (graduate students only) In the Cartesian coordinates, prove the vector identity        u u u    2 6. Three motions are applied to a material in the following sequence (a) A translational rigid motion moving origin C0 to C by uXC and uYC along X and Y , respectively. There is no motion along Z (b) Then stretching of    1 0 2 0 3 0    L L H H H H X Y Z , , and along , , and , respectively Y Y Z Z (c) rigid rotation  (positive counter clockwise CCW) in the X Y C , about Questions: 1) Write the combined motion in the matrix form 2) Write the deformation and displacement gradients in the matrix form 3) Change the sequence of (a)(b)  (c) to i) (a) (c) (b), i.e. translation-rotation-stretching ii) (c)(b)(a), i.e. rotation-stretching-translation Then, solve the corresponding questions in 1) and 2) and see if there is any difference.