Write the components of b tsa or b tsa in both index

Question

Midterm Exam

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

Question 1. Write both the expanded form and matrix form of

a) σ_ij = C_ijklε_kl with σ_ij = σ_ji (stress),ε_kl = ε_lk(strain),C_ijkl = C_klij (Stiffness tensor)

(hint: for Cijkl it has 81 components,

σ_ij = σ_ji (this symmetry will reduce from 81 to 54),
ε_kl = ε_kl (this symmetry will further reduce from 54 to 36), )
C_ijkl = C_klij (this symmetry will further reduce from 54 to 21)

Question 2. a) is not required for undergraduate students but is a bonus problem for undergraduates.

b) T_ij.S_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.

ε₁₁ = 1/E[σ₁₁ - v(σ₂₂ + σ₃₃), ε₁₂ = (1+v)/E.σ₁₂

ε₂₂ = 1/E[σ₂₂ - v(σ₁₁ + σ₃₃), ε₂₃ = (1+v)/E.σ₂₃

ε₃₃ = 1/E[σ₃₃ - v(σ₁₁ + σ₂₂), ε₃₁ = (1+v)/E.σ₃₁

Question 3. Write the components of b = T.S.a (or b =(TS)a ) in both index notation and matrix form, where a and b are vectors, T and S are 2nd order tensors.

Question 4. Prove that

(a) u x(v x w) = (u.w)v - (u.v)w

(b) u.(v x w) = w.(v x v) = v.(w x u)

(c) δ_ijδ_jkδ_ki = 3

where u, v, and w are vectors, "." and "x" are scalar products (or inner products or dot products) and cross product, respectively. δ_ij is the Kronecker Delta function.

Is u x(v x w ) equal to (u x v )x w, why?

Question 5. (graduate students only) In the Cartesian coordinates, prove the vector identity

∇x(∇x u) = ∇(∇.u) - ∇²u

Question 6. Three motions are applied to a material in the following sequence

(a) A translational rigid motion moving origin C₀ to C by u_XC and u_YC respectively. There is no motion along Z

(b) Then stretching of

λ₁ = L/L_o, λ₂ = H_z/H_Yo and u_YC along X and Y, and Z, respectively

(c) rigid rotation ψ (positive counter clockwise CCW) in the X , Y , and Z, respectively

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.