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C: Answer ONE of the questions in this [3] C.1 a) Give the definition of the components of the stress tensor dij in an elastic medium. b) Consider a cylinder stretched by two equal and opposite forces applied to its ends along the cylinder's axis. Derive the boundary condition for the stress tensor at the [5] cylinder's surface. with its axes along the lattice directions. The free energy density corresponding to a Consider an elastic crystal with a cubic lattice. Let (x, y, z) denote the coordinate system deformation given by the deformation tensor uj is given by F = = A₁ (u²z + Uzy + U²x) + √2 (UzzUlry + Uzelize + UylUlze) +213 (²y +uz+uzz). [5 c) Using the free-energy density given above, calculate the components of the stress tensor oj in terms of uij. [2 d) Invert this relationship and find ui; as functions of ij. e) Let the cylinder from b) be cut from the same crystalline material, with the cylin- der's axis being along the x-direction. A pair of equal and opposite forces is applied to its ends. Calculate the components of the deformation tensor uij for this situa- [9 tion. f) Express Young's modulus and Poisson's ratio for this material in terms of A₁, A2 and A3. Under what conditions is the material incompressible? i