Give the definition of the components of the stress tensor σij in an elastic medium. [3] (ii) Consider a two-dimensional

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Give the definition of the components of the stress tensor σij in an elastic medium. [3] (ii) Consider a two-dimensional elastic crystal with a square lattice. Let (x, y) denote the coordinate system with its axes along the lattice directions. Assume that the free energy density corresponding to a deformation given by the deformation tensor is given by Uij 1 F =√₁ (Uxx + U²³y) + A2UxxUyy + 2A3U«y (a) Using the free-energy density given above, calculate the components of the stress tensor oij in terms of uij. (b) Invert this relationship and find Uij as functions of Tij. (c) Let a rectangle be cut from this crystalline material, with the rectangle's longer side being parallel to the x-axis. A pair of equal and opposite forces is applied to its short sides in the x-direction. Calculate the components of the deformation tensor U ij for this situation. (d) Express the Young's modulus and the Poisson's ratio for this material in terms of A₁, A2 and X3. When is the material incompressible? [3] [3] [5] [3]