Consider an elastic crystal with the cubic lattice. Let (x, y, z) denote the coordinate system with its axes along the l

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Consider An Elastic Crystal With The Cubic Lattice Let X Y Z Denote The Coordinate System With Its Axes Along The L 1 (724.02 KiB) Viewed 40 times

Consider an elastic crystal with the cubic lattice. Let (x, y, z) denote the coordinate system with its axes along the lattice directions. The free energy density corresponding to a deformation given by the deformation tensor uij is given by 1 F ==⁄A₁ (U²z+ U³y + U²₂2) + √2 (UzzUyy + UzzUzz + Uyy Uzz) +2A3 (ury + urz + U²³₂). b) Using the free-energy density given above, demonstrate that the diagonal components of the stress tensor jare given by Oxx (A1-A2) Uxx + λ₂ Ukk, 1 where ukk is the trace of the deformation tensor, while the off-diagonal elements are given by Try = 2X3 Ury, and similar expressions for the other components. c) Invert this relationship and find ui; as functions of Tij.