- Please Answer D E F And G 1 (103.18 KiB) Viewed 38 times
please answer d e f and g
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please answer d e f and g
please answer d e f and g
a) Give the definition of the components of the stress tensor σij in an elastic medium. [3] 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 F = ½⁄ A1 (U²x + U³y + U²z) + A2 (UxxUyy + UzsUzz + UyUzz) + 2X3 (u²y +U²+U²³₂). xy b) Using the free-energy density given above, demonstrate that the diagonal components of the stress tensor oij are given by Orr = (A₁-A₂) Urx + A₂Ukk, where ukk is the trace of the deformation tensor, while the off-diagonal elements are given by Oxy = 2X3Uxy, and similar expressions for the other components. [3] c) Invert this relationship and find ui; as functions of σij. [2] Consider a cylinder cut from the same crystalline material, with the cylinder's axis being along a unit vector n. A pair of equal and opposite forces with the surface density p is applied to its ends. d) Argue that oij = pnn, satisfies the boundary conditions at the cylinder's surface. [4] e) Calculate the components of the deformation tensor Uij. f) Calculate the extension of the cylinder along its axis u = uinn; and determine Young's modulus of this material in terms of A₁, A₂ and X3. [5] g) For which n does Young's modulus reach its minimum value and why? [4]
a) Give the definition of the components of the stress tensor σij in an elastic medium. [3] 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 F = ½⁄ A1 (U²x + U³y + U²z) + A2 (UxxUyy + UzsUzz + UyUzz) + 2X3 (u²y +U²+U²³₂). xy b) Using the free-energy density given above, demonstrate that the diagonal components of the stress tensor oij are given by Orr = (A₁-A₂) Urx + A₂Ukk, where ukk is the trace of the deformation tensor, while the off-diagonal elements are given by Oxy = 2X3Uxy, and similar expressions for the other components. [3] c) Invert this relationship and find ui; as functions of σij. [2] Consider a cylinder cut from the same crystalline material, with the cylinder's axis being along a unit vector n. A pair of equal and opposite forces with the surface density p is applied to its ends. d) Argue that oij = pnn, satisfies the boundary conditions at the cylinder's surface. [4] e) Calculate the components of the deformation tensor Uij. f) Calculate the extension of the cylinder along its axis u = uinn; and determine Young's modulus of this material in terms of A₁, A₂ and X3. [5] g) For which n does Young's modulus reach its minimum value and why? [4]
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