The deformation in a solid can be described by a vector field u(x) = (u1, u2, u3)
Question:
The deformation in a solid can be described by a vector field u(x) = (u1, u2, u3) describing the displacement of the solid away from the equilibrium point x. This is similar to the displacement vector s used to analyze waves in ?31.2. For small deformations, the strain tensor ? is a 3 ? 3 matrix with elements ?i j = (?ui/?xj + ?uj/?xi)/2. For example, a rod compressed uniformly to 99% of its original length in the x-direction would have ?11 = ?0.01. The energy of compression is given by eq (33.13)
where ? is the shear modulus, K is the bulk modulus, and ?i j = 1 if i = j and 0 otherwise.?
(a) Show that for uniform hydrostatic compression, where u11 = u22 =u33 = u, and all other components vanish, the energy of compression depends only on K.?
(b) Show that for P- wave compression, where u11 = u sin (kx) and all other components vanish, the energy of compression is proportional to the elastic modulus ? = K + 4?/3.?
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