Answer Happy

a) The free energy density change associated with small deformations in a homogeneous and isotropic elastic medium is given by F = 1/ Ku ³ + x (x - 156³) ²³₁ Ալլ where uab are Cartesian components of the deformation tensor. Explain the physical meaning of the terms in this expression and state the dimensions of K and μ. [4] b) Use the principle of virtual work to show that the components of the stress tensor Tij are given by OF Jużj σij [4] c) The equation of force balance for an elastic medium is given by doij + f(ext) = 0, əxj where f(ext) is an external body force density. Introduce Young's modulus E and Poisson's ratio v as E E K= = μ = 3(1-2v)' 2(1+v)' and show that the equation of force balance in terms of Cartesian components of the displacement vector u for an isotropic and homogeneous elastic medium is given by ²ui + -f(ext) 1 2² - Uk = 1 - 2v əx₂0xk 20 f(ext) 2(1 + v) [5] E