a) In linear elasticity the relationship between the stress tensor ; and the deformation tensor u is given by Tij = Cijk

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A In Linear Elasticity The Relationship Between The Stress Tensor And The Deformation Tensor U Is Given By Tij Cijk 1 (81.8 KiB) Viewed 39 times

a) In linear elasticity the relationship between the stress tensor ; and the deformation tensor u is given by Tij = Cijkt ukl, where Cijk is an elasticity tensor. Give its rank and the number of components. [2] b) For an isotropic and homogeneous material Cijk can be expressed only in terms of 2 independent coefficients A and so that Cijkl Adijoki + 2µdikdj. Show that the constitutive relation is given by : = Tij = λdijukk + 2 μujj. [2] c) The constitutive relation can be inverted such that : Uij = Mijkt kl. Find the expression for Mijkl. [3] d) Consider a spherical shell of inner radius ro and outer radius ₁. The pressure inside the shell is equal to po and outside the shell p₁ (See figure). Due to the system symmetries, the displacement vector u,(r) can be written as: b u₁ (r) = (ar + 2) e₁ Show that the different components of the stress tensor can be written as B OTT = A 2B p3 000 = 0 = A + and show that constants A and B are given by: 3 3 Poro³ - pir₁³ A = and 3 3 3 r₁³-r³ p3 B = ½ (Po − P₁); r₁ 3 To [7]