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

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1 a) In linear elasticity the relationship between the stress tensor σij and the deformation tensor u is given by Tij = Cijklukl, where Cijkt 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 Adijokl + 2µdikdj. Show that the constitutive relation is given by : = Tij = №dijukk +2µużj. [2] c) The constitutive relation can be inverted such that : Uij = Mijklo kl Find the expression for Mijkl. [3] d) Consider a spherical shell of inner radius ro and outer radius 7₁. 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: u₁(r) = (ar + 2) e₁ Show that the different components of the stress tensor can be written as B OTT= A. 2B 7-3 Tee = 0 000 = A + and show that constants A and B are given by: 3 3 A = Poro³ — pir₁³ T₁³ - ro³ and B=(Po-P1)- 3 [7] e) Deduce the values of a and b. [2] Hint: The relevant components of the deformation tensor in spherical coordinates are given by du, Up Up Upp = U 00 Upo ==. Ər T T 7 3 3