A thin isotropic, linear elastic thin film with Young's modulus E, Poisson's ratio v and thermal...

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A thin isotropic, linear elastic thin film with Young's modulus E, Poisson's ratio v and thermal expansion coefficient a is bonded to a stiff substrate. The film is stress free at some initial temperature, and then heated to increase its temperature by AT. The substrate prevents the film from stretching in its own plane, so that &11 = 22 = 12 = 0, while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the stresses in the film in terms of material properties and temperature, and deduce an expression for the strain energy density in the film. Note that the Stress-strain relations for isotropic, linear elastic materials. Young's Modulus, Poisson's ratio and the Thermal Expansion Coefficient are given as: E11 €22 €33 2€23 2€13 2812. = 011 022 033 023 013 012. 1/2 1 -V 1 -V-V 0 0 0 0 0 0 The inverse relationship can be expressed as E (1+v)(1-2v) -V -V -V 1 0 0 2 (1+1) 0 0 0 0 Thin film Substrate (rigid) 1-v V V 0 0 0 V 1-v 0 0 V 0 0 0 V V 1-v 0 0 0 0 0 0 0 2 (1+ v) 0 0 0 0 (1-2) 2 0 0 0 2 (1+v). 0 0 0 0 0 0 0 0 (1-2v) 2 0 0 0 0 0 0 (1-2) 2 011 022 033 023 013 012 + a4T E11 €22 €33 2823 2013 2012. e₁ Ea AT 1-2v 1 0 1 1 1 0 0 A thin isotropic, linear elastic thin film with Young's modulus E, Poisson's ratio v and thermal expansion coefficient a is bonded to a stiff substrate. The film is stress free at some initial temperature, and then heated to increase its temperature by AT. The substrate prevents the film from stretching in its own plane, so that &11 = 22 = 12 = 0, while the surface is traction free, so that the film deforms in a state of plane stress. Calculate the stresses in the film in terms of material properties and temperature, and deduce an expression for the strain energy density in the film. Note that the Stress-strain relations for isotropic, linear elastic materials. Young's Modulus, Poisson's ratio and the Thermal Expansion Coefficient are given as: E11 €22 €33 2€23 2€13 2812. = 011 022 033 023 013 012. 1/2 1 -V 1 -V-V 0 0 0 0 0 0 The inverse relationship can be expressed as E (1+v)(1-2v) -V -V -V 1 0 0 2 (1+1) 0 0 0 0 Thin film Substrate (rigid) 1-v V V 0 0 0 V 1-v 0 0 V 0 0 0 V V 1-v 0 0 0 0 0 0 0 2 (1+ v) 0 0 0 0 (1-2) 2 0 0 0 2 (1+v). 0 0 0 0 0 0 0 0 (1-2v) 2 0 0 0 0 0 0 (1-2) 2 011 022 033 023 013 012 + a4T E11 €22 €33 2823 2013 2012. e₁ Ea AT 1-2v 1 0 1 1 1 0 0