Basic Equations of Thermoelasticity
In this chapter the basic governing equations of thermoelasticity for three-dimensional bodies are recalled. The equilibrium equations of stresses, Cauchy’s relations between the tractions and stresses, and the compatibility equations of strains in Cartes
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Basic Equations of Thermoelasticity
In this chapter the basic governing equations of thermoelasticity for three-dimensional bodies are recalled. The equilibrium equations of stresses, Cauchy’s relations between the tractions and stresses, and the compatibility equations of strains in Cartesian coordinates are presented. The formulae for coordinate transformation of stress, strain and displacement components are included. A solution of Navier’s equations is carried out wherein Goodier’s thermoelastic potential is used in conjunction with harmonic functions of various types. The equilibrium equations, stress, strain, the compatibility equations, Navier’s equations in cylindrical and spherical coordinates are also presented. [see also Chaps. 2, 3, 6, and 7.]
16.1 Governing Equations of Thermoelasticity 16.1.1 Stress and Strain in a Cartesian Coordinate System Stress The equilibrium equations of the elastic body from Eq. (2.21) are σji,j + Fi = 0 (i, j = 1, 2, 3)
(16.1)
where σji denote the components of stress, Fi mean the components of body force per unit volume. The components of stress satisfy symmetry relations σij = σji (i, j = 1, 2, 3)
(16.2)
Cauchy’s fundamental relations are σji nj = pni (i, j = 1, 2, 3)
M. Reza Eslami et al., Theory of Elasticity and Thermal Stresses, Solid Mechanics and Its Applications 197, DOI: 10.1007/978-94-007-6356-2_16, © Springer Science+Business Media Dordrecht 2013
(16.3)
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392
16 Basic Equations of Thermoelasticity
where nj denote the direction cosines between the external normal of the surface and each axis. The formulae for coordinate transformation of stress components between the components of stress (σxx , σxy , . . .) referred to an old Cartesian coordinate system (x, y, z) and the components of stress (σx x , σx y , . . .) referred to a new Cartesian coordinate system (x , y , z ) are (i , j = 1, 2, 3)
σi j = li k lj l σkl
(16.4)
where li k denote the direction cosines between the axis xi of the new Cartesian coordinate system (x1 , x2 , x3 ) and the axis xk of the old (x1 , x2 , x3 ). Strain The strains are from Eq. (2.5) ij =
1 (ui,j + uj,i ) 2
(i, j = 1, 2, 3)
(16.5)
where ui are the components of displacement. The components of strain are symmetric ij = ji
(i, j = 1, 2, 3)
(16.6)
The transformation of coordinates between the the new Cartesian coordinate system (xi ) and the old system (xk ) are xi = li j xj , xi = lj i xj
(i, i = 1, 2, 3)
(16.7)
The relationship between the components of the displacement in each coordinate system are ui = lj i uj (i, i = 1, 2, 3) (16.8) ui = li j uj , The coordinate transformation of strain components is i j = li k lj l kl
(i , j = 1, 2, 3)
(16.9)
16.1.2 Navier’s Equations, Compatibility Equations and Boundary Conditions Navier’s equations The constitutive equations for a homogeneous, isotropic body which are known as the generalized Hooke’s law are
16.1 Governing Equations of Thermoelasticity
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1 ν σij − Θδij + ατ δij (i, j = 1, 2, 3) ij = 2G 1+ν
(16.10)
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