Thermoelasticity
Thermoelasticity
Outline· Heat Conduction Equation? General 3-D Formulation: Combined Plane Hooke's LawStress Compatibility and Airy Stress Function: Displacement Equilibrium and Displacement PotentialsThermal Stresses in Thin-Plates?Summary of Solution StrategyPolar Coordinate: Airy Stress Function Polar Coordinate: Displacement PotentialsAxi-symmetric Problems - Direct SolutionThermal Stresses in Circular Plates2
Outline • Heat Conduction Equation • General 3-D Formulation • Combined Plane Hooke’s Law • Stress Compatibility and Airy Stress Function • Displacement Equilibrium and Displacement Potentials • Thermal Stresses in Thin-Plates • Summary of Solution Strategy • Polar Coordinate: Airy Stress Function • Polar Coordinate: Displacement Potentials • Axi-symmetric Problems – Direct Solution • Thermal Stresses in Circular Plates 2
Heat Conduction Eguation: Flow of heat in solids is associated with temperaturedifferences For isotropic case, the heat flux is related totemperature gradient through thermal conductivityqi=-kT From the principle of conservation of energy, theuncoupled heat conduction equation is given byaTVToh=pCap: mass densityc: specific heat capacity at constant volumeh: prescribed energy source term3
q kT i i , Heat Conduction Equation • Flow of heat in solids is associated with temperature differences • For isotropic case, the heat flux is related to temperature gradient through thermal conductivity 3 • From the principle of conservation of energy, the uncoupled heat conduction equation is given by 2 :mass density :specific heat capacity at constant volume :prescribed energy source term. T k T c h t c h
Heat Conduction Eguation. For zero heat sources and steady state, the heat onductionbecomes Laplace equationVT=0.With appropriate thermal BCs, i.e. specifiedtemperature or heat flux, the temperature field can bedetermined independent of the stress-field calculations. Once the temperature is obtained, elastic stress analysisprocedures can then be employed to complete theproblem solution. For us, the temperature distribution is usually a givencondition4
• For zero heat sources and steady state, the heat onduction becomes Laplace equation 2 T 0. • With appropriate thermal BCs, i.e. specified temperature or heat flux, the temperature field can be determined independent of the stress-field calculations. • Once the temperature is obtained, elastic stress analysis procedures can then be employed to complete the problem solution. • For us, the temperature distribution is usually a given condition. Heat Conduction Equation 4
General Formulation of Thermoelasticity - 3D? Strain-displacement relations: ,=(u, +uj:)j, +&uj -Sk,jlSilik =0Strain compatibility:· Equilibrium:j, +F=0Thermoelastic Hooke's Law:12GVV-3T)S, +2G(, -αT8, =26% 2(1+ ou +aTe,,Oj=.1-2VT=0Steady state heat conduction equation:16 equations for 16 unknowns (3 displacements, 6 strains6 stresses and T):f(u,G,Of; 2,G, F,T)=0 3-D thermoelastic problems are way too difficult.S
General Formulation of Thermoelasticity – 3D , , 1 2 ij i j j i u u ij kl kl ij ik jl jl ik , , , , 0 ij j i , F 0 • Strain-displacement relations: • Strain compatibility: • Equilibrium: • Thermoelastic Hooke’s Law: 1 2 , 3 2 2 2 1 1 2 ij ij kk ij ij ij kk ij ij ij G T T G T G G • Steady state heat conduction equation: 2 T 0 • 16 equations for 16 unknowns (3 displacements, 6 strains, 6 stresses and T): { , ; , , , } 0 i ij ij i , f u G F T • 3-D thermoelastic problems are way too difficult. 5