东南大学：《弹性力学》课程教学课件（英文讲稿）13 Thermoelasticity

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

Formulation of Thermoelasticity - 2D. Plane strain thermoelastic Hooke's lawVOuo+aTS2G2G(1+v)0=8, =→> 0; =v(, +o,)-2G(1+v)αT =→ Ou =(1+v)(α, +α,-2GαlG[(1-v)o, -vo, ]+(1+v)αT,S[(1-v)g, -vo, ]+(1+v)αT, &g, =02GV-3αT), +2G(sg-αT2,1-2v1+v1-vVQ, =2GT-211-2v1+v-VV0, =2Y1-2v1-2v-216

Formulation of Thermoelasticity – 2D                      1 2 2 1 0 2 1 1 2 1 1 1 , 2 1 1 1 1 , 2 2 2 3 2 1 2 1 1 2 1 2 1 2 1 2 ij ij kk ij ij z z x y kk x y x x y y y x xy xy ij kk ij ij ij x x y T G G G T G T T G T G G G T G T G T                                                                                                         , 1 1 2 , 2 1 2 1 2 1 2 y y x xy xy G T G                               • Plane strain thermoelastic Hooke’s law 6

Formulation of Thermoelasticity - 2D. Plane stress thermoelastic Hooke's law2Gv -3αT), +2G(eg-αT8,)1-21一I+V+α=&0=0, = 8,2G& +V,-(1+v)αT ],a2G, +VE,-(1+v)aT , T, =2Ge,91L2G%-2G(1+) ug +αTg11V+oT?O2G1+11+V11V1+o0x2G2G1+v1+V

Formulation of Thermoelasticity – 2D • Plane stress thermoelastic Hooke’s law               2 3 2 1 2 1 1 2 1 0 1 1 1 1 2 1 , 1 2 1 , 2 1 1 2 2 1 1 1 2 1 1 ij kk ij ij ij z z x y kk x y x x y y y x xy xy ij ij kk ij ij x x y G T G T T T G T G T G T G G G                                                                                                              , 1 1 1 , 2 1 1 2 y y x xy xy T T G G                        7

Formulation of Thermoelasticity - 2D. Combined plane thermoelastic Hooke's law: Define two material constants that are related to y3-KFor plane strain: K=3-4v or n=v43-V3-Kn=0.For plane stress: KO1+V1+k3-K+(1+n)αTaB,=2GOofoBF2(1-K)41-K(1+x)13-K+(1+n)αT)OxOx42G1+K1 (1+x)3-K+(1+n)αT,cmoyO2G42G1+K-x)& +(3-x)s, -4(1+n)αT1+x)e, +(3-k)&, -4(1+n)αT ,tx,=2Ge,8

                      1 3 3 2 1 1 , 2 2 4 2 1 1 1 3 1 1 , 2 4 1 1 3 1 1 1 , . 2 4 1 2 1 3 4 1 , 1 1 1 x x y y y x xy xy x x y y y T G T G T G T G G G T G                                                                                                                               3 4 1 , 2 .            x xy xy   T G Formulation of Thermoelasticity – 2D • Combined plane thermoelastic Hooke’s law 3 For plane strain: 3 4 or , ; 4 3 3 For plane stress: or , 0. 1 1                          • Define two material constants that are related to ν 8

Stress Formulation - 2D: Beltrami-Michell Equation:ePo.&ara?axay(1+x)3-K1(1+x)3-KExgyOx2GO2G42G41+K1+Ka3-x(1+x1+x3-, +2G(1+n)αlα, +2G(1+n)αTxgy44aax44axayoP.dodaPav(a, +α,)+2G(1+n)aV-T:Addto both sides:axaaaxayay(aFaF)Using Eqiliumonthe RHS: (g,+,)+2G(1+n)VT=atay44OFF8G(1+n)aV-TV(o, +o,)+11+Kaxay1+K3-K3-K3-vn=0For plane strain: x=3-4v orFor plane stress: Kn=viOr41+x1+vaF.F.aF1aF(o, +o,)+EoV-T=-(E-oVTyV (a, +o,)+1-vaxayaxay

• Beltrami-Michell Equation:             2 2 2 2 2 2 2 2 2 2 2 2 2 2 1 3 1 3 1 1 1 1 , 1 , . 2 4 1 2 4 1 2 1 3 1 3 2 1 2 1 2 4 4 4 4 Add x y xy x x y y y x xy xy xy x y y x x y y x x y T T G G G G T G T y x x y x                                                                                                                          2 2 2 2 2 2 2 2 2 2 2 2 1 to both sides: 2 1 2 4 1 Using Equilibrium on the RHS: 2 1 4 1 1 4 1 8 x y xy x y x y x y x y x y G T y x y x y F F G T x y F y G T F x                                                                                     2 2 2 2 3 3 3 For plane strain: 3 4 or , For plane stress: or , 0 4 1 1 1 1 1 1 x y x y x y x y E T E T F F F F x y x y                                                                      Stress Formulation – 2D

Stress Function Formulation without Body ForcesAir Stress Function SolutionyyQy0y=aT,=-axoya where y= y(x,y) is an arbitrary form called Airy's stressfunction. This stress form automatically satisfies theequilibrium equation: Beltrami-Michell Equation:8G(1+n)OVT=0Vy+1+K3-VFor plane strain K=3-4v, n=vFor plane stress: K1+vHoVT=0Vy+Eo-T=010

Stress Function Formulation without Body Forces • Air Stress Function Solution 2 2 2 2 2 , , x y xy y x x y                 • where  = (x,y) is an arbitrary form called Airy’s stress function. This stress form automatically satisfies the equilibrium equation. • Beltrami-Michell Equation:   2 4 4 2 2 4 8 1 0 1 For plane strain: 3 4 , 3 For plane stress: , 0 1 1 0 0 E T T T G E                                        10