Chpte 6 Themoelsticity Intoduction When theml enegy is dded to n elstic mteil it expnds. Fo the simple unidimensionl cse of b of length L, initilly t unifom tempetue T 0 which is then heted to nonunifom tempetue T nd thus gows in length by n mount L, the eltive uni-xil stetching due to theml expnsion is L L = ɛ = α(t T 0) whee ɛ is the stin nd α is the theml expnsion coefficient. Fo n isotopic cube of side L the (noml) themoelstic stins e ɛ x = ɛ y = ɛ z = α(t T 0 ) It is conventionl but not necessy to tke T 0 =0. Since the heted egion is joined to, nd constined by igid suoundings, it cn not expnd feely but becomes subjected to compessive stesses. At the sme time the colde potion is subjected to the pull exeted by of the djcent hot potion nd it is thus unde tension. Although Hooke s lw is still pplicble, due ccount must be tken of the dditionl stesses ceted by theml expnsion. Govening Equtions of Themoelsticity fo n Isotopic Solid The govening equtions fo the isotopic themoelstic solid include the equilibium equtions σ ij x j + X i = σ ij,j + X i =0
whee i, j =,,3, the genelized themoelstic stess-stin eltions σ ij = C ijkl ɛ kl β ij (T T 0 )=λɛ kk δ ij +Gɛ ij βδ ij θ whee θ = T T 0 is the excess tempetue distibution, nd β = αe/( ν) whee α is the theml expnsion coefficient. Expessed s stin-stess eltionships the bove e ɛ ij = +ν E σ ij ν E σ µµδ ij + αθδ ij The smll displcement stin-displcement eltions e, s befoe ɛ ij = (u i,j + u j,i ) Finlly, the comptibility equtions must lso be stisfied. The tempetue distibution θ must be detemined by solving the enegy consevtion eqution du dt = T S t + ρ σ ijv ij whee U is the intenl enegy, S the entopy nd V ij = ( v i x j + v j x i ) is the te of defomtion tenso whee v i is the velocity. One cn show tht the following fom the diffeentil theml enegy blnce eqution cn be deived fom the bove H t + θβ ɛ ij ij t = (k θ)+ whee H is the enthlpy, β ij e expeimentlly detemined numeicl coefficients nd is the te of intenl enegy genetion. The enegy eqution bove must be solved subject to suitble boundy nd initil conditions in ode to detemine the tempetue field θ. Fo stedy stte conditions in medium of constnt conductivity nd without intenl het genetion θ =0 i.e. solutions to stedy stte het conduction poblems e hmonic functions.
In uncoupled, qusi-sttic themoelstic theoy, the mechnicl coupling tems in the enegy nd the het conduction equtions e neglected. Theefoe, the het conduction poblem nd the themoelstic defomtion poblem e hndled septely. By substituting the genelized themoelstic stess-stin eltions nd the smll displcement stin-displcement eltions into the equilibium eqution one obtins the genelized Nvie s eqution Gu i,µµ +(λ+g)u µ,µi + X i βθ,i =0 The thee themomechnicl equilibium equtions togethe with the enegy equtions nd the six stess-stin eltions constitute set of ten equtions fo the ten unknowns u i,τ ij nd θ. One cn show this system is complete, yields n unique solution unde suitble boundy conditions nd the esulting stin stisfies the comptibility eltions. 3 Displcement Potentil nd Stess Functions Goodie intoduced the displcement potentil function φ s u = φ = u i = φ x i this, when substituted into the genelized Nvie eqution nd integted yields φ,µµ = (P +βθ) λ +G whee P is the potentil fo the ssumed consevtive body foces (i.e. X = P ). The solution of the bove is the sum of pticul solution nd the complementy solution of Lplce s eqution ( φ =0). Fo plne stin conditions, on x y plne in ectngul Ctesin coodintes, combintion of the equilibium equtions nd the comptibility condition yields yields (σ xx + σ yy )= β ν θ Intoducing the stess function Φ defined by σ xx = Φ y + βθ σ yy = Φ x + βθ σ xy = Φ x y 4 Φ= αe ν θ 3
4 Theml Stesses in Thin Plte Conside n infinitely long plte of vey smll thickness nd width c. Let the long diection be ligned with the x xis nd the width with y. Assume tht T 0 = 0 nd tht the tempetue in the plte is only function of y, (i.e. θ = T (y)). Wht would be the themoelstic sttes of stin nd stess esulting fom this tempetue field? The nswe is obtined using the pinciple of supeposition. Fist, one must detemine the mount of compessive stess tht would hve to be pplied to keep the plte fom stining ltogethe in the longitudinl (x) diection. Fom the bove, the equied stess would be σ x = αet (y) Since one is inteested in the theml stess in n expnding plte, to the bove stess one must supeimpose the stess geneted in the plte when unifomly distibuted tensile foce of mgnitude y=+c αet (y)dy c y= c is pplied t the x ± boundies. Theefoe, the ctul theml stess in the plte is y=+c σ x = αet (y)dy αet (y) c y= c Assume now tht T (y) is qudtic in y, T (y) =T y=0 ( y c ) I.e. the cente of the plte is t tempetue T y=0 while the edges y = ±c e t 0. Substituting this into the expession fo σ x gives σ x = 3 αet y=0 αet y=0 ( y c ) Clely, the stess is qudtic in y. The mximum compessive stess is t y =0nd it is equl to σ x,y=0 = αet 3 y=0, while the mximum tensile stess is t y = ±c nd it is σ x,y=±c = αet 3 y=0. The stess is zeo t y = ±c/ 3. 5 Theml Stess in Disks nd Cylindes Conside thin disk (dius b) with hole of dius t the cente. Assume the tempetue in the disk θ = T () is only function of the dil position mesued fom the cente of the hole. 4
If plne stess conditions e ssumed, mechnicl equilibium equies dσ d + σ σ φ =0 whee nd φ e the dil nd zimuthl diections, espectively. The stin-displcement eltions e ɛ = du d ɛ φ = u whee u is the dil displcement. Finlly, fo line themoelstic mteil the stess-stin eltions e σ = E ν [(ɛ + νɛ φ ) ( + ν)αt ] σ φ = E ν [(ɛ φ + νɛ ) ( + ν)αt ] Combintion of the stin-displcement eltions with the bove nd substitution into the mechnicl equilibium eqution yields d u d + du d u =(+ν)αdt d with the genel solution u =(+ν)α Td+C +C whee C,C e constnts. The ssocited stesses e σ = αe σ φ = αe Td+ Td αet + E C ν E C +ν E C ν + E C +ν Since no dil stesses ct t the inne nd oute the oute dius of the disk (σ () = σ (b)=0), ( ν)α C = b ( + ν)α C = b 5 b b Td Td
nd the xil stin is ɛ z =(+ν)αt να b Td b If plne stin conditions e ssumed insted ( good ppoximtion in the cse of tll hollow cylinde with its bses estined fom movement long the xil diection), the coesponding esults e, fo the displcement u = α +ν ν [ Td+ ( ν) + b b nd fo the ssocited stesses e σ = αe ν [ Td+ b b σ φ = αe ν [ T + Td+ + b b nd σ z = αe ν b [ T + ν b The solution to the cse of tll hollow tube unestined fom movement in the xil diection is given by u = α ν [( + ν) Td+ ( 3ν) +(+ν) b b the ssocited stesses e σ = αe ν [ Td+ b b nd σ φ = αe ν [ T + Td+ + b b σ z = nd the longitudinl stin is αe ν [ T + b b ɛ z = α b Td b 6
Specificlly, fo thin disk with dil stedy stte tempetue distibution T () =T b (T b T ) ln(b/) ln(b/) whee T = T (),T b =T(b). the stesses e σ = αe(t b T )[ (/) (/b) ln(/) ln(b/) ] σ φ = αe(t b T )[ +(/) (/b) +ln(/) ] ln(b/) with σ z = 0. The coesponding stesses fo the long hollow cylinde e obtined dividing the bove by ν. but in this cse with σ z = αe(t b T ) ( ν) [ (/b) +ln(/) ] ln(b/) Conside finlly the specific exmple of quenching long fee cylinde, initilly t unifom tempetue T () =T 0 by mintining its sufce tempetue t zeo (T ( = b) = 0). The solution of the homogeneous line tnsient D het conduction poblem is (see fo exmple Conduction of Het in Solids, nd ed, by Cslw nd Jege, Clendon, Oxfod, 959, p. 99): T () =T 0 n= β n J (β n ) J 0(β n βn b )e( κ whee κ is the theml diffusivity, J 0 nd J e the Bessel functions of fist kind, of odes zeo nd one, espectively nd β n e the eigenvlues of the poblem, which e the oots of J 0 (β n )=0 Substituting the expession fo T () into the stess equtions one obtins σ () = αet 0 ν n=[ β n β n b J (β n (/b)) J (β n ) b t) ]e ( κ β n b t) nd σ φ () = αet 0 ν n=[ β n + β n σ z () = αet 0 ν b J (β n (/b)) J 0(β n (/b)) βn J (β n ) β n J (β n ) ]e( κ n=[ J 0(β n (/b)) βn βn β n J (β n ) ]e( κ b t) b t) 7
If one is inteested only in the mximum vlue of the stesses (which occu when t 0 t the sufce), the esults e σ (b) =0 σ θ =σ z = αet 0 ν I.e. the sufce of the cylinde is unde cicumfeentil (hoop) nd xil tensions of equl mgnitudes. The (cold) sufce lyes of the cylinde wnt to contct but e pevented fom doing so by the (still hot) coe. If insted of quenching, cold cylinde is heted, the initil stess stte t the sufce is compessive. 6 Theml Stesses in Sphee Conside sphee of dius b in which the tempetue is only function of. The diffeentil mechnicl equilibium eqution is dσ d + (σ σ t )=0 whee σ nd σ t e, espectively, the dil nd tngentil stess. The stess-stin eltions e: ɛ = αt + E (σ νσ t ) nd ɛ t = αt + E [σ t ν(σ + σ t )] nd Finlly, the displcement-stin eltionships e ɛ = du d ɛ t = u The solution in this cse is u() = +ν ν α T d 0 8
σ () = αe ν [ b T d T d ] b 3 0 3 0 nd σ t () = αe ν [ b T d + T d T ] b 3 0 3 0 If T () is known, stesses e edily computed. Fo instnce, if cold solid sphee, initilly t T 0, is heted by mintining its sufce t tempetue T, the mximum compessive stess (occuing t the sufce t the vey beginning of the pocess) is σ (b) =σ t (b)= αe(t T 0 ) ν 9