101 297 THS m T1 EEMCLLAEYEC EfifiaiE EN THE TWF-$?ACE Thes's for H 9 Degree of DB. D. WKCEICAW STAE . EN YETERSET Y James Leland Bailey 1958 This is to certify that the thesis entitled vvr—n-sy ,' O A TILLLUuOELASTIC PROBLEM Iii THE HALF-SPACE presented by James Leland Bailey has been accepted towards fulfillment of the requirements for IhD degree in NlatthatiCS Major professor Egan/M 3AM. Date W8 0-169 *___. hnka-‘ __.._ LIBRARY Michigan State University A THENHOELASTIC PROB,EE IN THE HfiLF-SPACE By James Leland Bailey AN ABSTRACT Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Kathematics Year 1958 = i l ‘ 2.! L Approved + - :9 ”iwf x )Uinwu; M an. fl: u: v '_ ABSERAC'I' This thesis is concerned with the solution of the thermoeJastic problems resulting from the sudden application and maintenance of the following tanperature distribution on the surface of a half-space initially at zero temperatureutbe temperature equals a constant within a circle and zero elsewhere. The tanperature distribution within the half-space exposed to the aforanentioned surface temperature distribution is derived. Then under the assumption of a quasi-static condition, the stresses and dis- placenents inside the half-space and on its boundary are determined. This quasi -static solution is obtained by transformation of the problem into the Laplace subsidiary space and solving it there by the introduction of a thermoelastic potential and the use of the Galerkin-Westergaard method specialized in the case of axial symetry to determination of Love's function and then transforming the solution back into the original space. The quasi-static solution so derived is physically descriptive for all time with the exception of the first manent. To canplete the physical description of stresses in the half-space, dynamic effects due to the stress wave emitted because of the suddeness of the application of this surface temperature distribution are taken into account in the stress solution for small wines of time. This asymptotic dynamic solution is produced by means of the Iaplace transformation and a themoelastic potential. Finally, sane numerical results pertinent to the steady-state solution (a special case of the quasi -static solution) and asymptotic dynamic solution are canputed and tabulated. A THSELIOELASTIC PROBLEM IN ‘13": ifi‘iLF—Si‘jzgjij By James Leland Bailey A THESIS Submitted to the School of Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 1958 AWOWIEDGIENTS The author wishes to express his sincere thanks to Dr. Henry Parkus, under whose resourceful guidance this investigtion was undertaken, for origination of the problan, competent advice, and kindly encouragement. The results of this thesis are here- with dedicated to him. The writer is also indebted to Dr. Gerard P. Weeg those hard work and constant direction in the numerical phase of the problem made possible the use of the automatic computer. In addition the writer thanks the Canputer Department for allowing his use of the Mistic. TABIB OF CONTENTS Chapter Page I. Introduction......... ..... . ...... .. ......... ...... 1 II. General Considerations...” ..... ................. ’4 III . The Temperature Distribution. . . . ........... . ............. 9 IV. The Quasi -static Solution. . . . . . . . . . . . ................... 1’: v. The Dynamic Effects at the First Mment ................. 30 VI. MericalResults...... ......... ..... .. ........ ...37 VII. Conclusion.......... ..... . .............. . ............ ...Mt Bibliography I . Introduction. In 1950 v. I. Danilovskaya [3]+solved the one-dimensional themoelastic problem with dynamic effects inclined which results frcn the sudden application of a constant temperature to the bounding plane of a half-space and in 1952 she [h] generalized this problen by maintenance of the medium instead of the surface at constant temperature. In 1955 M. A. Sadowsky [21] adopted a simplified model an! neglected dynamic effects to obtain a solution for mail values of time of the axially symetric themselastic problem in the half-space which results from the application of a constant surface tamerature within a circle and zero surface temperature outside this circle. Still more recently in January, 1957, s. Sternberg and E. L. McDowell [21:] using the method of Green have solved the steady-state thermoelastic problem in the half-space and obtained explicit solutions for three tauperature distributions: surface temperature equals a constant within a circle and zero outside, surface temperature equals a constant within a rectangle an! zero. outside, and surface tanperature equals a hemispherical. distribution within a circle and zero outside. Earlier in 1937 J. N. Goodier [10] had utilized the thermoelastic potential by means of which he reduced the general threedimensional themoelastic problem in the entire space to determination of the Newtonian potential for a mass distribution whose density coincides with the given tenperature field. For domains other than the entire space, this potential yields only a particular solution of We numbers In square brackets‘ refer to bibliography entries. - 2 - the thermoelastic equations which in general will not satisfy prescribed boundary conditions on the surface. Since for small deformations the themoelastic equations are linear and the super- position principle is valid, it renains to find a solution of the classical homogeneous elasticity equations to superpose in order to adjust the boundary values to those prescribed. This latter solution can be obtained either by the Neuter -Papkovich [22, p. 328 if] or Galerkin-Westergaani [29 , p. 119 ff.] methods. In problems with rotational symmetry the Galerkin-Westergaazd method simplifies to solution for Love's displacement function [15, 1). 27km] . Within this framework, the results derived in this thesis are the following: (1) Given that the surface of a semi-infinite three-dimensional Hookean body bounded by a plane is stridenly subjected to a temperature distribution T = To within a circular area and T = 0 outside this circular area and then maintained at that tauperature distribution, the quasi -static solution is obtained. That is, the bony is assuned to progress through successive equilibrimn states slowly enough so that dynamic effects can be neglected. This assmption is plausible except at the first moment. From the quasi -static solution the corresponding steady- Btate solution is then derived. (2) Given the same physical situation as in (l) , the solution taking into account the dynamic terms is obtained for mall values of time. That is, in the second problem due regard is given the stress wave emitted at the first manent. It is known that the dynamic effects may - 3 - be of great importance at the first moment but rapidly cease to be of consequence after that. (3) Numerical values of the steady-state stresses and displacements for thirty-one points on the axis of symmetry and the surface and of the asymptotic stresses for a twenty-seven point grid in a representative plane containing the axis of symmetry are computed and tabulated. The first result is obtained by use of the Laplace transfor- mation, a thermoelastic potential, and the Galerkin-l-Iestergaard method specialized in this case of rotational symetry to derivation of the proper Love's displacanent function. For the second result, the thermoelastic potential in the Laplace subsidiary epace giving the desired solution directly is exhibited. The numerical evalua- tion of the integral in the third result is accomplished by use of Simpson's rule and the computation is done on the Michigan State Illiac Computer, the Mistic. Before initiating solution of the problems listed as (1) , (2), and (3) , the basic equations of thermoelasticity are reviewed, the boundary value problem is formulated, and certain general methods for solving it are discussed. -14.. II. Gmeral Considerations. Thermoelastic problems deal with the derivation of the stresses and displacements developed within bodies subjected to given heat dis- tributions. If the tenperature is not unifom throughout the body, the hotter parts of the body tend to expand more than the cooler parts, but since the body remains solid, this expansion is restricted. The restriction causes themal stresses. 'niese stresses and the resulting strains are assmed to be related through Hooke's law and displacanent derivatives are assumed mall. Problems in thermoelasticity involve sixteen unlmomsusix independent strains, six independent stresses, three displacements, an]. tauperature. For these sixteen unknowns, there are the following sixteen equations: 0 = 00 on _ 1+, " u: u a?! 2“ [EW‘ 1‘57" u 1:27“T‘§‘J] ““071 ‘57: .0 - a“. 311' .0- ,o 367 30:. 36“. Na; ‘1) x,+Tz';*Txi’-J°W' 3T- a" T, 3.5.... V where 1,.) = 1,2,3, rectansfl-r comm-mt.“ (1,1,!) = (119x013), 0% = the me” on the surface 2,- = constant in the direction of in- creasing xj, G = the modulus of shear, £ 2 the strain on the sur- ‘J face 1; a constant in the direction of increasing 1', 7 = the Poisson's ratio, e = the cubical dilatation, J}; = the Kronecker delta, o( g the coeff Lcient of linear expansion, T = the absolute tanperature, u- -.-.-. the displacanent in the direction of increasing 1; , f a the dmsity, t = the time, a = the diffusion constant, -5- I. 3" ‘ " and the bod forces normal a e ri in the third set a? equations are neglected. In addition three other conditions of compatibility must be satisfied. These equations express mathenatically the fact that the displacements are single-valued. The counpatibility equations are usually written in the form of six second-order differential equations. a. )‘ a 3‘ 9 new?“ 2%: 555‘? 22E rf’*:>?" i’ 395‘] . as. 3‘: 3‘5 2* _) a: a: e ‘8’ fievxffmi’ fip‘ifi‘abfi’+#*§ff” I 3 3 l l e 1.3 x, 1: xi Althougi it has been shown impossible to reduce these six equations to a lesser number of second-order differential equations, it is possible to write them as three differential equations of higier order. The set of sixteen equations and the compatibility equations can be reduced to four differential equations, the Navier equations and the heat equation, in the three displacements and temperature and the condition of single-valuedness of the displacements respectively? This set of four equations is Vim, 1 3e _.P at _2(i+r) 3 («'13) (3) ‘ 17-27 TT- 5 3?“ 1-27 Xx; = a" "T. 5— V We couplfng efleflis neglected. The change in heat of the body due to its deformation is of second order with respect to the temperature induced by the temperature distribution on the surface. -6- where 1 = 1,2,3 and e = 31;:- + 3‘; ... gig- . Fran these differen- tial equations one can make three pertinent observations: all equations are linear so the superposition principle is valid, the fom'th equation is independent of the preceding three, and each of the first three equations contains all three displacements since the cubical dilatation e contains than. The formulation of the thermoelastic boundary value problan is completed by specification of boundary conditions. These boundary conditions may be given directly in terms of the displacements on the surface, indirectly through specification of the stresses on the surface, or as a mixttu‘e by specification of the displacement over part of the surface and the stresses over the rest of the surface. The first step in the solution of this boundary value problem is the determination of the temperature distribution within the body Rom the heat equation. With the temperature distribution known, the four differential equations are reduced to three nonhcmogeneous differential equations for the three displacements. Because of the complicated nature of these three linear equations, a thermoelastic potential ¢ defined by the equations 20a; = 51f 1i is introduced in order to obtain a particular solution of the equations and simplify the remaining boundary value problem to one with hanogeneous differential equations. By substitution of the appropriate ¢ expressions into the three equations for the displacements, it can be sham that ¢ will produce particular solutions if ¢ satisfies -7- (4) V“? _ (l-ZHE bigg— 21+ ”(MT 2(1-r)c By use of Hooke's law in terms of the displacements, the definition of ¢ in terms of the displacements, and equation (in) , the following expressions are derived for the stresses correspondent to ¢. (5) a§=§%— v*¢£;,.+£,§-Z4’c§, with 1,.) = 1,2,3. In general these stresses will not be those specified in the boundary coalitions. To adjust the stresses to those prescribed, a solution of the hanogeneous equations is superposed. Since in the case of temperature independence the thermoelastic equations reduce to the classical equations of elasticity, the latter solution is obtained fran the classical theory. That the final solution is unique up to a rigid motion, even with the neglected coupling effects included, has been shown by Weiner [28] . After utilization of the thermoelastic potential to reduce the remaining boundary value problem to one independent of temperature, many physical problems of interest will admit a plane stress or strain condition making the Muskhelishvili technique and the powerful theory of canplex variable applicable. If the ’ temperature-independent problem will not admit either of these simplifying assumptions, the Neuber-Papkovich or Galerkin-Westergaard methods can be anployed. Both of these techniques reduce the -3- Navier displacanent equations to more familiar equations «the Neuber-Papkovich, to potential equations and the Galerkin-Westergard, to biharmonic equations. In problems emibiting axial symetry, the latter method can be simplified to solution for love's displacement function. The problem in this thesis, being of the axially symetric type, is solved by use of Love's function. -9- III. The Temperature Distribution. The bounding plane surface of a half-space initially at zero tenperature is subJected to the suiden application of the temperature distribution: T = To within a circle of radius b and T = 0 outside this circle. After applied, the temperature distribution is maintained for all succeeding values of time. In this section the resulting temperature distribution in the half-space is determined. With the introduction of cylindrical coordinates (r, (’52) and use of the rotational symetry of the problem to eliminate the dependence on 9:, the boundary value problem in T = T(r,z,t) can be mitten 3.: -_-_ a" v‘r for t,z>o, T(r,z,0) = O for all r,z>0, (6) T(r,o,t) = To for r0, = O for r)b and t) 0, |T(r,z,t)l ( M for all r,z, and t vhere M is a sufficiently large positive number. By means of a formal application of the Laplace transformation L {T(t)} :e TWP): [13“)th dt, this boundary value problem transforms into the following one in T* = T*(r,z,p) . v’iw - grim - o, (7) T*(r,0,p) = $9. for r(b, = for rpb, ‘T*(r,z,p)l ( M for all r,z, and p. -10- The differential equation, a wave equation, can be separated in cylindrical coordinates to yield the solution T*(r,z,p) =fA(p,;\) J. (Ar) e'deA, I I 1' where X(p,}\) = -L;-441 and where A(p,A) is to be determined from the boundary conditions. Use of an integal relation [26, p. 1406] fJ,(/\b) J°(}\r) dA = % for rb, and the boundary conditions on 2 = 0 gives A(p, A ) = Wag—45b) Therefore 0’ 8-82 (9) T*(r,z,p) = n. f J, m) J. (Ar) 1, as. and[7, p. 2146] G Hr fJ (Ab) J (Ar) (10) T(r,z,t) == 12. o l o l _ z ~2a At .[e‘z mrcéggAt) + e 1“ Erfc( )] dA. It renains to show that this formal solution is the actual solution of the boundary value problem (6) . By proving T to be represented by an integral uniformly convergent in t and z for t,z Z O and then interchanging the limiting processes as t goes to zero and as 2 goes to zero with integration, the initial and boundary conditions can be verified. To show the differential equation satisfied, the uniform convergence of integral representations -11- of T together with the first and second time derivatives and the first, second, and third space derivatives for t,z>0 can be shown, differentiations and integrations can then be interchanged [1, p. 1:810, and the solution T can be verified by direct substitution. The integral for T can be shown uniformly convergent with respect to z for z .>. 0 by the Abel Theorem: If ffbl) dz converges and if for every value of z for 2?.0, the function v( A,z) is non-negative, bounded for all z, A and never increasing with A, then ffbl) v().,z) dA is uniformly convergent with respect to z for 2?.0. Taking fol) = J,(/lb) J.(Ar) and v(}L,z) " see + e"“wc<‘—s§€e = e z Erfc( one notes fran : (8) that Ifu) dA converges, that for t>0 v(}(,z) is positive, bounded, and tending to zero as A increases if 2 f O or equal 2 if 2 =0, arni that for t = O v(/\,z) = 0. It remains yet to show v()\,z) is a monotone function in A for z>0 and t>0. Writing mm) = Wum) + y/wx ,2) where you) _g. = M maW) and using the relation Erfc(§ )2 7%}— .[l - 27].”: 1- &3), - %iufl [16, p. 126] valid for large §>0 yields W: cm [aft‘g-g-Erch) -|- z Erfc(§)] , A; ‘23-'W/Ee-g‘ 2: 3; _ l t...)]’ :8 =e*(7%rf+a‘-Xt) [_ 2aE+ gfxflg; _ ajtuJ], -12- where E: W vhich shows 3%(252) (0 for all sufficiently large A. And Wit/l , z) 512 [#f/E 3’7‘- 2 Erfc(7)] , = S” [3.379; J7 - 2(1 - Erf(7))]<0 for all large /\ where 7: W because h‘f (:7)<0 for 7< 0. Therefore v(,\ ,z) is a monotone decreasing function after a sufficiently large A for z>0 and t>0. Hence the integral for T is uniformly convergent with respect to z for a Z 0. Repetition of a similar argunent with 2 replaced by t shows uniform convergence with respect to t for t _>. O in the region :20. The integral for T can be shown uniformly convergent with respect to r for r _>_ O in the region z>0 by the Dirichlet-Hardy theorem: If if(/\,r) dA, is bounded for all 3>a and for r?.. O, and if v(A) is bounded, positive, non-ingeasing, and tends to zero as 2. approaches infinity, then a! f(A,r) v(A) CIA is uniformly convergent for r?. 0. It will suffice to show IJ,(Ab) J°()r) 2.. - a. 0 [ext Meg-1.5%) + e“ Erfc(3-§97T¢£)] d). uniformly convergent where a is an arbitrarily large poisétive number. Here r().,r) a JAM) Jami-M and ml): 1 Mai—W—E) - I. + 3A“ Errc(z_éaa;f’,\t ). Use of the asymptotic expansions of the Bassel functions gives J,(/\ b) J°(1\r) A. = 77%;" [sin A (1)-1‘) - cos A.(b +r)] + 0(751’7) [16, P- 32] - Then “A,” all = 1 _ cos )Jb-r) _ sin A (b+r)]x+ 0( 1 ) 18 7T? E b-r —_b+r a 7k. - 13 - bounded for all x>a. For all z>0, v(A) is bounded, positive, non-increasing, and tending to zero with increasing A by an argmnent similar to the one used previously in connection with the application of the Abel theorem. Therefore .£ J , (Ab) J°(/lr) . [eAz MMW) 4- e4” Erdefl d/l is uniformly convergent in r for r2 0 in the region z>0. With the inclusion of the fact fJ, (Ab) J , (Ar) ERA.- converges [16, p. 50] , the other integrals may be shown uniformly convergent in manners analogous to those already demonstrated. -lh- IV. The Quasi-static Solution. The temperature distribution derived in section III causes stresses ani displacements in the half-space. In this section a quasi-static condition is assured and the solution for those stresses and displacanents is obtained. The assumption of the quasi -static condition allows the neglection of the dynamic terms appearing in the displacanent equations (3). Under this asswnption uith the cylindrical coordinates introduced as in section III, the displacement equations take the form .u u + 1 be __ 2(1w) Bur) ‘9 1275“ “1-3733—— (ll) v=0, 1 3_ 21y3'r vw+mz§w52¥akh where u = displacement in the radial direction, v = displacement in the 7’ direction, w = displacement in the z direction, v1~=§t+:§_+g_:,e =§E+i~E +3_:.,ardTisglvenby equation (10). Since the surface plane is free from external loading, the normal stress 0;; and the shear stress 0;: must vanish on that boundary. Using the asstmption from the linear theory that the boundary conditions are applied to the undefomed body and writing these boundary conditions on the surface 2:: 0 gives 0;,J=o:= 1%_[(1_3y)3_:+ 7(3-3 + 3-) - mama] , (12) Pa G3“ 3w 2:0 “74"- ° Gr* 3- . Zao =0 -15- Although the latter equalities expressing the stresses in terms of the displacanents do not enter into the solution of the boundary value problem in that form, they are included for completeness. The differential equations (11) and the boundary conditions (12) canprise the boundary value problem to be solved in this section. As a first step in the solution, a thermoelastic potential 4: —._-_ 4>(r,z,t) is introduced in order to derifl a particular solution ani thus reduce the differential equations to hcmogeneous form. The themoelastic potential 4> defined through the relations 2Gu; = 3-1? yields particular solutions to the i differential equations (11) if 4> satisfies the equation (13) v‘cs = “1335‘“ T . Because of the simpler form of T in the Laplace subsidiary space, the equation (13) is transformed there with L{<}>(t)} =¢(P) a f¢(t) e’Pt dt. 0 (124) “P“ a may}; T*. According to the heat equation, 3% z a" W and pT* a a‘v”T*. Substituting fran this relation into (lit) yields 21+ 0 ‘ '1‘* V‘fi" = V1 A 1,974“ 31' 2. Then a particular solution of (it) is 95;? = 2(111-33G4a ?. Fran use of Hooke's law and the defining relation for ¢, the following equations relating the stresses and the thermoelastic -16- potential are derived. 3‘ L 3 = 332;: “Vi“: _. l 3 a. * 0;: = r35: “’4’ . _ 3‘ a"*=rz’ 0?: =5§€i-V‘¢*- Substituting cfi into equations (15) gives the stresses in the Jfil (15) form of integrals divergent on the surface 2:0. For this reason a harmonic function correspondent to the steady-state temperature distribution must be superposed to secure convergence. .12 43* _.__ -aglgyl’cx‘nganflAb) J°(,\r) 25? dz is such a 2. ' 0 function}, The solution for the themoelastic potential in the subsidiary space is °° e-Xz_ e-Az (16) ¢‘r’z’1°) =.- cafe wt) gar) -—5;-— as, §1+Y)Ga(a"hro 1-7 ° space due to this potential are avian by equations (15). there C, = The stresses in the subsidiary Wis selection of $f Is motivated by solution of the steady- state problem not included separately here since it is obtained subsequently as a special case of the quasi-static solution by letting t approach infinity. -17- __e-Az 2- 42- 1 -/\z a;:=qjgym)-L,%— J ” 3———- “ell-JAM) gar) 1° Pf" M. - -Az e-Xz 5:- =-. —c,][ alum .MEl g3... 11... Juli) J(Ar) —.f—p. a). , (l7) ill a an aclfJKXb) JMI') Xe-x 2;Ac.’1 2A (1A, a _ O. e-Xz__ e-Az z. 0;; =0, f J,(/\b) JJAr) p; A «1/1. 0 The physical problal demands that the stresses be continuous in the half-space and that the normal and shear stresses on the plane a = 0 vanish. That these 5:" and consequently the 0—; are continuous in the half-space can be shown by proving the integrals uniformly convergent in the same manner as that used in the latter part of section III. , Having shown the uniform convergence with respect to z, the limiting process as 2 goes to zero can be interchanged with integration to note 65:] = o and ‘67:} = C, I J,(Ab) J,(,\r) lf/l dA . M‘s-i: example of the“ method for showing these integrals unifomly convergent and also to show 5:: exists on the boundary 2 = O, 5:: is proved a continuous function of r and z in the half-space and on the boundary. This preposition includes 6:: hailing a finite value on the bounding surface 2 =.- 0. The Dirichlet-Hardy theoran as stated in section III with f(A,r) =-. J,(;tb) J,(Ar)A and v(A) J z -A2 a Xe - Ae is applicable; and as in section III, since the part of the integral firm 0 to a need not be considered, the -18- integral from a to co where a is an arbitrarily large positive number is used. The asymptotic expansion of f(/1, r) is W [cos A(b-r) -sin/\(b +r)]1+ 0(A ---) [16, p. 32] i A b- and the integral fflf r) dA-a: W [Bb—r n ( r) + £08 ALbi-r) 8+ “1—) is bounded for all x) a. v().) is b+r bomded, monotone, and tends to zero with increasing A for z 2 0. Therefore the integral representing 6?: is uni fomly convergent in r for r 2 O in the region 2 2.. 0 and E: is continuous in r for r 2. O and z 2. 0. The Abel test as stated in section III with WU = J,(Ab) Mir) and “A,” = Ask-h _ e-Az) show A uniform convergence with respect to z for 2 Z 0. Therefore 83* is a gontinuous function of z for z 2 0 having the value I J, (Ab) J,(Ar) 1%}5‘ A dA on the surface 2 = 0. Since the homal and shear stresses in the original space and thus the same stresses in the subsidiary space must vanish, the subsidiary shear stress derived iron the thermoelastic potential must be adjusted to zero by the superposition of stresses obtained fun a Love's function L. Because of the axial symetry of the problem, Love's function gives a general solution of the hunc- geneous elasticity equations and therefore such a function is known to exist. The boundary value problem in Love's function L :.-.. L(r,2,t) can be written in the subsidiary space in L* = L*(r,z,p) as V41,“ = 0 , 5?; - = airs-[32") Vi:6'}y] ’ (18) 2‘0 7,80 0:31 = - W) J,0\r) = fg-EEU-I’) V11" z-JaitLif] 3 Z‘O _. no where CT": f— *' (M. This function in the subsidiary space can 0 be obtained by canbination of two functions, L,’ = 2 {‘Ar) e"1 .12 an! L! .1: Jo“ r) e . (19) L*(r,z,p) == A(p,A) Lr(r,z,p) + B(p,A) L:(r,z,p), where A(p,}L) and B(p,).) are to be determined such that the boundary conditions are satisfied. Substitution of L* into the boundary con- ditions gives the following two equations for the two mknowns A and B. (20) O ’1'2327 [(I'NHfiB] gar), -M.A)%£y(/Lb) J,(,ir) = 13:37 X’Ee/A-tas] J,(Ar). Fran equations (20) _ 1H) 1.27%" Li— A) glut) Minx) — L L 11'9- an (21) B(p,/\) = --(-J=-';21-Z)-A(p,/1) . The stresses in the subsidiary space due to Love's function are given by [19, p- 73]. -20.- E: 8 ,‘M- 9a” J (22) as; = 3-;[YV‘V - fig-11 f:=:.§—[(1-Y)V v&*-%_fi _ 2G ,) a.* 3‘; azm-fi (2-7)VL -W 0 Use of equations (19), (21), and (22) and integation over 7\ yields the stresses necessary to adjust the normal and shear stresses from the thermoelastic potential to zero on the unloaded plane 2 = O. C m J(}\ r) n 351: [(x- “Am: 122/) J, (Ab) 47—— +(x- mz- 112) mm JW] ”Add, Erie f B"“‘"’"‘”” “N” “fig 0 + 27(Y-A) J,(Ab) J.(Ar)] e'A‘AJ/x. C 3:9: I (x ANAz-l) am) mm e'A/id/l, O 55* 9&1 (J-m was) Jaw) e *‘i‘dx. P a The final stresses in the subsidiary space are obtained by addition of the stresses derived from the potential and those fran Love's function. These final stresses in the subsidiary Space are -21- 0;: = C, f{J,(Ab)M 49.1.). [1.2.3. + (10.21.2142) (I—A-Afi-r—J + J,(1b) tariff +<,1(2.,\,) (1.1)”) J }M, (21:) r: c 'I{Jlub)Jr fifti: + (Maya-21) A(g.,g+,1>g::] + JP“) ‘19:)[- 15— +27A(X—A) fez-:1} (1A q ll .33 = c,f J,(Ab) flur)[::'Tfi +éuz-1) a.» {>14 M, = c, flab) gar)[J‘-‘-;1— + (250-2.) '2‘); 4:] cu In order to casplete the stress detemination, the final stresses in N N the subsidiary space must be transformed back into the original space. This is done by interchanging the inverse Iaplace werator with integration. In order to Justify this interchange of limiting processes, the integrals can be shown uniformly convergent in p by the Abel theoren in a manner analogous to that used in the latter part of section III . This inverse transformation involves terms of five types. These transformations are listed below and derived subsequently. ('0 L129“ = M 47.75? -I _L"{B- p: 1}: H%[ 22 mo (3+2. LAt) + e A” Eddm-fl a (o) 17%;; .-. i:- L"' {L35 }= fir’a‘ " t mmm+- -F .451”, _Vp+a"A'z (25) {1:17, (‘1) 31“} {—1—— —,—.—-— _ “3.;th a“ ”0‘” 23‘1") _ 2%é3 (’1‘ Om¢(-ME%¢‘E) , -22- (e) L {grfgiLq Euro }____,_ “gig-Mt A: - quW—h‘a‘l ‘) 4- 1.112531% e ”A n _ ._ 2‘ .m-még'fil)... %g e (a? AI-t +$¢ Transformticn (a) is an elementary transformation tabulated directly. Transformation (b) can be obtained fran the tabulated transformation for ape-9).! («"57" [7,1). 2116] by means of the shifting theorem 1’"(p-e) in the subsidiary space corresponis to e'tf(t) in the original space. Transformation (c) can be derived by use of the tabulated transformations for % and mtg-‘5' [7, p. 235] and the convolution theorau: ff(p) o f*(p) in the subsidiary space corresponds to ftht-t) fit) df in the original space. With the substitution of J‘A’t 4t)=L '1{ }=H1and4t)=17 ¥E=Ep— +sAErf(~W'5), {Wk—3.}? _j}§n df=m£*%€dt2‘ +r.& 22-]:th '5 (if. The first of these integrals can be evaluated by a change of variable f: 6f? 3the second, by the Dirichlet forumls: fb dz ff(x,y) dy = idy, J: f(z, y) d1. In the eveluation of the second integral, jdff gfi=~ffd 7] 2:11:30 7.3.215. a S 4— '15}: 7 ’l aa;°;_ 1:]? 1&0 = .175: ’1 ‘ 7= Fri aa/F_ L :EZ"; Erf(e “SW "T‘i f 5e dg . The latter integral, 55%-: d5 , can be evaluated by inserting a parameter n EL 61"? to define Na): £ 3" e 3d; , observing F(z) = dz) where -23- all? ‘ f(z)== -1; {‘3 a: = - gmfium), and noting 3(1) to be the integral in question. Canbination and simplification yields transformation (0). Use of the shifting theorem and changes of variable = a‘A" and x s :- reduces the evaluation of trans- formation (d) to evaluation of L-'{ 65"? By writi f*( ) W ° “3 1’ e45“: 9'51! 2 1 e41?! 81 e45.1 = 57:75-37 "' °‘ 3 r5725; °‘ 5:13:29, “PM“ W - I 3,, [7, p. 2’45] , and using the 1 operational formula: 5;,- f*(p) in the subsidiary space corresponds tabulated transformation for t to ( L d1: )n fit) in the original space, the evaluation of the simplified transformation (d) is reduced to solution of the following differential equation: f'(t) -— 2qf‘(t)+ (fit) -x/‘fi. _ I e an differential __ W 7t L. The general solution of s 4t equation is the sum of the hanogeneous solution, f(t) z: (A +Bt) e where A and B are thus far arbitrary, and the particular solution, ““5 — I [2 sinhE‘x - e-Rxfih'fl/«t' - ‘ fa?) =3 (014313) e where C— W :7?) .. e'q.x Erf(f:?+ 2717)] and D = ooshFZ'z + §[e-Ex oErf( J x t' - fig) - em‘ MHZ? + 1%] . Instrumental in the derivation of the particular solution by the method of variation of parameters is Horenstein's evaluation [12] of two integrals; I'=Lx-%en(- if. ‘b2_1) dz = __a_ 00811 28b + .[g— . [e-zab mfib Nair) - eubErflb J? 4’ firfl. * 1 dl 1 Laf£%exp(- a: - be) dz 3. - fi W‘vmere I, 18 given n O the preceding equation. To find the initial conditions {(0) and -2h- f‘(0) in order to evalmte A and B, the theorem [5, vol. 2, p. 1&5] --if lim f(t) exists then lim p f*(p) = lim f(t)--is applicable. By employing the fact that the solution f(t) a: (A+Bt)e a“ + (C +Dt) ext there A and B are arbitrary and where C and D are known functions bounied as t approaches zero, one can note that lain f(t) exists. This theorem yields the result f(0) = f'(o)s0. ’0 Use of these initial conditions shows A = B a O. The evaluation of 7—D- )‘ theorem yields the transformation (d). The last of the transforma- L.' }is thus ccmpleted. A second application of the shifting tions, transformation (e), is obtained through differentiation of z transformation (6.) with respect to '3- - The inverse Laplace transformation of the stresses in the subsidiary space from equations (214) by means of equations (25) canpletes the stress determination in the quasi-static case. ¢r3C1.[{J,(I\b) -éE-' [Avg-L“ ‘1': (3+ AQ:+2/-2)e""‘1, L——'{ :1} 42 -AM-(Az+211)e"'1,"1,i’§."{ f] + Jpn?) «Ii/11') [' 2'72- L’Ifi- } -1 L {51- Jr} '- lu"2)°.l ‘1‘“{5} (26) +A‘(A¢-1)°'M1fl{1l,zfl} ‘1" a.q.,....:._clJ:D{4(,\-n)A 21113.). {L 1‘1," { 5:? - Auti‘W-QW ”II-fit} -{z +JL”(Az+2”-1)°'Ln-%'{ Q'f Jab) Jur)[-:L - {5-} mun-:23; -222s-Az2-n{;.] } 22, - 25 - {Ignacio}, J().b) J.()~r)[7LL" {L°:h } + MM De '1,” 1,493,} 0 - X’s e‘M'L-W $3 02-50, [4(a) .mr) [A L-'{§2 a} + x: a” $1} .. z-(lz+1)e‘ kl." {$31 «1).. That the shear and normal stress on the plane 2:: 0 vanish can ' be verified by substitution 2 =0 in equations (26). By use of equations (8), (25), and (26), one can observe that for t approaching zero on the surface 2 = 0 were: 2.22.. 0;: 0;: l-y 2: «- BBQG“ Ti for 1‘: b, (27) = o for r) b, 0:1: Q: 0 . This result is of interest because it was conjectured earlier on the basis of classical work on the quenching of spheres and cylinders. The stress solution of the corresponding steady-state Problem is obtained by letting t approach infinity in the quasi- B’Oatic stress solution given by equations (26) . After a Justi- fiable interchange of limiting processes, the steady-state solution becomes a;'=.2(1+Y)G«Hr.f[-I,(l~b) £01”) " JP“) 9‘fo (28) ° Ar) 3-112 (111 24(1+’)G4W0 flow) a -26- 6;=.2(l+I)G«br. I J;(Ab) $2. 942 M, 0.7..” 0250- The displacements are canputed as the sum of displacements due to the thermoelastic potential plus those due to Love's function. As differentiation under the integral sign can be permitted in this case, the displacements in the subsidiary space due to the potential are - a c m (A. > “I” - 3"" 11‘" = E1 r = - zfi IJK :5 r P]- A M. (29) 4: '11! 3 == 221: a: = - gain/1b) gar) 1" 5,3” M, O which, after Justification of the interchange of the inverse Laplace operator and integration, yield in txhe original space - ‘ ' - - 1 t = - 3,5 from .y(1:)[ L ' {5;- } - 6M1. {PU/1M, _. Ag: flan) gar) [ L"{€;;h}-ze"‘L"{%*U 2m, 0 (30) fl \ In the subsidiary space the displacements due to Ipve's function [19, p. 73] are as _ ”’1' is = - 3:57 3* = 3d 4m») Wt) (2‘2"3‘) mg' “=1, (31) is: I127[2(1-10Vi*-§;¥ - 42 ~_-. 33 fJMb) gm (am-m 54%: cu, -27- which, after valid interchange of limiting processes, yield in the original space g = 3‘5 J,(A_b) Jpx) (2--2)’-/11=)°"1 “(Wig-J #1177 it!) (32) ta. 5%: 2m) 4111‘) (1-2Mz)e*‘(r‘{$} sill-1,20 dz. Canbination of results (30) and (32) determines the final dis- placements in the quasi-static case. u=5+g= (My)? a‘vrorJub) Jar) [(2.2%)th "1.)" "{P 1 } ~11. q; “r A(l-2/-Az)e"1zf' {1 pd] a1, (33) 2 t 2. s- Witch) 2‘22) [ ('1+2”'*‘>°’”L”{ 5‘ .. 12—h } + A. (22.1.22).” 171,2) ] u. The displacements in the corresponding steady-state problem by letting t approach infinity are obtained from equations (33) u = (1+nsvr2fwb) 31‘”) ‘4': 7" (31!) 71! M w =: -(l+/) «WI. JIQb) JJAI) e -28.. On the axis of symmetry and on the surface with f = g. , r 7 =-- b' , and x = A2, for r=0 the steady-state stresses and displacements of equations (28) and (3b) can be written [16, p. 147] In", '-E 0:05:01 euguil ’ _ J +1- (35) 0'7 C’ 2s ' 0:2: 05:01 a: o, W=C3(§"V;+": and for z =0 [15, PP. 1'9-50] 9 l .2. for 751' arr—=0: 233,1 for 7>1, 1 for 7(1, 2' q": C, for7sl, 3,22?qu>1 (35) 01;: 0:230 3 for'7sl, u =03 2.13:] for’)71, :le 1) 2222,22 v: fog-37:1, 2— 11E(%:2'2 r51)!” 7)]? -29- where cz =.2(1+)’)cour° , c3: (1 +V)e*+ 32%- 4% " at 53:51:31. - (7‘494-325-6'5, (”1) -32- 5:? =3¥—v‘¢*+1’2§f 4;: there again use has been made of the fact that ¢= 3?: O at t = 0. As in the quasi-static case, #3,” yields stresses in the form of integrals divergent on the surface 2:30. For this reason it is necessary to superpose on 4)? a solution of the homogeneous wave equation V3¢*- Cfp" ¢*= O in order to make the stress integrals convergent for z = 0. Separation of the wave equation in cylindrical coordinates in the case of axial symetry yields a solution of the form ¢3:— =rA(p,/\) J°(,{r) 3'8” M where A(p,A) is arbitrary and ,6: 0JG“) ‘+A" - With a convenient choice of A(p,A “‘2’ ¢* = 22; +42: = foam (.4: - e'P‘) n, 2. where c __ Kbr. Jl(,\b)_ h(l+)’)G:ata 1&5?ng Use of equations (111) gives the stresses in the subsidiary space result- ing fran the potential ¢" ° 0'3: f0: [4 xtg-N'h-F 15+ #290 a” J(Ar) + A (e — 2 '1”) tom] dA, (143) E;=f°: [(A- [flagser‘ (A- (34?): F).g(;1r) —é/€,‘z. The final stresses in the original space for small t corresponding to large values of p in the subsidiary space are g 306 Hr. 0,72%?- _.21+ at [m°(§.;¥) (1.6) o t < 46:: ]I (r,z), ’ "’ f“ 7.23;, t) (2,: -35- 4'23 0, :{O,y t<\/—'z ]I(r,z). 1, 1:) 0+2 At this point it can be observed that the asymptotic normal and (“6) shear stresses are zero on the surface 2 =- 0. Since, within the approximations made forlarge p, the stresses due to the thermo- elastic potential satisfy the boundary conditions, no stresses need to be superposed as in the quasi-static solution. In this case the thermoelastic potential has produced the asymptotic stress solution for mall t to boundary value problem (35) . With 53 «E, '73 %, and: =Az, forr=Othedynamic stresses for small values of time given by equations (16) can be written [16, p. 147] _a;'2 % E(§, t): 0:130, ‘1... - fizg‘JTrTj‘H 1" 5") for Z: O [16, P. 50], (1'7) ‘0 l , *7 <1, 0:.” 0;“ CL-rvl-fiy % ’ ’1‘1 ’ 4" bo’ (he) 0 , V1: 6,1230, Q80 ,for t>0 ; .. 36 - and for neither r nor 2 equal zero , 6;": $42 Esta-1311“”), (1:9) 07?“: @flji‘flng :7 )1 where 0‘ = -- W , M; ,t) = c,‘ [mfiéfig o , t<fi2b§ b o ,t thS i,t>./E;hg Ind 1(5a’2)= Jfl‘x) $0?!) 9-: dx . It can be noted that equations 0 (1‘8) describing the surface stresses in the dynamic case for mall values of time are the same as equations (27) describing the surface stresses in the quasi -static case for time approaching zero. That is, the surface stresses for time approaching zero are given correctly by the quasi -etatic solution. This result is in agreenent with the work of perineum: [3 and h] . - 37 - VI. Numerica1.Results. In.this section the steadyastate stresses and displacements of equations (35) and (36) are tabulated for thirty-one equally Spaced points on the axis of symmetry and on the surface in a representative plane containing the axis of symmetry, the integral I - :J, (2L1) "Jot?- x)?! dx is evaluated for fourteen of twenty-seven grid points in this plane, and the dynamic stresses for small values of time given by equations (M7), (NB), and (M9) are tabulated for these twenty-seven grid points in terms of E( f, t), and F( f, t) . Because of axial symmetry the evaluation of I and the asymptotic dynamic stresses actually involves canputation at only seven and sixteen points respectively. The sixteen grid points for which the asympto- tic dynamic stresses are computed are shown below in figure 1. The seven points in this grid neither on the axis of symmetry nor on the surface are those at which I is computed. 6 IO 7’ C) C) Figure I -38- The values of the steady-state stresses arrl displacements for the indicated points on the axis and on the surface are given to four -digit accuracy in table 1. Since the steady-state normal and shear stresses are zero throughout, they are not included in this tabulation. -39- $3.- enno... mmoo... ZS... ammo: omoo... Hmma... mama: 6mm... «:2. 884.. o o o o o o o o o o o 9.8. :8. $8. omoo. $8. Koo. 28. Rmo. ammo. éfi. ooom. 5 53-16 SIJ‘ 3k? nuoo. amoo. mmoo. onoo. $8. Koo. 9.8. Rmo. memo. 6i. ooom. its 3 m w u e W $ m a e 0 ”mass on» go 3.8.- 3.8.. memo... 35.- mmmo: mood... 8%... 33.- 5mm... memo: 88.1% 080. mono. ammo. Eb. mmmo. oooa. onma. 58. comm. 8cm. 0 mm omoo... «000.. 2.00.- «0.8... mmaof oomof mamo... mmno... onmaf 0 000m. much onoo. «moo. 2.8. moao. mmao. oomo. mamo. mono. ohms. ooom. coon. “now ya 0;. ms ms Na. ems um.“ *1“ ms .Nfl D" Q «scourge can no a 0.33. :odpsaom opauuazoaoum -);Q_ The value of I is approximated for the seven inner points of figure 1 by terminating the integation at x = 12 with a maximum error less than (3.73156 since IJ, (.].§'. 1) Jocé 1)| (,4. and J’s-x dx < (6.1) 10". The integral terminated at x =12 is then apgroximated by Simpson's rule: ”" h nh'r w Lin) axe3-(r,+ur,+2r,+ur,+. . .+14f».;+2f,,.1rhfw*fi,)— 186‘? (3) where h = the length of one subinterval = .03, n = the even number designating the number of subintervals = 1:00, fK’ f (1+kh) , 0 < f < 12. The maxinmm error in approximating Hg ,7) can be bounded in the following manner. let the integrand f(x) ==JS(X) 0(3§_ x) where 8(1): J,(.__. x) e”x . Since, for all the inner points marked in figure 1, the absolute values of both J, (_x and e as well as all their derivatives are bounded by one, 82;"? < 2": nh5_]3ecause ’a—fi Jo(x)l < l, 'd—J Jo(-7- I), < (1)" Therefore 186 f"(x) < (5. 1010" [16+ 32 34- 2M1) + (sq—h- (1)3 for all x) 0. Hence one can easily show that at all the marked inner points of figure 1 except 5 = l and ‘7: 6 the maximmn total error inherent in this approximating scheme for I( g, 1’) is less than (1010.: and that at g = l and 31 = 6 this error is less than (2.3) 10-1 Round-off error can enter the computation fran the approximation in the tabulation of the Bessel and exponential functions and from the operations performed within the computer. The round -off error due to approximating the Bessel and exponential functions is held to a minimum by use of a minimum of nine digits in their tabulation [23] . It is easily sham that for these seven grid -: points this round-off error is less than 10 . The round-off error -111- within the computer has been checked by instructing the computer to make the calculation once rounding down and again rounding up. Canparison of these results ascertains the maximum round-off error within the computer also to be less than 10-:at all seven points. In other words the round-off error cannot be of any consequence in the four -digit tabulations which follow. The value of I at the seven aforementioned grid points has been computed with the Michigan State Illiac Computer, the Mlstic, by the following program: Programt 1. Four hundred values of e“X for x = .03 to x = 12 in intervals of .03 are canputed and stored in the Mistic. 2. Four hundred values of J, (3; I) for I = .03 to I = 12- in intervals of .03 and for 5 fixed are input. 3. (he value of Job?" x) for x = .03 then x = .06 . . . up to 1 =12 and rm. .ELflgad is input and one f,,, f,,. where f“ 2 f(x+.03k) 18 canputed. r“. . . rm 14. The Simpsozis rule routine which evaluates the integral up to the if" sumnand is "Jumped into”. 5. The program returns to step three fourhinired times. Then step six is executed. 6. The result is output. 7. Thevalue of ‘7 is raised to the next higher value across the grid . 1' This program was written by Dr. Gerard P. Weeg. -142- 8. If all values of 3, for a given 5 are not yet used, the program returns to step three; if they are all used, then the program continues to step nine. 9. The S value is raised to the next higher value down the grid. If all g values in the grid are used, the program stops the computer; if not, then the program takes 17 at its lowest value and returns to step two. The values of I determined from this program are written in table 2, All digits tabulated are significant. ’18! 7:3 i=6 gs: I = .1787 .01724 .002 5:3 I = .1316 .0595 5:4 I = .0785 .0295 Table 2 By use of table 2 and equations (‘47), (148), and (119), the asymptotic dynamic stresses are written in table 3. In table 3 E(n,t) and F(n,t) defined on page 36 are abbreviated as E... and F,1 respectively. The t in the arguments of Bend F' is taken as positive in table 3. Because the shear stress vanishes throughout, it is not included in table 3. As in tables 1 and 2, all digits tabulated are significant. -13- A :1 q :4 am _ ,7 o o Fizz Era: 1.0000 .5000 0 0 E. 0 0 0;: ~ 0 O O ‘E‘.’ " I %8%3 .2929 .178! .0174 .002 . 787 out .002 EL 3:. .2929 .1 . E 3 £35: .0513 .0M9 .0198 E, a" 01m 0198 22;, .0513 . 9 . F3 15530-5: 2 .0136 .0131 .00119 6 E‘ E‘ 0049 0;; ~ .0136 .0131 . ‘F‘. .— IO [m 3 £32 .0050 Etc 5,. girl: .0050 i 6. Table 3 E - us - VII . Conclusion. The basic theory of the thermoelastic problem has been reviewed and the temperature distribution within a semi -infinite, three-dimensional half-space bounded by a plane due to a tempera- ture distribution of T = constant within a circular area and T8 0 outside maintained on the surface has been obtained. The original results of this thesis include the derivation of the quasi -static stress and displacement distributions within the half-space and on the boundary and the derivation of the dynamic stress dis- tribution within the half-space and on the boundary for small values of time due to the previously indicated temperature dis- tribution. In addition nunerical results relevant to the steady- state and asymptotic dynamic solutions have been tabulated. Bibliography 1. Branwich, T.J.I'a, An Introduction to the Theory of Infinite Series, Macmillan-r NEw York, 1955': 2. Carslaw, H. S. and Jaeger, J. C., Conduction of Heat in Solids, Oxford, London, 19117. 3. Danilovskaya, V. I., “Thermal Stresses in the Elastic Half- space Caused by the Sudden Heating of the Surface" (In Russian), Prikl. Mat. Mekh., in, 316, (1950). 1:. Danilovskaya, V. I., "On the Dynamic Problem of Thermoelasticity" (In Russian), Prikl. Mat. Mekh., 16, 31:1, (1952). 5. Doetsch, 0., Handbuch der Iaplace-Transformation, vols. l, 2, 3, Birld'iluser, Basel, 7556. 6. Doetsch, G. , Anleitung zun praktischen Gebrauch der Laplace- Transformation, Ufiefibourg, Munich, W- 7. Erdélyi, A. and others, Tables of Integal Transfoms (Bateman Manuscript Project), voI. Ifficfi‘awmill, New York, 1951). 8. Gill, S. and others, Illiac Prommming, Univ. Illinois, Urbans, 1956. 9. Goldstein, 8., ”Some Two-dimensional Diffusion Problems with Circular Symmetry", Proc. London lath. Soc. (2), 3H, 51, (1932)- 10. Goodier, J. 11., 93 the Integration 9}; the Theme-elastic Equatime, Phil. Mag., 23, 16327“, T1937)? 11. Hildebrand, F. B., Introduction to Numerical Analysis, McGraw- Hill, New York, 13756." l2. Horenstein, W. , "0n Certain Integrals in the Theory of Heat Conduction“, Quart. Appl. Math., 3, .103, (19145). 13. Jahnke, E. and mine-3,111, Tables of Functions, 14th ed., Dover, New York, 19145. .. 11). Jeffreys, H. and Jeffreys, B. 8., Methods of Mathematical Physics, 3rd ed., Cambridge Univ., Londom 195 ."" 15. Love, A. E. H., A Treatise 9.9. the Mathematical Them of Elasticity, WSW, over,‘NEw‘YT19’27'or , . "’ lb. 17. 18. 2h. 25. 26. 27 . 28. 29. Magnus, w. and Oberhettinger, H., Formeln und Sa’tze it'r die spe_z_i_ellen Eunictionen der mathematischeg—P'Kysik, 2E-ed. , Springer, GOttIFgen, 19778. Mura, T., 'Thermal Strains and Stresses in Transient State“, Proc. 2:th Japan Nat. Cong. Appl. Mech., 9, (1952). Parkus, H., "Stress in a Centrally Heated Disc“, Proc. 2nd U. S. Cong. Appl. Mech., Ann Arbor, Mich., June 111-18, 195“, p. 307. Parkus, H. and Nolan, H., Wé'rmespannungen, Springer, Vienna, 1953 - Parkus, H., Chapter on thermal stress in Handbook of Engineering Mechanics (in preparation), McGraw-Hill, New Tor? Sadowsky, M. 11., ”Thermal Shock on a Circular Surface of Emosure of an Elastic Half-space", J. Appl. Mech., 22, 2, 177, (1955)- . _Sokolnikoff, I. 8., Mathematical Theory of Elasticity, 2nd ed., Mcmw-HiII, New York", I955. The staff of the Harvard chnputation laboratory, Tables of the Bessel Motions of First Kind 93 Orders Zero and (he, Harvard UEiv. , Cambridge, 19?]. Sternberg, H., and McDowell, E. L., "0n the Steady-state Thermoelastic Problen for the Half-space”, Quart. Appl. Math., ll), 381, (1957). Timoshenko, S. and Goodier, J. N., Theory of Elasticity, 2nd ed., McGraw-Hill, New York, 1951. "'- Watson, G. N. , A Treatise on the Theory of Bessel Functions, and ed., Maddll‘a‘W’n, ew $571955". '- Weeg, G. P. , unpublished notes on canputer coding. Weiner, J. H. , "A Uniqueness Theorem for a Coupled Thermo- elastic Problsn", Quart. Appl. Math., 15, 102, (1957). Vestergaard, H. M. , The 93 Elasticity and Platicity Wiley at Harvard Un v. Press, NE? YorE,_I§5 . ROOM USE ONLY «UV!!! 03E CRLY TY LlB “ICIIW'ICHWININETIWVIUIINWIVIIEIWI'Iml! WI)!“ 3 119 3 0 3 0 8 2 5 800