Survey of recent developments in the fields of heat conduction in solids and thermo-elasticity

Paper L

NUCLEAR ENGINEERING AND DESIGN 18 (1972) 377-399 NORTH-HOLLAND PUBLISHING COMPANY

SURVEY OF RECENT DEVELOPMENTS IN THE FIELDS OF HEAT CONDUCTION IN SOLIDS AND THERMO-ELASTICITY * B.A. B O L E Y Cornell University, Ithaca, New York 14850, USA Received 6 July 1971

The developments of the last few years in both the fields of heat conduction in solids and of thermo-elasticity are reviewed. Attention is centered in both instances on solutions, results and techniques of direct engineering application, with comparatively less emphasis on those of primarily theoretical interest. Thus in the calculation of either the temperature or of the consequent mechanical response both exact and approximate approaches are treated; in the latter case this includes an assessment of the error involved, and in some problems the determination of bounds to the exact solution. The paper starts with a brief review of the basic theory, including the effects of thermo-mechanical coupling and the conditions under which these effects may safely be ignored in practice. The problem of heat conduction is taken up next, including a summary of recent solutions for rods, cylinders and other structural members, and of recent advances in numerical techniques. Some extensions of known approximate analytical procedures are indicated, and the conclusions to which they lead are compared with pertinent exact results. The status of heat-conduction calculations involving changes of phase is summarized; results pertaining to plates, cylinders and spheres are quoted, and analytical short-time solutions and numerical long-time calculations are discussed, and results in twoand three-dimensional melting are presented. Recent thermoelastic results pertaining to rods, beams, plates, cylinders and spheres are then reviewed. The importance of thermally induced vibrations is assessed, and an approximate formula for its evaluation is presented. Bounds for the stresses, deflections and the torsional rigidity of beams and plates under arbitrary temperature distributions are calculated by means of a simple analogy. The calculation of thermal stresses in indeterminate structures such as rings is critically examined, and the proper use of energy methods in such cases is indicated. Some new results concerning the mechanical response of melting plates are quoted. Developments concerning thermalstress calculations in which an extension to visco-elastic materials is readily carried out, are briefly referred to. The paper includes a comprehensive bibliography of recently published papers in heat conduction and thermoelasticity.

1. I n t r o d u c t i o n Research on the behavior o f solids and structures at high temperatures and the d e v e l o p m e n t o f appropriate m e t h o d s o f analysis are some o f the m o s t lively areas o f current technical activity, principally because o f i m p o r t a n t application, n o t only in connection w i t h nuclear reactors, b u t also in fields such as * This work was performed under sponsorship of the Office of Naval Research.

those o f high-speed flight, space travel and atmospheric re-entry, p o w e r plant design, d e f o r m a t i o n control o f large optical devices and so forth. This is evident f r o m even a superficial glance at the large n u m b e r o f papers related to this topic which are published each year; the above s t a t e m e n t b e c o m e s even m o r e true if one includes in this field (as indeed is done in the present survey) not only thermal-stress analysis proper but also the associated t h e o r y o f heat c o n d u c t i o n in solids. In view o f this, if the present w o r k is to be even m o d e r a t e l y successful in its goal

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B.A. Boley, Heat conduction in solMs and thermoelasticity

- namely that of providing a reasonably up-to-date bibliography and some guidance through it for the benefit of the reader - it is essential that some general information be given at the outset regarding the overall plan of presentation and the arrangement of the bibliographical references. This will now be done. This paper is divided into three main sections, of which the present introductory one is the first, the remaining two dealing respectively with heat conduction and with thermal-stress calculations. The introduction will present some general observations on the field, under the following four headings: 1.1. Thermal and mechanical analyses; 1.2. Heat conduction; 1.3. Thermal stresses; 1.4. Coupled thermomechanical problems. The references will be similarly grouped, and will include texts, surveys and other works of general applicability. The treatment of temperature determination will be treated under six articles in section 2, namely 2.1. General methods of analysis; 2.2. Numerical and experimental methods; 2.3. Special solutions and problems; 2.4. Composite materials; 2.5. Joint and contact resistance; 2.6. Melting problems. Lastly, section 3 will treat of thermal stresses under the following article headings: 3.1. General methods in thermo-elasticity; 3.2. Numerical and experimental methods; 3.3. Dynamic effects; 3.4. Beams, rods and simple structures; 3.5. Plates; 3.6. Cylindrical shells and disks; 3.7. Shells other than cylindrical; 3.8. Composites and laminates; 3.9. Stresses and deformation in melting bodies; 3.10. Inelastic effects. Bibliographical references will be listed, for each of the above subsections, alphabetically by firstauthor initial; thus, for example, the notation [3.3; B2] will refer to the publication on dynamic thermal effects (article 3.3) listed second under those pertaining to the initial B. Although at first sight of cumbersome appearance, it is believed this notation will prove to be a convenient one. References relevant to more than one article will be listed only once, but

some cross-listings will be provided. Extensive coverage of publications appearing earlier than 1966 has not been attempted, in order to maintain the desired emphasis on recent developments; the reader is referred to the works listed in section 1, and to the bibliographies they ~n turn contain, for information concerning earlier researches. It is only fair to add, however, that even the extensive list of works cited is by no means complete, and the references of the papers cited may have to examined by readers wishing to probe more deeply into particular topics. It is nevertheless hoped that a reasonable overview of the current state of the field has been provided, and that the comments given, however necessarily brief, will give a useful indication of which facets of the field have been more_extensively studies as well as those which are in need of further attention. 1.1. Thermal and mechanical analyses The analysis of structural elements at elevated temperature requires the determination of, first, the temperature distribution itself, and then, on the basis of that temperature distribution, of the subsequent mechanical response. Some works which treat both these aspects of the calculation are listed in the bibliography under this heading. Except as noted in subsection 1.4, the calculations of temperature and stresses may be performed separately as has been just mentioned. It is nevertheless at times useful to keep both problems simultaneously in mind, because of the resulting possibility of some more general conclusions. Among these we might mention: (a) the overall Saint-Venant principle predicting [1.1 ; B2] a rapid decay of thermal stresses in a rod under self-balancing and heat-inputs; (b) the special simplifications possible for slowly heated beams or plates [ 1.1 ; B1 ] ; (c) the facts that plane steady-state temperature distributions will cause no stresses in two-dimensional simply connected bodies, and in doubly connected bodies save for the first few Fourier-expansion terms of the temperature [ 1.1 ; B 1], [ 1.3; M 1] ; (d) the discussion (cf. subsection 3.4 and [3.4; B3]) on the accuracy of elementary beam theory for steady-state temperatures; (e) Goodier's method [ 1.1 ; B 1] for the construction of a particular solution of the thermo-elastic equations. Note that all but the first of the above examples refer to problems in which the termal properties are constant.

B.A. Boley, Heat conduction in solids and thermoelasticity

Ref. [ 1.1 ; Z 1] is particularly useful for its large number of detailed examples solved and for its emphasis on results of direct application in nuclear reactors.

379

previously calculated temperature, cf. subsection 1.1) is permissible whenever inertia effects (see subsection 3.3) are small and for values of the coupling parameter

1.2. Heat conduction

The standard text of Carslaw and Jaeger [1.2; C1] may be supplemented by Crank's book [ 1.2; C2] on the mathematically analogous problem of diffusion, particularly with regard to cases with temperature-dependent properties, and by more recent books such as [1.2; A1]. It may be remarked here that much effort has been spent on studies of the general theory of partial differential equations of the parabolic type, of which the Fourier heat-conduction equation is of course the most common example. These researches are as a rule of a more mathematical character than that usually required for engineering analyses; nevertheless it has been pointed out [1.2; B1] that many of the mathematical results are often of rather direct usefulness in actual applications. Examples of results of this type are those pertaining to the location of the maximum temperature in a body under transient or steady-state heating conditions, to the validity of a heat-conduction equivalent of Saint-Venant's principle [2.1; B2], and to the determination of bounds in many problems with and without change of phase. 1.3. Thermal stresses

In addition to the references listed under this heading, all those given under 1.1 should be consulted. Thermo-elastic analyses occupy a considerable portion of all the works indicated, but plastic and viscoelastic analyses are also treated. In the present paper emphasis will be again on thermo-elastic analyses, since the others once are treated elsewhere in this conference. Refences to important solutions in which the effects of either plastic or viscous behavior are included will nevertheless be discussed as they arise in subsections 3.1 to 3.9, while some additional works of general relevance to these topics are given under subsection 3.10.

~=

(3X + 2ta)2a2T0 (X + 2/~)0%

small compared to unity. Here X and/a are the Lam~ elastic constants, a is the coefficient of thermal expansion, O is the density, c o the specific heat at constant deformation, and To a reference absolute temperature. For most engineering applications the above conditions are met, and coupling can be disregarded. For the sake of completeness, however, a few coupled solutions are given under this heading. An examination of these will reveal that the phenomenon of coupling is indeed unimported except in problems in which the details of stress-wave propagation in the structure must be known. Nevertheless, it has been shown by Blot that a variational method of solution of the coupled thermo-elastic equation is useful even when consideration of coupling is not needed ([1.I; B1], [1.2; B1] for further references, [1.4;P1]). A further discussion of this and other analytical methods of approximate solution of heat-conduction problems will be found in subsection 2.2. A specific solution may be usefully given here to make more specific the role played by coupling, and its lack of importance in problems in which inertia effects are small. Consider in fact a one-dimensional problem for the half-space x > 0, governed by the field equations [1.1; B1] k T " - p c o J ' - (3X+ 2~)aT0ti' = 0 ,

(X + 2/a)u" - (3X + 2 ~ ) a T ' = 0 , when inertia terms are neglected. Here k is the thermal conductivity, t is time, T(x, t) is the temperature and u (x, t) the displacement. Primes and dots indicate differentiation with respect to x and t respectively. Integration of the second of the above equations gives

1.4. Coupled thermomechanical problems

It is well known [1.1 ; B1] that uncoupling of the temperature and stress problems (i.e. the calculation of the temperature independently if the stresses, and then the calculation of latter on the basis of the

(X+ 2~)u' - (3X+ 2 g ) a T = o x = f ( t ) , where the arbitrary function f ( t ) must vanish identically if the surface x = 0 is traction free. In this case

380

B.A. Boley, Heat conduction in solids and thermoelasticity

then the first field equation reduces to k T " = pco(1 + 8 ) T ,

which is identical in form to the ordinary heat-conduction equation. Thus the solution to any one problem is immediately derived from the corresponding uncoupled one by replacement of the diffusivity K = k / p c v by the quantity K/(1 +6). For example, if T= T O on the surface, the usual solution, namely r = TO erfc

[x/(2n/-AT)l

becomes, when coupling is introduced but inertia is neglected, T = T O erfc [xx,/1-+6/(2X/-k--t-)] . Clearly the difference between the two solutions is small when 8 << I.

2. Heat conduction 2.1. General m e t h o d s

The theory of heat conduction is of course well established, and various classical approaches are available for the solution of problems in this field. In general these are valid only when the properties are constant, while comparatively little analytical work is available for temperature dependent properties. Among the published research in that area, we might mention the variational method of [2.1 ; H I ] , which is valid for any temperature-dependence of the conductivity k, and is based (for either prescribed boundary temperature or insulated boundaries) on the variation of the functional

Il -

~ k L\Ox I + \3y ] + p c T ~-~

~3z ] ]

dV dt .

A perturbation analysis for the latter case, for quadratic temperature-dependence of the thermal properties, may be found in [2.1; O1], and the steady-state problem with internal heat sources is discussed in

[2.1 ; P1] by means of Kirchhoff's transformation of variables. Earlier solutions are treated in [ 1.2; C 1] and [1.2; C2], but usually solutions of this type are more readily obtained by numerical or approximate analytical techniques, as discussed later. Some attention has been given to the extension of the method of separation of variables to domains to which the method in its ordinary sense cannot be applied. Indeed, with any given differential equation is associated only a finite collection of domains for which the classical method of separation of variables is possible, the collection being exclusively a property of the differential operator. The number of such separable domains is generally small, with the result, for example, that for the two-dimensional heat equation the domain consisting of the geometrically simple shape of a trapezoid is not amenable to solution by this or, in fact, any other classical analytical method. The general question of separability for Laplace's and Hemholtz's equations for example was considered in [2.1; L1]. The conclusion reached is that complete separation of the Helmholtz equation is possible only for the well-known cases of degenerate ellipsoidal or paraboloidal coordinates - elliptical domains or rectangular parallelepipeds, spheres, circles, segments of circles, etc. Even when the domain is a member of the collection associated with the operator, the application of the method of separation of variables may be restricted because the solution may consist of nonelementary functions whose properties have not been studied or tabulated. The solution then becomes essentially numerical in nature, as for example in the analysis of [2.1 ; K1] for elliptic cylinders, although separability in this case was shown to exist in [2.1 ; M 1]. A more thorough treatment of this question may be found in [2.1 ; D1], where several three- and two-dimensional domains are studied in detail. Among general methods of solution must be mentioned that of [2.1; S1], which is applicable to two or three-dimensional problem, and usos, in the former, a complex variable technique and in both a series expansion in suitable functions. The solution for layers adjacent to the surface (or to a moving boundary) was treated in [2.1 ; L3]. Nonlinear boundary conditions were discussed by a semianalytical procedure in [2.1 ; C 1].

B.A. Boley, Heat conduction in solids and thermoelasticity Variational methods have been widely used for the approximate analytical solution of heat-conduction problems. Various approaches may be grouped under this category, such at Biot's work, the paper [2.1 ;HI] already cited, and [2.1;G1], [2.1 ;L1], [2.1 ;R1], [2.1 ;R2], [2.1 ;Z1], [2.2;H2]. The general problem of establishing variational formulations of in heat conduction was discussed in [2.1 ; F 1], with a rather pessimistic conclusion that they were not needed in practice for non-self-adjoint and nonlinear systems; some of the works cited earlier show however that nonlinearity is conveniently taken into account by procedures of this type. Furthermore, as will be pointed out in the next article, variational formulations are essential in the derivation of finite-element equations. Rel:,'ed to the variational methods in spirit (and often in theory as well) are Galerkin-type solutions (e.g. [2.1 ; J1]), whose convergence for some cases has been considered in [2.1 ;M1] and :[2.1 ; D1], and also the heat-balance method, in which satisfaction of the heat equation integrated over the volume of the body is deemed sufficiently accurate. Applications of the latter to plates with variable surface temperature, may be found in [2.6; T1] and [2.6; S1], and other applications appear for example in [2.6; C2]. When seeking an approximate analytical solution by any of the above methods the use of Duhamel's theorem [ 1.1 ; B 1] in connection with variable heating problems, although possible in theory, is probably more cumbersome than direct application of the approximate technique for the specific heating history in question [2.1; B1]. When any of the methods described in the preced, ing paragraphs is employed in the solution of surfaceheating problems, it is often convenient to use the concept of "penetration depth", i.e. a depth beyond which the surface heating is assumed to have no effect. When the penetration depth becomes large enough to reach a surface point removed from the point of heat application (i.e. at the "transit time") a new form for the solution must be adopted (cf. for example, [1.1; B1]). It has recently been pointed out [2.1; B1] that, in order to obtain solutions valid in the latter regime, the methods of superposition and of imaging may be employed, on the basis of the approximate solution found in the first phase of the solution. As an example of the above, let the approximate

381

solution for the half-space x > 0, initially (t = 0) at T = 0 and raised on x = 0 to T = T0, be

( To(1-x2/q 2) Tl(X, t) = {

forx<~q(t) ;

0

for x ~> q (t),

where q is the penetration depth (=x/(147/13)r t, x/-i-6-~, ~ for the Blot, Galerkin and heat-balance methods respectively). Then the solution for a slab of thickness L, under the same conditions as the above half-space and insulated at x =L, for example, is T 2(x, t) = T 1(x, t) oc

+ ~ ( - 1 ) n [Tl(2nL+x, t ) - T l ( 2 n L - x , t)] . n=l Among the papers bearing special emphasis on problems of internal heat generation, which are of particular interest in nuclear applications, we might mention [2.1;M3] (where the steady-state problem is treated by means of a change of variables reducing it to a problem with no internal generation), [2.1 ; M4] (where useful estimates of the maximum temperature for several bodies are given) and [2.1;P1] (where the temperature-dependence of properties is taken into account). 2.2. Numerical and experimental methods in heat

conduction 2.3. Special solutions Finite-difference methods are widely used in nuclear reactor temperature analyses, and indeed most of the numerical solutions are obtained in this manner. In general, both the space and time variables are considered at discrete nodal points. When solutions are obtained numerically by means of a differential analyzer, however, time is considered as a continuous variable, while all variables are continuous when an electrical analogue is employed. A discussion of these techniques may be found, with several examples, in [1.1;Z1] and [1.2;A1]. Several variations of the above mentioned basic techniques have been introduced. A numerical solution in the Lpplace-transform domain, coupled with

B.A. Boley, Heat conduction in solids and thermoelasticity

382

numerical inversion was given in [2.2; R2]. This paper employs the method of Green's functions in the transform domain; a numerical approach in which Green's functions are used in the real domain (with finite differences in space and time) may be found in [2.2; L2]. Another approach of this type, in which however the numerical solution of some integral equations of Volterra's type is required, rather than of partial differential equations, is presented in [2.2; T4]. The "isotherm migration" method, in which the variable T(x, t) is replaced by the isotherm position x(T, t), is adopted (together with numerical solution of the new differential equations) in [2.2; D1 ]. A point-matching technique for boundary and initial conditions was introduced in [2.2; L1 ]. Monte-Carlo techniques have been treated, for twodimensional problems, in [2.2; K1] and in [2.2; T2], where comparisons with earlier probability solutions [2.2;H4], [2.2;E1 ] of the heat conduction are given. An interesting variation of the finite-difference technique was introduced in [2.3; $2] under the name of "rational approach", in which the coefficients of a series expansion are determined by suitably adjusting the boundary conditions. Other finite-difference solutions are found in [2.2; B1] and [2.2; C1], an iterative procedure with temperature-dependent conductivity in [2.2; C2], and are improvement on the Crank-Nicholson procedure in [2.2; W2]. Finite-element solutions have been successfully employed in many problems (e.g. [2.2; Wl] and [3.2; O1]) and are in fact becoming increasingly more popular. The basic equations are derived from some variational principle, as has been mentioned. Several of these are available, of which one was specifially indicated in the preceding article. Another, used in fact in [2.2; Wl], is based on the following functional, whose Euler equations are indeed the heat-conduction equation and its associated boundary and initial conditions: 12 = 1 f ( p c T * T +

T i . k i / . T,j

V

- 2O*p'T- 2oeT~T)dV- fQini*TdS, S where TO is the initial temperature, the conductivity-

matrix elements are ki/, the density/9 and the specific heat c may depend on position but not on temperature, p is the internal heat generation and Q the applied surface-heat input. The convolution integral is denoted by t

u*v = f u(x, t-r)v(x, r ) d r . 0 Discontinuities of the meeting points of neighboring finite elements are permitted in the analysis of [2.2; T3]. Solutions for special plate configurations, which may be useful in these types of analyses were given for example in [2.3 ;R1 ] (for triangular plates) and in [2.3; L1] (for polygonal plates). 2.4. Composite materials 2.5. Contact resistance The references listed under this heading include fiber and particulate reinforced composites as well as laminates and multi-material members. The problems presented by such materials range from those concerned with the determination of macroscopic thermal properties ( [2.4;A 1], [2.4 ;B 1], [2.4;B2], [2.4;H2], [2.4;S1], [2.4;$2], [2.4;$3], [2.4;T2], [2.4;Z 11, [3.8 ;B1], [3.8;R2], and the review of Soviet research in this field [2.4;L1] to solutions of the heat-conduction problem proper [ 1.1 ;Z 1], [2.4;H 1], [2.4;O 1], [2.4;T 1], [2.4;Y 1], [3.5 ;S 1] ; the latter include solutions for clad cylinders and plates, and other configurations of importance in reactors. The analysis of composite members is often carried out on the basis of an anisotropic idealization; references to such analyses, particularly as referred to thermal-stress calculations, will be listed later. The thermal contact resistance either between elements of a multiphase composite, or at the joining of two structural elements, is an important problem which has received comparatively little recent attention. A few papers are listed here; for earlier references see for example [1.1; B1] and [1.2; B1]. 2.6. Melting problems One of the important critical design conditions in nuclear reactors concerns melt-down, as indicated for example in [1.1;Z1] ,[2.6;F2] ,[2.6;UI], or in the detailed calculation of melt-down conditions of

B.A. Boley, Heat conduction in solids and thermoelasticity

[2.6; T2]. Unfortunately, most of the available work in which change of phase is considered deals with problems of surface heating rather than of internal heat generation, as is needed in most nuclear reactor applications. The literature or the former is extensive (see for example the bibliographies of [ 1.2; B 1] and [2.6; M1]) and has bearing on the latter type of problem because of numerous similarities in the two, both in possible methods of solution and in the results themselves, as was pointed out in [2.6; B1]. For this reason several typical papers dealing with melting (or the mathematically analogous solidification) under surface heat application are collected under this headingl Works dealing with change-of-phase problems but emphasizing the ensuing stresses and deformations are discussed in subsection 3.8. The temperature distribution and the rate of melting can be obtained in closed form only in the classical Neumann solution [1.2; C1], although many extensions of that solution have been treated, such as [2.6; R1] (where thermal interaction with surroundings at surfaces other than the heated one is taken into account by perturbation and numerical techniques), [2.6; B3] (an iterative solution including the temperature dependence of properties), [2.6; T1] and [2.6; C2] (where the surface temperature is allowed by vary with time; heat-balance solutions), [2.6;P2] for the case in which the melted portion is in contact with a solid mold. In all these papers, as in Neumann's solution, the temperature at the surface is allowed to change suddenly from a value below the melting point to one above it, with the characteristic variation of melt-dpeth proportional to ( t - t m ) ~A,for any time t after the time t m of start of melting. If such a jump in temperature is not allowed, then the melt-depth will usually be proportional to ( t - tm)% for shor~ times after t m, but a general starting solution, in which all possible cases are catalogued, can be found for a plate in [2.6; B7]. The solution for short times can be found analytically by means of the "embedding technique", introduced in [2.6; B8] for the case of a plate, and subsequently employed in many other problems (such as [2.6; B6] and [2.6; Y1] for melting under a hot-spot, [2.6; B2] for cylinders and spheres, [2.6; G3] for a body with a spherical cavity, [2.6; Wl] for the problem of ablation, [2.6; L3] for arbitrary initial and surface conditions.

383

For longer times, a numerical solution has to be resorted to, either by the embedding technique again ([2.6;G1] for spheres and [2.6;L2] for cylinders) or by finite-difference techniques [2.6;K 1], [2.6;B4], [2.6 ;E 1], [2.6 ;B5 ] or by approximate analytical methods [2.6;L 1], [2.6;S 1], [2.6;C2], [2.6;C3]. A few studies have appeared in which melting occurs because of contact with a hot fluid [2.6;B5], [2.6;G2], [2.6;L4] and the results of some experimental researches have been published [2.6;H 1], [2.6;$2], [2.6;T3], [2.6;B9] ; A conformal mapping procedure for steady-state melting problems was introduced in [2.6; $3]. The general formulation of the interface conditions in three-dimensional problem was stated in a convenient way in [2.6; P1].

3. Thermal stresses and deformations 3.1. General thermo-elastic analyses

As in the case of heat conduction, the basic theory underlying thermo-elastic analyses is well understood, and methods of solution have been well established in a general way (although, as we shall see for example in subsection 3.4, some basic questions on the validity and accuracy of simplified theories exist). The practical application of these methods, however, with bodies of complex geometries or under complicated loading conditions, is quite often a matter of considerable difficulty. Grouped as references under the present article, therefore, are not only works which present new methods or approaches, over those, say, of [1.1;B1] ,[I.3;M1] ,[1.3;NI] or [3.1 ; P1 ], but rather papers which present solutions of complex problems of a general nature, and not therefore referred to very particular bodies such as beams, plates, shells and the like. A number of the papers listed [3.1 ;KI ],[3.1 ;L1 ], [3.1 ;R1 ] consider the important problem of crack behavior in a heated solid. The effect of inclusions of different material than the parent one is taken up in [3.1 ; H2], for example, and in other papers listed under subsection 3.3. The reduction of a thermoelastic problem to an isothermal one with equivalent mechanical loadings is discussed in [3.1 ; H3], with particular examples referred to cylindrical shapes under various heating and loading conditions. An iterative integral-equation formulation is introduced

384

B.A. Boley, Heat conduction in solids and thermoelasticity

in [3.1 ;H4], where an approximate analytical solution by means of Laguerre polynomials is employed for an axisymmetric problem. Some basic problems in which the material properties are temperature dependent are investigated in [3.4; T1] where, for example, the temperatures which cause zero stress in materials of this type are briefly described.

considered in [3.3;J 1 ] ,[3.3;H21 ,[3.3;S 1 ] ,[ 1.1 ;B 1], and shells in [3.3;M2] ,[3.3;H1 ] ,[3.3;K 1] ,[3.7;$3] ,[3.3;R1 ] (including visco-elasticity), [3.3; Z 1]. In general, the magnitude of this dynamic affect depends on the parameter

3.2. Numerical and experimental methods

where t T is the characteristic thermal time (i.e. that measuring the rapidity of temperature rise in the body) and t M is the mechanical characteristic time (usually the lowest natural period). In problems of surface heating one can usually take t T = h 2/K, where h is a characteristic length of the body and K the thermal diffusivity; in cases of internal nuclear heat generation, however, the temperature distribution can be established in the body much more quickly than in surface heating cases, and in fact is measured by the characteristic time of the actual heat input, independently of the material diffusion characteristics [3.3; L1 ]. This results in the possibility of a very small value of B, corresponding, as will now be seen, to a marked dynamic effect. As a measure of this effect we may take the ratio R of the maximum deflection including inertia terms to the maximum quasi-static deflection (i.e. without inertia effects). It was shown in [3.3; BI] that one can often write, in very good approximation

Analytical techniques cannot be relied upon for the solution of many problems of practical importance and as a consequence numerical techniques of various kinds have been extensively studied. The large majority of these employ the finite-element method, because of its superiority over the finite-difference approach used for example in [3.2; T3]. The entire literature of isothermal finite-element techniques is useful here, since the temperature is readily introduced in the fundamental variational principle from which the finite-element equations are derived; indeed this is done in [3.2; W1] and in such applications as [3.2; O1] and [3.2; F2]. Some direct thermal applications pertaining to non-linear problems are given in [3.2; L1], while a visco-elastic analysis is presented in [3.2;T1]. 3.3. Dynamic effects in thermo-elasticity The importance of including inertia effects in thermo-elastic problems was first recognized in [3.3; B2], and its study was then extended to many other practically important cases. The reasons for its significance in problem of nuclear internal heat generation are discussed below and were pointed out in [3.3; L1], although the importance of this effect is not restricted to that case alone. Two types of problems may be distinguished, as far as dynamic effects are concerned. The first, of greater importance with massive bodies, studies the propagation of elastic waves set up by the thermal disturbance; recent examples of this type of investigation are [3.3;M 1] ,[3.3;F2] ,[3.3;FI] ,[3.3;H3] and [3.3; R2]. Of more importance in connection with members such as beams, plates or shells, (or in other words those which possess one cross-sectional dimension much smaller than the others) are the gross thermally induced vibrations of the member in question. Studies pertaining to beams are found in [3.3 ;B2], [3.3 ;F4], [3.3 ;Y 1], [3.3 ;F3], while plates are

B = tT/t M ,

R=I+

-

1

so that ifB << 1 (that is if the thermal disturbance is rapidly applied) the effect is important, and decreases in magnitude as B increases. The preceding formula was obtained on the basis of an assumed onemode response of the structure, and will therefore be accurate whenever this assumption holds; justifications for it for most beam and plate problems were given in the last cited reference. The formula may be extended to cases in which viscous damping is present, in which case it becomes

1-~B exp ( - n T / X / 1 ,),2), for V ~< 1 ~ 1 +B 2 - 2B3' and B7 ~< 1 simultaneously;

R={ 1-~1

in all other cases,

B.A. Boley, Heat conduction in solids and thermoelasticity where 7 is the ratio of the damping coefficient to the critical value of that coefficient. As expected, damping thus decreases the dynamic effect. On the other hand, beam-column effects increase it, since they tend to increase the value of the characteristic mechanical time, and thus decrease B. Denoting by O/Oct the ratio of the applied stress to its buckling value, one then finds that, approximately, 1

R=I+

X/1 + (1 -alacr)B 2 so that large dynamic magnifications will result if o/acr approaches unity. An estimate of the dynamic stresses can also be obtained in a similar way. For a statically determinate structure, the resultant moment or force across any cross-section must be zero if dynamic effects are neglected, and hence the ratio of the dynamic to the quasi-static values of these quantities cannot be used. The ratio of dynamic to quasi-static strains, however, is still R; hence an estimate of the stresses may be obtained, once R is known, by introducing the strains into the stress-strain relation appropriate to the par, ticular problem. Thus, for example, for a beam we can write, in terms of the curvatures K, R = Kmax'dyn = EI(M+MT)max Kmax,st

EI (MT )max

'

or

stresses near the ends [3.4;C 1], [3.8 ;I-I2] with inelastic behavior or the effect of temperature-dependent properties [3.4;G 1], [3.4;T 1], [3.4;H 1], with the determination of influence functions for thermal loadings [3.4;K 1], and with specific analyses of deformations of fuel elements [3.4;N 1], [3.4;N2], [3.8 ;H4]. The analysis of a surface-reinforced laterally heated rod is presented in [3.4; I I ] , but most other works concerned with heterogeneous members are discussed under composite materials in subsection 3.8. A basic problem regarding the calculation of thermal stresses in beams concerns the use of the wellknown formula

o = - a E T + P T I A + MTYlI, where the thermal force and moment are respectively

A

3.4. Beams, rods and simple structures A great deal of work has been done in the last few years, (and, for that matter, before that as well) in the analysis of these basic structural members under conditions of thermal loadings. Nevertheless, two important types of investigations have been and continue to be carried out, namely that concerned with the practical solution of specific problems, and that concerned with basic theoretical questions. As examples of the former, as regards beams and rods, we may mention problems concerned with the

A

and the other symbols have the usual meanings. The assumptions underlying this formula are well-known (e.g. [1.1 ;B1] ); suffice it for present purposes to mention here that no loss in generality is incurred in its use. What is important, however, is that, from the standpoint of the three-dimensional theory of thermoelasticity, the correct formula for the axial stresses in a beam is

ox* =-aET*

Mma x = (MT)max(R- 1), where M is the bending moment and M T the thermal moment in the beam, defined more precisely in the following article.

385

+P~IA +M~ylI

where T* = T - (v/aE)V2¢ ;

A

A

with v Poisson's ratio, ~72 the two-dimensional Laplacian operator in the plane of a cross-section, and are stress function satisfying the plane-strain equation

V4¢ = -

ctE V2 T

1-v

386

B.A. Boley, Heat conduction in solids and thermoelasticity

under the traction-free boundary conditions ¢ = a¢/an = 0 with n the boundary normal. The importance of performing the calculations on the basis of T* rather than of Thas not been emphasized in the literature, although it is sometimes of considerable importance. The question has been studied in [3.4; B3], with the following principal conclusions (in addition to the obvious one that T* = T and thus o* = o if T is plane harmonic as in the steady state): (a) in all cases P~ = PT and M~ = M T, so that O* = 0 + p V 2 O .

(b) cases exist (e.g. a solid or hollow circular rod under axisymmetric temperature distributions) in which

Estimates of the maximum thermal stress and deflection which may arise in a given beam under arbitrary temperature distributions are useful in many cases, one of which was just cited in the preceding paragraph. Other uses involve, for example, the estimate of errors in the stress when an approximate temperature distribution is used. The answer to this question is important because of the existence of certain temperature distributions (namely, linear ones, denoted by TL) for which o ~-0, and which may therefore be added to any other temperature distribution without altering the stresses. Suppose in fact that, in a given problem, the temperature is known with a certain accuracy; i.e., the exact and approximate temperature distributions T E and TA satisfy the relation TA,ma x -- A T A < [ T E - TA[ < T A , m a x .

and that the error in the temperature is small, or

o* = o/(1-v) AT A < TA,ma x .

or

(o* - o ) / a = 1)/(1 - v )

.

In such cases clearly the omission of the correction term dependent on ~ loads to considerable error for usual values of Poisson's ratio. (c) for solid-section beams, the results quoted under (b) above often represent the maximum error that can be expected, i.e.

I o * - o Imax/I °max I ~< v/(1 - v ) . Examples pertaining to rectangular and elliptic cross-sections were given, as well as some consideration of arbitrary sections with monotomic rate of temperature change OT/Ot. (d) for thin-walled sections simpler results can be written; indeed in this case 4v [ o * - o 1 ~< 3(---~_v)a E A T , where AT is the maximum temperature difference across the thickness. The factor 4/3 arises from consideration of the bounds on thermal stresses which are discussed next.

It does not follow that the error in o is similarly small; in fact, if it happened that TE = T L + T 1 ;

TI<
then the error in T L is small, while the error in the only stress-producing part, namely T1, may be quite large, as was indeed shown in [1.2; B1] to be quite possible in a practical example. If it were known, however, that in any beam in which the temperature is bounded, i.e. TM - A~r-<
the stress could not exceed the value given by

lal <<-kaEAT , then the error in the stress could not exceed the value k a E A T A ; thus the accuracy of the stress calculation based on the approximate temperature could be estimated. A relation of the form of the last equation is indeed known, and was in fact derived in [3.4; BI], where it was shown that the coefficient k depends only on the cross-sectional shape, and where a simple

B.A. Boley,Heatconductionin solidsand thermoelasticity rule for its calculation was given. For example, it was shown there that k = 4/3 for a rectangular section, k = 2/3 + (X/3)fir for a thin-walled circular tube, k = l for any beam for which M T = 0, etc. The calculation of bounds on the deflections of heated beams was also given in that reference, while similar calculations for composite beams may be found in [3.8; B3] and [3.8; T4]. The calculation of bounds on the torsional rigidity of heated beams may be performed by similar techniques [3.4; B4]. References pertaining to some common types of structures are also collected under this heading; among these may be mentioned studies of concrete vessels [3.4; El, 2, 3] and shields [3.4; T2], the study of the joint between a plate and a cylindrical shell in [3.4; V2], and the general investigation of thermal stresses and deflections in indeterminate structures such as rings and frames of [3.4; B2]. The results obtained in the latter work are of some importance since they raise some basic questions concerning the thermo-elastic analysis of indeterminate structures. To fix ideas, however, only rings will be specifically discussed here. The analysis of thin rings under thermal loadings (and, for that matter, mechanical loadings as well) is usually carried out by means of the principle of stationary complementary energy (thermo-elastic Castigliano's theorem), and leads to results which are well known, cf. [1.1; B1]. The accuracy of such a calculation is shown in [3.4; B2] to differ in rings composed entirely of straight members or containing also curved segments, and to depend in general upon the two parameters

13=h/a and fll = h / r , where h is the thickness, a is an overall characteristic length, and r a representative radius of curvature. The following principal conclusions may be drawn: (a) The analysis of rings may be accurately performed on the basis of the conventional theory, i.e. with the complementary energy given by

s (M+MT)2 U'= f 2EI ds o in all cases, except the one indicated under (b) below. Stresses will be accurate to first-order

387

terms in an expansion in powers of/3 and/31, i.e. the error is of O03+/31). By this is meant that the terms proportional to/3 and/31 in the expansion will not be known, while those of lower order (e.g. independent of/3) will be calculated by the procedure outlined. If, in particular, however,

OT-------MT - 0 and S

S

f

TsinOds= f

0

0

Tcos0ds=0

then o - 0, with possible error of O(fl2+/31). Here 0 is the slope of the ring at any point, and s the distance along the median line. (b) If again oT = M T = 0, but at least one of the above integrals is not zero, then the conventional theory gives a = 0, but with an error of O(fl+/31). It is at times necessary to obtain the stress in this case; this can be done by introducing extensional terms in the complementary energy. The error in the stress is then of O(fl2+/31), but care must be exercised that curvature terms must be omitted even if curved segments are present. (c) In contrast to the isothermal theory, it is not possible in general to improve here the accuracy with regard to/31 with an elementary theory; hence corrections of 0(/3) are only meaningful if /31<
or

M+MT=0,

in which case extensional terms must be added to the complementary energy. The deflection calculations will lead to results with an error of O(3+/31); again, improvements can be obtained as before with regard to/3, but not with regard to31.

388

B.A. Boley, Heat conduction in solids and thermoelasticity

3.5. Plates

3.6. Cylinders and disks 3.7. Shells other than cylindrical Several papers dealing with thermal re~onses of the structures listed above are collected here, and are of particular importance in nuclear structural mechanics because of the very fundamental nature of the geometries involved. It is clearly impossible to discuss here in but the most cursory fashion their contents, and so only a few general remarks will be given. Of the papers which discuss approaches of some generality, we may mention [3.5 ;F 1], where the analogy (similar in spirit to the body-force analogy well-known in thermoelasticity) between three-dimensionally heated plates and the generalized plane-stress problem is discussed. A strain energy expression for plastic shells is given in [3.7; B2], which may also be employed in the analysis of buckling and post-buckling behavior since large deformations are included. Specific problems of buckling are treated for conical shells in [3.7 ;B1], [3.7 ;C 1], [3.7 ;L 1], for plates in [3.5 ;J 1] (post-buckling), and [3.5;D1] (with temperaturedependent modulus), [3.6; P1] (experimentally)and for other shell configurations in [3.7; H 1] and [3.7; S 1]. Papers in which the temperature dependence of the material properties are taken into account are [3.5; T 1] (where various comparisons with constantproperties solutions are given for various panforms and edge conditions), [3.6; V1] (where visco-elastic behavior is also included, and where time averagedvalues of Poisson's ratio are employed). The possibility of different modulus values in tension and compression is allowed in [3.6; A1] in a plane-stress analysis, while a general theory of this type of material was earlier given in [3.10; A1]. Problems in which internal heat generation is emphasized, rather than surface heating, may be found in [3.6;V2] ,[3.6;V3], [3.6 ;V4], [3.6;M2], [3.6;S 1], [3.6 ;T 1], [3.7 ;H2] while solutions pertaining to particular geometries and heating conditions arising in nuclear reactor applications are discussed in [3.5; M1] (where an axisymmetric multi-plate structure under steady-state conditions is considered), [3.5; Y1] (a plane-strain solution for arbitrary temperature variations), [3.6; R1] (a variational solution for cylindrical fuel elements), [3.7; T 1] (a semi-analytical solution). Reference

[3.6; H 1] treats cyclic loadings on cylindrical and spherical visco-elastic pressure vessels, while other examinations of the thermal response of various pressure vessels may be found in [3.6;L 1], [3.6;M 1] and [3.6;T3]. 3.8. Composites and laminates

Some treatments of thermal effects in heterogeneous and in anisotropic materials are included here, since they are often used as good idealizations of actual composites. Determinations of the effective properties of composites were presented in [3.8; B2] (for isotropic particulate composites), [3.8; C 1] (for laminates under conditions of rapid heating), [3.8; R2] and [3.8; S1] (for effective thermal-expansion coefficients). The latter work is based on energy considerations; a study of variational energy principles for thermo-elastic composite material is given in [3.8; R3]. The important end-problem of laminated strips is discussed in general terms in [3.8; H1] and in more detail for a two-strip model in [3.8; H2]. Sandwich plates are treated in [3.8 ;B 1], [3.4;I 1]. Thermal stresses and deformations in beam and rods are discussed in [3.8; B3] (including calculations of bounds on the stress of the type discussed in subsection 3.4), [3.8; T4] (generalizations of the preceding reference, and an analysis of the buckling of the reinforcing fibers) and [3.8; H4] (for cylindrical rods). Cylindrical shells are discussed in [3.6 ;B1], [3.6;F2], [3.6 ;L2], [3.6 ;R 1], [3.6 ;$4], [3.8 ;R 1] and plates in [3.8 ;$2 ]. Various papers discussing the effect of solid inclusions in larger bodies may be mentioned here, namely [3.8;T1] ,[3.8;T2] ,[3.1;D1] and [3.1;H2]. Inelastic analyses are carried out in [3.8; T3] and [3.8; Y1] : the former for plane strain in a rectangular rod with a rectangular core, presenting a numerical iterative technique based on a non-linear temperature-dependent (but with constant Poisson ratio) stress-strain law, the latter for a numerical solution for a hollow cylinder composed of several concentric shells of differing materials. 3.9. Thermal stresses and deformations in melting bodies

The determination of temperature distributions and melting rates in problems including changes of phase was discussed in subsection 2.6; the present remarks refer to papers dealing primarily with the

B.A. Boley, Heat conduction in solids and thermoelasticity

calculation of the consequent stresses and deformations. This is a difficult subject, primarily because of the complexity of determining accurately what the properties of the material are at temperatures comparable with the melting temperature. In one instance at least this is not too serious a difficulty, since it has been fortunately found that reasonable accuracy will result in displacement calculations pertaining to plates when inelastic effects are neglected and the analysis is carried out on purely elastic considerations. This was shown in [3.9 ;F 1] for surface heating and in [2.6; B1] for internal heat generation, and it was found that in the former the elastic analysis leads to conservative estimates for the deflections, while it leads to non-conservative estimates in the latter case. The calculation of the thermal stresses is however greatly affected by the particular type of material behavior considered, as can be seen for example by comparison of the elastoplastic plate analysis of [3.9; Wl] (where a yield stress linearly varying with temperature, and vanishing at the melting temperature, was assumed) or [3.9;M 1] and the elastic analysis of [3.9; T2], or by comparison of the elastoplastic (under the same assumptions are just stated) and elastic calculation in [3.9; F1]. A visco-elastic analysis has been given in [3.9; T3]. Reference to earlier work with either elastic, visco-elastic or plastic material behavior may be found in [1.2; B1]. 3.10. Inelastic effects

Papers dealing with analyses of various structural components including either viscous or plastic effects have been discussed in the preceding articles. There remains to list, as is done in the present group, some general references in which inelastic effects are treated paying particular attention to the elevated temperature regime.

[ 1.1; Z1]

389

Pergamon Press, New York (1964). Z. Zudans, T.C. Yen and W.H. Steigelmann, Thermal Stress Techniques in the Nuclear Industry, American Elsevier Publ. Co., Inc., New

York (1965). 1.2. Heat conduction

[ 17; A1] [ 1.2; B 1]

[1.2; C1] [ 1.2; C2]

V.S. Arpaci, Conduction Heat Transfer, Addison-Wesley, Reading, Mass. (1966). B.A. Boley, The Analysis of Problems of Heat Conduction and Melting, in: High Temperature Structures and Materials, Pergamon Press, New York (1964) pp. 260-315. H.S. Carslaw and J.C. Jeager, Conduction of Heat in Solids, 2nd Edition, Clarendon Press, Oxford, England (1959). J. Crank, The Mathematics of Diffusion, Clarendon Press, Oxford (1957).

1.3. Thermo-elasticity

[ 1.3; G 1] [ 1.3; M1] [ 1.3; N1] [ 1.3; P1]

B.E. Gatewood, Thermal Stresses, McGraw-Hill Co., New York (1957). E. Melan and H. Parkus, Wi~rmespannungenlnfolge Stationarer Temperaturfelder Springer (1970) pp. 467-474. W. Nowacki, Thermoelasticity, Addison-Wesley Publ., Reading, Mass. (1962). H. Parkus, Instationiire Witrmespannungen, Springer Verlag, Vienna (1959).

1.4. Coupled thermomechanical problems B.A. Boley and l.S. Tolins, Transient Coupled Thermoelastic Boundary-Value Problems in the Half Space, J. Appl. Mech. 29 (1962) pp. 637-

[1.4;BI]

[ 1.4; B2]

646. B.A. Boley and R. Hetnarski, Propagation of Discontinuities in Coupled Thermoelastic Problems, J. Appl. Mech. vol. 35 (1968) pp. 489-

[ 1.4; B3]

494. K.J. Baumeister and T.D. Hamill, Hyperbolic Heat-Conduction Equations - A Solution for the Semi-infinite Body Problem, J. Heat Trans.

[1.4;C1]

ASME 91, C (1969) 543-548. S.T. Chang, Thermomechanical CouplingProblem for an Axially Symmetric Region of a Viscoelastic Medium, J. Appl. Mech., vol. 37

I. General references 1.1. Thermal and mechanical analyses [ 1.1; B 1] B.A. Boley and J.H. Weiner, Theory o f Thermal Stresses, J. Wiley and Sons, New York (1960). [ 1.1; B2] B.A. Boley, Thermal Stresses, in: Structural Mechanics, edited by J.N. Goodier and N.J.

[ 1.1 ; F 1]

Hoff, Pergamon Press, New York (1960). A.M. Freudenthal, B.A. Boley and H. Liebowitz, High Temperature Structures and Materials,

[ 1.4; H1]

(1970) pp. 1195-1198. N.C. Huang, Dissipation Function in Thermomechanical Phenomena of Viscoelastic Solids,

[ 1.4; K1]

ZAMP, vol. 19 (1968) pp. 492-501. R.T. Knops and L.E. Payne, On Uniqueness and Continuous Dependence in Dynamical Problems of Linear Thermoelasticity, Intern.

[ 1.4; M1]

J. Solids and Struct., vol. 6 (1970) pp. 11731182. M.F. Mc Carthy, Constitutive Equations for Thermomeehanical Mechanical Materials with

390

[1.4;Nl1

[1.4;Oll

[ 1.4; 02]

[1.4;Pl1

B.A. Boley, Heat conduction in solids and thermoelasticity Memory, Intern. J. Engin. Science, vol. 8 (1970) pp. 467-474. R.E. Nickell and J.L. Sackman, Approximate Solutions in Linear, Coupled Thermoelasticity, J. Appl. Mech., vol. 35 (1968) pp. 255-266. J.T. Oden, Finite-Element Analysis of Nonlinear Problems in the Dynamical Theory of Coupled Thermoelasticity, Nucl. Eng. Design, vol. 10 (1969) pp. 465-475. J.T. Oden and G.A. Ramirez, Formulation of General Discrete Models of Thermomechanical Behavior of Materials with Memory, Intern. J. Solids and Struct., vol. 5 (1966) pp. 10771095. I.S. Podstrigach and P.R. Shevchuk, Variational Form of the Theory of Thermodiffision Process in a Deformable Solid, Applied Mathematics and Mechanics (Engl. transl, of: Prikl. Mat. i Mekh.) 33, 4 (1969) 753.

[2.1;Kl]

[2.1;L1]

[2.1;L2] [2.1;L3]

[2.1;M1]

[2.1;M2]

2. Heat conduction [2.1;M3] 2.1. General methods o f analysis [2.1;A1] D.E. AmosandP.J. Chen, TransientHeatConduction with Finite Element Wave Speeds, J. Appl. Mech., vol. 37 (1970) pp. 1145-1146. [2.1; B1] B.A. Boley, On Some Approximate Solution in Heat Conduction, Tech. Report No. 7, Dept, of Theoretical and Applied Mech., CorneU University, in preparation ( 1971). [2.1; B2] B.A. Boley, Upper Bounds and Saint-Venant's Principle in Transient Heat Conduction, Quarterly of Appl. Math., vol. 18 (1960) pp. 205207. [2.1;Cl1 A.L. Crosbie and R. Viskanta, A Simplified Method for Solving Transient Heat Conduction Problems with Non-linear Boundary Conditions, J. Heat Transfer, Trans. ASME, vol. 90, C (1968) pp..358-359. [2.1;D1] D. Dicker and M.B. Friedman, Non-Separable Solutions of the Transient Heat Conduction Equations, J. Appl. Mech., vol. 30 (1963) pp. 493-499. [2.1;Fl] B.A. Finlayson and L.E. Scriven, On the Search for Variational Principles, Intern. J. Heat and Mass Trans., vol. 10 (1967) pp. 799-821. [2.1;Gl] M.E. Gurtin, Variational Principles for L inear lnitial Value Problems, Quarterly of Appl. Math., vol. 22 (1964) W252. [2.1;Hl1 D.F. Hays and H.N. Curd, Heat Conduction in Solids: Temperature Dependent Thermal Conductivity, Intern. J. Heat and Stress Trans., voL 11 (1968) pp. 285-295. [2.1;J1] D.J. Johns, Some Heat Conduction Analyses using Kantorovich "sMethod, AIAA J., vol. 5 (1967) p. 191.

[2.t;M4]

[2.1;Oll

12.1;P1]

[2.1;P21

[2.1;R1]

[2.1;R2l

[2.1;R3]

[2.1;Sl1

E.T. Kirkpatrick and W.F. Stokey, Transient Heat Conduction in Elliptical Plates and Cylinders, J. Heat Transfer, Trans. ~:SME, vol. 8l, C (1959) p. 54. N. Levinson, B. Bogert and R.M. Redheffer. Separation of Laplace's Equation, Quarterly of Appl. Math., vol. 7, No. 3 (1949) p. 241. T.J. Lardner, Biot's Variational Principle in Heat Conduction, AIAA J., vol. 1 (1963) p. 196. J.S. Letcher Jr., An lmproved Theory for Heat Conduction in Thin Surface Layers, J. Heat Trans., Trans. ASME, vol. 91, C (1969) pp. 585-587. N.W. McLachlan, Heat Conduction in Elliptical Cylinder, and in Analogous Electromagnetic Problem, Philosophical Magazine, vol. 36 (1945) p. 600. S.G. Michlin, Some Sufficient Conditions for Convergence of Galerkin 'sMethod, trans, from Uzhjonyje Zapiski LGU, No. 135, by the Institute of Math. Sciences, New York University (1950). G.B. Melese-d'Hospital, J.E. Wilkins Jr., Heat Conduction in One-dimensional Geometries with Nonuniform Internal Heat Generation, Proc. Fourth Intern. Heat Trans. Conf., Paris vol. 1 (1970). M.J. Moran, A New Approach to Steady Heat Conduction in Regions with Generation, Proc. Fourth Intern. Conf., Paris, vol. 1 (1970). V. Olsson, Application of a Perturbation Method to Heat Flow Analysis in Materials Having Temperature-dependent Properties, vol. 8 (1970) pp. 1902-1903. J. Pfann, Determination of the Steady Temperature Distribution in Solids with Internal Heat Sources and Temperature-dependent Thermal Conductivity, Nucl. Eng. Design, vol. 4 (1966) pp. 121-128. W.P. Pilkey, Comments on Heat Conduction in a bounded, anisotropic Medium, AIAA J., vol. 7 (1969) p. 380. P. Rafalski and W. Zyszkowski, Lagrangian Approach to the Nonlinear Boundary Heattransfer Problem, AIAA J., vol. 6 (1968) p. 1606. P. Rafalski and W. Zyszkowski, On the Variational Principles for the Heat Conduction Problem, AIAA J., vol. 7 (1969) p. 606. H. Reissmann, Heat Conduction in a Bounded, Anisotropic Medium, AIAA J., vol. 6 (1968) p. 744. E.M. Sparrow and A. Haji-Sheikh, Transient and Steady Heat Conduction in Arbitrary Bodies with Arbitrary Boundary and lnitial Conditions, J. Heat Trans., Trans. ASME, vol. 90, C (1968) pp. 103-108.

B.A. Boley, Heat conduction in solids and thermoelasticity [2.1;Zl]

W. Zyszkowski, The Transient Temperature Distribution in One- DimensionaI Heat Conduction Problems with Non-linear Boundary Conditions, J. Heat Transf., Trans. ASME, vol. 91 (1969) pp. 77-82.

2.1. N u m e r i c a l and experimental m e t h o d s [ 2.2; A.1] T.S. Ashley, L. Carruth and H.A. Blum, The Thermal Conductivities of Some 400 Series Stainless Steels, J. Heat Transf., Trans. ASME vol. 92, C (1970) pp. 179-180. [2.2; B1] H.Z. Barakat and J.A. Clark, On the Solution of the Diffusion Equation by Numerical Methods, J. Heat Transf., Trans. ASME, vol. 88, C (1966) pp. 421-427. [2.2;C11 C.L. Chow, Explicit Heat Conduction Equations o f Thermally Insulated Surface, J. Heat Transf., Trans. ASME, vol. 91, C (1969) pp. 446-447. [2.2;C2] B.M. Cohen, Heat Transfer by Conduction and Radiation with Temperature-dependent Thermal Conductivity, J. Heat Transf., Trans. ASME, vol. 91, C (1969) pp. 159-160. [2.2;D1] R.C. Dix, J. Cizek, The Isotherm Migration Method for Transient Heat Conduction Analysis, Proc. Fourth Intern. Heat Trans. Conf., Paris, vol. 1 (1970). [2.2;El1 A.F. Emery and W.W. Carson, A Modification to the Monte Carlo Method - the Exodus Method, J. Heat Transl., Trans. ASME, vol. 90, C (1968) pp. 328-332. [2.2;H1] M.A. Heaslet and B. Baldwin, Close Analogy Between Radiative and Conductive Heat Flux in a Finite Slab, AIAA J., vol. 3 (1964) p. 2180. [2.2; H2] R.P. Heinisch and R. Viskanta, Transient Combined Conduction - Radiation in an Optically Thick Semi-infinite Medium, AIAA J., vol. 6 (1968) p. 1409. [2.2;H31 Z.J. Holy, A Synthesis Technique for Variable Surface Heat Transfer, Nucl. Eng. Design, vol. 8 (1968) pp. 289-298. [ 2.2; H4] A. Haji-Sheikh and E.M. Sparrow, The Solution of Heat Conduction Problems by Probability Methods, J. Heat Transf., Trans. ASME, vol. 89, C (1967) pp. 121-131. [2.2;Kl1 C.N. Klahr, A Monte Carlo Method for the Solution of Elliptic Partical Differential Equations, in:Mathematic Methods for Digital Computers, eds. A. Ralston and H. Wief, Wiley, New York (1960). [2.2;L1] C.F. Lo, Numerical Solutions o f the Unsteady Heat Equation, AIAA J., vol. 7 (1969) p. 973. [2.2;L21 J. Le Qu~r~, Application de la Th~orie des Fonctions de Green aux Mdthodes de Calcul Num&iques et Analogiques des Champs Thermiques, Proc. Fourth Intern. Heat Transfer

[2.2;R1]

[2.2; R2]

[2.2;T1]

[2.2;T21

[2.2;T3]

[2.2;T41

[2.2;W1]

[ 2.2; W2]

391

Conf., Paris, voL 1 (1970). W.J. Rivers, Transient Conduction Method of Evaluating Thermal Protection Materials, J. of Rockets and Spacecraft, vol. 5 (1968) p. 741. F.J. Rizzo and D.J. Shippy, A Method of Solution for Certain Problems o f Transient Heat Conduction, AIAA J., vol. 8 (1970) pp. 20052009. O.E. Tewfik, Measurements of Thermal Conductivity o f Porous A nisotropic Materials, AIAA J., vol. 1 (1963) p. 919. J.J. Thompson and P.Y.P. Chen, Heat Conduction with lnternal Sources by Modified Monte Carlo Methods, Nucl. Eng. Design, vol. 12 (1970) pp. 207-214. J.J. Thompson and P.Y.P. Chen, Discontinuous Finite Elements in Thermal Analyses, Nucl. Eng. Design, vol. 14 (1970) pp. 211-222. R.H. Thaler and W.K. Mueller, A New Computational Method for Transient Heat Conduction in Arbitrarily Shaped Regions, Proc. Fourth Intern. Heat Transfer ConL, Paris, vol. 1 (1970). E.L. Wilson and R.E. Nickell, Application of the b~'nite Element Method to Heat Conduction Analysis, Nucl. Eng. Design, vol. 4 (1966) pp. 276-286. M.E. Weber, Improving the Accuracy of Crank-Nicholson Numerical Solutions to the Heat Conduction Equation, J. Heat Transl., Trans. ASME, vol. 91, C (1969) pp. 189-191.

2.3. Special solutions and problems [ 2.3; D1] D. Dicker and M.B. Friedman, Heat Conduction in Elliptical Cylinders and Cylindrical Shells, AIAA J.,vol. 1 (1963) pp. 1139-1145. [ 2.3; E l i R.J. Eby and R.D. Karam, Solar Heating of a Rotating Cylinder with a Conduction Discontinuity, J. of Rockets and Spacecraft, vol. 7 (1970) pp. 1140-1142. [2.3; HI] T.R. Hsu, Thermal Shock on a Finite Disk due to an Instantaneous Point Heat Source, J. Appl. Mech., vol. 36 (1969) pp. 113-120. [2.3; L1] P.A. Laura and A.J. Faulstich Jr., Unsteady Heat Conduction in Plates o f Polygonal Shape, Intern. J. of Heat and Mass Transf., vol. 11 (1968) pp. 297-304. [2.3; M1] B.M. Ma, Transient Temperature Distributions in End Closures of Annular Fuel Elements, Nucl. Eng. Design, vol. 4 (1966) pp. 129-137. [ 2.3; M2] R.A. Matula, Transient Temperature Distribution in a Spherical Region Subjected to a Variable Surface Heat F/ux, J. Heat Transf., Trans. ASME, vol. 89, C (1967) pp. 278-279. [2.3; M3] N. Malmuth, M. Kascic and H.F. Mueller, Asymptotic add Numerical Solutions for Nonlinear Conduction in Radiating Heat Shields, J. Heat Transfer, Trans. ASME, vol. 92, C

392

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B.A. Boley, Heat conduction in solids and thermoelasticity (1970) pp. 264-268. N.Y. Olcer, Unsteady Temperature Distribution in a Sphere Subjected to lime-dependent Surface Heat Flux and Internal Heat Source, J. Heat Transf., Trans. ASME, vol. 91, C (1969) pp. 45-50. R.C. Pfahl Jr., Transient Radial Temperature Distribution in Cylindrical Shells, Intern. J. Heat and Mass Transf., vol. 12 (1969) pp. 17041706. W.P. Reid, Steady-State Temperature in a Triangle, J. Heat Transf., Trans. ASME, vol. 90, C (1968) pp. 365-367. M. Svoboda, Numerische Berechnung Eindimensionaler Instationiirer Temperatur-und Spannungsfelder, Nucl. Eng. Design, vol. 14 (1970) pp. 258-270. F.S. Shih, The Temperature Field Surrounding Coolant Holes in a Heat-Generating Solid, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). C.L. Tien, P.S. Jagannathan and B.F. Armarly, Analysis of Lateral Conduction and Radiation Along Two Parallel Long Plates, AIAA J., vol. 7 (1969) p. I806. J.H. van Sant, The Spot-Insulated Plate, Nucl. Eng. Design, vol. 8 (1968) 247- 250. E.M. Winkler, R.L. Humbrey, M.T. Madden and J.A. Koenig, Substructure Heating on Cracked Ablative Heat Shields, AIAA J., vol. 8 (t970) pp. 1895-1896. L. Wolf and K. Johannsen, Nichtlinearisierte Behandlung azimutaler StOrungen bei der Berechnung yon Brennelementtemperaturen, Nucl. Eng. Design, vol. 12 (1970) pp. 16-26. L. Wolf and K. Johannsen, An Analysis of the Two-dimensional Temperature Distribution in Eccentrically Mounted Cylinders with Special Application to Sodium-bonded Reactor Fuel Elements, Proc. Fourth Intern. Heat Transfer Conf., Paris (1970) vol. 1. W.R. Wells, Re-entry Heat Conduction of a Finite Slab with a Non-constant Thermal Conductivity, AIAA J., vol. 2 (1964) p.379.

[2.4; C1]

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[ 2.4; H 1]

[2.4; H2]

[2.4; L1]

[ 2.4; O 1]

[2.4;$1]

[2.4; $2]

[2.4;$3]

[2.4; T1]

[ 2.4; T2]

[2.4;Yll

[2.4; Zl] 2.4. Composite materials [2.4; A1] J.G. Androulakis and R.L. Kosson, Effective Thermal Conductivity Parallel to the Laminations, and Total Conductance of Multilayer Insulation, J. of Rockets and Spacecraft, vol. 6 (1969) p. 841. [2.4; Bt] E. Behrens, Thermal ConductivitiesofComposite Materials, J. Composite Mat., vol. 2 (1968) p. 2-17. [2.4; B2] M. Ben-Amoz, The Effective ThermalProperties of Two.Phase Solids, Intern. J. Eng. Sciences, vol. 8 (1970) pp. 39-48.

R.D. Casagrande and C.N. Shen, Nonuniform Temperature Distribution Analysis for Long Coaxial Cylinders, J. Heat Transf.,. Trans. ASME vol. 92, C (1970) pp. 564-548. G. D'Andrea and F.F. Ling, On Thermal Conductivity of Composites, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). H.J. Harris, Bondline Temperature of a Twolayer Slab Subjected to Aeroheating-Graphical Solution, J. of Rockets and Spacecraft, vol. 7 (1970) pp. 998-1001. Z. Hashin, Assessment of the Self Consistent Scheme Approximation: Conductivity of Particulate Composites, J. Comp. Mat., vol. 2 (1968) pp. 284-300. G.N. Lul'new and Y.P. Zarichnyak, A study of the Generalized Conductivity Coefficients in Heterogeneous Systems (Review) Heat TransferSoviet Research, vol. 2 (1970) pp. 89 107. N.Y. {31cer,A General Unsteady Heat Flow Problem in a finite Composite Hollow Circular Cylinder Under Boundary Conditions of the Second Kind, Nucl. Eng. Design, vol. 7 (1968) pp. 97-112. G.S. Springer and S.W. Tsai, Thermal Conductivities of Unidirectional Materials, J. of Composite Mater., vol. 1 (1967) pp. 166-173. S.V. Stepanov, Concerning the Thermal Conductivity of Two-phase Systems, Heat Transfer, Soviet Research, vol. 2 (1970) pp. 18-23. D.W. Sundstrom and S.Y. Chen, Thermal Conductivity of Rein]brced Plastics, J. Composite Mater., vol. 4 (1970) pp. 113-117. Y. Takeuti, Steady Temperature Distribution in a Heat-Generating Multi-bore Cylinder, Nucl. Eng. Design, vol. 11 (1970) pp. 41-56. J.D. Thornburg and C.D. Pears, Prediction of the Thermal Conductivity of Filled and Reinforced Plastics, J. Heat Transf., Trans. ASME, vol. 87, C (1965). M.M. Yovanovich, On the Temperature Distribution and Constriction Resistance in Layered Media, J. Composite Mater., vol. 4 (1970) p. 567. G.E. Zinsmeister and K.S. Purocht, Comments on Springer and Tsai's Method of Predicting Effective Thermal Conductivities of Unidirectional Materials, J. of Composite Mater., vol. 4 (1970) pp. 278-279.

2.5. Joint and contact resistance [2.5;B1] J.V. Beck, Determination of Optimum Transient Experiments for Thermal Contact Conductance, Int. J. Heat and Mass Trans., vol. 12 (1969) pp. 621-634. [ 2.5 ; F 1] N.R. Folkman and T.G. Lee, Thermodynamics Design Fundamentals of High Performace In-

B.A. Boley, Heat conduction in solids and thermoelasticity

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[2.5;M21

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[2.5;Vl1

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[2.5;V31

[2.5;V4]

[2.5;Wl1

[2.5;Y1]

sulation, J. of Rockets and Spacecraft, vol. 5 (1968) p. 954. D.R. Jeng, Thermal Contact Resistance in Vacuum, J. Heat Transl,, Trans. ASME, vol. 89, C (1967) pp. 275-276. V.A. Mal'kov, Thermal Contact Resistance of Machined Metal Surfaces in a Vacuum Environment, Heat Transfer, Soviet Research, vol. 2 (1970) pp. 25-33. B. Mikic and G. Carnasciali, The Effect of Thermal Conductivity of Plating Material on Thermal Contact Resistance, J. Heat Transf., Trans. ASME, vol. 92, C (1970) pp. 475-482. T.R. Thomas and S.D. Probert, Thermal Contact Resistance: the Directional Effect and Other Problems, Int. J. Heat and Mass Transf., vol. 13 (1970) pp. 789-807. E.D. Veilleux, Use of Thermal Greases to Conduct Heat Across Sheet-metal Interfaces, J. of Rockets and Spacecraft, vol. 5 (1968) p. 1238. E. Veilleux and M. Mark, Thermal Resistance of Bolted or Screwed Sheet Metal Joints in a Vacuum, J. of Rockets and Spacecraft, vol. 6 (1969) p. 339. T.N. Veziroglu and S. Chandra, Direction Effect in Thermal Contact Conductance, Proc. Fourth Int. Heat Transfer Conf., Paris, vol. 1 (1970). N. Vutz and S.W. Angrist, Thermal Contact Resistance of Anisotropic Materials, J. Heat Transf., Trans. ASME voi. 92, C (1970) pp. 1720. A. Williams and C.V. Madhusudana, Heat Flow Across Cylindrical Metallic Joints, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). M.M. Yovanovich and M. Tuarze, Experimental evidence of Thermal Resistance at Soldered Joints, J. Rockets and Space, vol. 6 (1969) p. 855.

2.6. Melting problems [ 2.6; 1311 B.A. Boley, Temperature and Deformations in Rods and Plates Melting under Internal Heat Generation, Proc. First Intern. Conf. on Struct. Mechanics in Reactor Tech., Berlin (1971) Sept. 20-24. Paper L2/3. [2.6; B2] B.A. Boley, J. Lederman and P.B. Gdmado, Radially Symmetric Melting of Cylinders and Spheres, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). [2.6; B3] B.A. Bole)', On a Melting Problem with Temperature-dependent Properties, W. Nowacki Anniversary Vol., Trends in Elasticity and Thermoeleasticity,Wolters-Noordhoff Publ., (1971). [2.6; B4] H.J. Breaux, A numericalMethodfora Stefan Problem, AIAA J.,vol. 6 (1968) p. 1821. [ 2.6; B5] J.A. Bilenas and L.M. Jiji, Numerical Solution of a Nonlinear Free Boundary Problem of

[ 2.6; B6]

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[2.6;El1

[2.6;Fl1

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393

Axisymmetric Fluid Flow in Tubes with surface Solidification, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). B.A. Boley and H.P. Yagoda, The Three.Dimensional Starting Solution for a Melting Slab, Proc. Royal Soc., vol. 323, No. A.1552 (1971). B.A. Boley, A General Starting Solution for Melting and Solidifying Slabs, Intern. J. Eng. Science, vol. 6 (1968) p. 89. B.A. Boley, A Method of Heat Conduction Analysis of Melting and Solidification Problems, J. Math. Phys., vol. 40 (1961) pp. 300-313. D.V. Boger and J.W. Westwater, Effect of Buoyancy on the Melting and Freezing Process, H. of Heat Transl., Trans. ASME, vol. 89, C (1967) pp. 81-89. C.J. Chen and S. Ostrach, Melting Ablation for Two-dimensional and axisymmetric Blunt Bodies with a Body Force~ Progress in Heat and Mass Transfer, vol. 2, Pergamon Press, pp. 195211. S.H. Cho and J.E. Sunderland, Heat Conduction Problems with Melting or Freezing, J. Heat Transf., Trans. ASME, vol. 91, C (1969) pp. 421-426. S.H. Cho and J.E. Sunderland, Phase Change of Spherical Bodies, Intern. J. Heat and Mass Transf., vol. 13 (t970)pp. 1231-1233. R. Eppes, A Finite-Difference Heat Conduction Method Applicable During Surface Recession, AIAA J., vol. 5 (1967) p. 1679. H.A. Friedman and B.L. McFarland, Twodimensional transient Ablation and Heat Conduction Analysis for Multimaterial Thrust Chamber Walls, J. of Rockets and Spacecraft, vol. 5 (1968) p. 753. M.H. Fontana, Core Melt-Through as a Consequence of Failure of Emergency Core Cooling, Nucl. Safety, vol. 9 (1968) pp. 14-24. P.B. Grimado and B.A. Boley, A numerical Solution for the Symmetric Melting of Spheres, Intern. J. Numerical Methods in Engr., vol. 2 (1970) pp. 175-188. O.M. Griffin and A.A. Szewczyk, An Analytical and Experimental Study of the Melting of Bulk Solids on an Inclined Plane Heated Surface, Proc. Fourth Intern. Heat Transf. Conf., Paris (1970) vol. 1. A.N. Guzelsu and A.S. Cakmak, Starting Solution for an Ablating Hollow Cylinder, Intern. J. Solids and Structures, vol. 6 (1970) pp. 1087-1102. E.K. Halteman, R.W. Gerrish and E.M. Krokosky, Unsteady-state Heat Conduction in a 1~vo Phase, Two Dimensional Solid Domain, Proc. Fourth Intern. Heat Transfer Conf., Pads, vol. 1 (1970). G.V. Isakhanov, V.V. Vengzhen and V.A.

394

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B.A. Boley, Heat conduction in solids and thermoelasticity Terletskii, Temperature Field in a Melting Hollow Cylinder, Strength of Mat., vol. 1 (1968) p. 296. J.C.Y. Koh and J.F. Price and R. Colony, On a Heat and Mass Transfer Problem with Two Moving Boundaries, Progress in Heat and Mass Transfer, Pergamon Press, vol. 2, pp. 225-247. T.J. Lardner, Approximate Solutions to Phase-change Problems, AIAA J., vol. 5 (1967) p. 2079. J.M. Lederman and B.A. Boley, Axisymmetric Melting or Solidification of Circular Cylinders, Intern. J. Heat and Mass Transf., vol. 13 (1970) p. 413. V.V. Lebedev and V.I. Antipov, Solution of Direct and lnverse Stefan Problem for Slab with Transient Boundary Conditions o f Second Kind, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). G.S.H. Lock, R.D.J. Freeborn and R.H. Nyren, Analysis of lce Formation in a Convectivelycooled Pipe, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). J.C. Muehlbauer and J.E. Sunderland, Heat Conduction with Freezing or Melting, Appl. Mech. Reviews, vol. 18 (1965) pp. 951-959. P.D. Patel, Interface Conditions in Heat-conduction Problems with Change of Phase, AIAA J., vol. 6 (1968) p. 2454. P.D. Patel and B.A. Boley, Solidification Problems with Space and glme Varying Boundary Conditions and Imperfect Mold Contact, Intern. J. Engr. Science, vol. 7 (1969) pp. 1041-1066. P.R. Pujado, F.J. Stermole and J.O. Golden, Melting of a finite Paraffin Slab as Applied to Phase-change Thermal Control, J. of Space and Rockets, vol. 6 (1969) p. 280. K.A. Rathjen and L.M. Jiji, Transient Heat Transfer in Fins Undergoing Phase Transformation, Proc. Fourth Intern. Heat Transf. Conf., Paris, vol. 1 (1970). S.R. Robertson and H. Schenck Jr., Correcting the London-Seban Equation for the Case of Molten Metal Solidification, Intern. J. Heat and Mass Transf., Trans. ASME, vol. 89, C (1967) pp. 118-119. O.P. Sharama, M. Rothenberg and S.S. Penner, Phase-change Problems with Variable Surface Temp., AIAA J., vol. 5 (1967) p. 677. J.M. Savino, J.F. Zumdieck and R. Siegel, Experimental Study of Freezing and Melting of Flowing Warm Water at a Stagnation Point on a Cold Plate, Proc. Fourth Int. Heat Transfer Conf., Paris, vol. 1 (1970) R. Siegel, M.E. Goldstein and J.M. Savino, Conformal Mapping Procedure for Transient and Steady-state Two-dimensional Solidification,

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Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). R.H. Tien, Freezing of Semi-infinite Slab with Time-dependent Surface Temperature An Ex. tension of Neumann's Solution, Trans. of Metall. Soc. of AIME, vol. 233 (1965) pp. 1887-1891. L.S. Tong, Core Cooling in a Hypothetical Loss of Coolant Accident; Estimate o f Heat Transfer in Core Meltdown, Nucl. Eng. Design, vol. 8 (1968) pp. 309-312. L.C. Tien and J.O. Wilkes, Axisymmetrical Normal Freezing with Convection Above, Proc. Fourth Intern. Heat Transfer Conf., Paris, vol. 1 (1970). U.S. Atomic Energy Commission, Research and Development in Reactor Safety, U.S. Government Printing Office, Washington, D.C. (1959). T.S. Wu and B.A. Boley, Bounds in Melting Problems with Arbitrary Rates of Liquid Removal, SIAM J., vol. 14, No. 2 (1966) pp. 306-323. H.P. Yagoda and B.A. Boley, Starting Solutions for Melting of a Slab under Plane or Axisymmetric Hot Spot~, Quart. J. Mech. and Appl. Math., vol. 23 (1970) pp. 225-246. S.W. Zelazny and L.A. Kennedy, Ablation of a Spherical Body due to a Nonuniform Surface Heat-flux Distribution, J. of Space. and Rockets, vol. 5 (1968) p. 874.

3. T h e r m o - e l a s t i c i t y

3.1. General m e t h o d s in thermoelasticity [ 3.1; D 1 ] B. Das, Stresses due to a Nucleus o f Thermo. elastic Strain in a Infinite Elastic Solid with Two Ridig Circular lnserts, AIAA J., vol. 7 (1969) p. 755. [ 3.1 ; G 1] P.B. Grimado, On the Thermal Stress Induced in a Semi.infinite Body as a Result of a Suddenly Applied Step Heat Input, J. Appl. Mech., vol. 37 (1970) pp. 1149-1150. [3.1;H1] Z.J. Holy, The lnversion Transformation for the Solution o f Static Thermoelastic Problems with Variable Heat Transfer, Nucl. Eng. Design, vol. 9 (1969) pp. 1-14. [3.1 ; H2] M.A. Hussain and S.L. Pu, Thermal Stresses near a Prolate Spheroidal Inclusion, J. Appl. Mech., vol. 37 (1970) pp. 4 0 3 - 4 0 8 . [ 3.1, H3] K.A. Holsapple, W.F. Schmidt and M.E. Foumey, Some Pressure-Temperature Equivalences in Elasticity, J. Appl. Mech., vol. 37 (1970) pp. 1158-1160. [ 3.1 ; H4] Z.J. Holy, Temperature and Thermoelastic Stress Fields in an Infinite Half-space due to Axisymmetric Surface Heat Transfer, Nucl.

B.A. Bole),, Heat conduction in solids and thermoelasticity

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[3.[:, R1]

Eng. Design, vol. 5 (1967) pp. 255-267 and pp. 367-374. Z.J. Holy, Temperature and Thermoelastic Stress Fields in a Heat Producing Sphere due to Axisymmetric Surface Heat Transfer, Nucl. Eng. Design, vol. 4 (1966) pp. 433-445. Z.J. Holy, An Efficient Iterative Method for the General Static Thermoelastic Sphere Problem, Nucl. Eng. Design, vol. 7 (1968) pp. 503516. M.K. Kassir and G.C. Sih, Thermal Stresses in a Solid ICeakeneed by an External Circular Crack, Intern. J. of Solids and Structures, vol. 5 (1969) pp. 351-368. R.A. Lucas, A quasi-static Thermoelastic Analysis of a Propagating Crack, Intern. J. of Solids and Struct., vol. 5 (1969) pp. 175-190. V.C. Mow and H.S. Cheng, Thermal Stresses in an Elastic Half-space Associated with an Arbitrary Distributed Moving Heat Source, ZAMP, vol. 19 (1968) pp. 500-507. H. Parkus, Methods of Solution of Thermoelastic Boundary Value Problems, contained in Ref. [1.1,F1] pp. 317-347. H.J. Petroski and D.E. Carlson, Some Exact Solutions to the Equations of Nonlinear Thermoelasticity, J. Appl. Mech., vol. 37 (1971) pp. 1151-1154. L. Rubenfeld, Non-axisymmetric, Thermo. elastic Stress Distribution in a Solid Containing an External Crack, Intern. J. of Eng. Science, vol. 8 (1970) pp. 499-510.

3.2. Numerical and experimental methods [3.2; B1] D.W. BrasweU, W.F. Ranson and W.F. Swinson, Scattered Light Photoelastic Thermal Stress Analysis of a Solid Propellant Rocket Motor, J. of Space. and Rockets, vol. 5 (1968) p. 1411. [3.2;F1] I. Fried, Finite-ElementAnalysisof~medependent Phenomena, AIAA Jol., vol. 7 (1969) p. 1170. [ 3.2; F2] T. Fujino and K. Ohsaka, The Heat Conduction and Thermal Stress Analysis by the FiniteElement Method, Proc. Second Conf. on Matrix Methods in Structural Mech., Report AFFDL-TR-68-150, Wright-Patterson AF Base, Ohio (1969). [3.2; L1] W. Lansing, I.W. Jones and P. Ratner, Nonlinear Analysis of Heated, Cambered ICings by the Matrix Force Method, AIAA J., vol. 1 (1963) p. 1619. [3.2; L2] J. Larrain and C.F. Bonilla, Cross-conduction Errors in Thermocouples. Correction of Long Swaged Thermocouples at High Temperatures, Nucl. Eng. Design, vol. 8 (1968) pp. 251-272. [3.2; O1] J.T. Oden and D.A. Kross, Analysis of General Coupled Thermoelasticity Problems by the

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Finite Element Method, Proc. Second Conf. on Matrix Methods in Structural Mech., Report AFFDL-TR-68-150, Wright-Patterson AF Base, Ohio (1969). R.L. Taylor and T.Y. Chang, An Approximate Method for Thermoviscoelastic Stress Analysis, Nucl. Eng. Design, vol. 4 (1966) pp. 21-28. J.J. Thompson and Z.J. Holy, The Analysis of Thermoelastic Transients due to Local Heat Transfer Coefficient Perturbations, Nucl. Eng. Design, vol. 12 (I970) pp. 297-312. J.J. Thompson, Z.J. Holy and P. Chen, Numerical Studies of Temperature and Thermoelastic Stress due to Local Heat Transfer Coefficient Transients, Nucl. Eng. Design, vol. 12 (1970) pp. 313-325. K. Washizu, Variational Methods in Elasticity and Plasticity, Pergamon Press (1968).

3.3. Dynamic effects [3.3; B1] B.A. Boley ,Approximate Analysis of Thermally lnduced Vibrations, J. Appl. Mech. (1971). [ 3.3; B2] B.A. Boley, Thermally lnduced Vibrations of Beams, J. Aeronautical Sciences, vol. 23 (1956) pp. 179-181. [ 3.3; FI] P.H. Francis and U.S. Lindholm, Effect of Temperature Gradients on the Propagation of Elastoplastic Waves, J. Appl. Mech., vol. 35 (1968) pp. 441-448. [3.3; F2] P.H. Francis, Elastoplastic IcavePropagation in a rate Sensitive Finite Rod Having a Thermal Gradient, J. Appl. Mech., vol. 37 (1970) pp. 315-323. [3.3;F3] H.P. Frisch, Thermallylnduced Vibrations of Long Thin-walled Cylinders of Open Section, J. Space. and Rockets, vol. 7 (1970) pp. 897904. [3.3; F4] P.H. Francis, The Response of a Thin Elastic Rod to Combined Thermal and Mechanical 1repulse, ZAMP, vol. 19 (1968) pp. 113-127. [ 3.3; H 1] E. Heel Thermal Excitations of Thin Elastic Shells, AIAA J., vol. 4 (1966) pp. 2234-2236. [ 3.3; H2] T.R. Hsu, Thermal Shock on a finite Disk due to an Instantaneous Point Heat Source, J. Appl. Mech., vol. 36 (1969) pp. 113-120. [3.3; H3] G.A. Hegemier and F. Tzung, Stress-wave Generation in a Temperature-dependent Absorbing Solid by Impulsive Electromagnetic Radiation, J. Appl. Mech., vol. 37 (1971) pp. 339344. [3.3;J1] B.M. Johnston, Uncoupled Dynamic Thermal Stresses in Infinite Flat Plates with Instantaneous and Transient Internal Heating, Nucl. Eng. Design, vol. 9 (1969) pp. 327-348. [3.3; K1] H. Kraus, Thermally Induced Vibrations of Thin Nonshallow Spherical Shells, AIAA J., vol. 4 (1966) pp. 500-505.

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B.A. Boley, Heat conduction in solids and thermoelasticity

W.C. Lyons, Comments on Heat lnduced Vibrations o f Elastic Beams, Plates, and Shells, AIAA J., vol. 4 (1966) pp. 1502-1503. L.W. Morland, Generation o f Thermoelastic Stress Waves by Impulsive Electromagnetic Radiation, AIAA J., vol., 6 (1968) p. t063. E.J. McQuillen and M.A. Brull, Dynamic Thermoelastic Response o f Cylindrical Shells, J. Appl. Mech., vol. 37 (1970) pp. 661-670. P. Rafalski, Dynamic Thermal Stresses in Reactor Shells, Nucl. Eng. Design,vol. 1 (1965) pp. 265-275. P.J. Rausch, The Effect o f Heating Time on Thermally lnduced Stress Waves, J. Appl. Mech., vol. 36 (1969) pp. 340-342. Y. Stavsky, Thermoelastic Vibrations o f Heterogeneous Membranes and Inextensional Plates, A1AA J., vol. 1 (1963)p. 722. Y.Y. Yu, Thermally induced Vibration and Flutter o f a Flexible Beam, J. Space. and Rockets, vol. 6 (1969) p. 902. Z. Zudans, Dynamic Response o f Shell Type Structures Subjected to Impulsive Mechanical and Thermal Loadings, Nucl. Eng. Design, vol. 3 (1966) pp. 117-137.

3.4. Beams, rods a n d s i m p l e s t r u c t u r e s [ 3.4; B 1] B.A. Boley, Bounds on the Maximum Thermoelastic Stress and Deflection in a Beam or Plate, J. Appl. Mech., vol. 33 (1966) pp. 881887. [ 3.4; B2] B.A. Boley, On Thermal Stresses and Deflections in Thin Rings, Intern. J. of Mech. Sci., vol. 11 (1969) pp. 781-789. [3.4; B3] B.A. Boley, On Thermal Stresses in Beams, Intern. J. of Solids and Structures (forthcoming) (197!). [ 3.4; B4] B.A. Boley, Bounds for the Torsional Rigidity o f Heated Beams, AIAA J., vol. 9 (1971) pp. 524-525. [3.4; B5] J.E. Brock,A Temperature Chart and Formulas Useful with the USASI B31.7 Code for Thermal Stress in Nuclear Power Piping, Nucl. Eng. Design, vol. 10 (1969) pp. 79-82. [ 3.4; C 1] W.H. Chu and F.T. Dodge, End Thermal Stresses in a Long Circular Rod, J. Appl. Mech., vol. 35 (1968) pp. 267-273. [3.4;C2] S.K.R. Choudri, Note on the Thermoelastic Stress and Displacement in a Thin Rod o f Finite Length due to a Point Source o f Heat Moving with a Constant Velocity along the Rod, J. Appl. Mech., vol. 38 (1971) pp. 277279. [ 3.4; E 1] G.L. England, Steady-state Stresses in Concrete Structures Subjected to Sustained Temperatures and Loads 1, Nucl. Eng. Design, vol. 3 (1966) pp. 54-65.

[ 3.4; E 2]

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G.L. England, Steady-state Stresses in Concrete Structures Subjected to Sustained Temperature and Loads, H, Nucl. Eng. Desig~a, vol. 3 (1966) pp. 246-255. G.L. England and M. Phok, Time-dependent Stresses in a Long Thick Cylindrical Prestressed Concrete Vessel:Subfected to Sustained Temperature Crossfall, Nucl. Eng. Design, vol. 9 (1969) pp. 488-495. F.A. Florio and A.T. Josloff, Thermo/structural Analysis o f a Large Flexible Paraboloid Antenna, J. Space. and Rockets, vol. 5 (1968) p. 1417. F.A. Florio and R.B. Hobbs Jr., A n analytical Representation o f Temperature Distributions in Gravity Gradient Rods, AIAA J., vol. 6 (1968) p. 99. B.E. Gatewood and R.W. Gehring, Deflections o f Inelastic Beams with Non-uniform Temperature Distribution, AIAA J., vol. 1 (1963) p. 217. R. Hibbeler and T. Mura, Viscous Creep Ratchetting o f Nuclear Reactor Fuel Elements, Nucl. Eng. Design, vol. 9 (1969) pp. 131-143. G.V. lsakhanov, Y.M. Rodichev and V.V. Vengzhen, Thermal Stresses in Straight Reinforced Plastic Bars Heated on One Side, Strength of Met., vol. 2 (1969) p. 174. J. Kowalewski, Influence Functions ]br Displacements and Stresses from Temperature and Heat Loads, AIAA J., vol. 5 (1967) p. 1694. L.A. Lys, Einige Probleme der Thermischen Berechnung des Kernes eines Dampfgekiihlten Druckrohr-Leistungsreaktors, Nucl. Eng. Design, vol. 5 (1967) pp. 123-133. J.B. Newman, Inelastic Analysis o f Bowing in Multispan Fuel Road Subjected to Axial Thrust, Nucl. Eng. Design, vol. 9 (1969) pp. 81 104. R. Nijsing, Temperature and Heat Flux Distribution in Nuclear Fuel Element Rods, Nucl. Eng. Design, vol. 4 (1966) pp. 1-20. J. Reynen, Temperature and Thermal Stress Distribution in Smooth and b~'nned Cannings Due to Axial Flux Variations, Nucl. Eng. Design vol. 9 (1969) pp. 144-154. L.D. Russell and A.J. Chapman, A naly tieal Solution for Transient Flow o f Energy in a One-dimensional Radiating Fin, A1AA J., vol. 6 (t968) p. 90. S. Tang, Some Problems in Thermoelasticity with Temperature-dependent Properties, J. of Space. Rockets, vol. 6 (1969) p. 217. D.R. Thomas, Temperature and Thermal Stress Distributions in Concrete Primary Shields for Nuclear Reactors, Nucl. Eng. Design, vol. 1 (1965) pp. 368-384. B. Vahidi amd N.C. Huang, Thermal Buckling o f Shallow Bimetallic Two-Hinged Arches, J.

B.A. Boley, Heat conduction in solids and thermoelasticity

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D.H. Van Campen, On the Stress Distribution in an Arbitrarily Loaded Nozzle-to flat Plate Connection, Nucl. Eng. Design, vol. 11 ( 1970) pp. 495 -516. P.E. Wilson, Large Thermal Deflection of a Cantilever Beam, AIAA J., vol. 1 (1963) Pt 1451.

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F.S. Brunschwig, Computing Temperature Perturbations on Thin-skin Panels, AIAA J., vol. 1 (1963) p. 2163. M.H. CoNe, Transient Thermal Stresses in Plates Having a Distributed Source and Arbitrary Tgme-dependent Surroundings, J. Appl. Mech., vol. 36 (1969) pp. 348-350. B.R. Dewey and G.A. Costello, Thermal Buckling Nonhomogeneous Plates, Nucl. Eng. Design vol. 7 (1968) pp. 249-261. M. Forray and M. Newman, Analogy Between Three-dimensionally Heated Plates and Generalized Plane Stress, AIAA J., vol. 2 (1964) p. 165. H.J. Harris, Thermal Resonse of a Rotating Isothermal Plate in Space, J. Space. and Rockets, vol. 5 (1968) p. 340. G.S. Johnston, Post-buckling Behaviour o f Heated Skin Panels, AIAA J., vol. 2 (1964) p. 141. A.H. Marchertas, Strength-deformation Considerations o f a Reactor Core Support Structure, Nucl. Eng. Design, voL 9 (1969) pp. 45-62. S. Tang, Thermal Stresses in Temperaturedependent Isotropic Plates, J. Space. and Rockets, vol. 5 (1968) p. 987. J.H. Van Sant, The Spot-insulated Plate, Nucl. Eng. Design, vol. 8 (1968) pp. 247-250. C.K. Youngdahl, Thermoelastic Stresses and Deformations in Reactor Fuel Plates, Nucl. Eng. Design, vol. 3 (1966) pp. 205-222.

3.6. Cylindrical shells and disks [ 3.6; A 1] K.M. Antonenko, Temperature Stresses in Thin Disks Model of Materials with differing Tensile and Compression Strength, J. of Strength of Mat., vol. 1 (1969)p. 183. [ 3.6; A2] T. Adman, L.H.N. Lee and R.E. Hoffman, Thermal Stresses in Open Cylindrical Shells with Circular Cutouts, Technical Report THEMISUND-68-2, University of Notredame (1968) [3.6; B 1] D.E. Boyd and B.R. Kishore, Thermal Stresses in Axially Loaded Orthotropic Cylinders, AIAA J., vol. 6 (1968) p. 980. [3.6;Fl1 M.J. Forrestal, W.E. Alzheimer and H.W. Schmitt, Thermally Induced Membrane Stress in a Circular Elastic Shell, AIAA J., vol. 6

[3.6;I2]

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(1968) pp. 946 and 2464. A.M. Freudenthal, Temperature and Shrinkage Stresses in Orthotropic Cylinder Bonded to a Rigid Case, AIAA J., vol. 6 (1968) p. 376. J. Hult, Cyclic Thermal Loading in Viscoelastic Pressure Vellels, Contained in ref. [ 3.10;B 1 ] pp. 102-119. M. Iqbal and B.D. Aggarwala, Solar Heating of a Long Circular Cylinder with Semigray Surface Properties, J. of Space. and Rockets, vol. 5 (1968) p. 1229. K.T.S.R. lyengar and K. Chandrashekhara, Thermal Stresses in a Finite Hollow Cylinder Due to an Axisymmetric Temperature Field at the End Surface, Nucl. Eng. Design, vol. 3 (1966) pp. 382-393. M. Kurashige and A. Atsumi, Thermal Stresses in a Circular Cylinder with a Ring of Holes, J. Appl. Mech., vol. 35 (1968) pp. 604-606. P. Launay, G. Charpenet and C. YouiUon, Prestressed Concrete Pressure Vessels: Twodimensional Thermoelasticity Computer Program, Nucl. Eng. Design, vol. 8 (1968) pp. 443-459. B. Longcope, M.J. Forrestal and W.E. Warren, Thermal Stress in a Transversely Isotropic, Hollow, Circular Cylinder, AIAA J., vol. 7 (1969) p. 2174. B.M. Ma, Thermal and Pressure Stresses in Cylindrical Reactor Pressure Vessels, Nucl. Eng. Design, vol. 11 (1970) pp. 416-426. J.R. Matthews, T,~ermal Stresses in a Finite Heat Generating Cylinder, Nucl. Eng. Design, vol. 12 (1970) pp. 291-296. W.H. Miller, Measurement of Solid Rocket Motor Thermally Induced Radial Bond Stresses, J. of Space. and Rockets, vol. 6 (1969) p. 1253. N.Y. Olcer and J.E. Sunderland, Unsteady Heat Conduction in Finite Hollow Circular Cylinders Under Time-dependent Boundary Conditions of the Second Kind, Nucl. Eng. Design, vol. 8 (1968) pp. 201-223. R. Papirno, Experimental Plastic Buckling of Aluninium Cylinders at 50OF, AIAA J., vol. 5 (1967) p. 2266. R.C. Petrof and S. Raynor, Thermal Stresses in a Rotating Cylinder Heated by Solar Radiation, AIAA J., vol. 7 (1969) p. 744. P. Rafalski and J. Szczurek, Transient Heat Conduction in Multiregion Systems with Non. linear Boundary Conditions with an Application to Nuclear Reactors, Nucl. Eng. Design, vol. 9 (1969) pp. 123-130. W.A. Shaw, G.T. Yahr and F.J. Witt, Irradiationinduced Stresses in Solid Semi-infinite and Short Graphite Cylinders Subjected to an Axi-

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B.A. Boley, Het conduction in solids and thermoelasticity symmetric Axial Flux Gradient, Nucl. Eng. Design, vol. 5 (1967) pp. 391-404. Y.U. Shemegan and V.A. Terletskii, Estimation of the Stress Level in Cylindrical Samples at Large Heating Rates, J. Strength of Mat., vol. 1 (1969) p. 180. S. Sikka, M. lqbal and B.D. Aggarwala, Temperature distribution and Curvature Produced in Long Solid Cylinders in Space, J. of Space. and Rockets, vol. 6 (1969) p. 911. Y. Stavsky and I. Smolash, Thermoelasticity of Heterogeneous Orthotropic Cylindrical Shells, Intern. J. Solids Struct., vol. 6 (1970) pp. 1211-1232. K.T. Sundra Raja lyengar and K. Chandrashekhara, Thermal Stresses in a Finite Solid Cylinder due to an Axisymmetric Temperature FieM at the End Surface, Nucl. Eng. Design, vol. 3 (1966) pp. 21-31. Y. Takeuti, Approximate Formulae for Temperature Distribution in a Heat-Generating Polygonal ~linder with Central Circular Hole, Nucl. Eng. Design, vol. 8 (1968) pp. 241-246. S. Tang, Thermal Stresses in Hollow Graphite Cylinders with Asymmetric Heating, J. of Space. and Rockets, vol. 5 (1968) p. 611. R.L. Taylor, Methods for Thermoviscoelastic Stress Analysis in Concrete Reactor Vessels, Nucl. Eng. Design, vol. 1 (1965) pp. 385-394. K.C. Valanis and G. Lianis, Thermal Stresses in a Viscoelastic Cylinder with Temperature Dependent Properties, AIAA J., vol. 2 (1964) p. 1642. R.A; Valentin, Steady-state Thermal Stresses in Circular Cylinders due to Abrubt Axial Variations in lnternal Heat-generation, Nucl. Eng. Design, vol. 7 (1968) pp. 59-72. R.A. Valentin and J.J. Carey, Thermal Stresses and Displacements in Finite, Heat-generating Circular Cylinders, Nucl. Eng. Design, vol. 12 (1970) pp. 277-290. R.A. Valentin and D.F. Schoeberle, Transient Thermal Stresses Associated with Sudden Initiation of Internal Heat Generation in an Axial Segment of Circular Cylinder, Nucl. Eng. Design, vol. 9 (1969) pp. 63-80. F.J. Witt, Thermal A nalysis of Cylindrical Shells, Nucl. Eng. Design, vol. 1 (1965) pp. 276-284.

3.7. Shells other than cylindrical [3.7; BI] C. Bendavid and J. Singer, Buckling of Conical Shells Heated Along a Generator, AIAA J., vol. 5 (1967) p. 1710. [3.7; B2] A.P. Boresi and I.C. Wang, Strain Energy Expression for Large Deformations of lsotropic Elastic Shells Subjected to Arbitrary Tempera-

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ture Distribution, Nucl. Eng. Design, vol. 5 (1967) pp. 443-464. L.K. Chang and S.Y. Lu, Non-linear Thermal Elastic Buckling of Conical Shells, Nucl. Eng. Design, vol. 7 (1968)pp. 159-169. F.A. Cozzarelli and R.P. Vito, Creep of Conical Membrane Shells Under Aerodynamic Heating and Loading, AIAA J., vol. 7 (1969) p. 412. H.M. Haydl, Elastic Buckling of Heated Doubly Curved Thin Shells, Nucl. Eng. Design, vol. 7 (1968) pp. 141-151. S.Y. Lu and L.K. Chang, Thermal Buckling of Conical Shells, AIAA J., vol. 5 (1967) p. 1877. A.P. Sinitsyn, Vibrations and Stability of Shells due to Loads and Temperature, Intern. J. Solids and Struct., vol. 5 (1969) pp. 1333-1337. G.J. Sova and N.D. Malmuth, Asymptotic Solutions for Heat Conduction in Radiating Shells Subject to Discontinuous Solar Flux, AIAA J., vol. 7 (1969) p. 1631. C.L. Sun and S.Y. Lu, Non-linear Dynamic Behaviour of Heated Conical and Cylindrical Shells, Nucl. Eng. Design, vol. 7 (1968) pp. 113-122. J.J. Thompson and Z.J, Holy, Axisymmetric Thermal Response Problems for a Spherical Fuel Element with Time Dependent Heat Transfer Coefficients, Nucl. Eng. Design, vol. 9 (1969) pp. 39-44. W.E. Warren, A Transient Axisymmetric Thermoelastic Problem for the Hollow Sphere, AIAA J., vol. 1 (1963) p. 2569. H.E. Williams, A "membrane" solution for axisymmetric Heating of dome-shaped shells of Revolution, AIAA J., vol. 2 (1964) p. 1491. H.E. Williams, Axisymmetric Thermal Stresses in Sandwich Shells of Revolution, AIAA J., vol. 5 (1967) p. 981. F.J. Witt, Thermal Stress Analysis of Conical Shells, Nucl. Eng. Design, vol. 1 (1965) pp. 449-456. J.C. Yao, Thermoelastic Differential Equations for Shells of Arbitrary Shape, AIAA J., vol. 1 (1963) p. 479.

3.8. Composites and laminates [3.8; B1] E.R. Bruun, Thermal Deflection of a Circular Sandwich Plate, AIAA J., vol. 1 (1963) p. 1213. [ 3.8; B2] B. Budianski, Thermal and Thermoelastic Properties of lsotropic Composites, J. of Comp. Mat., vol. 4 (1970) p. 286. [ 3.8; B3] B.A. Boley and R.B. Testa, Thermal Stresses in Composite Beams, Intern. J. Solids and Struct., vol. 5 (1960) pp. 1153-1169. [ 3.8; C1] A. Ching and W.E. Welsch Jr., Strength and Stress-strain Properties of Rapidly Heated Laminated Ablative Materials, AIAA J., vol. 6

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(1968) p. 2312. M.S. Hess, The End Problem for a Laminates Elastic Strip L The GeneralSolution, J. of Comp. Mat., vol. 3 (1969) pp. 262-280. M.S. Hess, The End Problem for a Laminated Elastic Strip - H Differential Expansion Stresses, J. of Composite Mat., vol. 3 (1969) pp. 630-641. Edited by L. Holliday, Composite Materials, Elsevier Publ. Comp. (1966). B.L. Hunter, Y.Y. Yu, Thermoelastic Bowing of Composite Cylindrical Fuel Rods, Nucl. Eng. Design, vol. 37 (1968) pp. 38-43. D.R. Otis, Thermal Damping in Gas-filled Composite Materials During lmpact Loading, J. Appl. Mech.,vol. 7 (1968) pp. 487-502. B.W. Rosen, Thermal Stresses in Nonhomogeneous Thin Shells, AIAA J., vol. 1 (1963) p. 1700. B.W. Rosen and Z. Hashin, Effective Thermal Expansion Coefficients and Specific Heats of Composite Materials, Intern. J. Eng. Science, vol. 8 (1970) pp. 157-174. B.W. Rosen, Thermoelastic Energy Functions and Minimum Energy Principlesfor Composite Materials, Intern. J. Eng. Science, vol. 8 (1970) pp. 5 - i 8 . D. Rubin, Mechanical and Thermodynamic Considerations ofan Assemblage of Homogenous Elastic-Plastic States, J. Appl. Mech., vol. 35 (1968) pp. 596-616. R.A. Schapery, Thermal Expansion Coefficients of Composite Materials Based on Energy Principles, J. of Composite Mat., vol. 2 (1968) pp. 380-404. Y. Stavsky, Cross-thermoelastic Phenomenon in Heterogeneous A elotropic Plates, AIAA J., vol. 1 (1963) p. 960. T.R. Tauchert, Thermal Stresses at Spherical Inclusions in Uniform Heat Flow, J. of Composite Mat., vol. 2 (1968) pp. 478-486. T.R. Tauchert, Thermal Stress Concentration in the Vicinity of Cylindrical lnclusions, J. of Comp. Mat., vol. 3 (1969) pp. 192-195. I.S. Tuba and D.P. Wei, Thermo-elastic Plastic Stress Distribution in Composite Media with Nonuniform Temperature Distribution and Temperature Dependent Material Properties, vol. 5 (1967) pp. 43-51. R.B. Testa and B.A. Boley, Basic Thermoelastic Problems in Fiber-Reinforced Materials, Proc. Intern. Conf., on the Mech. of Composite Mat.,

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Philadelphia, Pa., Pergamon Press, New York (1970) pp. 361-385. J.P. Yaleh and J .E. McCo nnelee, Plane Strain Creep and Plastic Deformation Analysis of a Composite Tube, Nucl. Eng. Desigla, vol. 5 (1967) pp. 52-62.

3.9. Stresses and deformations in melting bodies [3.9; F1] E. Friedman and B.A. Boley, Stresses and Deformations in Melting Plates, J. Space, and Rockets, vol. 7 (1970) pp. 324-333. [3.9; M1] Y. Masuko and S. Matsunaga, On the Temperature Distribution in Solidifying Ingots and the Residual Stresses, Sumitomo Kinzoku J., vol. 18 (1966)p. 4151. [ 3.9: T1] I.G. Tadjbakhsh, Thermal Stressis in an Elastic Half-space with a Moving Boundary, AIAA J., vol. 1 (1963) p. 214. [ 3.9; T2] R.H. Tien and V. Koump, Thermal Stresses During Soh'dification on the Basis of an Elastic Model, J. Appl. Mech., vol. 36 (1969) pp. 763767. [3.9; T3] E.C. Ting, Stress Analysis for a Nonlinear Viscoelastic Cylinder with Ablating lnner Surface, J. Appl. Mech., vol. 37 (1970) pp. 44-47. [3.9; Wl] J.H. Weiner and B.A. Boley, Elasto-Plastic Thermal Stresses in a Solidifying Body, J. of Mech. and Phys. of Solids, vol. 11 (1963) pp. 145-154. 3.10. Inelastic effects [3.10; All S.A. Ambartsumian, Equations of the Theory of Thermal Stress in Double-Modulus Materials, contained in ref. [3.10; B1] pp. 17-32. [3.10; B1] B.A. Boley, Thermoinelasticity, Proc. IUTAM Symp., East Kilbride, Scotland 1968, SpringerVerlag, Vienna (1970). [3.10; F1] A.M. Freudenthal, Aspects of Reliability Under Conditions of Elevated Temperature Creep and Fatigue, contained in ref. [ 1.1; F1] pp. 399412. [3.10;K1] L.M. Kachanov, Analytical Methods of Creep Design, Especially Within the Non-linear Range, contained in ref. [1.1;F1] pp. 383-398. [ 3.10; O 1] W. Olszak and P. Perzyna, On Thermal Effects in Viscoplasticity, ZAMP, vol. 20 (1969) pp. 676-680. [3.1] ;S1] E. Sternberg, On the Analysis of Thermal Stresses in Viscoelastic Solids, contained in ref., [1.1;F1] pp. 348-382. [3.10;TI] T.H. Lin, Theory of lnelastic Structures, J. Wiley and Sons, New York (1968).