Plane Thermoelasticity of Inhomogeneous Solids
Synonyms
Definitions
 Plane problem

That addresses a problem formulated under either the planestress or planestrain hypothesis, which allows for representing a general threedimensional formulation in two inplane dimensions within a plane section of a considered elastic solid.
 Elasticity, thermoelasticity

That addresses the geometrically and physically linear theories of elasticity and thermoelasticity.
 Material inhomogeneity

That addresses the variation of material properties (elastic and thermophysical moduli) within a macrovolume of an elastic solid.
Introduction
Plane problems of the elasticity and thermoelasticity theories attract widespread attention for over a hundred years. Being derived from general threedimensional problems by making use of either the planestrain hypothesis for long prismatic (cylindrical) solids or the (generalized) planestress hypothesis for thin plates with plane profile, these problems appeared to be more straightforward in their formulation and, consequently, analysis (Timoshenko and Goodier, 1951). On the other hand, they have numerous practical applications and also provide fertile ground for the development of new methods and techniques of the continua mechanics. A significant contribution to the general methodology of the mathematical physics has been made, e.g., with an idea of reducing specific plane problems to harmonic or biharmonic equations (Meleshko, 2003) by making use of potential functions (Airy, 1863; Maxwell, 1862) with further implementation of the complex variable method (Goursat, 1898; Kolosov, 1935; Muskhelishvili, 1977). This idea has been then efficiently adopted in many branches of solid and fluid mechanics. More detail can be found in exhaustive reviews of the pioneering research on plane (twodimensional) problems (Michlin, 1935; Meleshko, 2003).
It is worth noting that the classical theory of elasticity implies a deformable elastic solid to be acted upon by external force loadings and/or volumetric body forces. Within the framework of the thermoelasticity theory, an elastic body is subjected to thermal impact induced by external heating, internal heat sources, and, when accounting for the effect of mechanicalthermal coupling, the heat generation due to mechanical work. Thus, when neglecting the coupling terms and using linear theory for an elastic solid subjected to both thermal and mechanical impacts, the superposition principle can be used so that the elasticity and thermoelasticity problems are treated individually, and, after obtaining their solutions, they are to be superposed in order to obtain a solution to the original problem. Moreover, implication of specific potential functions (Kovalenko, 1969; Hetnarski and Eslami, 2009) allows for conversion, nominally, of thermoelasticity problems into elasticity ones and vice versa.
This essay addresses the unified formulation of plane static elasticity and thermoelasticity problems for inhomogeneous materials (the ones exhibiting variation of their properties within a macrovolume). The governing equations for these problems contain variable coefficients expressed through the derivatives of the material moduli by the inplane coordinates. This implies certain degree of smoothness for the functions representing the dependences of material moduli on the coordinates. The dominant methods and trends in analysis of the plane problems for inhomogeneous solids are outlined.
Formulation and Basic Assumptions
General ThreeDimensional Formulation of Elasticity and Thermoelasticity Problems
Three equations of equilibrium (1), six constitutive equations (2), and six either straincompatibility (4) or Cauchy equations (3) present a complete system of 15 equations for determination of 15 unknown functions: six stresstensor components σ_{ξ}, σ_{ξη}, six straintensor components ε_{ξ}, ε_{ξη}, and three displacements u, v, w. Here, ξ, η = x, y, z, ξ ≠ η. In order to reduce the number of questedfor functions, the formulation of elasticity or thermoelasticity boundaryvalue problems can be given in terms of either stresses or displacements, which is usually motivated by the boundary conditions (5) or (6), respectively. In the case when the boundary \(\mathscr {S}^\ast \) is exposed to external force loadings P, it is convenient to formulate all the governing equations in terms of stresses by substituting the constitutive equations (2) into the compatibility ones (4) which allows for representing the latter equations in terms of stresses, which are known as the BeltramiMichell equations (Teodorescu, 2013). If the components of the displacement vector u^{∗} are imposed on the boundary, then substituting (2) and (3) into (1) yields the Lamé equations for determination of vector u.
Although the formulation of a boundaryvalue problem in terms of either six stresses or three displacements appears to be simple in comparison to the general formulation (1)–(4), and (5) or (6), it still presents a challenge for both analytical and numerical modes of attack. In some cases, the problems can be simplified when the considered solids and loadings exhibit certain types of symmetry with respect to some of the spatial coordinates or the loadings show no variation in some spatial directions. One of such simplifications can be made when implementing the planestrain or planestress hypotheses, under which the elasticity and thermoelasticity problems can be reduced to a plane (twodimensional) formulation with lower number of independent equations and questedfor functions of two spatial variables only.
Plane Strain
In such manner, the problem on determination of stresses in a solid \(\mathscr {S}\) is reduced to determination of three stresstensor components σ_{x}, σ_{y}, σ_{xy}; three straintensor components ε_{x}, ε_{y}, ε_{xy}; and two displacements u and v as functions of x and y from a set of eight Eqs. (7), (9), and (10) under the conditions (5) or (6) imposed on the boundary \(\mathscr {D}^\ast \) of domain \(\mathscr {D}\).
Substitution of the constitutive equations (9) into the compatibility one (11) allows for deriving a compatibility equation in terms of stresses, which along with two equations of equilibrium (7) present a closed system of three equations for determination of three stresstensor components. After the inplane stresses are computed, the outofplane stress σ_{z} can be found by formula (8).
Substitution of the Cauchy equations (10) into the constitutive ones (9) allows for representing the equilibrium equations (7) in terms of displacements, which present a closed system of two governing equations for determination of u and v.
Plane Stress
Similar to the planestrain case, the generalized planestress hypothesis allows for reduction of a problem to three equations for inplane stresses σ_{x}(x, y), σ_{y}(x, y), and σ_{xy}(x, y) or two equations for displacements u(x, y) and v(x, y) with the outofplane strain being expressed through the inplane normal stresses by formula (14).
Unified Formulation of Plane Problems in the Cartesian and Polar Coordinate Systems
Due to the fact that the material moduli for the formulated plane problems are involved only into the constitutive equations (15) or (20), further implementation of the latter ones in order to derive a compatibility equation in terms of stresses (on the basis of (15) or (19)) or equilibrium equations in terms of displacements (on the basis of (7) or (18)) is to be performed with accounting for possible variation of the material properties (16) within the inplane coordinates x and y or r and θ, which is in the case of material inhomogeneity.
Material Inhomogeneity
The classical elasticity theory assumes elastic solids to be homogeneous (mechanical properties are the same at any point of an arbitrary macrovolume and resemble the properties of the entire solid). However, comprehensive analysis of the mechanical performance of reallife materials requires more accurate hypotheses to be introduced for covering the effect of structural imperfections or defects, whose consolidation within a macrovolume causes significant disturbances of the mechanical fields and variation of the material properties within the macrovolume. This effect is known as macroinhomogeneity or inhomogeneity (also, nonhomogeneity, heterogeneity Maugin 1993). There is no stable terminology with regard to the model of inhomogeneous material (Kupradze, 1979; Podstrigach et al., 1984; Muskhelishvili, 1977; Lekhnitskii, 1968); therefore, in the relevant literature, the term is often used along with additional explanation of what kind of material inhomogeneity is actually addressed in the study (Olszak, 1959). There are also different meanings of the term involving the microinhomogeneity, stochastic inhomogeneity, etc. (Kolchin, 1971). Herein, the macroscopic continuous inhomogeneity is addressed, which allows for representation of the material properties by smooth (in some cases, nonsmooth or discontinuous) functions of the spatial coordinates (Olszak et al., 1962).
The material inhomogeneity, as a vital issue of the mechanical performance, can be induced by impacts of environmental (mechanical, physical, chemical influences, etc.), technological (specific conditions of production, exploitation, etc.), or intentional (achievement of certain types of inhomogeneity by means of special material design, such as elastic composites, functionally graded and reinforced materials, etc.) origin. In many cases, when the material inhomogeneity is caused by the impacts of certain nature, it may receive a unidirectional character, which makes the material properties to vary, predominantly, in one of the spatial directions. This type of inhomogeneity is known as the inductive inhomogeneity (Kolchin, 1971) and is presented by two basic types: multilayer or piecewisehomogeneous solids (the material properties are rather constant within some layers or segments of the body so that they can be modeled by piecewiseconstant functions) and continuously inhomogeneous bodies (the variation profiles of the material properties are continuous but not necessarily smooth). Inhomogeneous solids can often exhibit an anisotropic behavior (Olszak, 1959; Hashin, 1964; Lekhnitskii, 1968).
A specific kind of inhomogeneity is concerned with a response of certain materials to nonuniform temperature distributions in the way that the material properties in a macrovolume can vary with the temperature change (Hilton, 1952; Noda, 1991). This effect is known as the material thermosensitivity and presents a challenge for analysis as many of the relevant heat conduction and thermoelasticity problems for the thermosensitive materials appear to be nonlinear.
In the 1980s, a significant progress in the analysis of inhomogeneous materials was concerned with the concept of the functionally graded (gradient) materials (FGM), which, basically, are specific composites combining metal and ceramic phases and exhibiting continuous or almost continuous variation in material properties from one phase to another (Rabin and Shiota, 1995). The FGM are widely used for improving operational performance of structures subjected to significant mechanical and thermal impacts (Mortensen and Suresh, 1995; Suresh and Mortensen, 1997; Miyamoto et al., 1999).
When considering the formulation of plane problem for isotropic inhomogeneous material, the material properties (16) are assumed to be functions of the inplane (either Cartesian, x and y, or polar, r and θ) coordinates. Thus, when representing all governing equations of the plane problem in terms of either stresses or displacements, the variation of the material moduli causes extra terms with arbitrarily variable coefficients involving derivatives of functions representing the material moduli by the spatial coordinates.
Governing Equations of Plane Elasticity and Thermoelasticity Problems for Inhomogeneous Materials
Compatibility Equation in Terms of Stresses
Lamé Equations for Inhomogeneous Material
Dominant Analytical Methods
Due to the fact that the governing equations of the plane problem either in terms of stresses, (21), (24), or displacements, (29), (30), contain variable coefficients involving the material properties, the analysis of such problems presents a major challenge for both analytical and numerical means (Tanigawa, 1995). In a major part, this is rather because the presence of arbitrarily variable coefficients in the governing equations makes it nearly impossible to employ the classical solution techniques developed for the problems with constant coefficients. Moreover, if within the framework of a twodimensional problem, the material inhomogeneity is also twodimensional, it appears to be impossible to employ the classical separation of variables in a general case. Therefore the proper analysis of inhomogeneous solids calls for the development of specific solution methods that allow for capturing basic features of inhomogeneity effect in the mechanical performance of elastic solids and, in general, are more complex in comparison to the ones sufficient for homogeneous materials.
An important issue in the analysis of inhomogeneous solids is the substantiation of the solution existence and correctness (Muskhelishvili, 1977) which is strongly connected with ensuring the variation of material moduli within the allowances of the elastic material model (Radu, 1966). This problem becomes extremely important when substantiating the numerical solution procedures (Kupradze, 1979).
It has also to be noted that the implementation of a certain solution method strongly depends on the shape of a solid under consideration. For example, the methods those are sufficient for infinite (unbounded) domains (planes, halfplanes, strips, etc.) appear to be insufficient for finite domains with corner points (semistrips, rectangles, annuli, etc.) and vice versa. The variety of methods for analysis of inhomogeneous solids is also persuaded by a type of inhomogeneity, a chosen or required coordinate system, a solution goal, a number of dimensions, a type of loading, etc. There are, however, certain trends in the solution techniques, which determine the dominant analytical modes of attack.
Note that representation of the Young modulus in the form of an exponential function of both inplane coordinates \(E=E_0\exp (c_xx+c_yy)\), where c_{x}, c_{y} = const, allows for reducing Eq. (23) to the one with constant coefficients, which then can be treated by a variableseparation method.
Another trend in analysis of inhomogeneous bodies implies the representation of a continuously inhomogeneous solid by an assembly of perfectly connected homogeneous layers of the same shape in the way that the original dependences of the material properties can be approximated by a piecewiseconstant function. Having solved the problems formulated for each homogeneous layer by making use of a proper classic method, the constructed solutions then are tailored through the interface conditions in order to obtain a solution for an entire solid satisfying the original boundary conditions on its surface. This method is known as the discretelayer approach (Ramirez et al, 2006). Obviously, the solution constructed in such manner is more accurate if its layerwise model represents the original continuous variation of the elastic moduli sufficiently. In many cases this implies the growing number of layers while narrowing their thickness down accordingly. In order to optimize the convergence (reduce the number of layers with no significant loss in accuracy) and avoid the mismatch of the properties on the layer interfaces, this method can be combined with the previous one in the way that the individual layers are assumed to be inhomogeneous with elastic moduli given by elementary functions of the thickness coordinate, e.g., linear, exponential, etc. (Guo and Noda, 2007).
More discussion on the available analytical and numerical methods for the analysis of inhomogeneous solids can be found in a number of reviews, e.g., (Swaminathan et al., 2017; Swaminathan and Sangeetha, 2017; Dai et al., 2016; Thai and Kim, 2015; Jha et al., 2013; Tokovyy and Ma, 2019).
CrossReferences
References
 Airy BD (1863) On the strains in the interior of beams. Philos Trans R Soc Lond 153:49–79CrossRefGoogle Scholar
 Ambatsumyan SA (1970) Theory of anisotropic plates. Strength, stability, and vibration. Technomic, StamfortGoogle Scholar
 Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New YorkzbMATHGoogle Scholar
 Carslaw HS, Jaeger JC (1959) Conduction of heat in solids. Clarendon Press, OxfordzbMATHGoogle Scholar
 Dai HL, Rao YN, Dai T (2016) A review of recent researches on FGM cylindrical structures under coupled physical interactions, 2000–2015. Compos Struct 152:199–225CrossRefGoogle Scholar
 Goursat M (1898) Sur l’équation ΔΔu = 0. Bull Soc Math France 26:236–237zbMATHGoogle Scholar
 Guo LC, Noda N (2007) Modeling method for a crack problem of functionally graded materials with arbitrary properties – piecewiseexponential model. Int J Solids Struct 44:6768–6790CrossRefGoogle Scholar
 Hashin Z (1964) Theory of mechanical behavior of heterogeneous media. Appl Mech Rev 17(1):1–9Google Scholar
 Hetnarski RB, Eslami MR (2009) Thermal stresses – advanced theory and applications. Springer, DordrechtzbMATHGoogle Scholar
 Hilton H (1952) Thermal stresses in bodies exhibiting temperaturedependent elastic properties. J Appl Mech 19:350–354Google Scholar
 Jha DK, Kant T, Singh RK (2013) A critical review of recent research on functionally graded plates. Compos Struct 96:883–849Google Scholar
 Kolchin GB (1971) Computation of the structural elements from elastic nonhomogeneous materials. Kartya Moldovenyasce, Kishinau [in Russian]Google Scholar
 Kolosov GV (1935) Application of complex diagrams and the theory of functions of a complex variable to the theory of elasticity. ONTI, Moscow, LeningradGoogle Scholar
 Kovalenko AD (1969) Thermoelasticity: basic theory and applications. WoltersNoordhoff Pub, GroningenzbMATHGoogle Scholar
 Kupradze VD (1979) Threedimensional problems of the mathematical theory of elasticity and thermoelasticity. NorthHolland Publ. Co., AmsterdamGoogle Scholar
 Lekhnitskii SG (1968) Anisotropic plates. Gordon and Breach, New YorkGoogle Scholar
 Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. Mir Pub., MoscowzbMATHGoogle Scholar
 Maugin GA (1993) Material inhomogeneities in elasticity. Chapman and Hall, LondonCrossRefGoogle Scholar
 Maxwell JC (1862) Report on a paper by George Biddel Airy on stress in beams. In: Harman PM (ed) The scientific letters and papers of James Clerk Maxwell, 1995, vol 2. Cambridge University Press, Cambridge, pp 62–69Google Scholar
 Meleshko VV (2003) Selected topics in the history of the twodimensional biharmonic problem. Appl Mech Rev 56(1):33–85. https://doi.org/10.1115/1.1521166 CrossRefGoogle Scholar
 Michlin S (1935) Le problème plan de la théorie statique d’élasticité. Publ Ints Séism 65:1–82Google Scholar
 Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally graded materials: design, processing and applications. Springer, BostonCrossRefGoogle Scholar
 Mortensen A, Suresh S (1995) Functionally graded metals and metalceramic composites: part 1 processing. Int Mater Rev 40(6):239–265CrossRefGoogle Scholar
 Muskhelishvili NI (1977) Some basic problems of the mathematical theory of elasticity. Noordhoff, LeydenCrossRefGoogle Scholar
 Noda N (1991) Thermal stresses in materials with temperaturedependent properties. Appl Mech Rev 44(9):383–397CrossRefGoogle Scholar
 Olszak W (1959) Nonhomogeneity in elasticity and plasticity. Pergamon Press, New YorkzbMATHGoogle Scholar
 Olszak W, Rychlewski J, Urbanowski W (1962) Plasticity under nonhomogeneous conditions. Adv Appl Mech 7:131–214MathSciNetCrossRefGoogle Scholar
 Podstrigach YS, Lomakin VA, Kolyano YM (1984) Thermoelasticity of bodies of nonhomogeneous structure. Nauka, Moscow [in Russian]zbMATHGoogle Scholar
 Rabin BH, Shiota I (1995) Functionally gradient materials. MRS Bull 20(1):14–18CrossRefGoogle Scholar
 Radu A (1966) Problema lui Saint–Venant pentru bare neomogene. An Şti Univ “Al. I. Cuza” Iaşi, Secţ. 1a Mat 12:415–428Google Scholar
 Radu A (1968) Sur la déformation plane d’un corps élastique isotrope nonhomogène. Bul L’Acad Polon Sci. Sér Scie Tech 16(2):91–100Google Scholar
 Ramirez R, Heyliger PR, Pan E (2006) Static analysis of functionally graded elastic anisotropic plates using a discrete layer approach. Compos B Eng 37(1):10–20CrossRefGoogle Scholar
 Southwell RV (1938) Castigliano’s principle of minimum strainenergy, and the conditions of compatibility for strain. In: Timoshenko S (ed) 60^{th} anniversary volume. The Macmillan Company, New York, pp 211–217Google Scholar
 Suresh S, Mortensen A (1997) Functionally graded metals and metalceramic composites: part 2 thermomechanical behaviour. Int Mat Rev 42(3):85–116CrossRefGoogle Scholar
 Swaminathan K, Sangeetha DM (2017) Thermal analysis of FGM plates – a critical review of various modeling techniques and solution methods. Compos Struct 160:43–60CrossRefGoogle Scholar
 Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E (2017) Stress, vibration and buckling analyses of FGM plates – a stateoftheart review. Compos Struct 120:10–31CrossRefGoogle Scholar
 Tanigawa Y (1995) Some basic thermoelastic problems for nonhomogeneous structural materials. Appl Mech Rev 48(6):287–300CrossRefGoogle Scholar
 Teodorescu PP (2013) Treatise on classical elasticity. Theory and related problems. Springer, DordrechtCrossRefGoogle Scholar
 Thai HT, Kim SE (2015) A review of theories for the modeling and analysis of functionally graded plates and shells. Compos Struct 128:70–86CrossRefGoogle Scholar
 Timoshenko SP (1953) History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. McGrawHill, New YorkzbMATHGoogle Scholar
 Timoshenko SP, Goodier JN (1951) Theory of elasticity. McGrawHill Book Co., Inc., New YorkzbMATHGoogle Scholar
 Tokovyy YV (2014) Direct integration method. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 2. Springer, Dordrecht, pp 951–960CrossRefGoogle Scholar
 Tokovyy YV, Ma CC (2019, accepted) Elastic analysis of inhomogeneous solids: history and development in brief. J Mech, Springer, DordrechtGoogle Scholar
 Tokovyy YV, Kalynyak BM, Ma CC (2014) Nonhomogeneous solids: integral equations approach. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 7, pp 3350–3356CrossRefGoogle Scholar