Encyclopedia of Continuum Mechanics

Living Edition
| Editors: Holm Altenbach, Andreas Öchsner

Plane Thermoelasticity of Inhomogeneous Solids

  • Yuriy Tokovyy
Living reference work entry
DOI: https://doi.org/10.1007/978-3-662-53605-6_361-1



Plane problem

That addresses a problem formulated under either the plane-stress or plane-strain hypothesis, which allows for representing a general three-dimensional formulation in two in-plane 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 macro-volume of an elastic solid.


Plane problems of the elasticity and thermoelasticity theories attract widespread attention for over a hundred years. Being derived from general three-dimensional problems by making use of either the plane-strain hypothesis for long prismatic (cylindrical) solids or the (generalized) plane-stress 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 (two-dimensional) 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 mechanical-thermal 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 macro-volume). The governing equations for these problems contain variable coefficients expressed through the derivatives of the material moduli by the in-plane 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 Three-Dimensional Formulation of Elasticity and Thermoelasticity Problems

Within the framework of the uncoupled theory of thermoelasticity, the state of inner strains and stresses in an elastic isotropic solid \(\mathscr {S}\) is governed by certain sets of equations which may have different appearances in different coordinate systems. In the Cartesian coordinate system (x, y, z), the balance of inner forces is described by the equations of motion, which can be given in a local differential form (Timoshenko and Goodier, 1951) as
$$\displaystyle \begin{aligned} \mathbf{div}\hat{\boldsymbol{\sigma}} + \rho\left(\mathbf{F}-\mathbf{f}\right)=0, {} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{rcl} \hat{\boldsymbol{\sigma}}&\displaystyle =&\displaystyle \left( \begin{array}{ccc} \sigma_x \hfill &\displaystyle \sigma_{xy}\hfill &\displaystyle \sigma_{xz}\\ \sigma_{xy}\hfill &\displaystyle \sigma_y \hfill &\displaystyle \sigma_{yz}\\ \sigma_{xz}\hfill &\displaystyle \sigma_{yz}\hfill &\displaystyle \sigma_z \end{array}\right), \quad \mathbf{F}=\left( \begin{array}{c} F_x\\ F_y \\ F_z \end{array} \right),\\ \mathbf{f}&\displaystyle =&\displaystyle \frac{\partial^2\mathbf{u}}{\partial t^2},\quad \mathbf{u}= \left( \begin{array}{c} u\\ v \\ w \end{array} \right), \end{array} \end{aligned} $$
σξ, σξη, (ξ, η = x, y, z, ξ ≠ η), and u, v, and w are the components of the symmetric stress tensor and the displacement vector, respectively, which are functions of the coordinates x, y, z and time t, Fξ, is a volumetric force projection onto the axis ξ = x, y, z, ρ is the material density, and \(\mathbf {div}=\left ( \partial /\partial x,\partial /\partial y,\partial /\partial z\right )\). In the case when the inertia terms are negligibly small, f = 0, Eq. (1) becomes static and presents the equations of equilibrium. In what follows, only the latter case of equilibrium equations (within either static or quasi-static formulation) is addressed.
The strain-tensor components εξ, εξη, (ξ, η = x, y, z, ξ ≠ η) are expressed through the stresses by the constitutive equations of the generalized Hooke (elasticity) or Duhamel-Neumann (thermoelasticity) law (Boley and Weiner, 1960):
$$\displaystyle \begin{aligned} \begin{array}{rcl} E\varepsilon_{x} &\displaystyle =&\displaystyle \sigma_x - (\sigma_y+\sigma_z)\nu + \alpha E T,\\ E\varepsilon_{xy}&\displaystyle =&\displaystyle 2(1+\nu)\sigma_{xy},\\ E\varepsilon_{y} &\displaystyle =&\displaystyle \sigma_y - (\sigma_x+\sigma_z)\nu + \alpha E T,\\ E\varepsilon_{yz}&\displaystyle =&\displaystyle 2(1+\nu)\sigma_{yz},\\ E\varepsilon_{z} &\displaystyle =&\displaystyle \sigma_z - (\sigma_x+\sigma_y)\nu + \alpha E T,\\ E\varepsilon_{xz}&\displaystyle =&\displaystyle (1+\nu)\sigma_{xz},{} \end{array} \end{aligned} $$
where E and ν are the Young modulus and the Poisson ratio, respectively, α is the coefficient of linear thermal expansion, and T = T(x, y, z;t) is a temperature distribution, which can be measured, approximated, or found from a relevant heat conduction problem (Carslaw and Jaeger, 1959); herein it is considered to be a given function. Note that the constitutive equations for anisotropic materials may have more complex form involving a greater number of independent coefficients expressing the dissimilar material properties in different spatial directions (Ambatsumyan, 1970; Lekhnitskii, 1981).
The strains are expressed in terms of displacements through the Cauchy (geometric) equations
$$\displaystyle \begin{aligned} \begin{array}{ccc} \varepsilon_x & =\displaystyle\frac{\partial u}{\partial x},\hfill\qquad \varepsilon_{xy} & =\displaystyle\frac{\partial u}{\partial y} + \displaystyle\frac{\partial v}{\partial x},\\ \varepsilon_y & =\displaystyle\frac{\partial v}{\partial y},\hfill\qquad \varepsilon_{yz} & =\displaystyle\frac{\partial w}{\partial y} + \displaystyle\frac{\partial v}{\partial z},\\ \varepsilon_z & =\displaystyle\frac{\partial w}{\partial z},\hfill\qquad \varepsilon_{xz} & =\displaystyle\frac{\partial u}{\partial z} + \displaystyle\frac{\partial w}{\partial x}. \end{array} {} \end{aligned} $$
By eliminating the displacements from the above equations, the strain-compatibility equations
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \frac{\partial^2\varepsilon_y}{\partial x^2} &\displaystyle +&\displaystyle \frac{\partial^2\varepsilon_x}{\partial y^2}= \frac{\partial^2\varepsilon_{xy}}{\partial x\partial y },\hfill\qquad 2\frac{\partial^2\varepsilon_z}{\partial x \partial y} = \frac\partial{\partial z}\left( \frac{\partial\varepsilon_{yz}}{\partial x}+ \frac{\partial\varepsilon_{xz}}{\partial y}- \frac{\partial\varepsilon_{xy}}{\partial z} \right),\\ \frac{\partial^2\varepsilon_z}{\partial y^2} &\displaystyle +&\displaystyle \frac{\partial^2\varepsilon_y}{\partial z^2}= \frac{\partial^2\varepsilon_{yz}}{\partial y\partial z },\hfill\qquad 2\frac{\partial^2\varepsilon_x}{\partial y \partial z} = \frac\partial{\partial x}\left( \frac{\partial\varepsilon_{xz}}{\partial y}+ \frac{\partial\varepsilon_{xy}}{\partial z}- \frac{\partial\varepsilon_{yz}}{\partial x} \right),\\ \frac{\partial^2\varepsilon_x}{\partial z^2} &\displaystyle +&\displaystyle \frac{\partial^2\varepsilon_z}{\partial x^2}= \frac{\partial^2\varepsilon_{xz}}{\partial x\partial z },\hfill\qquad 2\frac{\partial^2\varepsilon_y}{\partial x \partial z} = \frac\partial{\partial y}\left( \frac{\partial\varepsilon_{xy}}{\partial z}+ \frac{\partial\varepsilon_{yz}}{\partial x}- \frac{\partial\varepsilon_{xz}}{\partial y} \right) \end{array} \end{aligned} $$
can be derived. Being obtained by Barré de Saint-Venant (Timoshenko, 1953), the strain-compatibility equations, also known as the continuity equations, in the form (4) can be regarded as the classic ones and are included into the vast majority of the elasticity textbooks. There exist, however, a number of different ways on eliminating the displacements from the Cauchy equations (3), which result in various systems of independent compatibility equations in terms of stresses. This matter is directly related to a dispute on the sufficient number of the strain-compatibility equations and their most efficient form, which was started with the so-called Southwell paradox (Southwell, 1938) and remains open (Tokovyy, 2014).
The formulation of the static (or quasi-static) elasticity and thermoelasticity boundary-value problems is to be concluded with imposing boundary conditions for the stress vector
$$\displaystyle \begin{aligned} \hat{\boldsymbol{\sigma}}\cdot\mathbf{n}=\mathbf{P} {} \end{aligned} $$
or the displacement vector
$$\displaystyle \begin{aligned} \mathbf{u}=\mathbf{u^\ast} {} \end{aligned} $$
or either the combinations of certain components of both stress and displacement vector (mixed-type boundary conditions) on the boundary \(\mathscr {S}^\ast \) of body \(\mathscr {S}\). Here, n is the vector of outer normal to the surface \(\mathscr {S}^\ast \), P is a vector of external forces applied to the surface \(\mathscr {S}^\ast \), and u denotes the vector of displacements given on the boundary\(\mathscr {S}^\ast \).

Three equations of equilibrium (1), six constitutive equations (2), and six either strain-compatibility (4) or Cauchy equations (3) present a complete system of 15 equations for determination of 15 unknown functions: six stress-tensor components σξ, σξη, six strain-tensor components εξ, εξη, and three displacements u, v, w. Here, ξ, η = x, y, z, ξ ≠ η. In order to reduce the number of quested-for functions, the formulation of elasticity or thermoelasticity boundary-value 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 Beltrami-Michell 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 boundary-value 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 plane-strain or plane-stress hypotheses, under which the elasticity and thermoelasticity problems can be reduced to a plane (two-dimensional) formulation with lower number of independent equations and quested-for functions of two spatial variables only.

Plane Strain

If an elastic deformable solid \(\mathscr {S}\) is of the form of a long prismatic (cylindrical) body (the body extension in one of the spatial directions, let it be z, is significantly larger than the ones in x and y directions) and the distributions of volumetric forces F in Eqs. (1), temperature T in (2), and external forces P or boundary displacements u in conditions (5) and (6) are independent of the coordinate z, then, at a sufficient distance from the end-faces of body \(\mathscr {S}\), (i) the stress and displacement fields are irrespective of coordinate z and vary only with coordinates x and y within a plane (z =  const) cross-section \(\mathscr {D}\) of body \(\mathscr {S}\); (ii) the shearing strains εxz, εyz and, consequently, stresses σxz, σyz are equal to zero; and (iii) the axial strain is, in general, constant: εz = e0 = const. If both end-faces of body \(\mathscr {S}\) are confined between two smooth rigid planes, then, obviously, e0 = 0. If one or both of the end-faces are free, then, due to (2), the constant axial strain can be determined as
$$\displaystyle \begin{aligned} e_0=\int\limits_{\mathscr{D}}\left( \frac{\sigma_z}E-\frac\nu E(\sigma_x+\sigma_y) + \alpha T \right)dxdy. \end{aligned}$$
Note that the latter formula implies, however, that if domain \(\mathscr {D}\) is unbounded with respect to x, y, or both, then the plane-strain hypothesis requires the improper integral in this formula to be either zero or constant. If e0 ≠ 0, then the Cauchy equations (3) imply the displacement w to be a linear function of z.
In view of the foregoing assumptions, the equilibrium equations (1) take the following form:
$$\displaystyle \begin{aligned} \frac{\partial\sigma_x }{\partial x} + \frac{\partial\sigma_{xy}}{\partial y} + X=0,\qquad \frac{\partial\sigma_{xy}}{\partial x} + \frac{\partial\sigma_y }{\partial y} + Y=0. {} \end{aligned} $$
Here, X = ρFx and Y = ρFy are in-plane components of the body forces in directions x and y, respectively.
Making use of the constitutive equations (2) allows for expressing the out-of-plane stress σz through the in-plane normal ones, σx and σy, as
$$\displaystyle \begin{aligned} \sigma_z=Ee_0+\nu(\sigma_x+\sigma_y)-\alpha ET. {} \end{aligned} $$
This formula along with the foregoing assumptions on the absence of the out-of-plane shearing strains reduces the number of independent constitutive equations (2) to only the three ones:
$$\displaystyle \begin{aligned} \begin{array}{rcl} E\varepsilon_x&\displaystyle =&\displaystyle (1-\nu^2)\sigma_x -\nu(1+\nu)\sigma_y\\ &\displaystyle &\displaystyle -E\nu e_0+\alpha(1+\nu)ET,\\ E\varepsilon_y&\displaystyle =&\displaystyle (1-\nu^2)\sigma_y -\nu(1+\nu)\sigma_x\\ &\displaystyle &\displaystyle -E\nu e_0+\alpha(1+\nu)ET,\\ E\varepsilon_{xy}&\displaystyle =&\displaystyle 2(1+\nu)\sigma_{xy}. {} \end{array} \end{aligned} $$
Similarly, under the plane-strain hypothesis, the Cauchy equations (3) take the form
$$\displaystyle \begin{aligned} \varepsilon_x=\frac{\partial u}{\partial x},\qquad \varepsilon_y=\frac{\partial v}{\partial y},\qquad \varepsilon_{xy}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}, {} \end{aligned} $$
and the strain-compatibility equations (4) are represented only with one equation
$$\displaystyle \begin{aligned} \frac{\partial^2\varepsilon_x}{\partial y^2}+\frac{\partial^2\varepsilon_y}{\partial x^2}= \frac{\partial^2\varepsilon_{xy}}{\partial x\partial y}. {} \end{aligned} $$

In such manner, the problem on determination of stresses in a solid \(\mathscr {S}\) is reduced to determination of three stress-tensor components σx, σy, σxy; three strain-tensor components εx, εy, εxy; 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 stress-tensor components. After the in-plane stresses are computed, the out-of-plane stress σz 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

A similar simplified two-dimensional formulation can be presented in the case when an elastic solid \(\mathscr {S}\) is a thin plate of constant thickness h (−h∕2 < z < h∕2), which is significantly smaller in comparison to both its in-plane dimensions in x and y directions, \((x,y)\in \mathscr {D}\). Assume that (i) the faces z = ±h∕2 are free of force loadings, (ii) the circumference of the plate is loaded by forces symmetric with respect to the midplane z = 0 and is parallel to it, and (iii) the component Fz of volumetric force vector is absent, and component Fx, Fy and the temperature T are symmetric about the midplane z = 0. The assumed symmetry of loadings implies the vertical displacement w to be zero at the midplane z = 0. This fact along with the smallness of h implies that w = 0 for arbitrary z and the variation of u and v with z are insignificant so that they can be sufficiently substituted with their averaged values
$$\displaystyle \begin{aligned} \tilde{u}=\frac 1h\int\limits_{-h/2}^{h/2}udz,\qquad \tilde{v}=\frac 1h\int\limits_{-h/2}^{h/2}vdz, {} \end{aligned} $$
which are, obviously, the functions of x and y only, \((x,y)\in \mathscr {D}\).
By averaging the stress-tensor components in a similar to (12) fashion, the foregoing assumptions allow for concluding that σz = 0, σxz = σyz = 0, εxz = εyz = 0 and Eqs. (7), (10), and (11), where tildes are omitted for the simplicity sake, are valid for the considered case. This is the case of generalized plane stress, for which the constitutive equations (2) take the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} Ee_x&\displaystyle =&\displaystyle \sigma_x-\nu\sigma_y+\alpha ET,\\ Ee_y&\displaystyle =&\displaystyle \sigma_y-\nu\sigma_x+\alpha ET,\\ Ee_{xy}&\displaystyle =&\displaystyle 2(1+\nu)\sigma_{xy}. {} \end{array} \end{aligned} $$
In this case, the averaged axial strain appears as
$$\displaystyle \begin{aligned} E\varepsilon_z=-\nu(\sigma_x + \sigma_y) + \alpha ET. {} \end{aligned} $$

Similar to the plane-strain case, the generalized plane-stress hypothesis allows for reduction of a problem to three equations for in-plane stresses σx(x, y), σy(x, y), and σxy(x, y) or two equations for displacements u(x, y) and v(x, y) with the out-of-plane strain being expressed through the in-plane normal stresses by formula (14).

Unified Formulation of Plane Problems in the Cartesian and Polar Coordinate Systems

Despite the fact that the plane-strains and plane-stress hypotheses are in use for solids of utterly different shapes (long prisms and thin plates, respectively); in both cases, however, the elasticity and thermoelasticity problems imply the determination of stresses σx(x, y), σy(x, y), and σxy(x, y) and/or displacements u(x, y) and v(x, y) from Eqs. (7) and (11) or (10) under the constitutive equations in the form (9) or (13). Note that Eqs. (9) and (13) can be given in a unified form
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} E^\ast\varepsilon_{x} &\displaystyle =&\displaystyle \sigma_x - \nu^\ast \sigma_y - \nu E^\ast e_0^\ast + \alpha^\ast E^\ast T,\\ E\varepsilon_{xy} &\displaystyle =&\displaystyle 2(1+\nu)\sigma_{xy},\\ E^\ast\varepsilon_{y} &\displaystyle =&\displaystyle \sigma_y - \nu^\ast \sigma_x - \nu E^\ast e_0^\ast + \alpha^\ast E^\ast T, \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \begin{array}{ll} E^\ast=\left\{ \begin{array}{l@{\quad }l@{\quad }l} E,\hfill & \mathrm{plane stress,}\\ E/(1-\nu^2), \hfill & \mathrm{plane strain,} \end{array}\right.\qquad \qquad &\nu^\ast=\left\{ \begin{array}{l@{\quad }l@{\quad }l} \nu,\hfill & \mathrm{plane stress,}\\ \nu/(1-\nu), \hfill & \mathrm{plane strain,} \end{array}\right.\quad \\ \alpha^\ast=\left\{ \begin{array}{l@{\quad }l@{\quad }l} \alpha,\hfill &\quad \, \mathrm{plane stress,}\\ \alpha(1+\nu), \hfill &\quad \, \mathrm{plane strain,} \end{array}\right.\quad &e_0^\ast=\left\{ \begin{array}{l@{\quad }l@{\quad }l} 0,\hfill &\quad \qquad \, \mathrm{plane stress,}\\ e_0, \hfill &\quad \qquad \, \mathrm{plane strain.} \end{array}\right.\quad \vspace{6pt} \end{array} {} \end{aligned} $$
When the polar coordinates \((r,\theta )\in \mathscr {D}\) are introduced instead of the Cartesian ones, the in-plane displacements ur and uθ can be expressed through the ones in the Cartesian coordinates as
$$\displaystyle \begin{aligned} u_r+iu_\theta=(u+iv)\exp{(-i\theta)},\qquad i^2=-1. \end{aligned}$$
Then the Cauchy equations (10) for the plane problem take the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varepsilon_r&\displaystyle =&\displaystyle \frac{\partial u_r}{\partial r},\quad \varepsilon_\theta=\frac 1r\left(\frac{\partial u_\theta}{\partial\theta}+u_r\right),\\ \varepsilon_{r\theta}&\displaystyle =&\displaystyle \frac 1r\frac{\partial u_r}{\partial\theta}+r\frac{\partial}{\partial r}\left(\frac{u_\theta}{r}\right), {} \end{array} \end{aligned} $$
and the equilibrium equations (7) appear as
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \frac{\partial}{\partial r}(r\sigma_r) + \frac{\partial\sigma_{r\theta}}{\partial\theta}-\sigma_\theta+rR=0,\\ &\displaystyle &\displaystyle \frac 1r\frac{\partial}{\partial r}(r^2\sigma_{r\theta}) + \frac{\partial\sigma_\theta}{\partial\theta} +rS=0. {} \end{array} \end{aligned} $$
Here, σr, σθ, and σ and εr, εθ, and ε are the in-plane radial, circumferential, and shearing (tangential) stress- and strain-tensor components, respectively, and R and S are the body forces in the radial and circumferential directions.
By eliminating the displacements from (17), the strain-compatibility equation can be obtained in the form
$$\displaystyle \begin{aligned} \frac\partial{\partial r}\left( r^2\frac{\partial\varepsilon_\theta}{\partial r} \right)-r\frac{\partial\varepsilon_r}{\partial r} +\frac{\partial^2\varepsilon_r}{\partial\theta^2}=\frac{\partial^2(r\varepsilon_{r\theta})}{\partial r\partial\theta}. {} \end{aligned} $$
In polar coordinates, the constitutive equations (15) appear as
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} E^\ast\varepsilon_r &\displaystyle =&\displaystyle \sigma_r - \nu^\ast \sigma_\theta - \nu E^\ast e_0^\ast + \alpha^\ast E^\ast T,\\ E\varepsilon_{r\theta} &\displaystyle =&\displaystyle 2(1+\nu)\sigma_{r\theta},\\ E^\ast\varepsilon_\theta &\displaystyle =&\displaystyle \sigma_\theta - \nu^\ast \sigma_r - \nu E^\ast e_0^\ast + \alpha^\ast E^\ast T, \end{array} \end{aligned} $$
where the material properties are given by (16).

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 in-plane 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 macro-volume and resemble the properties of the entire solid). However, comprehensive analysis of the mechanical performance of real-life materials requires more accurate hypotheses to be introduced for covering the effect of structural imperfections or defects, whose consolidation within a macro-volume causes significant disturbances of the mechanical fields and variation of the material properties within the macro-volume. This effect is known as macro-inhomogeneity or inhomogeneity (also, non-homogeneity, 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 micro-inhomogeneity, 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, non-smooth 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 piecewise-homogeneous solids (the material properties are rather constant within some layers or segments of the body so that they can be modeled by piecewise-constant 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 macro-volume 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 in-plane (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

Putting the constitutive equations (15) into the compatibility equation (11) in view of the dependences of the material properties on the in-plane coordinates x and y and making use of Eqs. (7) yield the compatibility equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \nabla^2\left(\frac 1{E^\ast}(\sigma_x + \sigma_y) - \nu e^\ast_0 + \alpha^\ast T \right) \\ &\displaystyle &\displaystyle \quad =2\sigma_{xy}\frac{\partial^2}{\partial x\partial y}\left(\frac{1+\nu}{E}\right) +\sigma_x\frac{\partial^2}{\partial x^2}\left(\frac{1+\nu}{E}\right)\\ &\displaystyle &\displaystyle +\sigma_y\frac{\partial^2}{\partial y^2}\left(\frac{1+\nu}{E}\right)-\frac{1+\nu}{E}\left( \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} \right)\\ &\displaystyle &\displaystyle -2\left( X\frac{\partial}{\partial x}\left(\frac{1+\nu}{E}\right) +Y\frac{\partial}{\partial y}\left(\frac{1+\nu}{E}\right) \right),{} \end{array} \end{aligned} $$
which along with two equations of equilibrium (7) present a system of three equations for determination of three stress-tensor components σx, σy, and σxy. Here,
$$\displaystyle \begin{aligned} \nabla^2=\frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial x^2} {} \end{aligned} $$
is the Laplace differential operator in the Cartesian coordinate system. In the case of constant material properties, the obtained equation presents well-known compatibility equation
$$\displaystyle \begin{aligned} \nabla^2(\sigma_x+\sigma_y)=-\alpha^\ast E^\ast\nabla^2T-(1+\nu^\ast)\left( \frac{\partial X}{\partial x}+\frac{\partial Y}{\partial y} \right) \end{aligned}$$
of plane thermoelasticity for homogeneous solids.
Introduction of a potential function φ by formulae
$$\displaystyle \begin{aligned} \begin{array}{rcl} \sigma_x&\displaystyle =&\displaystyle \frac{\partial^2\varphi}{\partial y^2}-\int X dx,\quad \sigma_y=\frac{\partial^2\varphi}{\partial x^2}-\int Y dy,\\ \sigma_{xy}&\displaystyle =&\displaystyle \frac{\partial^2\varphi}{\partial x\partial y} \end{array} \end{aligned} $$
allows for identical satisfaction of the equilibrium equations (7), while the compatibility one and (21) yields the following fourth-order differential equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \nabla^2\left(\frac{\nabla^2\varphi}{E^\ast} - \nu e^\ast_0 + \alpha^\ast T \right) -2\frac{\partial^2\varphi}{\partial x\partial y}\frac{\partial^2}{\partial x\partial y}\left(\frac{1+\nu}{E}\right) -\frac{\partial^2\varphi}{\partial y^2}\frac{\partial^2}{\partial x^2}\left(\frac{1+\nu}{E}\right) \\ &\displaystyle &\displaystyle \quad -\frac{\partial^2\varphi}{\partial x^2}\frac{\partial^2}{\partial y^2}\left(\frac{1+\nu}{E}\right) =\nabla^2\left(\frac 1{E^\ast}\left(\int X dx +\int Ydy \right) \right) \\ &\displaystyle &\displaystyle \quad -\frac{\partial^2}{\partial x^2}\left(\frac{1+\nu}{E}\right)\int X dx -\frac{\partial^2}{\partial y^2}\left(\frac{1+\nu}{E}\right)\int Y dy \\ &\displaystyle &\displaystyle \quad -\frac{1+\nu}{E}\left( \frac{\partial X}{\partial x} + \frac{\partial Y}{\partial y} \right) -2\left( X\frac{\partial}{\partial x}\left(\frac{1+\nu}{E}\right) +Y\frac{\partial}{\partial y}\left(\frac{1+\nu}{E}\right) \right). {} \end{array} \end{aligned} $$
Function φ is similar to the Airy biharmonic function in case of homogeneous material (Timoshenko and Goodier, 1951). If the material properties in the latter equation are assumed to be constant, then it yields a classical biharmonic equation for the plane thermoelasticity problem.
In a similar fashion, putting constitutive equations (20) into the compatibility equation (19) in the polar coordinate system with account for dependences of the material properties (16) on ρ and θ yields the compatibility equation
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \varDelta\left( \frac 1{E^\ast}(\sigma_r+\sigma_\theta) - \nu e_0^\ast + \alpha^\ast T \right)=\sigma_r\frac{\partial^2}{\partial r^2}\left( \frac{1+\nu}E \right) +\frac{\sigma_\theta}r\left( \frac\partial{\partial r}\left(\frac{1+\nu}E \right)\right. \\ &\displaystyle &\displaystyle \quad \left.+ \frac 1r\frac{\partial^2}{\partial\theta^2}\left(\frac{1+\nu}E \right) \right) + 2\sigma_{r\theta}\frac\partial{\partial r}\left( \frac 1r\frac\partial{\partial\theta} \left(\frac{1+\nu}E \right) \right) \\ &\displaystyle &\displaystyle \quad -\frac 1r\left( \frac\partial{\partial\theta}\left( \frac{1+\nu}E S \right) + S\frac\partial{\partial\theta}\left(\frac{1+\nu}E \right)\right) \\ &\displaystyle &\displaystyle \quad -\frac 1r\frac\partial{\partial r}\left(\frac{1+\nu}E rR\right) - R\frac\partial{\partial r}\left(\frac{1+\nu}E \right) {} \end{array} \end{aligned} $$
in terms of stresses, which along with the equilibrium ones (18) present a system of three equations for determination of three stress-tensor components σr, σθ, and σ. Here,
$$\displaystyle \begin{aligned} \varDelta = \frac 1r\frac\partial{\partial r}\left(r\frac\partial{\partial r}\right) + \frac 1{r^2}\frac{\partial^2}{\partial\theta^2}. {} \end{aligned} $$
In the case of constant material properties, Eq. (24) takes the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} \varDelta\left( \sigma_r+\sigma_\theta \right)&\displaystyle =&\displaystyle - \alpha^\ast E^\ast\varDelta T - \frac{1+\nu^\ast}r\\ &\displaystyle &\displaystyle \left( \frac{\partial(rR)}{\partial r}+\frac {\partial S}{\partial\theta} \right). {} \end{array} \end{aligned} $$
Introduction of potential function ϕ by the following formulae:
$$\displaystyle \begin{aligned} \begin{array}{rcl}{} \sigma_r &\displaystyle =&\displaystyle \frac 1r\frac{\partial\phi}{\partial r} + \frac 1{r^2}\frac{\partial^2\phi}{\partial\theta^2} -\frac 1r\iint r\left( S + \frac{\partial R}{\partial\theta}\right)drd\theta,\\ \sigma_\theta &\displaystyle =&\displaystyle \frac{\partial^2\phi}{\partial r^2}-r\int Sd\theta,\quad \sigma_{r\theta}= -\frac\partial{\partial r}\left( \frac 1r\frac{\partial\phi}{\partial\theta}\right)\\ \end{array} \end{aligned} $$
allows for transforming the equilibrium equations (18) into identical equalities and yields the expression
$$\displaystyle \begin{aligned} \sigma_r+\sigma_\theta=\varDelta\phi-\phi_0, {} \end{aligned} $$
$$\displaystyle \begin{aligned} \phi_0=\int\left( rS+\frac 1r\int r\left( S + \frac{\partial R}{\partial\theta} \right)dr \right)d\theta. \end{aligned}$$
Putting (27) and (28) into (24) yields a partial differential equation of fourth order with variable coefficients for determination of function ϕ within the framework of the plane problem in polar coordinates. If the material properties are constant, then putting (28) into (26) yields the well-known biharmonic equation in polar coordinates.

Lamé Equations for Inhomogeneous Material

In order to derive the governing equations of the plane problem for inhomogeneous material in terms of displacements, the stresses can be expressed through the displacements by making use of the constitutive and Cauchy equations, (15) and (10), respectively. Then, putting the obtained expression into equilibrium equations (7) yields
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \nabla^2u+\frac{1+\nu^\ast}{1-\nu^\ast}\frac\partial{\partial x}\left(\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) +2\frac{1+\nu}E\left( \frac\partial{\partial x}\left(\frac E{(1+\nu)(1-\nu^\ast)} \right)\frac{\partial u}{\partial x}\right. \\ &\displaystyle &\displaystyle \quad +\left. \frac\partial{\partial x}\left(\frac {\nu^\ast E}{(1+\nu)(1-\nu^\ast)} \right)\frac{\partial v}{\partial y} \right) +\frac\partial{\partial y}\left(\ln\frac E{1+\nu} \right)\left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right) \\ &\displaystyle &\displaystyle \quad =2\frac{1+\nu}E\left(\frac{\partial}{\partial x}\left( \frac{\alpha^\ast E^\ast}{1-\nu^\ast} T \right) - e_0^\ast\frac\partial{\partial x}\left( \frac{\nu E^\ast}{1-\nu^\ast} \right) -X \right), \\ &\displaystyle &\displaystyle \quad \nabla^2v+\frac{1+\nu^\ast}{1-\nu^\ast}\frac\partial{\partial y}\left(\frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right)+2\frac{1+\nu}E\left( \frac\partial{\partial y}\left(\frac E{(1+\nu)(1-\nu^\ast)} \right)\frac{\partial v}{\partial y}\right. \\ &\displaystyle &\displaystyle \quad +\left. \frac\partial{\partial y}\left(\frac {\nu^\ast E}{(1+\nu)(1-\nu^\ast)} \right)\frac{\partial u}{\partial x} \right) +\frac\partial{\partial x}\left(\ln\frac E{1+\nu} \right)\left( \frac{\partial u}{\partial y} + \frac{\partial v}{\partial x} \right) \\ &\displaystyle &\displaystyle \quad =2\frac{1+\nu}E\left(\frac{\partial}{\partial y}\left( \frac{\alpha^\ast E^\ast}{1-\nu^\ast} T \right) - e_0^\ast\frac\partial{\partial y}\left( \frac{\nu E^\ast}{1-\nu^\ast} \right) - Y \right). {} \end{array} \end{aligned} $$
Here, ∇2 is given by formula (22). Two Eqs. (29) can be used for determination of two displacement vector components u(x, y) and v(x, u) within the framework of a plane problem in the Cartesian coordinate system in the case of inhomogeneous material. If the material properties are constant, then these equations yield the well-known Lamé equations for homogeneous isotropic material (Timoshenko and Goodier, 1951).
In a similar manner, by substitution of the stress-tensor components σr, σθ, and σ, preliminary expressed through the displacements ur and uθ by making use of (20) and (17), into the equilibrium equations (18), the equations
$$\displaystyle \begin{aligned} \begin{array}{rcl} &\displaystyle &\displaystyle \varDelta u_r + \frac{1+\nu^\ast}{1-\nu^\ast}\left( \frac 1r\frac\partial{\partial r}\left(r\frac{\partial u_r}{\partial r}\right) +\frac\partial{\partial r}\left( \frac 1r\frac{\partial u_\theta}{\partial\theta} \right) \right) -\frac 2{r^2}\left( \frac{\partial u_\theta}{\partial\theta} + \frac{u_r}{1-\nu^\ast}\right) \\ &\displaystyle &\displaystyle \quad +2\frac{1+\nu}E\left( \frac\partial{\partial r}\left(\frac E{(1+\nu)(1-\nu)^\ast}\right)\frac{\partial u_r}{\partial r}+ \frac 1r\frac\partial{\partial r}\left( \frac{\nu^\ast E}{(1+\nu)(1-\nu^\ast)}\right)\left(\frac{\partial u_\theta}{\partial\theta}+u_r \right) \right) \\ &\displaystyle &\displaystyle \quad +\frac\partial{\partial\theta}\left(\ln\frac E{1+\nu}\right)\left( \frac 1{r^2}\frac{\partial u_r}{\partial\theta} + \frac\partial{\partial r} \left(\frac{u_\theta}r\right)\right) =2\frac{1+\nu}E\left( \frac\partial{\partial r}\left(\frac{\alpha^\ast E^\ast}{1-\nu^\ast}T \right)- \frac\partial{\partial r}\left(\frac{\nu E^\ast}{1-\nu^\ast}\right)e_0^\ast - R\right), \\ &\displaystyle &\displaystyle \quad \varDelta u_\theta + \frac{1+\nu^\ast}{1-\nu^\ast}\frac 1{r^2}\frac\partial{\partial\theta}\left( \frac\partial{\partial r} (ru_r) +\frac{\partial u_\theta}{\partial\theta} \right) +\frac 1{r^2}\left( 2\frac{\partial u_r}{\partial\theta} + u_\theta\right)\\ &\displaystyle &\displaystyle \quad +\frac 2r\frac{1+\nu}E\left( \frac\partial{\partial\theta}\left(\frac {\nu^\ast E}{(1+\nu)(1-\nu)^\ast}\right)\frac{\partial u_r}{\partial r} +\frac 1r\frac\partial{\partial\theta}\left( \frac{ E}{(1+\nu)(1-\nu^\ast)}\right)\left(\frac{\partial u_\theta}{\partial\theta}+u_r \right) \right) \\ &\displaystyle &\displaystyle \quad +r\frac\partial{\partial r}\left(\ln\frac E{1+\nu}\right)\left( \frac 1{r^2}\frac{\partial u_r}{\partial\theta} + \frac\partial{\partial r} \left(\frac{u_\theta}r\right)\right) =\frac 2r\frac{1+\nu}E\left( \frac\partial{\partial\theta}\left(\frac{\alpha^\ast E^\ast}{1-\nu^\ast}T \right)- \frac\partial{\partial\theta}\left(\frac{\nu E^\ast}{1-\nu^\ast}\right)e_0^\ast - rS\right)\\ {} \end{array} \end{aligned} $$
can be derived for the determination of ur(r, θ) and uθ(r, θ) within the framework of the plane problem for inhomogeneous material in the polar coordinate system. Here, Δ is given by formula (25).

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 two-dimensional problem, the material inhomogeneity is also two-dimensional, 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, half-planes, strips, etc.) appear to be insufficient for finite domains with corner points (semi-strips, 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.

One of such mainstream trends implies the construction of general solutions to the problems for inhomogeneous solids under the minimum possible restriction on the variation of the material moduli. Some of the methods here are based on the generalization of the complex variable method by introducing
$$\displaystyle \begin{aligned} z=x+iy,\quad \bar{z}=x-iy,\quad i^2=-1 {} \end{aligned} $$
and reducing the governing equations to new variables (31) in the complex mapping domain (Radu, 1968). An alternative approach to the construction of general solutions to the problems of elasticity and thermoelasticity for arbitrarily inhomogeneous solids rests upon the reduction to the integral equations of second kind with further implementation of either numerical or numerical techniques (Tokovyy et al., 2014).
Another trend in construction of analytical solutions for inhomogeneous materials consists in assuming the material moduli to be given by certain elementary functions (e.g., the linear, polynomial, exponential, etc.) so that it allows for the separation of variables in the governing equations and, evidently, yields comparatively simple solutions by making use of the classical technique. For instance, when considering a plane problem for solids with properties ν = const, E(y) = E0𝜗(y), and E0 = const, in the absence of temperature and body forces (T = X = Y  = 0), Eq. (23) takes the form
$$\displaystyle \begin{aligned} \begin{array}{rcl} \nabla^4\varphi &\displaystyle -&\displaystyle \frac 2{\vartheta^\ast}\frac{d\vartheta^\ast}{dy}\frac{\partial}{\partial y}\left(\nabla^2\varphi\right) + \left( \frac 1{\vartheta^\ast}\frac{d^2\vartheta^\ast}{dy^2}\right.\\ &\displaystyle -&\displaystyle \left. 2\left(\frac 1{\vartheta^\ast}\frac{d\vartheta^\ast}{dy}\right)^2 \right) \left(\nu^\ast\frac{\partial^2\varphi}{\partial x^2}-\frac{\partial^2\varphi}{\partial y^2}\right)=0.{}\\ \end{array} \end{aligned} $$
$$\displaystyle \begin{aligned} \vartheta^\ast=\left\{ \begin{array}{ll} \vartheta,\quad &\mathrm{plane stress,}\\ \vartheta/(1-\nu^2),\hfill &\mathrm{plane strain.} \end{array} \right. \end{aligned}$$
$$\displaystyle \begin{aligned} \frac{1}{\vartheta} \frac{d\vartheta}{dy}=c_0=\mathrm{const}, \end{aligned}$$
or, which is the same,
$$\displaystyle \begin{aligned} \vartheta=\vartheta_0\exp c_0y,\qquad 0<\vartheta_0=\mathrm{const}, \end{aligned}$$
then Eq. (32) appears to be with constant coefficients and can be treated by classical methods.
When imposing
$$\displaystyle \begin{aligned} \frac 1\vartheta\frac{d^2\vartheta}{dy^2}-2\left(\frac 1\vartheta\frac{d\vartheta}{dy}\right)^2=0 \end{aligned}$$
or, which is the same,
$$\displaystyle \begin{aligned} \vartheta=\frac 1{c_1+c_2y},\quad c_1+c_2y>0\qquad c_1,c_2=\mathrm{const}, \end{aligned}$$
then Eq. (32) yields
$$\displaystyle \begin{aligned} \left\langle (c_1+yc_2)\nabla^2+c_2\frac\partial{\partial y} \right\rangle \nabla^2\varphi=0. \end{aligned}$$
The latter equations can be easily integrated under the assumption that φ is a harmonic function ∇2φ = 0. The simplified equations can be derived for various kinds of representations of the material moduli by elementary functions in a similar fashion.

Note that representation of the Young modulus in the form of an exponential function of both in-plane coordinates \(E=E_0\exp (c_xx+c_yy)\), where cx, cy = const, allows for reducing Eq. (23) to the one with constant coefficients, which then can be treated by a variable-separation 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 piecewise-constant 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 discrete-layer approach (Ramirez et al, 2006). Obviously, the solution constructed in such manner is more accurate if its layer-wise 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).



  1. Airy BD (1863) On the strains in the interior of beams. Philos Trans R Soc Lond 153:49–79CrossRefGoogle Scholar
  2. Ambatsumyan SA (1970) Theory of anisotropic plates. Strength, stability, and vibration. Technomic, StamfortGoogle Scholar
  3. Boley BA, Weiner JH (1960) Theory of thermal stresses. Wiley, New YorkzbMATHGoogle Scholar
  4. Carslaw HS, Jaeger JC (1959) Conduction of heat in solids. Clarendon Press, OxfordzbMATHGoogle Scholar
  5. 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
  6. Goursat M (1898) Sur l’équation ΔΔu = 0. Bull Soc Math France 26:236–237zbMATHGoogle Scholar
  7. Guo LC, Noda N (2007) Modeling method for a crack problem of functionally graded materials with arbitrary properties – piecewise-exponential model. Int J Solids Struct 44:6768–6790CrossRefGoogle Scholar
  8. Hashin Z (1964) Theory of mechanical behavior of heterogeneous media. Appl Mech Rev 17(1):1–9Google Scholar
  9. Hetnarski RB, Eslami MR (2009) Thermal stresses – advanced theory and applications. Springer, DordrechtzbMATHGoogle Scholar
  10. Hilton H (1952) Thermal stresses in bodies exhibiting temperature-dependent elastic properties. J Appl Mech 19:350–354Google Scholar
  11. Jha DK, Kant T, Singh RK (2013) A critical review of recent research on functionally graded plates. Compos Struct 96:883–849Google Scholar
  12. Kolchin GB (1971) Computation of the structural elements from elastic nonhomogeneous materials. Kartya Moldovenyasce, Kishinau [in Russian]Google Scholar
  13. 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
  14. Kovalenko AD (1969) Thermoelasticity: basic theory and applications. Wolters-Noordhoff Pub, GroningenzbMATHGoogle Scholar
  15. Kupradze VD (1979) Three-dimensional problems of the mathematical theory of elasticity and thermoelasticity. North-Holland Publ. Co., AmsterdamGoogle Scholar
  16. Lekhnitskii SG (1968) Anisotropic plates. Gordon and Breach, New YorkGoogle Scholar
  17. Lekhnitskii SG (1981) Theory of elasticity of an anisotropic body. Mir Pub., MoscowzbMATHGoogle Scholar
  18. Maugin GA (1993) Material inhomogeneities in elasticity. Chapman and Hall, LondonCrossRefGoogle Scholar
  19. 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
  20. Meleshko VV (2003) Selected topics in the history of the two-dimensional biharmonic problem. Appl Mech Rev 56(1):33–85. https://doi.org/10.1115/1.1521166 CrossRefGoogle Scholar
  21. Michlin S (1935) Le problème plan de la théorie statique d’élasticité. Publ Ints Séism 65:1–82Google Scholar
  22. Miyamoto Y, Kaysser WA, Rabin BH, Kawasaki A, Ford RG (1999) Functionally graded materials: design, processing and applications. Springer, BostonCrossRefGoogle Scholar
  23. Mortensen A, Suresh S (1995) Functionally graded metals and metal-ceramic composites: part 1 processing. Int Mater Rev 40(6):239–265CrossRefGoogle Scholar
  24. Muskhelishvili NI (1977) Some basic problems of the mathematical theory of elasticity. Noordhoff, LeydenCrossRefGoogle Scholar
  25. Noda N (1991) Thermal stresses in materials with temperature-dependent properties. Appl Mech Rev 44(9):383–397CrossRefGoogle Scholar
  26. Olszak W (1959) Non-homogeneity in elasticity and plasticity. Pergamon Press, New YorkzbMATHGoogle Scholar
  27. Olszak W, Rychlewski J, Urbanowski W (1962) Plasticity under non-homogeneous conditions. Adv Appl Mech 7:131–214MathSciNetCrossRefGoogle Scholar
  28. Podstrigach YS, Lomakin VA, Kolyano YM (1984) Thermoelasticity of bodies of nonhomogeneous structure. Nauka, Moscow [in Russian]zbMATHGoogle Scholar
  29. Rabin BH, Shiota I (1995) Functionally gradient materials. MRS Bull 20(1):14–18CrossRefGoogle Scholar
  30. Radu A (1966) Problema lui Saint–Venant pentru bare neomogene. An Şti Univ “Al. I. Cuza” Iaşi, Secţ. 1-a Mat 12:415–428Google Scholar
  31. Radu A (1968) Sur la déformation plane d’un corps élastique isotrope non-homogène. Bul L’Acad Polon Sci. Sér Scie Tech 16(2):91–100Google Scholar
  32. 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
  33. Southwell RV (1938) Castigliano’s principle of minimum strain-energy, and the conditions of compatibility for strain. In: Timoshenko S (ed) 60th anniversary volume. The Macmillan Company, New York, pp 211–217Google Scholar
  34. Suresh S, Mortensen A (1997) Functionally graded metals and metal-ceramic composites: part 2 thermomechanical behaviour. Int Mat Rev 42(3):85–116CrossRefGoogle Scholar
  35. 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
  36. Swaminathan K, Naveenkumar DT, Zenkour AM, Carrera E (2017) Stress, vibration and buckling analyses of FGM plates – a state-of-the-art review. Compos Struct 120:10–31CrossRefGoogle Scholar
  37. Tanigawa Y (1995) Some basic thermoelastic problems for nonhomogeneous structural materials. Appl Mech Rev 48(6):287–300CrossRefGoogle Scholar
  38. Teodorescu PP (2013) Treatise on classical elasticity. Theory and related problems. Springer, DordrechtCrossRefGoogle Scholar
  39. 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
  40. Timoshenko SP (1953) History of strength of materials: with a brief account of the history of theory of elasticity and theory of structures. McGraw-Hill, New YorkzbMATHGoogle Scholar
  41. Timoshenko SP, Goodier JN (1951) Theory of elasticity. McGraw-Hill Book Co., Inc., New YorkzbMATHGoogle Scholar
  42. Tokovyy YV (2014) Direct integration method. In: Hetnarski RB (ed) Encyclopedia of thermal stresses, vol 2. Springer, Dordrecht, pp 951–960CrossRefGoogle Scholar
  43. Tokovyy YV, Ma CC (2019, accepted) Elastic analysis of inhomogeneous solids: history and development in brief. J Mech, Springer, DordrechtGoogle Scholar
  44. 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

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  • Yuriy Tokovyy
    • 1
  1. 1.Pidstryhach Institute for Applied Problems of Mechanics and MathematicsNational Academy of Sciences of UkraineLvivUkraine

Section editors and affiliations

  • Sergey Alexandrov
    • 1
  1. 1.Institute for Problems in MechanicsRussian Academy of SciencesMoscowRussia