TOKOVYY Yuriy，BOIKO Dmytro，GAO Cunfa

1.Pidstryhach Institute for Applied Problems of Mechanics and Mathematics，National Academy of Sciences of Ukraine，Lviv 79060，Ukraine；2.Department of Applied Mathematics，Institute of Applied Mathematics and Fundamental Sciences，Lviv Polytechnic National University，Lviv 79000，Ukraine；3.State Key Laboratory of Mechanics & Control of Mechanical Structures，Nanjing University of Aeronautics and Astronautics，Nanjing 210016，P.R.China

Abstract: By making use of the direct integration method，an exact analysis of the general three-dimensional thermoelasticity problem is performed for the case of a transversely isotropic homogeneous half-space subject to local thermal and force loadings. The material plane of isotropy is assumed to be parallel to the limiting surface of the halfspace. By reducing the original thermoelasticity equations to the governing ones for individual stress-tensor components，the effect of material anisotropy in the stress field is analyzed with regard to the feasibility requirement，i.e.，the finiteness of the stress field at a distance from the disturbed area. As a result，the solution is constructed in the form of explicit analytical dependencies on the force and thermal loadings for various kinds of transversely isotropic materials and agrees with the basic principles of the continua mechanics. The solution can be efficiently used as a benchmark one for the direct computation of temperature and thermal stresses in transversely isotropic semi-infinite domains，as well as for the verification of solutions constructed by different means.

Key words：three-dimensional problem；analytical solution；transversely isotropic composites；semi-infinite model；force and thermal loadings；finite stress distributions

0 Introduction

Transversely isotropic materials can be regard?ed as the simplest example of materials exhibiting spatial anisotropy. Due to the specific micro-struc?ture（i.e.，the structure of molecular lattice or specif?ic features of the material composition），the elastic and thermo-physical properties of a macro-volume remain the same within a certain plane，which is known as the plane of isotropy［1-2］，but are different from the ones in the direction that is perpendicular to the plane mentioned. The symmetry of such kind is typical for a number of natural and composite ma?terials，e.g.，the ones with lattices of hexagonal syn?gony［3］，as well as angle-ply laminates［4-5］or fiber composites with hexagonal packing［6］，etc.，which can be regarded as homogeneous transversely isotro?pic solids after utilization of certain homogenization techniques［7-10］.

Despite the apparent simplicity of the trans?versely isotropic material in comparison with the ma?terials of rather general anisotropy，the dissimilarity of effective properties presents a certain challenge for the analysis of the relevant three-dimensional problems of mechanics. Being involved in the consti?tutive equations of the corresponding mathematical model，the elastic and thermoelastic moduli of a transversely isotropic material are thereby affecting the coefficients of the governing equations of ther?moelasticity theory［11］. Thus，the form of a solution to the corresponding governing equations strongly depends on the interrelations between the material moduli. This presents a challenge for the general analysis of thermal stresses in transversely isotropic solids due to the fact that a solution is to cover inter?relations between material moduli of any kind.

This problem becomes even more involved when analyzing local effects of force or thermal load?ings. One of the basic models frequently used for such analysis deals with an elastic half-space whose surface is acted upon by locally distributed im?pacts［12-13］. Besides the satisfaction of the boundary conditions，the solutions of such elasticity and ther?moelasticity problems are to exhibit asymptotic be?havior that is vanishing at a distance from the loaded zones，which meets the feasible requirements of the Saint-Venant’s principle［14］. Because of uncertain in?terrelation between the coefficients of the governing equations（which，in turn，depend on the material moduli）for any given kind of a transversely isotro?pic material，ensuring the required asymptotic be?havior of the solution remains an important and yet unanswered challenge（e.g.，the reviews［15-18］）espe?cially for analytical methods based on the application of potential functions of higher differential rate.

An efficient technique for the analysis of aniso?tropic and inhomogeneous solids has been devel?oped on the basis of the direct integration meth?od［19-20］. This technique has also been extended onto the cases of three-dimensional problems for trans?versely isotropic solids［15-17］. It implies a reduction of the original thermoelasticity equations to a set of governing equations for individual stress-tensor com?ponents with accompanying local and integral bound?ary conditions. The fact of getting an individual equation for a stress-tensor component can be effec?tively used for the more accurate evaluation of the stress asymptotic for transversely isotropic semi-infi?nite solids.

This paper presents an attempt towards the construction of analytical solutions to a three-dimen?sional thermoelasticity problem for a transversely isotropic half-space，which meets the original equa?tions along with the given boundary conditions and is vanishing at a distance from the loaded zones of the limiting surface or inner heat-sources.

1 Formulation of the Problem

Consider a three-dimensional problem of the thermoelasticity theory for a transversely isotropic half-space (x，y，z)∈R2×R+in the dimensionless Cartesian coordinate system. Let the plane of isotro?py be parallel to the limiting surface z=0. Within the framework of the quasi-static formulation in the absence of body forces， the problem is gov?erned［1，2，14］by the equilibrium equations

the strain-compatibility equations

and the constitutive ones

where σξη=σηξ，εξη=εηξare the stress- and straintensor components，ξ，η={ x，y，z }；E，Ezand G，Gzare the in- and out-of-plane（with respect to the plane of isotropy）Young and shear moduli，respec?tively；ν，νzand α，αzare the transversely isotropic Poisson ratios and the linear thermal expansion coef?ficients；symbolsimply obtain?ing two and one more equation from the one they follow by the cyclic and mutual，respectively，per?mutations of indices and variables.

The stationary temperature field T(x，y，z) can be determined from the following heat-transfer equa?tion［21］

under the general boundary condition

where c and czare the heat conductivity coefficients within the plane of isotropy and transversely；Q(x，y，z)and T0(x，y)are the given densities of in?ternal heat sources and thermal boundary function，both vanishing at x2+y2+z2→+∞ and x2+y2→+∞，respectively. Constant parameters a and b indicate the type of boundary condition（5）：If a ≠0 and b=0，Eq.（5）is the Dirichlet condition imposing the temperature on the boundary；if a=0 and b ≠0，Eq.（5）is the Neumann condition impos?ing the heat flux through the boundary；if a ≠0 and b ≠0，Eq.（5）is the third-kind boundary condition covering，for example，the heat-exchange through the boundary［22］.

Our intent is to construct an analytical solution to the formulated thermoelasticity Eqs.（1—3）un?der the temperature field determined from the heatconduction problem Eqs.（4，5）and the force load?ings

imposed on the boundary z=0，where p(x，y) and qξ(x，y) are given functions vanishing at x2+y2→+∞. Making use of equilibrium equation（1）and boundary conditions（6）for the tangential stress yields the condition

for the partial derivative by z of the normal stress.For the complete determination of the stress field，the strain-compatibility condition in the following in?tegral form［11］is to be used.

2 Solution Method

To separate variables in the foregoing thermo?elasticity and heat conduction equations and bound?ary conditions，we employ the Fourier double-inte?gral transform［23］

where sxand syare the transform parameters with re?spect to x ，y and i is the imaginary unit.

Making use of transform（9）allows for solving the problems（4）and（5）in the Fourier mapping domain in the form as follows

By implementing the technique［15，24］，we can reduce the formulated thermoelasticity problem（1—3）to the following system of governing equations in terms of stresses

where

and

Having applied transform（9）to Eq.（11）and conditions（6）and（7），we obtain the following boundary-value problem

where

Note that the elastic moduli involved in expres?sions（18）meet the following physical constraints［25］

The form of a solution to Eq.（16）along with conditions（17）on the limiting plane z=0 and the decreasing condition at the points of infinity z →+∞，strongly depends on the interrelations be?tween the coefficients（18），as the eigenvalues of Eq.（16）can be given in the following form

In the context of constraints（19）and expres?sions（18），we can conclude that a2＞0 for all physically allowed transversely isotropic moduli so that the eigenvalues（20）can be：（A）real and dis?similar（forand a1＞0），（B）real and mul?tiple（forand a1＞0，which is the case，e.g.，of isotropic materials），（C）imaginary multi?ple（forand a1＜0），and（D）complex and dissimilar. Note that if in the latter case the ei?genvalues（20）were represented in the following form

Then due to the obvious equalitiesand，we necessarily conclude that，and

The complete analytical solution to the formu?lated thermoelasticity problem is，obviously，to cov?er all of the foregoing cases A—D of the eigenvalues which results，particularly，in the character of the so?lution’s asymptotic behavior at the points infinity.

For example，in case A of dissimilar real eigen?values（20），an analytical solution to Eq.（16），which meets boundary conditions（17）and is limit?ed at z →+∞，can be given in the following form

where λj∈R are the eigenvalues computed by Eq（.20）atand a1＞0，and

where j=1，2.

To determine the normal stressin the map?ping domain of transform（9），we use Eq.（13），which takes the following form

in the mapping domain of transform（9）. By differ?entiating the latter equation twice and making use of Eq.（16），we can obtain the following

Now，putting Eqs.（25，26）into Eq.（24）yields

where

An analytical solution to Eq.（27）can be given in the following form

where

Substituting Eq.（23）into Eq.（28）yields the following

where

and A is an arbitrary constant of integration. The lat?ter one can be determined by means of condition（8），which takes the following form

in the mapping domain of transform（9）. Making use of Eq.（3）along with Eq.（15）and boundary conditions（6）and（7）yields

where

To derive an expression for the total stressappearing in condition（30），we substitute Eq.（23）into Eq.（25），which yields

where

Now，using Eqs.（29，31）together with condi?tion（30）allows for eliminating constant A. As a result，the stresscan be given in the following form

where

In such a manner，the transversal and normal stresses，and，are found in the map?ping domain of transform（9）in Eqs.（23，32），re?spectively. The stresscan be found in a simi?lar form by making use of formula（A1）presented in Appendix.

In order to derive the tangential stress-tensor components，we use Eq.（14），which can be pre?sented as

Making use of Eqs.（33，34） along with Eqs.（15，25）yields the expressions of the tangential stresses in terms of the key ones in the forms given by formulae（A2—A4）in Appendix.

In case B of the multiple real eigenvalues（20），coefficients（18）of Eq.（16）are to meet the following conditions：and a1＞0. In view of Eq.（18），these conditions imply

and

Hence，a solution to Eq.（16）which meets con?ditions（17）and vanishes at infinitely distant points can be given in the following form

where

and j=1，2.

In case C of the multiple imaginary eigenvalues（20），coefficients（18）meet the conditions=a2=a4and a1＜0.Then Eq.（35）holds，while

Although Eq.（37）may seem unrealistic，there is however a practical possibility of the existence of such materials in view of Eq.（19）.

A solution to Eq.（16）in this case can be given as

where

The constants of integrationin Eq.（38）can be eliminated by making use of the van?ishing conditions at the points of infinity and the boundary conditions（17）. As a result，the stress takes the following form

In case D of the dissimilar complex eigenvalues（20），their real ljand imaginary ?jparts indicated in Eq.（21），meet necessary conditions（22）. By de?noting λ=|lj| and μ=|?j|，we can construct the van?ishing solution to Eq.（16）with conditions（17）in the form as follows

where

and

Then the rest of the stresses for cases C and D can be found by the routine used for the foregoing cases A and B.

After the stress-tensor components are found in the mapping domain of the Fourier integral trans?form（9），they can be restored in the physical do?main by making use of the inverse transform

realized either numerically or by an analytical mean.Similarly，the physical value of the temperature can be computed by applying Eq.（39）to Eq.（10）.

3 Numerical Examples and Discus?sion

To verify the efficiency of the constructed solu?tion，we compute the stresses in various transverse?ly isotropic materials（Table 1）under the local nor?mal force loading Eq.（6）of the following profile

where p0is a constant in the dimension of stresses.Note that the properties of materials presented in rows 1—3 of Table 1 correspond to case A，while the material in the 4th row corresponds to case D.

Table 1 Elastic moduli of the considered transversely isotropic materials[16, 26]

Fig.1 presents the full-field analysis of the stress σzznormalized by the parameter p0at the cross-section x=0. As we can observe，the com?puted stress exactly satisfies the boundary condi?tions（6，40），and vanishes at a distance from the loaded segment of the surface z=0. It is also nota?ble that the material properties play a crucial role in the quantitative behavior of the stress.

Fig.1 Full-field distributions of normal stress σzz/p0 at x=0 for different transversely isotropic materials under loading (40)

Fig.2 Normal stress σzz/p0 at x=0, y=0 for carbon fiber(curve 1), ceramic PZT-4 (curve 2),composite 60%fiber (curve 3), and hexagonal zinc (curve 4)

The effect of material properties in the stress distribution is clearly pronounced in Fig.2 present?ing the same stress under the center of the loaded zone：x=y=0. It is also notable that the curves presented in this figure are orthogonal（in the mean?ing of differential geometry）to the surface at z=0，which in view of the equilibrium Eq.（1）indicates the zero boundary conditions for the tangential stress. The latter conclusion agrees with the bound?ary conditions（40）and Eq.（7）.

Consider the results of computation of thermal stresses in a transversely isotropic half-space made of hexagonal zinc（Table 1，the 4th row），for which［16］：c=cz=124[W/(K ?m)]，α=5.818×10-6[1/K ]， and αz=15.350×10-6[1/K ]. As?sume the surface z=0 of the half-space to be free of force loadings，i.e. p=qx=qy=0. Instead，the surface undergoes the thermal impact（5），where

and T0is a constant parameter in the dimension of temperature.

Fig.3 presents the full-field distribution of tem?perature （10） in the physical domain of inverse transform（39）under condition（41）and the corre?sponding thermal stress σzz. For the considered steady-state case，the disturbance of temperature occurs in the vicinity of the heated zone of the sur?face and decreases rapidly when moving away from it. The stress meets homogeneous boundary condi?tion and is also locally disturbed over the area under the heated segment of the surface. In Fig.4，stress σzzis shown in some characteristic cross-sections of the half-space in the vicinity of the thermally affect?ed area.

Fig.4 Distributions of the stress σzz/T0 in the characteristic cross-sections of the thermally loaded half-space(x=0)

4 Conclusions

This paper presents an analytical technique for exact thermoelastic analysis of a transversely isotro?pic half-space subject to local thermal and force load?ings. The technique is based on the application of the direct integration method which allows for the reduction of the original thermoelasticity problem to a set of governing equations for the individual stresstensor components. The equations are accompanied by the corresponding local and integral boundary conditions.

Making use of the proposed solution technique allows for capturing explicit dependencies between the applied thermal and force impacts and the in?duced stress field for any possible case of interrela?tions between the effective moduli of transversely isotropic material. Special attention is given to the correct asymptotic of the constructed solutions when moving away from the zones where the loadings were applied.

The constructed solutions can be used for the analysis of thermal and force impacts on the elastic semi-infinite composites made of materials exhibit?ing transversal isotropic properties. Due to its explic?it form，it may serve as an efficient tool in solving inverse problems［19］，as well as the verification of re?sults gained by either numerical or semi-analytical means.

AppendixFormulae for computation of the normal and tan?gential stresses by the known key stresses

The normal stress

The tangential stresses