## Abstract

This paper presents fundamental solutions for an infinite space of one-dimensional hexagonal quasi-crystal medium, which contains a penny-shaped or half-infinite plane crack subjected to two identical thermal loadings on the upper and lower crack lips. In view of the symmetry of the problem with respect to the crack plane, the original problem is transformed to a mixed boundary problem for a half-space, which is solved by means of a generalized method of potential theory conjugated with the newly proposed general solutions. When the cracks are under the action of a pair of point temperature loadings, fundamental solutions in terms of elementary functions are derived in an exact and complete way. Important parameters in crack analyses such as stress intensity factors and crack surface displacements are presented as well. The underlying relations between the fundamental solutions for the two cracks involved in this paper are discovered. The temperature fields associated with these two cracks are retrieved in alternative manners. The obtained solutions are of significance to boundary element analysis, and have an important role in clarifying simplified studies and serving as benchmarks for computational fracture mechanics can be expected to play.

## 1. Introduction

In 1984, Shechtman, the Nobel Prize winner of chemistry 2011, discovered quasi-crystals (QCs), which are aperiodic but ordered structural forms between crystals and glasses [1]. In the past three decades, the study of QCs has been one of the focuses of condensed matter physics [2]. Owing to the desirable properties, such as low friction coefficients, low adhesion and high wear resistance [3], QCs have a promising potential as effective reinforcement particles in a ductile matrix [2,4]. As a matter of fact, the tribological properties and storage modulus of polymer matrix composites reinforced by AlCuFe powder (a kind of QC) have been enhanced significantly compared with those of the polymer matrix, as evidenced by Belin-Ferre *et al.* [5] and Brunet *et al.* [6]. Furthermore, QCs can be employed to fulfil sealing purpose due to their low level of porosity [7]. Consequently, an engineering application of QC can be expected, to be used as components in nuclear storage facilities, where the thermal effect should be taken into consideration [2].

According to Landau theory [8], a phonon field and a phason field are needed to describe the elasticity of QCs, which are generally anisotropic. Furthermore, if the thermal effect is taken into account, the temperature and elastic fields are coupled in the deformation of QCs, hence an extra mathematical challenge to scientific studies. For instance, one-dimensional QCs with point groups 6*mm*, 62_{h}2_{h}, and 6*m*_{h}*mm*, whose atoms are arranged quasi-periodically in one plane and periodically in the direction normal to the former plane, bear a property of transverse isotropy [9,10]. These intrinsic characteristics, namely multi-field coupling and anisotropy, of one-dimensional hexagonal QCs are reflected in their constitutive laws. Referred to the Cartesian coordinate system (*x*,*y*,*z*) with *z*-axis perpendicular to the isotropic plane, the constitutive relations are of the following forms [2,11]:
1.1and
1.2where *T* represents the change in temperature and *T*=0 corresponds to a state free of phonon stress components *σ*_{ii} (no summation); *τ*_{ij} and *H*_{ij} are, respectively, the phonon and phason stress components; *u*_{x}(*u*_{y},*u*_{z}) and *w*_{z} are phonon displacement in the *x*-(*y*-,*z*-) direction and phason displacement, respectively; *β*_{i} denotes the thermal constants; *c*_{ij}, *K*_{i} and *R*_{i} are phonon, phason and phonon–phason coupling elastic constants. It is interesting that the constitutive laws (1.1) and (1.2) are mathematically similar to these of thermo-electro-elastic media [12,13].

Crack analysis is a long-standing problem in solid mechanics. Up to now, various papers on crack analyses for QCs have been published. For example, Peng & Fan [14] derived the general solutions in terms of four quasi-harmonic functions for one-dimensional hexagonal QCs and presented the physical quantities on the crack plane for a penny-shaped crack by solving the dual integral equations; Peng & Fan [15] studied crack and indentation problems for one-dimensional hexagonal QCs using Fourier series and Hankel transform methods. By means of perturbation theory characterizing the coupling effect of phason and phonon fields, Peng & Fan [16] presented the analytical solutions for a circular crack embedded in a half-infinite two-dimensional decagonal QC. By means of complex variable method, the interaction between a half-infinite plane crack and a line dislocation in decagonal QCs were characterized in terms of local stress intensity factors, energy release rate (ERR) at the crack tip and the Peach–Koehler force acting on the line dislocation by Wang & Zhong [17]. For one-dimensional hexagonal QCs, Li & Fan [18], taking advantage of conformal transformation in complex variable analysis, developed analytical solutions for stress intensity factors. Modelling an elliptical hole or a crack by a plane problem in cubic QCs, Gao *et al.* [19] used complex potential to seek the explicit expressions for stress intensity factors, crack open displacements and strain ERR. Based on the Stroh-type formulism for anti-plane deformation, four cracks originating from an elliptical hole in one-dimensional hexagonal QCs were investigated by Guo & Lu [20] solving Cauchy integral equations. Based on the Dugdale model, the crack tip plasticity at the crack front were estimated by various scholars [21,22,23]. Recently, Fan *et al.* [24] presented fracture theory of QCs concerning with linear, nonlinear and dynamic fracture problems. The state of the art of the studies on crack problems can be founded in the review paper [25], the monograph [2] and the references cited there.

It should be pointed out that most of the previous works are planar [17,19,20] or axisymmetric [14–16], and all the physic quantities are therefore independent of one spatial coordinate. This may not reflect the three-dimensional nature associated with the problem in practice. To the best of the author's knowledge, no non-asymmetric solutions for crack problems, which are much more involved than planar and axisymmetric problems, have been reported yet. Furthermore, the thermal effect is also beyond the scope of the previous analyses. The difficulty in solving three-dimensional problems may result from the anisotropy and phonon–phason coupling effect of QCs.

This paper aims to develop fundamental solutions of non-axisymmetric crack problems in the framework of thermo-elasticity of one-dimensional hexagonal QCs. On the basis of newly developed three-dimensional general solutions [26], the potential theory method initially proposed by Fabrikant [27,28] is generalized to crack problems of QC. A new simple layer potential (SLP) is introduced to account for the effect of the phason field. For planar cracks of any configuration, one boundary integral equation and two boundary integro-differential equations are established by using the boundary conditions and the properties of SLP. For a penny-shaped crack and a half-infinite crack, which are subjected to two identical point temperature loads on the upper and lower crack lips, corresponding fundamental solutions are explicitly expressed in terms of elementary functions. The intrinsic links between these two sets of fundamental solutions are observed through a limiting procedure. Singularities in phonon and phason stresses are discussed and corresponding stress intensity factors are obtained. The present fundamental solutions are exact without introducing any hypothesis, and consequently they can play an important role in clarifying the simplified analyses and serving as benchmarks for computational fracture mechanics.

## 2. Static thermo-elastic general solutions of one-dimensional quasi-crystals

Assume that the aforementioned one-dimensional QC is in equilibrium and that the temperature field is in a steady state. Hence, the following equilibrium equations in terms of the generalized displacements and temperature variation should be satisfied [26]
2.1a
2.1b
2.1c
2.1d
and
2.1ewhere *k*_{11} (*k*_{33}) is the coefficient of thermal conductivity, and the operator Δ=∂^{2}/∂*x*^{2}+∂^{2}/∂*y*^{2} is the planar Laplacian.

It is seen that the influence of body forces in the phonon field, the conservative component of the inner self-action shown in the study of Mariano [29] to exist, associated with the phason field, and the sink and source in the temperature field are not taken into account, in the equilibrium equations (2.1). According to Mariano [29] and Colli & Mariano [30], the absence of a conservative self-action in the standard theory of quasi-crystal may lead to non-physical results. However, the topic is not discussed further here and the common assumptions are accepted, as in the recent studies for static problems [24,31] and dynamic problems [24,32].

For the partial differential equations in (2.1), Li & Li [26] recently developed the following general solutions:
2.2where *α*_{ij} are material constants specified in appendix A, and *Ψ*_{i}(*i*=0,1,…,4) are quasi-harmonic functions satisfying
2.3with *z*_{i}=*zs*_{i}. The scaling factors *s*_{i} (*i*=0,1,…,4) characterize the anisotropy degree of the one-dimensional hexagonal QC. Here, and *s*_{i} (*i*=1,2,3) with a positive real part are eigenvalues of the characteristic equation (A1) specified in appendix A.

Differentiation of generalized displacements and temperature with respect to the spatial coordinates gives rise to the extended stress components as following:
2.4where (without definition elsewhere), *Λ*=∂/∂*x*+*j*∂/∂*y* , and the constants *γ*_{ij} are listed in appendix A as well.

According to Hu *et al.* [10], one-dimensional hexagonal QCs with point groups 6*mm*, 62_{h}2_{h}, and 6*m*_{h}*mm* are transversely isotropic, i.e. *s*_{i} being distinct. The general solutions (2.2) and (2.4) are applicable for those QCs with transverse isotropy.

Worthy to mention is that the general solutions (2.2) express the generalized displacement and temperature simultaneously, instead of solving the temperature and displacements sequentially as in the traditional treatment. Owing to this merit, the potential theory method can be employed to seek non-axisymmetric solutions to some non-classic crack problems with thermal effect involved. This has been firstly point out by Chen *et al.* [33] and further clarified by Li *et al.* [34].

## 3. Potential theory method for planar crack problems

Consider a planar crack contained in an infinite space of one-dimensional hexagonal QC, with crack surfaces parallel to the isotropic plane. For simplicity, the cracked region (denoted by *S* hereafter) is assumed to be located in the plane *z*=0 (symbolized by *I*) and the origin of the coordinate system lies in the planar crack *S*. A pair of identical thermal loads *Θ* are applied on the upper and lower crack surfaces, as shown in figure 1. It is seen that the problem is symmetric with respect to the crack plane. Consequently, the problem can be converted to the following mixed boundary value problem of the half-space *z*≥0:
3.1

To extend the potential theory method to the crack problem of one-dimensional hexagonal QC, it is assumed that
3.2where *d*_{ji} are constants to be determined, and
3.3In (3.3), the kernels *Ξ*_{1}, *Ξ*_{2} and *Ξ*_{3} of the potentials *Φ*_{j}(*j*=1−3) denote the crack surface phonon displacement *u*_{z}(*x*,*y*,0), the crack surface phason displacement *w*_{z}(*x*,*y*,0) and the gradient of temperature ∂*T*/∂*z*|_{z=0}, respectively. Hereafter, *R*(⋅,⋅) represents the distance between two indicated points; for instance, *R*(*M*,*N*_{0}) is the distance between the points *M*(*x*,*y*,*z*) and *N*_{0}(*x*_{0},*y*_{0},0). Without specification elsewhere, the surface element d*S*_{0} is equal to d*x*_{0}d*y*_{0}. It should be pointed out that *Φ*_{1}, *Φ*_{2} and ∂^{2}*Φ*_{3}/∂*z*^{2} are SLPs with the following properties:
3.4

Following a similar procedure as shown in Chen *et al.* [12] for planar cracks in thermo-electro-elastic medium, the constants can be determined from (3.4) and the conditions prescribed in the last two lines in (3.1) as following:
3.5where *k* ranges from 1 to 3, and *δ*_{1k} is the Kronecker delta. Satisfaction of the boundary conditions in the first line of (3.1) gives rise to
3.6with (*m*=1−2,*k*=1−3), both *N*(*x*,*y*,0)∈*S* and *N*_{0}(*x*_{0},*y*_{0},0)∈*S*. With the help of the third of (3.6), the first two of (3.6) are recast to the following two integro-differential equations:
3.7where
3.8

It can be readily verified that 3.9

The third of (3.6) and (3.7) are, respectively, boundary integral and integro-differential equations governing the crack problem in thermo-elasticity of one-dimensional hexagonal QCs. It is noted that (3.6) has the same mathematical structure as that for the contact problem, whereas (3.7) is similar to the governing equation for the crack problem, in the theory of pure elasticity. For cracks with an irregular configuration, (3.6) and (3.7) can be generally solved by numerical methods, such as the boundary element method. When the crack has a special shape, analytical solutions to these equations can be derived with the help of the results available in the literature [27,33,35,36]. This is the topic of the next section, where a penny-shaped crack and a half-infinite plane crack are considered.

## 4. Fundamental thermo-elastic solutions for crack problems

### (a) Penny-shaped crack

Consider a penny-shaped crack of radius *a*, with its centre coincident with the origin of the cylindrical coordinate (*ρ*,*ϕ*,*z*). In this case, the cracked region is symbolized by
with two representative points *N*(*ρ*,*ϕ*,0) and *N*_{0}(*ρ*_{0},*ϕ*_{0},0), and the area element thus becomes d*S*_{0}=*ρ*_{0} d*ρ*_{0} d*ϕ*_{0}.

Solutions to (3.6) and (3.7) can be obtained as [27]
4.1where £(⋅) and *η*(⋅,⋅) are two operators specified by
4.2

Substituting (4.1) back to (3.3), one can obtain the potential functions in integral forms for any distributed thermal loads. When the crack is subjected to a point temperature at *N*_{0}, i.e. *Θ*(*N*)=*Θ*_{δ}*δ*(*ρ*−*ρ*_{0})*δ*(*ϕ*−*ϕ*_{0})/*ρ*, the derivatives of the potentials *Φ*_{j}(*j*=1−3), which are necessary to construct the fundamental thermo-elastic field of QCs, can be determined by referring to Chen *et al.* [33] and Fabrikant [27]. Owing to the mathematical structures of (3.2) and (3.3), the corresponding fundamental solutions are decomposed into two parts, the first of which results from *Φ*_{1} and *Φ*_{2}, and the latter attributes to *Φ*_{3}.

Without details, the first part of the fundamental field, denoted by a superscript (1), reads as
4.3where *U*=*u*_{x}+*ju*_{y}, *μ*_{i}=*d*_{1i}*M*_{1}+*d*_{2i}*M*_{2} and *f*_{i}(*z*) (*i*=1,2,…5) are given in appendix B.

The second part, which is indicated by a superscript (2), can be derived as
4.4where *g*_{i}(*z*) (*i*=1,2,…5) are defined in appendix B as well.

The complete thermo-elastic field in QC is the sum of physical quantities in (4.3) and (4.4). Furthermore, in view of the relations one may derive that 4.5

### (b) Half-infinite plane crack

For a half-infinite plane crack, which is symbolized by
the solutions to (3.6) and (3.7) turn out to be [35]
4.6where £*(⋅) is an operator defined as
4.7Inserting (4.6) into (3.3), one can derive the potentials associated with the half-infinite plane crack subjected to arbitrary thermal loads. For a point-temperature load exerted at *N*_{0}, i.e. *Θ*(*N*)= *Θ*_{δ}*δ*(*x*−*x*_{0})*δ*(*y*−*y*_{0}), one can obtain the corresponding fundamental solutions following a similar procedure described by Li [36] for half-infinite plane crack contained in elastic media. Again, the fundamental solutions are divided into two parts, identical to the case of the penny-shaped crack. The first part pertinent to *Φ*_{1} and *Φ*_{2} can be derived as
4.8where the functions (*i*=1−5) are listed in appendix B.

The second part associated with *Φ*_{3} turns out to be
4.9where the functions (*i*=1−5) are specified also in appendix B.

Summarizing the variables in (4.8) and (4.9) leads to the fundamental solutions of the half-infinite crack problem. Similarly, the temperature field in this case reads 4.10

## 5. Crack tip singularity and crack surface displacement

Crack tip singularity and crack surface displacement are important parameters in crack analyses. Since all the field variables have been explicitly expressed in terms of elementary functions, these parameters can be easily obtained for penny-shaped and half-infinite plane cracks.

### (a) Penny-shaped crack

By virtue of the following properties [27]: one can readily arrive at Furthermore, the following results can be deduced: with

From (4.3) and (4.4), one may derive that
5.1With the aid of (3.9), the following relations are derived:
5.2with *m*=1,2. Thus, the generalized stress components on the crack plane, specified in (5.1), can be simplified as
5.3It is seen that the boundary conditions *σ*_{zz}|_{z=0}=0 and *H*_{zz}|_{z=0}=0 on the crack surfaces have been satisfied as a posterior check. At the crack front (*ρ*>*a*), both *σ*_{zz} and *H*_{zz} are the sum of a singular term and a non-singular term. This has been also observed in the crack analyses for thermo-piezoelectric media [12].

On defining the stress intensity factors one can deduce that 5.4

Crack surface displacements *Ξ*_{1}(*ρ*,*ϕ*) and *Ξ*_{2}(*ρ*,*ϕ*), which are quantitatively equal to one half of the corresponding crack open displacements, can be directly derived from (4.1) by setting *Θ*(*N*)=*Θ*_{δ}*δ*(*ρ*−*ρ*_{0})*δ*(*ϕ*−*ϕ*_{0})/*ρ*
5.5which can be also retrieved from (4.3) and (4.4) with the help of (3.5) and (3.9).

### (b) Half-infinite plane crack

By means of the properties [35] the following relation can be derived: one can further obtain that with

With the aid of (5.2), the generalized normal stresses can be derived as
5.6As in the case of penny-shaped crack, satisfaction of the boundary conditions *σ*_{zz}|_{z=0}=0 and *H*_{zz}|_{z=0}=0 on the crack surfaces has been posteriorly checked. At the crack front (*y*=0), both *σ*_{zz} and *H*_{zz} are singular.

Defining the stress intensity factors
one can immediately obtain that
5.7Parallel to the penny-shaped crack, the crack surface displacements for half-infinite plane crack can be directly obtained from (4.6) wherein *Θ*(*N*_{0}) is taken to be *Θ*_{δ}*δ*(*x*−*x*_{0})*δ*(*y*−*y*_{0})
5.8

## 6. Intrinsic links between fundamental solutions

It is seen that the fundamental solutions (4.8) and (4.9) for half-infinite plane cracks have the same mathematical structures as (4.3) and (4.4) for penny-shaped cracks, respectively, with the exception of the functions as shown in appendix B. There must be some intrinsic links between the solutions for these two cracks of different geometries. In fact, solutions for a half-infinite plane crack can be extracted from those for a penny-shaped one, through a limiting procedure.

The first step of that procedure is to place the Cartesian coordinate system (*x*,*y*,*z*) at the penny-shaped crack edge such that
Then the radius of the crack, namely *a*, is treated as a variable. If the radius *a* approaches to infinite, a penny-shaped crack is converted to a half-infinite plane crack. This procedure is graphically shown in figure 2. In the limiting case, one can derive the following relations without any difficulty:
6.1

On the basis of (6.1), one can further obtain that 6.2

Starting from (6.1) and (6.2), one can come to the following conclusions: 6.3Consequently, the fundamental solutions for penny-shaped cracks tends to these for half-infinite plane crack in the limiting case .

It should be pointed out that such a treatment has been proved appropriate, as evidenced by various scholars studying the crack problem without thermal effect [37–39].

## 7. Verification of the temperature field

### (a) Penny-shaped crack

When the crack is subjected to a distributive temperature load, the corresponding coupled field can be obtained via integrating the fundamental solutions. For example, when the crack is under the action of thermal loads uniformly distributed over the crack surface, i.e. *Θ*(*x*,*y*)=*Θ*_{0}, one can derive the temperature by integrating (4.5)
This integral is very tedious but basic, and turns out to be
7.1

The temperature field given in (7.1) can be retrieved by means of Hankel transform method. In fact, the temperature field should be figured out first and a prior in the classic treatment by solving the following boundary value problem: 7.2subjected to 7.3

The Hankel transform of the order *n* of a function *f*(*ρ*) is defined [40]
7.4and its inverse by
7.5where *J*_{n}(⋅) is the Bessel function of the first kind of order *n*.

If *T* is assumed to be solution of (7.2), then its Hankel transform of order zero satisfies the ordinary differential equation
This means that
7.6where *A*(*ξ*) and *B*(*ξ*) are functions to be determined. Since the temperature and thermal flux approach zero at infinite, *B*(*ξ*) should vanish and (7.6) is reduced to
Therefore, the temperature field is of the form
7.7Substitution of (7.7) into (7.3) leads to the following dual integral equations:
7.8Equation (7.8) can be recast to the following form by setting *t*=*ρ*/*a*, *p*=*aξ* and *ψ*(*p*)=*pA*:
7.9

If one assumes that
7.10then the second integral equation in (7.9), which is the inverse Hankel transform of *ψ*(*p*) has been satisfied automatically [40]. Meanwhile, the first one becomes an Abel-type integral equation
whose solution is
7.11Inserting (7.11) into (7.10), one can obtain that
7.12Substituting (7.12) to (7.7) yields
7.13with *ζ*_{4}=*z*_{4}/*a*. According to Maugis [40], the integral in (7.13) is equal to
7.14Inserting (7.14) into (7.13) give rise to
which is exactly (7.1).

It should be pointed out that the non-axisymmetric temperature solution (4.5) cannot be retrieved via integral transform methods.

### (b) Half-infinite plane crack

The temperature field (4.10) associated with half-infinite plane crack can be derived in an alternative way. In contrast to the integral transform methods, the potential theory method is employed in the present case. To solve the mixed boundary value problem 7.15with 7.16assume that 7.17with .

It can be readily verified that the temperature in (7.17) satisfy (7.15) and the second condition in (7.16). To meet the first condition in (7.16), one has
whose solution reads
7.18Substituting (7.18) back to (7.17) and referring to appendix A of Fabrikant & Karapetian [35], one may obtain that
On letting *Θ*(*x*,*y*)=*Θ*_{δ}*δ*(*x*−*x*_{0})*δ*(*y*−*y*_{0}), it is ready to get
which is identical to (4.10).

## 8. Numerical results and discussions

The complete fundamental field in the cracked infinite spaces and some important physical parameters on the cracked planes have been explicitly expressed in terms of elementary functions. As a consequence, numerical analysis can be readily performed, once the values of the material constants are available. However, owing to the absence of the experimental data, especially those characterizing interactions between phonon and phason fields [2,32], such a goal seems not to be achieved. On the other hand, the values of the materials can be approximately estimated by referring to the results associated with isotropic QCs. Such a purpose can be fulfilled in view of the results available in the literature [32,41–46].

It has been reported that the Lamé constants of AlPdMn, a special kind of icosahedral QC, are reported to be *λ*_{0}=85 GPa and *μ*_{0}=65 GPa [42], and that the ratio between the phonon–phason coupling coefficient and the shear modulus *R*_{i}/*μ*_{0} is approximately 0.03 [44]. Furthermore, Richer *et al.* [42] also showed that the phason constants *K*_{1}=0.084 GPa and *K*_{2}=0.036 GPa. In addition, the coefficient of thermal conductivity is lower than those of conventional crystals [2,32]. In this paper, the material constants, tabulated in table 1, are adopted in the present analyses, by referring to the aforementioned works. It should be emphasized that the positive-definite conditions for the stiffness tensor, specified in table 1, have been checked. Correspondingly, the eigenvalues *s*_{i} (*i*=0,1,…,4) have the following numerical values *s*_{0}=0.9747, *s*_{1}=0.7485+0.5177*i*, *s*_{2}=0.7485−0.5177*i*, *s*_{3}=0.5070 and *s*_{4}=1.0000, which are different from each other.

For the sake of illustration, the following dimensionless quantities are introduced:
In other words, *a*, *Θ*_{δ}/*a*^{2}, *β*_{1}*Θ*_{δ}/*a*^{2} and *β*_{1}*Θ*_{δ}/*a*^{1.5} are chosen as the reference scales for spatial coordinates, temperature, stresses and stress intensity factors, respectively. It should be pointed out that the magnitude of the parameter *Θ*_{δ} should be smaller than a threshold such that the phase transition does not take place [41].

For simplicity, a pair of point thermal loadings are applied at the point (*ξ*,*η*,*ς*)=(*λ*,0,0^{±}) for the penny-shaped crack for illustration. In addition, the concern is confined only to the cracked plane, without statement elsewhere.

The validity of the present solution has been examined analytically by comparing of the temperature field in the half-space with a plane crack in §7 and by bridging the solutions for penny-shaped and half-infinite plane cracks in §6. Let us now proceed to numerically discuss the validity of the fundamental solutions. To this end, the phonon–phason coupling constants *R*_{i} are artificially set to zero. In this case, the coupling effect between the phonon and phason fields vanishes and the fundamental solutions for the penny-shaped crack should be reduced to those in the context of thermo-elasticity. This provides a useful way to check the appropriateness of the present fundamental solutions. In table 2, the present crack surface displacements are compared with their thermo-elastic counterparts predicted by Chen *et al*. [33]. An excellent agreement is observed, further validating the present solutions. It should be pointed out that, in the section, only the data in table 2 are obtained by setting *R*_{i}=0.

The variation of the dimensionless temperature *Γ* with the spatial coordinate is plotted in figure 3. It is seen that the temperature in the cracked region varies in a discontinuous way owing to the presence of the penny-shaped crack, as shown by the curve for *η*=0 in figure 3*a* and that for *ξ*=0 in figure 3*b*. In contrast, continuous variation is observed in the intact region. It is also interesting that the temperature changes significantly at the crack front. Furthermore, the temperature at the points, which are near to the thermal loading, is generally higher than that for the point far away from the external source. This makes sense from a physical point of view. Since the thermal load is exerted at the point (0.5,0,0), the temperature is symmetric with respect to *x*-axis (*y*=0), as shown in figure 3*b*. In figure 3*c* displaying the variation with the dimensionless coordinate *ζ*, comparison is made between the present solution with those proposed by Chen *et al.* [33]. The present solutions coincide with the latter. Furthermore, in the region with *ζ*=*z*/*a*≥10, the variation of temperature can be negligible (*Γ*≤0.01).

Figure 4 plots the distributions of the dimensionless normal stress *Σ*_{ζ} in the intact region of the crack plane *ζ*=0. The normalized stress *Σ*_{ζ}=−*σ*_{zz}*a*^{2}/*Θ*_{δ}*β*_{1} in the intact region is positive, which means that the decrease (increase, respectively) in temperature will lead to a tensile (compressive, respectively) stress at the crack front. This phenomenon is consistent with the previous studies [33,47]. As expected in (5.3), the normal stress *Σ*_{ζ} is singular at the crack tip as shown in figure 4*a,b,d,e*. The symmetric property inherent to the problem with respect to the *x*-axis (*η*=0) has been indirectly and directly checked (figure 4*c*,*f*). Furthermore, the normal stress is of the order of 10^{−3} for the point with *ξ*≥10 or *η*≥10.

As shown in figure 4*f*, the distribution of the normal stress *Σ*_{ζ} is very similar to that in the framework of thermo-elasticity. Indeed, after correcting the mistake involved in equation (45) in Chen *et al.* [33], the normal stress in the intact region of the crack plane *ζ*=0, in our notations, reads
8.1where the constant is defined in Chen *et al.* [33] and the superscript *e* denotes the physical quantities derived from theory of thermo-elasticity. From (5.6) and (8.1), it is evident that , *σ*_{zz}|_{z=0} and *H*_{zz}|_{z=0} are different from each other only by the coefficients. Hence, the following two parameters *α*_{1} and *α*_{2} are defined as:
8.2both of which characterize the influence of the phason field. For the particular one-dimensional QC considered in this section, *α*_{1}=0.8852<1.0 and *α*_{2}=2.2×10^{−4} . Hence, the distribution of the normal phason stress on the crack plane is similar to those plotted in figure 4, if the parameter *α*_{2} is viewed as a scaling factor. For brevity, these are not shown.

The dimensionless normal stress *Σ*_{ζ}|_{ζ=0} is singular at the crack tip. The singularity in *Σ*_{ζ}|_{ζ=0} is quantified by the dimensionless stress intensity factor as
8.3On letting *ϱ*_{0}=*x*_{0}/*a*=*λ* and *ϕ*_{0}=0, (8.3) is reduced to the following form under the current circumstance:
which are displayed in figure 5, as a function of *λ* and *ϕ*, in the polar coordinate system. It can be readily seen from (8.3) that is proportional by inversion, to the distance between the points on the edge of circle crack and the position of external loading (figure 5*b*–*d*). Hence, in particular for the parameters *λ*=0, is a circle concentric to the undeformed penny-shaped crack, as shown in figure 5*a*. The symmetry with respect to the horizontal line *ϕ*=0 and *ϕ*=*π* is satisfied as well. Worthy to mention is that the following relations also hold true:
from (5.4) and equation (47) in Chen *et al.* [33].

The dimensionless crack surface displacements is illustrated in figure 6. Similar to the normal stress in figure 4, the present crack surface displacement and its thermo-elastic counterpart varies in the same fashion. In fact, the expression of the elastic crack surface displacement can be obtained from (32) in Chen *et al.* [33] with the aid of the property of the Dirac-delta function. In our notation, it reads
8.4where is defined in Chen *et al.* [33], the superscript *e* and have been specified previously. It can be seen from (5.8) and (8.4) that these three quantities are of the same mathematical structures with the exception of the coefficients involved. Parallel to the case of the normal stress, another two parameters *ϑ*_{1} and *ϑ*_{2} are introduced
8.5which quantify the effect of the phason field as well. Numerically, *ϑ*_{1} and *ϑ*_{2} are equal to 0.9142 and 3.3542, respectively.

As pointed out in previous sections, the absence of a conservative self-action may result in non-physical results [30]. This crucial remark was theoretically foreseen firstly by Mariano [29] and then evidenced by Colli & Mariano [30], who investigated a plane of two-dimensional QCs subjected to an external load. The dimensionless displacement in the phason field at infinity would be divergent, as shown in Colli & Marino [30], a problem excluded in QC elasticity in the presence of a conservative phason self-action emerging from SO(3) invariance requirements of the external power of actions over a QC. To avoid falling into such a situation, the variations of the normal phason stress *H*_{zz} is displayed in figure 7. It is seen from figure 7*b* that *H*_{ζ} would vanish as *ζ* approaches to infinity. From figure 7*a*, the boundary conditions specified have been checked. As expected, the curve for *ξ*=0 will pass the origin (0,0), corresponding to the condition *H*_{zz}=0, and the tangential vectors of these for *ξ*=2.5 and 5 are horizontal, corresponding to the symmetry with respect to the crack plane associated with the problem. It should be pointed out that the phason displacement, similar to the normal phason stress *H*_{zz}, would be zero at infinity. Its variation is not presented in order to save space.

ERR, denoted by the symbol *Π* here, is an important parameter in fracture mechanics. For the penny-shaped crack, ERR is related to the generalized stresses (*σ*_{zz} and *H*_{zz}) and displacements (*Ξ*_{1} and *Ξ*_{2}) by [2]
where the superscript *s* denotes the asymptotic expression of the corresponding variables, and ⊗ represents the product of two functions, namely, *f*_{1}(*x*)⊗*f*_{2}(*x*)=*f*_{1}(*x*)*f*_{2}(*a*−*x*). With the help of (8.2) and (8.5), ERR is transformed to
where *Π*_{e}= represents the component of ERR due to the phonon field. For the materials concerned, , which means that *Π*≃*Π*_{e}. In other words, the component of ERR due to the phason field can be negligible. In addition, for static problems which is independent of time, the energy associated with the diffusion of phason activity certainly vanishes.

Numerical simulations for half-plane crack can also be made similar to the penny-shaped crack. To save the space, they are not shown. However, (8.2) and (8.5) still hold true for half plane cracks. Hence, *α*_{1}, *α*_{2}, *ϑ*_{1} and *ϑ*_{2}, independent of the crack geometry, are four intrinsic parameters associated with the planar crack.

## 9. Concluding remarks

In the framework of thermo-elasticity of QCs, fundamental solutions for penny-shaped and half-infinite plane cracks, which are subjected to point thermal loadings on the crack surfaces, are presented by virtue of the generalized potential theory method in conjunction with the newly developed general solutions in forms of quasi-harmonic functions [26]. Exact and complete fundamental solutions are in terms of elementary functions. Crack surface displacements and stress intensity factors, both of which are important parameters in crack analyses, are explicitly given as well.

The fundamental solutions for the penny-shaped crack and those for the half-infinite plane crack bear the same mathematical structures. As shown in §6, the latter solutions can be derived from the former through a limiting procedure. On the basis of the links between them, the validity of these two sets of solutions can be checked.

The temperature fields are derived in two different ways. For penny-shaped crack subjected to a uniform thermal load on the crack lip, the temperature field obtained by integrating the fundamental solution is retrieved by solving mixed boundary value problem via Hankel transform method. As regards half-infinite plane crack, the fundamental solution is directly verified by means of the basic result of the potential theory method.

As pointed out by Duffy [48], Ding *et al.* [49] and Gao & Ricoeur [31] for Green's functions of specific problems, the present fundamental solutions are of high significance: they themselves can serve as benchmarks for various numerical codes and simplified solutions; in addition, when distributive thermal loadings are exerted to the crack surfaces, the corresponding field variable can be obtained by integrating the present solutions.

It is noted that the present solutions make sense only for the case of distinct eigenvalues. If any two eigenvalues are identical (this is true for some isotropic QCs), the general solutions, instead of being (2.2) and (2.4), would take on a different form. Under such a circumstance (say *s*_{i}=*s*_{j}, *i*≠*j*), there are two feasible options of treating the corresponding crack problems. The first way is to apply the L’hôspital rule to the present results, by setting *s*_{i} . Such a limiting procedure can be realized either analytically and numerically as pointed out by Fabrikant & Karapetian [35]. The second approach is to seek five appropriate potentials relative to the corresponding general solutions. In view of the argument in Chen [50], the governing equations would be of the same mathematical structures as these in (3.6) and (3.7). The remaining analyses would follow a similar procedure as shown in §§4–7.

It should be pointed out that the potential theory method incorporating with the general solution newly proposed by Li & Li [26] can deal with more complicated problems, for instance, mode II and III problems, and crack subjected to heat flux. Under these circumstances, the SLPs would take in a totally different form. This will be reported in another journal paper.

## Acknowledgements

This work is supported by the National Natural Science Foundation of China (nos. 11102171 and 11242014) and by the Fundamental Research Funds for the Central University (nos SWJTU11CX069 and SWJTU11ZT15). The author is thankful to the reviewer for the constructive suggestions leading to the improvement of the former manuscript.

## Appendix A

This section is devoted to defining the constants involved in the general solutions in §2 proposed by Li & Li [26]. The eigenvalues *s*_{i}(*i*=1,2,3) are the roots with a positive real part, of the following the algebraic characteristic equation
A1where the constants involved are specified by
A2

In order to define the constants in (2.2), we need the following auxiliary quantities:
A3a
A3b
and
A3cand the parameters specified by
Thus, the constants *α*_{ij} in (2.2) are determined by
A4and *γ*_{ij} in (2.4) read
A5

## Appendix B

The functions *f*_{i}(*z*) (*i*=1,2,…,5) involved in (4.3) are defined as [27]
B1a
B1b
B1c
B1d
and
B1ewhere the over bar represents the complex conjugate of the indicated variable, and
B2

The functions *g*_{i}(*z*) (*i*=1,2,…,5) in (4.4) are specified by [27,33]
B3a
B3b
B3c
B3d
and
B3ewith *ς*=*ρ* e^{j(ϕ−ϕ0)}/*ρ*_{0}.

The functions involved in (4.8) read [35] B4a B4b B4c B4d and B4ewhere and

The functions in (4.9) are defined as [36] B5where denotes the real part of a complex variable.

- Received January 17, 2013.
- Accepted March 12, 2013.

- © 2013 The Author(s) Published by the Royal Society. All rights reserved.