The Analysis of a Cracked Material under Combined Unsymmetric Thermal Flux and Symmetric Linear Mechanical Loading

An improved partially permeable crack model is put forward to deal with the problem of a single crack embedded in an orthotropic or isotropic material under combined unsymmetric thermal ﬂux and symmetric linear mechanical loading. With the application of the Fourier transform technique (FTT), the thermoelastic ﬁeld is given in a closed form. Numerical results show combined unsymmetric linear thermal ﬂux, symmetric linear mechanical loading, and dimensionless thermal conductivity, and the coeﬃcient has inﬂuences on fracture parameters. For the improved partially permeable crack, the mode II stress intensity factor and the energy release rate might be zero or positive under combined unsymmetric thermal ﬂux and symmetric linear mechanical loading. Therefore, closure of the crack tip region need not be considered under combined unsymmetric thermal ﬂux and symmetric linear mechanical loading when making use of fracture parameters as a criterion.


Introduction
An elastic solid's expansion and contraction from changes in temperature is inhibited for the external constraints and the mutual constraints between the internal parts of the solid, resulting in thermal stress [1,2]. So, it is essential to address the thermoelastic field of cracked solid in practical engineering [3][4][5]. A great many of studies have been published to consider the fracture behaviors of various cracks which are embedded in an orthotropic solid under thermal loading [6][7][8][9][10][11][12][13][14]. By making use of the J-integral, Wilson and Yu obtained the stress intensity factor [11]. Chen and Zhang addressed the thermoelasticity problem of cracked orthotropic solid [12]. Rizk investigated the problem of the cracked orthotropic semifinite material [15]. Wu et al. applied the Fourier transform technique to compute the II stress intensity factors of two collinear cracks under antisymmetrical linear thermal flux [16]. Afterwards, Wu et al. gave the solution of a single crack in the closed form under quadratic thermomechanical loading by using the Fourier transform technique [17].
In the abovementioned studies, the partially permeable crack model becomes more widely available [18][19][20][21]. According to the mathematical intuition, an improved partially permeable crack model in Figure 1(a) can be raised as where Q 1c and Q 1 denote the heat flux per thickness to the crack surface and initial heat flux; h c stands for the thermal conductivity on the crack region. e coefficient ε is regarded as a constant and in line with the complex and real situation, i.e., the nonoptimal cases of complicated crack surface. In general, ε proves to be negative or positive relying on the thermoelastic field.
An improved partially permeable crack model is presented to consider the problem of a single crack embedded in an orthotropic or isotropic material under combined unsymmetric linear thermal flux and symmetric linear mechanical loading. With application of the Fourier transform technique, the thermoelastic partial differential equations (PDE) are converted to singular integral equations. e thermoelastic field is given in the closed form by solving singular integral equations. e obtained results reveal physical quantities that have influences on fracture parameters for cracked solid under combined unsymmetric linear thermal flux and symmetric linear mechanical loading. e constitute equations are given based on the state of plane stress [22].

Problem Statement
where where σ yy and σ xy are the components of stress, E xx and E yy represent Young's moduli, T is the temperature, v xx and v yy denote Poisson's ratios, u and v indicate the components of elastic displacement, c 66 � G xy represents the shear modulus, and α xx and α yy are the coefficients of linear expansion. Using the following equation, one obtains c 66 Based on the Fourier heat conduction, one has where Q x and Q y are the components of the heat flux; λ x and λ y are the coefficient of heat conduction. Furthermore, based on equilibrium equation, one has Using the thermal equilibrium equation, one has where In this study, the crack-face boundary conditions are used as follows.
e superscripts I and II denote the physical quantities of y > 0 and y < 0 region, respectively. Q 1 and σ 1 are the prescribed constants. By using the improved partially permeable crack model, the crack-face boundary conditions can be given as Based on the antisymmetry of thermal flux and symmetry of mechanical loading, only half of the field (i.e., x > 0) is considered under combined unsymmetric thermal flux and symmetric linear mechanical loading. where Moreover, factors such as stresses, elastic displacements, temperature, and thermal flux across the crack-free region of the horizontal x-axis comply with the following conditions.

Solution Procedure
3.1. Temperature Field. Due to equation (11) is irrelevant to the elastic strain, the temperature field can be solved out first.
Considering the antisymmetry of thermal flux, the solution of equation (11) is written with application of FTT.
where g ± (ξ) is unknown and will be solved. δ + � 1 and δ − � −1 denote y > 0 and y < 0, respectively. From (9), one gets Making use of the second relation in equation (21), one gets By the aid of the first relation in equations (21) and (17), one attains For the solvation of equations (25) and (26), the auxiliary function c(x) is defined as Using inverse Fourier transform, one has Substituting equations (28) into (26), one gets From the known result [23], Equation (29) can be expressed as follows: Mathematical Problems in Engineering 3 Equation (31) is a singular integral equation including the Cauchy kernel [24], and the solution can be depicted as Considering the following condition, After some computations, one has C � 0. Equation (32) can be given as In addition, with application of equations (27) and (34), the temperature change on the crack is given as which is consistent with that in [16] for a single crack of length 2a.

Elastic Field.
In order to solve equations (7) and (8), u I,II 1 (x, y) and v I,II (x, y) are written as [25] u I,II (x, y) � u I,II where u I,II 1 (x, y) and v I,II 1 (x, y) correspond to the general solutions under T I,II (x, y) � 0. u I,II 2 (x, y) and v I,II 2 (x, y) are the peculiar solutions under thermal flux. With application of Fourier transform technique, u I,II 1 (x, y) and v I,II 1 (x, y) can be written as and g ± j (ξ) will be solved. ρ j , (j � 1, 2)(Reρ j > 0) are chosen as the roots of the following character equation. where Furthermore, u I,II 2 (x, y) and v I,II 2 (x, y) are chosen as Substituting equations (41) and (42) into (7) and (8), one has g * ± (ξ) where Based on equations (2) To obtain the explicit solution of the problem, two parts are divided. One part is what is induced by mechanical loading (−σ 1 |x|/a). e other part is what is resulted from thermal flux (−Q 1 x/2a 2 ). e solution procedure is neglected for simplicity under mechanical loading (−σ 1 |x|/a). e elastic displacement on the crack is obtained with application of the Fourier transform technique (FTT) [17].
Furthermore, one can obtain the stresses field as follows: en, the elastic field under linear thermal flux will be given in the closed form. With application of thermal flux, it fits the condition as follows.
With application of equations (48) and (45b), one arrives at 2 j�1 c 12 With application of equation (15) and the first relation in equation (20), the dual integral equations are given: To give the solvation of equations (50) and (51), the auxiliary function χ(x) can be defined as Applying equation (52) and inverse Fourier transform, one has Utilizing equations (49) and (53), one obtains where with With the knowledge of equations (30), and (56) can be rewritten as With application of equation (22) and the inverse Fourier transform, one has (60) Based on the singular integral theory with the Cauchy kernel, equation (59) is obtained as

Crack Tip Field. Using equations (18) and (35), one has
In particular, when ε � 0, one has e quantity E c � h c /λ y stands for the dimensionless thermal conductivity. In equation (65), when E c ⟶ ∞ or e observations reveal εQ 1 can be considered to be created by the thermal flux or other large loading such as mechanical loading.
Furthermore, the stress intensity factor is of much importance for analysis of the behavior of cracked solid. e mode I stress intensity factor and the mode II stress intensity factor are given as Next, the corresponding intensity factors to characterize opening crack displacement are defined as With application of equations (47) e energy release rate (G) has an important effect on the analysis of crack growth, and the energy release rate is defined as [25] G � lim

Numerical Results
is section presents a research on the influence of combined unsymmetric linear thermal flux, symmetric linear mechanical loading, the dimensionless thermal resistance and the coefficient on the heat flux to the crack surface, the mode II stress intensity factor (K II ), and the energy release rate (G). For convenience, the isotropic material constants are chosen as shown in [26] (Table 1). Figure 2 shows Q 1 /Q 1c versus E c with x/a � 0.25, 0.5, 0.75, 0.99 for ε � 0.01. e cases of E c � 20, E c � 40, E c � 50, and E c � 60 indicate the heat conduction is 19 times, 39 times, 49 times, and 59 times, respectively, stronger than thermal conductivity of the orthotropic material. E c � 0 or E c ⟶∞ corresponds to the thermally fully impermeable or permeable case. Figure 3 presents Q 1 /Q 1c versus ε with x/a � 0.25, 0.5, 075, 0.99 for E c � 1. It is seen from Figure 3 when ε increases, Q 1 /Q 1c increases. e case of E c � 1 correlates with the situation that the crack region is identical to that of the isotropic material. Figure 4 presents K II versus Q 1 . In the case of purely unsymmetric thermal flux, K II remains zero. Figure 5 shows G, as computed using equation (72), for the improved partially permeable crack with a variety of crack lengths. In the case of purely unsymmetric thermal flux or symmetric tension stress, the value of G may turn zero or positive. It means that symmetric linear mechanical loading rather than unsymmetric thermal flux has an effect on G.
It is revealed that dimensionless thermal conductivity, the coefficient, and the size of crack have great effects on the heat flux per thickness to the crack surface, the mode II stress intensity factor, and the energy release rate. In the case of purely unsymmetric thermal flux, the mode II stress intensity factor and the energy release rate is zero. e mode II stress intensity factor and the energy release rate may be zero or positive under combined unsymmetric thermal flux and symmetric linear mechanical loading. e mode II stress intensity factor and the energy release rate may be zero or positive under combined unsymmetric thermal flux and symmetric linear mechanical loading. Contrary to the situations under symmetric thermal flux and symmetric mechanical loading [17], such consideration need not be taken when using the energy release rate as a criterion.

Conclusions
An improved partially permeable crack model is put forward to deal with the problem of a crack embedded in an orthotropic or isotropic solid under combined unsymmetric   thermal flux and symmetric linear mechanical loading.
Using the Fourier transform technique, the jumps of temperature, elastic displacements on the crack, and so on are obtained in the closed form. Numerical results show combined unsymmetric linear thermal flux, symmetric linear mechanical loading, dimensionless thermal conductivity, and the coefficient have great influences on fracture parameters. For the extended partially permeable crack, the mode II stress intensity factor and the energy release rate may be zero or positive under combined unsymmetric thermal flux and symmetric linear mechanical loading. erefore, closure of the crack tip region need not be considered under combined unsymmetric thermal flux and symmetric linear mechanical loading when making use of fracture parameters as a criterion.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interest.