Abstract
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.
1. 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 [3–5]. 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 [6–14]. 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 [18–21]. According to the mathematical intuition, an improved partially permeable crack model in Figure 1(a) can be raised aswhere and denote the heat flux per thickness to the crack surface and initial heat flux; stands for the thermal conductivity on the crack region. The coefficient 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, 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. The thermoelastic field is given in the closed form by solving singular integral equations. The 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.
2. Problem Statement
A single crack embedded in an orthotropic or isotropic material under combined unsymmetric linear thermal flux and symmetric linear mechanical loading is shown in Figures 1(a) and 1(b).
(a)
(b)
The constitute equations are given based on the state of plane stress [22].wherewhere and are the components of stress, and represent Young’s moduli, is the temperature, and denote Poisson’s ratios, and indicate the components of elastic displacement, represents the shear modulus, and and are the coefficients of linear expansion. Using the following equation,one obtains
Based on the Fourier heat conduction, one haswhere and are the components of the heat flux; and are the coefficient of heat conduction. Furthermore, based on equilibrium equation, one has
Using the thermal equilibrium equation, one haswhere
In this study, the crack-face boundary conditions are used as follows.
The superscripts I and II denote the physical quantities of y > 0 and y < 0 region, respectively. and 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.
3. 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 is unknown and will be solved. and denote and , 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 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:
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 aswhich is consistent with that in [16] for a single crack of length 2a.
3.2. Elastic Field
In order to solve equations (7) and (8), and are written as [25]where and correspond to the general solutions under . and are the peculiar solutions under thermal flux. With application of Fourier transform technique, and can be written asand will be solved. are chosen as the roots of the following character equation.where
Furthermore, and are chosen as
Substituting equations (41) and (42) into (7) and (8), one haswhere
Based on equations (2)–(4), (37), (38), (41), and (42), the stresses are rewritten as
To obtain the explicit solution of the problem, two parts are divided. One part is what is induced by mechanical loading . The other part is what is resulted from thermal flux . The solution procedure is neglected for simplicity under mechanical loading . The 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:
Then, 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
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 can be defined as
Applying equation (52) and inverse Fourier transform, one has
Utilizing equations (49) and (53), one obtains
With application of equations (53)–(55), one obtainswherewith
With the knowledge of equations (30), and (56) can be rewritten as
With application of equation (22) and the inverse Fourier transform, one has
Based on the singular integral theory with the Cauchy kernel, equation (59) is obtained as
After some calculations, E = 0. Equation (61) can be given as
With application of equation (52), one obtains
From equations (45c), (54), (55), and (63), one can obtain the shearing stresses as follows.
3.3. Crack Tip Field
Using equations (18) and (35), one has
In particular, when = 0, one has
The quantity stands for the dimensionless thermal conductivity. In equation (65), when or , or . The observations reveal 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. The 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), (64), (46), (63), (67), (68), (69), and (70), one has
The energy release rate (G) has an important effect on the analysis of crack growth, and the energy release rate is defined as [25]
4. Numerical Results
This 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 (KII), and the energy release rate (G). For convenience, the isotropic material constants are chosen as shown in [26] (Table 1).
Figure 2 shows versus with , 0.5, 0.75, 0.99 for = 0.01. The cases of = 20, = 40, = 50, and = 60 indicate the heat conduction is 19 times, 39 times, 49 times, and 59 times, respectively, stronger than thermal conductivity of the orthotropic material. = 0 or ⟶∞ corresponds to the thermally fully impermeable or permeable case.
Figure 3 presents versus with , 0.5, 075, 0.99 for = 1. It is seen from Figure 3 when increases, increases. The case of = 1 correlates with the situation that the crack region is identical to that of the isotropic material.
Figure 4 presents KII versus Q1. In the case of purely unsymmetric thermal flux, KII 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. 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. 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. 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.
5. 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. 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.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest.
Acknowledgments
The work was supported by Hebei University Scientific Research Foundation for Higher-Level Talents (521100221019), the Project Funded by China Postdoctoral Science Foundation (2019M662946), and the Ningxia Key Research and Development Program (Introduction of Talents Project) (2020BEB04039).