Heat Concentration around a Cylindrical Interface Crack in a Composite Tube

Cracks always form at the interface of discrepant materials in composite structures, which influence thermal performances of the structures under transient thermal loadings remarkably. The heat concentration around a cylindrical interface crack in a bilayered composite tube has not been resolved in literature and thus is investigated in this paper based on the singular integral equation method. The time variable in the two-dimensional temperature governing equation, derived from the non-Fourier theory, is eliminated using the Laplace transformation technique and then solved exactly in the Laplacian domain by the employment of a superposition method. The heat concentration degree caused by the interface crack is judged quantitatively with the employment of heat flux intensity factor. After restoring the results in the time domain using a numerical Laplace inversion technique, the effects of thermal resistance of crack, liner material, and crack length on the results are analyzed with a numerical case study. It is found that heat flux intensity factor is material-dependent, and steel is the best liner material among the three potential materials used for sustaining transiently high temperature loadings.


Introduction
As a kind of star material, fiber reinforced resin matrix (FRRM) composite tubes with adhesive layers have been extensively used in weight-sensitive engineering applications, such as automobile, aircrafts, and gun tube. FRRM composites are used to reduce the weight of a structure, while the adhesive layer can usually protect the structure from the severely thermal and mechanical loadings. However, due to the manufacturing issue and the discrepancy of material properties between the FRRM composite and adhesive layer, their interface will sometimes debond and form an interface crack which affects the structure performance remarkably. The continuity of the heat flow in the structure under transient thermal loadings is destroyed by the crack, which causes heat concentration around the crack tip. Various aspects of composite structures have been studied, such as the constitutive equations of composite structures [1], the thermoelastic responses under transient thermal loadings [2], and the way to enhance the rigidity of composite plates [3].
Back in 1965, Sih [4] studied the thermal disturbance problem in an infinite region with lines of crack by adopting the complex variable method and found that heat flux processed the inverse square root singularity around the crack tip. Tzou [5] confirmed the singularity behavior by introducing the intensity factor of a temperature gradient. Heat conduction of plates containing multiple insulated cracks under arbitrary Neumann thermal conditions was investigated by Chen and Chang [6]. As to the interface crack, Chao and Chang [7] obtained the exact temperature solutions for a composite media made of dissimilar anisotropic materials with a crack located at the interface. The similar problem with the consideration of interstitial medium filled between dissimilar anisotropic materials was investigated by Shiah and Shi [8]. Chiu et al. [9] calculated the temperature distribution in an infinitely functionally graded (FG) plane containing an arbitrarily oriented crack. Zhou et al. [10] considered a partially insulated interface crack of a homogeneous orthotropic substrate coated by a graded orthotropic layer. And, an interface crack in a bilayered magnetoelectroelastic material under heat flow loadings was analyzed by Gao and Noda [11]. Wang et al. [12] discussed the effect of thermal resistance of a cohesive zone on the thermal fracture of mode II crack under static thermal loadings.
Recently, non-Fourier heat conduction theories have found their applications on the thermal study of cracked structures. These new theories were created to eliminate the drawback of the classical Fourier one that the thermal wave propagates at an infinitely large speed. Among these theories, the heat conduction model proposed by Cattaneo [13] and Vernotte [14] (C-V model) has experimentally proved that this model could accurately predict the finite thermal propagation speed in processed meat and some materials with nonhomogeneous inner structures [15,16]. Some other non-Fourier heat conduction models including the commonly used dual-phase-lag (DPL) model were reviewed in Ref. [17]. The generalized thermoelastic theory, extended from the C-V model, has been used to study the thermomechanical coupling effects under various loading conditions, such as pulsed laser heating [18]. Chen and Hu [19] got the disturbed temperature field of an infinite half-plane coated by a thin layer containing an insulated interface crack using the C-V model, while the DPL model was used to conduct the crack problem in a half-plane by Hu and Chen [20]. The transient non-Fourier temperature and corresponding thermoelastic fields of a half-plane [21] or strip [22] with a crack parallel to the surface subjected to thermal shock were investigated using the singular integral equation method. Xue et al. [23] revisited the thermoelastic problem of a crack in a strip based on the memory-dependent heat conduction model. Fu et al. [24] conducted the thermal analysis of a sandwich structure with a cracked foam core based on the non-Fourier heat conduction model. Cracks lead to a kind of mathematical problem termed as mixed-boundary-value problem; another similar mathematical problem termed as mixed-initial-boundary-value problem can be found in [25][26][27], in which thermoelastic responses of micropolar porous bodies are considered.
As to cylindrical structures, Guo et al. [28] obtained the temperature field and stress intensity factor (SIF) for a penny-shaped crack using the DPL heat conduction theory. An insulated penny-shaped crack in an elastic half-space was handled by employing the fractional-order non-Fourier heat conduction model [29]. The transient thermoelastic fields of isotropic or transversely isotropic cylinders containing a circumferential crack were investigated based on different kinds of non-Fourier models, and the corresponding thermal SIFs were calculated therein [30][31][32][33][34]. It should be noted that the circumferential crack, spreading along the radial direction, did not disturb the temperature field as the temperature loading was applied on the lateral surfaces.
On the contrary, the cylindrical crack spreading along the axial direction would disturb the heat flow in the radial direction remarkably. However, the investigation on cylindrical cracks among most of the works published to date is limited to mechanical loading conditions such as tension and torsion [35][36][37][38][39], and the work on its thermal analysis is rare. The thermoelastic study of cylindrical cracks under static thermal loadings was conducted by Itou [40]. Recently, Fu et al. [41] analyzed the disturbed temperature field of a FG cylinder containing a cylindrical crack.
To the authors' knowledge, the heat conduction in the composite tubes affected by the cylindrical interface crack has not been resolved even using the classical Fourier theory. The current work is designed to study the heat concentration behavior of a cracked bilayered composite tube with the adoption of non-Fourier heat conduction theory. The C-V heat conduction model and the problem with corresponding boundary conditions are described in Section 2, and the solution procedure based on the singular integral equation is illustrated in Section 3 in detail. Section 4 gives the expressions of temperature field and heat flux intensity factor (HFIF). Moreover, the effects of crack face resistance, liner material, and crack length are analyzed in Section 5. Finally, the main conclusions of this paper are summarized in Section 6.

Problem Formulation
A bilayered composite tube with an inner radius R 0 and outer radius R 2 is located in the cylindrical coordinate O − rφz, as illustrated in Figure 1. The sudden change of external and internal environments of the tube will lead to a dynamic heat flow in the structure, which could be disturbed remarkably by the cylindrical crack occupying the region −c < z < c at the interface r = R 1 . The tube with uniform initial temperature T 0 is assumed infinitely long and will not deform under thermal loadings. The outer layer (2) is the fiber reinforced resin matrix composite with fibers parallel to the axis and randomly distributed in the resin, while the material of the inner layer (1) is a thermal protective liner and has significantly higher strength.
The heat flux in the radial and axial directions within the C-V model takes the form.
in which, q r and q z are the heat flux in the radial and axial direction, respectively, t is the time, and T is the temperature. λ r and λ z are, respectively, the thermal conductivity in the transverse plane and longitudinal direction for the  Advances in Mathematical Physics transversely isotropic composite layer, and λ r = λ z holds for the isotropic inner layer. Please note that the thermal properties of fiber reinforced composite material will take its macroscopically equivalent parameters. The phase lag of heat flux τ, which can be interpreted as the time lag needed to excite heat flow at a position when a temperature gradient loading is applied on that position, is an intrinsic material property with the unit of time and varies from 10 −14~1 0 2 s for different materials. It can be easily seen from Equation (1) that τ = 0 reduces the C-V model to the Fourier one. Without considering the heat source, the energy conservation equation reads where c P and ρ are specific heat capacity and mass density, respectively. The temperature governing equation can be obtained by eliminating the heat flux from Equation (2) and Equation (1) as where d = λ/ρc P is the thermal diffusivity. Equation (3) also indicates that the speed of thermal wave in the radial direction is C CV = ffiffiffiffiffiffiffiffi d r /τ p . The thermal boundary conditions for the cracked tube can be written as The convective heat transfer conditions between surfaces of the tube and its corresponding environments with temperature T a and T b are shown in Equations (4) and (5), where h i and h o are heat transfer coefficients. Equations (6) and (7) describe the thermal conditions at the interface. It can be easily seen that the mixed boundary condition, given by Equations (7) and (8), brings much mathematical complexity. It should be noted that a zero value of thermal resistance V represents a perfectly conductive crack, while an insulated crack generates for an infinitely large value of V.

Solution of the Mixed-Boundary-Condition Problem
In this section, the Laplace transform and Fourier transform as well as the superposition method are adopted to solve the problem. It should be mentioned that the singular integral equation method is originally proposed by Erdogan et al. [42] to study the mechanical fracture problems, and this paper extends the method to investigate the mixedboundary-condition problem in thermal fields. The mismatch of thermal properties of the material on the sides of crack improves the mathematical complexity and leads to a material-property-dependent HFIF.
In order to normalize the equations, the following nondimensional parameters are introduced: It should be mentioned that other dimensional parameters with the same unit as those in Equation (9) could be normalized accordingly. The parameters with the superscript "0" are reference properties.
Substituting the nondimensional parameters in Equation (9) into Equation (3), the governing equation can be rewritten as where υ ðiÞ2 = d ðiÞ z /d ðiÞ r . When zero initial temperature change and rate of temperature change are assumed, Laplace transformation is applied to Equation (10) to eliminate the time variable t ′ , as in which, k ðiÞ 1 = sð1 + τ ðiÞ′ sÞd ð0Þ /d r ðiÞ and s is the Laplace variable.

Advances in Mathematical Physics
Equations (4)-(8) are treated with similar nondimensionalization and Laplace transformation operations, and the boundary conditions are rewritten as The governing equation (11) under the boundaries (12)- (16) can be solved using the superposition method. The first problem (P1) can be described as inner and outer surfaces of an uncracked tube applied with inhomogeneous boundary conditions. And the second problem (P2) is a cracked tube without thermal loadings applied on the surfaces. The mathematical expressions of P1 and P2 are, respectively, as follows: P1: : The actual results of temperature and heat flux are addition of the solution of P1 and P2 asT ðiÞ′ =T The solution of P1 can be easily obtained as where k ðiÞ 2 = −ðλ ðiÞ /λ ð0Þ Þð1/ð1 + τ ðiÞ′ sÞÞ and I n ðÞ and K n ðÞ represent the nth-order modified Bessel functions of the first kind and the second, respectively. The unknowns S ðiÞ 1 and S ðiÞ 2 can be obtained from PS = Q, in which, S is defined as and the nonzero elements of 4 × 4 matrix P and 4 × 1 vector Q are Advances in Mathematical Physics The application of the Fourier transformation, to Equation (18) leads to where k ðiÞ 3 = υ ðiÞ2 ξ 2 + k ðiÞ 1 and ξ is the Fourier transformation variable. Thus, the temperature and heat flux in the doubletransformed domain can be solved as The corresponding temperature and the heat flux in the Laplace domain are then calculated using the Fourier inversion transform, where the imaginary number j = ffiffiffiffiffi ffi −1 p . Using the boundary equations (18)- (21), the unknowns in Equation (32) are linked as in which The last unknown coefficient S ð2Þ 4 can be obtained using the mixed boundary conditions (22) and (23). A function, similar to dislocation density function in mechanical fields, is defined as This gradient function is crucial for obtaining the twodimensional temperature field caused by the crack with the adoption of the singular integral equation method.
Thus, from Equation (22), one could have Equation (38), referred as the single-valuedness condition, is obtained from the zero value of temperature jump at the crack tips jz′j = 1. It also reflects the fact that the temperature jump starts from zero at z ′ = −1, then increases to a specific value within the crack, and finally decreases to zero again at the other tip z ′ = 1. Substituting Equations (32) and (34) into Equation (36) in which
The Gauss-Jacobi integration formulas in [31] can be adopted to solve Equations (41) and (38) numerically as in which After getting f ðη, sÞ, the four unknown coefficients in the general solution of P2 given by Equation (2)

Transient Temperature Field and Heat Concentration
Addition of Equation (24) and Equation (2) results in the transient temperaturẽ in which, S Similar to the stress intensity factor, heat flux intensity factor (HFIF) is introduced to judge the heat concentration degree around the crack tip quantitatively, as Using the parameter K q ′ = K q ffiffi c p /ðλ ð0Þ rr T 0 Þ, HFIF can be nondimensionalized in the Laplace domain as Substituting Equations (25) and (33) into Equation (56) leads toK in which, f ð1, sÞ is calculated from f ðη μ , sÞ using the interpolation method and Y = 1 It can be clearly seen from Equation (58) that the HFIF is dependent on thermal properties of the material on both sides of the crack. Finally, the transient temperature field and HFIF in the time domain can be numerically transformed using the Laplace inversion technique given in [43].

Numerical Results
In this section, the solution procedure is verified first. Then, the effects of crack face resistance, liner material, and crack length on the transient temperature field and HFIF of a 6 Advances in Mathematical Physics composite tube containing an interface crack are analyzed in the following three subsections. The composite tube with initial temperature T 0 = 300 K has the size R 0 = 0:006 m, R 1 = 0:008 m and R 2 = 0:01 m. By assuming the fully conductive inner and outer surfaces as h i = h o = ∞, the non-Fourier heat conduction of the cracked tube with sudden temperature rise applied on the inner surface T a ðtÞ = 330HðtÞ K is investigated based on the C-V model, in which, HðtÞ is the Heaviside step function. The inner layer can be made of steel, silicon carbide ceramic, or tantalum, while the outer layer is carbon fiber reinforced resin matrix composite. Table 1 lists properties of all the material. It should be mentioned that the accurate value of phase lag of the carbon fiber reinforced composite cannot be found in experimental tests, and it is assumed according to Ref. [41]. Phase lag of heat flux is in the order of picoseconds for most of metals due to the phonon-electron coupling effect. This parameter can be as large as 10 2 s [37] for nonhomogeneous materials, as the thermal flow needs time to circumvent inclusions or pass through the interface of different materials. The experimental evidence of non-Fourier heat conduction in some materials at room temperature can also be found [44,45].
The macroscopically equivalent material properties of the composite are calculated as [51] in which, the parameters with subscripts "f" and "m," respectively, refer to those of fiber and matrix. Moreover, the fiber fraction c 1 is taken to be 0:2 in this section. The structure profile described here is used in this whole section except Subsection 5.1 designed for validation with reference.

Validation of the Solution Procedure.
Before starting the analysis of the parameters, the solution procedure presented above should be validated. As stated in Introduction, the C-V heat conduction model is accurate to describe the temperature distribution in processed meat and other materials measured by especially designed experiments. However, these kinds of experiment are not available for tubes with or with-out cracks. The numerical calculation of the transient temperature field for cylindrical cracks using the Fourier or non-Fourier models is also absent among the literature. Thus, the temperature result for uncracked tubes based on the C-V model obtained by Babaei and Chen [52] is adopted for comparison. The same structure, material properties, thermal loading, and nondimensional parameters as those in Ref. [52] for homogeneous tubes are employed, while the cylindrical crack in the present model spreads along the middle plane R 1 ′ = 0:8 and is assumed insulated. Figure 2 illustrates the nondimensional temperature distribution along the radial direction at the instant t′ = 0:126 for different axial locations. The Saint-Venant principle indicates that the continuous temperature field at a position far away from the crack will not be affected, which means the same temperature distribution as that in uncracked tubes. This is verified by comparing the temperature distribution at z ′ = 10 for a cracked tube with the results in [52] for an uncracked tube. The dash-dot line depicts the temperature in the cross section across the crack face midpoints. As the crack is insulated, thermal energy will accumulate before the crack and a much higher temperature will then be obtained.

The
Effect of Thermal Resistance. The effects of crack resistance V on the temperature field and HFIF are shown in Figures 3 and 4, respectively. An infinitely large V corresponding to an insulated crack is compared with three other partially insulated cracks with V = 1 × 10 −3 km 2 sJ −1 , V = 1 × 10 −4 km 2 sJ −1 , and V = 1 × 10 −5 km 2 sJ −1 . The inner layer is made of steel, and a crack with a length of 0:006 m affects the temperature field of the tube remarkably. Figure 3(a) illustrates the temperature distribution along the interface for different values of V at the instant t = 3:184 s when the transient temperature field approaches to the equilibrium state. It is shown that the temperature jump across    Advances in Mathematical Physics the crack face exists for all the cases, while the temperature is continuous along other parts of the interface plane. Also, the temperature at the inner crack face is higher than that in the outer crack face. With the reduction of V, the temperature jump shrinks as well, and the perturbation effect of crack on the temperature fades away. Similar results were reported by Chen and Hu [19] for layered structures as well. Furthermore, the temperature histories at the crack face midpoints for the insulated crack case and almost conducting crack case (V = 1 × 10 −5 km 2 sJ −1 ) are shown in Figure 3(b). It can be seen that the temperature history performs in an obviously wave-like manner and oscillates around a steady value. Figure 3(b) also depicts the fact that the temperature jump across the insulated crack faces is much higher than that across the conducting crack faces all the while. The oscillating behavior of transient HFIF is shown in Figure 4. Similar to the temperature results, the effect of V on the HFIF indicates that the heat concentration degree reduces with the raise of thermal conduction capability of the crack.   Advances in Mathematical Physics insulated interface crack, while a tube with the same size made of pure steel is added for comparison. Please note that the phase lag of steel is so small that the non-Fourier effect is negligible for the steel tube considered in this paper, which has been verified by comparing with the obtained results when setting τ = 0 of steel manually. Thus, the results for steel tube can generally represent thermal behaviors predicted by the Fourier model. And, the crack length is taken to be c = 0:003 m. The temperature distribution along the interface of the four tubes at the instant t = 3:184 s is depicted in Figure 5, while Figure 6 shows the HFIF history. Besides the conclusions obtained from Figure 3(a), Figure 5 reveals that the composite tube with steel liner has the lowest temperature off the crack compared with the other two tubes with silicon carbide and tantalum liner. However, their temperatures at the crack midpoints show a negligible difference. Although the temperature of the steel tube is lowest, it has the highest weight as well as heat concentration degree as shown in Figure 6. It can also been observed from Figure 6 that HFIF of the steel tube increases rapidly to the maximum value without oscillating as other tubes. This difference comes from the significant non-Fourier effect of fiber reinforced resin matrix material. The oscillation behavior of the heat flux and HFIF predicted by the non-Fourier models is also described in [20,53]. Similar    Figure 7, which shows the temperature distribution along the interface at the instant t = 3:184 s, that the shrink of crack length reduces the temperature jump across the crack faces. It can be imagined that the temperature will be continuous across the crack faces when the crack shortens to a size as small as possible. Coincidentally, Figure 8 shows that the HFIF generally reduces as the crack length shrinks.
The histories of heat flux in the radial direction at the same point (r = 0:008 m, z = 0:0035 m) for different crack lengths are shown in Figure 9, in which the result at the point (r = 0:008 m, z = 10 m) far away from the crack (c = 0:003 m) is incorporated for comparison. The heat flux at the point z = 10 m will not be affected by the crack and represents the result for uncracked structure. It is seen that the heat flux from the liner to composite material decreases with the reduction of crack length, as the point picked to plot the result becomes away from the crack tip. The tiny discrepancy between the black line and red line shows that the crack c = 0:002 m is too short to affect the heat flux at the point z = 0:0035 m.
In order to show the influence of crack on the temperature field more clearly, Figure 10   10 Advances in Mathematical Physics instant when the maximum temperature happens. It is clear that heat gathers around the crack tip and temperature jump across the crack faces exists, while the temperature field approaches uniform at positions far away from the crack. Similar to the tendency shown in Figure 7, a shorter crack leads to a more homogeneous temperature field.

Conclusion
This paper studies the thermal concentration behavior of a bilayered composite tube containing a cylindrical interface crack. The two-dimensional temperature field disturbed by the crack is obtained using the singular integral equation method. By taking the crack as partially thermal insulated, the effects of crack resistance, liner material, and crack length on the temperature field and HFIF of the cracked tube are analyzed numerically. The following main conclusions are arrived: (1) Heat flux intensity factor serves as an effective parameter to describe the heat concentration degree caused by the crack quantitatively; (2) Steel is the best liner material among the three potential materials used for protecting the barrel from the transient temperature change of the inner environment, as the composite tube with steel liner has the lowest temperature and HFIF results compared with the other two liner materials; (3) The insulating degree of crack affects the heat concentration around the crack tip remarkably. The temperature jump across the crack faces is highest for fully insulated cracks, while the temperature jump and corresponding HFIF decreases when the crack becomes more conductive; (4) When a small crack grows along the interface, a severer temperature diagram with higher HFIF emerges. This indicates that the crack length in the tube should be controlled to a size as small as possible during the manufacturing process, and the crack detection of the tube in service should be conducted regularly to avoid overheating around the crack tip

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

Conflicts of Interest
The authors declare that they have no conflicts of interest.