Accepted Manuscript Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model Zhang-Na Xue, Zeng-Tao Chen, Xiao-Geng Tian PII: DOI: Reference: S0013-7944(18)30413-2 https://doi.org/10.1016/j.engfracmech.2018.08.018 EFM 6124 To appear in: Engineering Fracture Mechanics Received Date: Revised Date: Accepted Date: 19 April 2018 22 July 2018 15 August 2018 Please cite this article as: Xue, Z-N., Chen, Z-T., Tian, X-G., Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model, Engineering Fracture Mechanics (2018), doi: https:// doi.org/10.1016/j.engfracmech.2018.08.018 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Thermoelastic analysis of a cracked strip under thermal impact based on memory-dependent heat conduction model Zhang-Na Xue a a,b , Zeng-Tao Chen b* , Xiao-Geng Tian a** State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, PR China b Department of Mechanical Engineering, University of Alberta, Edmonton, AB T6G 1H9, Canada * Tel.: 1-780-48922620; zengtao.chen@ualberta.ca **Tel: +86 029 82665420; tiansu@mail.xjtu.edu.cn Abstract In this paper, by introducing memory-dependent derivative (MDD) into the Cattaneo and Vernotte (CV) heat conduction model, the thermoelastic problem of a thermally insulated crack parallel to the boundary of a strip under thermal impact loading is considered. Laplace and Fourier transforms are used to reduce the thermoelastic problem to a system of singular integral equations which are solved numerically. Numerical results are presented to illustrate the effects of time delay and kernel function on the temperature, intensity factors of temperature gradients, and stress intensity factors. The results are compared with those based on Fourier and CV model, which can be taken as two special cases of the present model. The results show that the responses are strongly affected by the two parameters, which can help understand the crack behaviors of materials under thermal impact loading. Keywords Memory-dependent derivative; Singular integral equations; Stress intensity factors; Intensity factors of temperature gradients 1 Introduction With the development of micro/nanoscale electromechanical systems and the wide applications of ultrashort pulse lasers, heat transfer at micro/nanoscale may have very different physical bases than that in macroscale. The Fourier law is the conventional approach that has been extensively used to study heat conduction in many engineering applications. Although well established, one limitation of this theory is that it implies that the speed of heat propagation is infinite [1]. This means that the temperature field in the medium will be affected by the initial and boundary conditions instantaneously, which is inaccurate for very low temperature or short-pulse thermal heating in micro temporal/spatial scales. Examples including the heat conduction in micro/nanoscale devices demonstrate lots of distinct phenomena such as size effect and wave phenomena, which cannot be captured by the Fourier law [2-5]. As a result, non-Fourier heat conduction schemes have been proposed, among which the first is formulated by Cattaneo [6] and Vernottee [7] (CV), in which a hyperbolic heat conduction equation was obtained by combining the CV model with the local energy equilibrium. Another refinement, proposed by Tzou [8] to better describe heat conduction in a transient case, the so-called dual-phase-lag (DPL) model, accounts for the microscale temporal and spatial effects of the heat transfer. Since then, considerable efforts have been devoted to the non-Fourier heat conduction problems. In aerospace and nuclear engineering applications, many structural components are subjected to severe thermal loadings, giving rise to intense thermal stresses in the components, especially near cracks and other kinds of defects. The high thermal stresses around defects may cause catastrophic damage to the cracked structure [9]. Heat conduction problems of cracked materials using the Fourier model have been investigated by many researchers [10-13]. Some investigations on crack problems have been carried out using the CV model to analyze the strength of cracked semi-infinite body [14, 15], cracked plate [16, 17], and cracked cylinder [18, 19], under sudden thermal impacts. Based on the DPL model, the transient temperature field around a partially insulated crack in a half-plane was obtained by Hu and Chen [20]; Fu et al. [21] carried out a theoretical study of non-Fourier heat conduction in a sandwich panel with a cracked foam core. The fracture behavior of a cylindrical bar containing a circumferential edge crack under transient thermal loading was analyzed in [22]. Although the non-Fourier models have been widely used to study cracked problems, a careful inspection of these models indicates that the differential and integral terms in their heat conduction equations always appear in an integer order. The existence of the anomalous diffusion of the material (such as amorphous solid and porous media) leads to the abnormal heat conduction [23, 24], whereas the anomalous diffusion possesses memory and path dependence as well as global correlation. Integer-order derivative cannot describe the history-dependent process, but the fractional derivative is very suitable for the description of the process of memory. Therefore, fractional calculus was introduced in the heat conduction equation [25]. In recent years, the following fractional heat conduction models have been established by introducing the fractional calculus to the CV model: Sherief et al. [26] modified the CV model by introducing the Caputo fractional derivative as: q q kT t 0 1 where q is the heat flux vector, is the thermal relaxation time, k is the thermal conductivity, is the spatial gradient operator, and T is absolute temperature. Influenced by Povstenko’s Riemann-Liouville fractional I 1 f t [27], Youssef [28] integral introduced operator the as: 0 2 q q kI 1T where work 1 t 1 2 t 0 f d ( is the gamma function). Ezzat q [29] adopted q kT t the fractional Taylor’s series as: 0 1 in which, the fractional exponent, , is used to describe memory effect of the material, which is essentially significant in practice. 1) Riemann-Liouville fractional integral: I a f t 1 t 1 t a f d , t a, b , 0 Accordingly, the fractional derivative of Riemann-Liouville is defined as: Da f t D m I am f t dm m dt t a m m 1 t f d 2) Caputo fractional derivative Da f t I am D m f t K t , f t a m d t K t , m m 1 in which, m 1 m , is gamma function, D m is the common m-order derivative. It can be seen that for a given real number , the kernel function K t , is a fixed function. But from the viewpoint of applications, different processes need different kernels to reflect their memory effects, so the kernel should be chosen accordingly. In addition, since the fractional derivative is defined on the interval a, t with a fixed real number a, it may be invalid in describing the memory effect for large time t. In fact, the memory effect of a real process basically occurs in a segment of time, that is, the delayed interval t , t ( 0 denotes the time delay). To better reflect the memory effect (instantaneous change rate depends on the past state), Wang and Li [30] recently proposed a memory-dependent derivative (MDD) and defined the first order MDD of function f as: D f x, t 1 t t K t , f x, d in which is the time delay and K t , is the kernel function that can be chosen arbitrarily. The definition indicates that the value interval of the weight K t , is 0, 1 for t , t , so that the magnitude of MDD, D f x, t is usually smaller than that of the common partial derivative, f x, t t . It is worth noting that the kernel function can be chosen as K t , 1 t , 0, 1, 2 , which may be more practical. When K t , =1 , we have: D f x, t 1 t t f x, d = f x, t f x, t f x, t t which means that as 0 D approaches the common partial derivative t . In 2014, Yu et al. [31] introduced the MDD instead of fractional calculus, into the rate of heat flux in the CV model, to denote memory-dependence. More applications based on the MDD heat conduction model can be found in [32-34]. All the studies mentioned above are limited to elastic media without defects. To the authors’ best knowledge, no work exists in the current literature that uses the memory-dependent CV model to investigate the transient thermoelastic responses of a cracked structure. In this paper, memory-dependent CV model will be implemented to investigate the transient thermoelastic behavior of a cracked strip under thermal impact loading. The crack lies parallel to the boundary of the strip and the crack faces are assumed to be completely insulated. First, the first order MDD is introduced into the rate of heat flux in the CV model to build the memory-dependent CV heat conduction equation. Then by using the Fourier and Laplace transforms, the thermal and mechanical problems are reduced to two systems of singular integral equations, which are solved numerically. Results are presented illustrating the influence of time delay and kernel function on the temperature, intensity factors of temperature gradients, and stress intensity factors. 2 Problem formulation and basic equations Fig. 1. Crack geometry and coordinates. As shown in Fig. 1, consider an infinitely long strip containing a Griffith crack of length 2c parallel to the free surfaces. Denote by (x, y) the rectangular coordinate system with its origin at the middle point of the crack face and x direction along the crack line. The initial temperature of the strip is assumed to be zero, and its free surfaces, y = h1 and y = h2 are suddenly heated to temperature T1 and T2, respectively. The crack surfaces are assumed to be completely insulated. It is noted that the effects of inertia and thermal-elastic coupling are neglected in this study. When the heat transfer situations include extremely high temperature gradients, extremely large heat fluxes, or extremely short, transient durations, the heat propagation speed becomes finite, and the non-Fourier heat conduction models, such as the CV and DPL models, should be adopted. On the other hand, since the existence of the anomalous heat conduction in the material, and the anomalous heat conduction possesses memory and path dependence, which cannot be described by the integer-order derivative. Applying the MDD to the CV model, the following heat flux law can be obtained [31]: q Dwq kT (1) Here, we call it memory-dependent CV model. In the absence of any inner heat source, the energy conservation equation reads: q = mcET (2) where m and cE are, respectively, mass density and specific heat capacity. Combining Eq. (1) with Eq. (2) leads to the heat conduction equation with MDD: k2T 1 D mcET (3) in which 2 is the Laplace operator. The equilibrium equations for isotropic, homogeneous, elastic bodies (in the absence of body forces): xy y x xy 0, 0 x y x y (4) The strain-displacement relations and the compatibility condition: x u v 1 u v , y , xy x y 2 y x (5) 2 2 xy 2 x y 2 2 y 2 x xy (6) 1 x y tT E 1 y y x tT E 2 1 xy xy E (7) The constitutive law: x where E , and t are the Young’s modulus, the Poisson’s ratio and the coefficient of linear thermal expansion, respectively. Let F(x, y) be the Airy stress function, then the stresses can be expressed as: x 2 F 2 F 2 F , , y xy y 2 x 2 xy (8) With the help of Eq. (7), substituting (8) into the Eq. (6) one obtains: 22 F E 2 tT 0 (9) For simplicity, the following non-dimensional quantities are introduced: x, y, h , h x, y, h , h c , u, v u, v cT T ,T T ,T T , E T , T 1 1 2 2 1 1 2 0 2 ij t , , k t , , m cE , t ij t 0 ij F F E tT0c 2 ij 0 t 0 (10) where T0 is the reference temperature. The governing Eqs. (3) and (9) can be rewritten as (here and in the following, the apostrophes of dimensionless quantities are omitted for brevity): 2T 1 D T (11) 22 F 2T 0 (12) The initial and thermal loading boundary conditions have the dimensionless forms: t 0 T T 0 T ( x, h1 ) T1 T ( x, h2 ) T2 (13) x x (14) T ( x, 0 ) T ( x, 0 ) 0 y y T ( x , 0 ) T ( x, 0 ) x 1 x 1 T ( x, 0 ) T ( x, 0 ) y y x 1 (15) The mechanical conditions read: x ( x, h ) ( x , h ) 0 x ( x, 0) ( x, 0) 0 x 1 (16) x 1 ( x, 0 ) ( x, 0 ) x 1 u ( x, 0 ) u ( x, 0 ) x 1 v( x, 0 ) v( x, 0 ) x 1 (17) xy ( x, h1 ) y ( x, h1 ) 0 xy xy 2 y 2 y xy ( x, 0 ) xy ( x, 0 ) y y 3 Temperature field Considering zero initial condition (13), the Laplace transform (p-plane, variables being noted with the asterisk) of Eqs. (11) and (12) becomes: 2T * 1 G pT * (18) 22 F * 2T * 0 (19) where G is defined in Appendix A. The boundary conditions (14) and (15) in the Laplace domain can be rewritten as: T * ( x, h1 ) T1 p T * ( x, h2 ) T2 p x x (20) T * ( x, 0 ) T * ( x, 0 ) 0 y y T * ( x, 0 ) T * ( x, 0 ) x 1 x 1 T * ( x, 0 ) T * ( x, 0 ) y y x 1 (21) Eq. (18) subjected to the boundary conditions (20) and (21) can be solved by decomposing the present problem into two cases: (a) a crack-free strip with inhomogeneous thermal boundaries and (b) a cracked one with homogeneous boundary conditions [35, 36]. As a result, the temperature field in the Laplace domain can be obtained by superposition: T * x, y D exp my D2 exp my exp ix d W * y y 0 T * x, y D3 exp my D4 exp my exp ix d W * y y 0 (22) where m 2 1 G p . The unknown coefficient W * y is determined by solving the temperature field of a strip without crack: d 2W * 1 G pW * 2 dy W * ( x, h1 ) T1 p W * ( x, h2 ) T2 p x x (23) (24) and W * y is given in the Appendix A. By substituting Eq. (22) into the homogeneous part of the boundary conditions (20) and (21) results in: D2 exp(2mh2 ) D, D3 12 exp(2mh1 ) D, D4 12 D (25) where 12 is given in Appendix A. The unknown coefficient D is obtained by employing the mixed boundary conditions (20)3 and (21). Introducing the density function x : T * x, 0 T * x, 0 x x x (26) From the boundary conditions (21), we obtain: x dx 0 1 1 x 0, (27) x 1 The substitution of (22) into (26) and applying Fourier inverse transform, one gets: D 1 i 1 exp(2mh1 ) t exp(it )dt 4 exp(2mh1 ) exp(2mh2 ) 1 (28) Substituting (22) into (20)3 and using the Eq. (28), the singular integral equation for x is derived as follows: 1 t t x k x, t dt 1 2 q T1 cosh qh2 T2 cosh qh1 1 p sinh q h1 h2 x 1 (29) in which q is given in Appendix A and the kernel k x, t reads: m 1 exp(2mh1 )1 exp(2mh2 ) k x, t 1 sin x t d 0 1 exp 2m h1 h2 (30) It is noted that integral Eq. (29) under the singled-value condition in (27)1 has the following form of solution [37]: x x 1 x2 , x 1 (31) in which x is bounded and continuous on the interval 1, 1 . From Eq. (27), x x . Following the numerical techniques in [37], Eqs. (29) and (27)1 can be approximated by the following system of n linear algebraic equations at n unknown discrete points of tk : 2q T1 cosh qh2 T2 cosh qh1 1 k xl , tk l 1, 2, p sinh q h1 h2 k xl n 1 n t t k 1 k , n 1 (32) n t 0 n k 1 (33) k where tk cos 2k 1 2n , k 1, 2, xl cos l n , l 1, 2, , n 1 ,n Once function t is obtained, the function D can be calculated by using the method of Chebyshev quadrature: D n 1 exp 2mh1 k tk sin tk 4 exp 2mh1 exp 2mh2 k 1 (34) where k cos n , k 1, 2, ,n Substituting (34) into (22), one can obtain the temperature in the Laplace domain. 4 Intensity factors of temperature gradients Tzou [38] proposed that the energy bearing capacity of a solid medium could be assessed by the singular behavior of the temperature gradients in the vicinity of a crack tip. In order to obtain the intensity factors of temperature gradients (IFTGs), we first derive their expressions. The temperature gradients in the Laplace domain can be obtained as: T, *y m D exp my D2 exp my exp ix d W, *y y 0 T, *y m D3 exp my D4 exp my exp ix d W, *y T, *x i D exp my D2 exp my exp ix d T i D3 exp my D4 exp my exp ix d * ,x y 0 (35) y 0 y 0 (36) The singularity of temperature gradients is only due to the asymptotic properties of the integrands in (35) and (36) for large values of the variable, . Considering this and following the steps in [39] and using the asymptotic formula in [40]: 1 j t 1 1 t2 2 exp i t dt j 1 exp i 4 0 s j 1 exp i 4 1 (37) 1 exp s sin , cos 2 2 2 1 sin tan s s 0, 0 1 , cos tan s (38) The singular temperature gradients around the crack tip in the Laplace domain can be obtained as: 1, p sin 2 2r 2 1, p T, *y r , , p cos 2 2r 2 T, *x r , , p T,r* r , , p (39) 1, p sin 2 2r 2 where the subscript “,j” (j = x, y, r) denote the temperature gradients in the x, y and r direction, respectively, r, are the polar coordinates which are measured from the crack tip: x 1 r cos , y r sin (40) The temperature gradients reach the maximum values at the crack tip at , and the IFTG in the Laplace domain around the crack tip can be defined as [38]: KT* p lim 2 rT,r* r , , p r 0 1, p 2 (41) It can be seen from Eq. (39) that the transient temperature gradients appear singularity at 1 r around the crack tip, and the dynamic effect of IFTG is due to its time-dependence as shown in Eq. (41). 5 Stress intensity factors From the temperature expressions (22), the governing equations for the Airy function F * can be read as: 2 2 F * 2 m2 D exp my D2 exp my exp ix d q 2W * y y 0 2 2 F * 2 m2 D3 exp my D4 exp my exp ix d q 2W * y y 0 (42) Considering the regular conditions at infinity, the general solutions of Eq. (42) can be expressed as: F * A1 A2 y exp y A3 A4 y exp y exp ix d C11 exp my C12 exp my exp ix d W * y q 2 y 0 F * B1 B2 y exp y B3 B4 y exp y exp ix d C21 exp my C22 exp my exp ix d W * y q 2 y 0 (43) in which Ai and Bi (i = 1, 2, 3, 4) are unknowns to be determined, and Cij (i, j = 1, 2) are defined in Appendix A. With the help of (10), substituting the Airy functions (43) into (8), the stresses are obtained: *y 2 A1 A2 y exp y A3 A4 y exp y exp ix d 2 C11 exp my C12 exp my exp ix d y 0 *y 2 B1 B2 y exp y B3 B4 y exp y exp ix d 2 C21 exp my C22 exp my exp ix d y 0 (44) x* 2 A 2 2 A1 A2 y exp y 2 A4 2 A3 A4 y exp y exp ix d m 2 C11 exp my C12 exp my exp ix d W * y x* 2B 2 y 0 2 B1 B2 y exp y 2 B4 2 B3 B4 y exp y exp ix d m 2 C21 exp my C22 exp my exp ix d W * y y 0 (45) xy* i A2 A1 A2 y exp y A4 A3 A4 y exp y exp ix d i m C11 exp my C12 exp my exp ix d y 0 xy* i B2 B1 B2 y exp y B4 B3 B4 y exp y exp ix d i m C21 exp my C22 exp my exp ix d y 0 (46) Substituting (5) into (7) and considering (10), one gets: u x y T x v y x T y (47) xy 2 1 xy The jumps of displacements along u u x,0 u x,0 the line y = 0 v v x,0 v x,0 , and is denoted as then from (47) and the boundary conditions (16) and (17), one may obtain: u x T x 2 v x 2 x y (48) Introducing two dislocation density functions 1 x and 2 x : 1 x u v , 2 x x y (49) From Eq. (17), we know that: 1 1 i x dx 0 i x 0, i 1, 2, x 1 (50) By substituting Eqs. (44)-(46) into the boundary conditions (16) and (17), it can be seen that i x (i = 1, 2) satisfy the following singular integral equations: 1 1 K x , t dt 11 1 t x 12 t K12 x, t dt 2 M1 x 1 1 1 t K x , t dt 1 21 1 12 t t x K 22 x, t dt 2 M 2 x 1 1 t (51) where Kij x, t and Wi x i, j 1, 2 are given in Appendix A. According to Erdogan et al. [37], the singular integral equations (51) under the singled-value conditions (50) have the following form of solutions: i x i x 1 x2 , i 1, 2, x 1 (52) To solve the functions i x (i = 1, 2), the integral equation (51) can be reduced to the following algebraic equations by using the integral algorithm in [41]: n 1 K11 x j , ti 1 ti i K12 x j , ti 2 ti 2 M 1 x j i 1 ti x j n i i 1 (53) n t 0 i 1 i 1 i 1 K 22 x j , ti 2 ti 2 M 2 x j ti x j i K 21 x j , ti 1 ti i n n i 1 i 1 (54) n t 0 i 1 i 2 i where ti cos i 1 n 1 , i 1, 2, , n x j cos 2 j 1 2 n 1 , j 1, 2, i 2 n 1 , i 1, n; i n 1 , , n 1 i 2, 3, , n 1 Considering the asymptotic properties of the integrands in Eqs. (44)-(46) for large values of the variable , following the steps in [39] and using the asymptotic formulas (37)-(38), the singular stresses in the Laplace domain around the crack tip are expressed as: 1 * 3 K1 p cos 1 sin sin 2 r 2 2 2 3 2 cos cos 2 2 x* r , , p * K 2 p sin 2 * 3 * K1 p cos 1 sin sin K 2 p sin 2 2 2 2 3 cos cos 2 2 1 2 r *y r , , p 1 * 3 K1 p sin cos cos 2 r 2 2 2 3 1 sin sin 2 2 xy* r , , p * K 2 p cos 2 (55) where r, are the polar coordinates as defined by Eq. (40), and the SIFs K1* p and K 2* p in the non-dimensional forms are defined as: K1* p = lim 2r *y r , 0, p 2 1, p , mode-I 4 1 1, p K 2* p = lim 2r xy* r , 0, p , mode-II r 0 4 r 0 (56) Thus far, all the variables in the Laplace domain have been obtained. To obtain the solutions in the time domain, numerical inversion of Laplace transform (NILT) is needed. In this work, we adopt an algorithm of NILT proposed by Miller and Guy [42]. 6 Result and discussions In this section, we focus on the effects of time delay and kernel function on the transient temperature, IFTGs, and SIFs of the cracked strip. 6.1 Validation and the thermal relaxation time effects First, the solutions should be validated. The temperature history of the midpoints of the crack at the upper and lower surfaces and SIFs history at the crack tip are compared with those reported in [17], as shown in Fig. 2. In Fig. 2(a), increasing the thermal relaxation time could increase the maximum temperature and extend the time required for reaching the maximum temperature at the midpoints of the crack. A similar observation can be made in Figs. 2(b) for SIFs. These observations are consistent with the results reported in [17]. It is worth noting that the non-zero temperature at the initial stages is derived from the numerical errors of the inverse Laplace transform used in this paper. (a) (b) Fig. 2. Effects of thermal relaxation time on (a) temperature at the crack surface midpoints and (b) SIFs at the crack tip. 6.2 Effects of time delay and kernel function on the temperature and IFTGs In this subsection, some parametric studies are performed to evaluate the effects of time delay and kernel function on the temperature and IFTGs. Here, three kernel functions: 1, 1 t , 1 t are considered. The crack length is 2c=2, 2 the thermal relaxation time is =0.5 , the geometric size is h1 h2 1 , and the temperatures at the top and bottom faces of the cracked strip are T2 2 and T1 1 , respectively. In order to evaluate the effects of the time delay on the temperature field under different kernel functions, comparisons with the CV model and Fourier model are made as shown in Fig. 3. It should be noted that when the kernel function K t , is 1, the current heat conduction model reduces to the CV model or Fourier model as the time delay approaches to infinitesimal (here = 1e 10 ) or a particularly large value (here = 100 ), respectively. Clearly, the temperature predicted by the Fourier model monotonically increases with time, whereas the results based on the CV and memory-dependent CV models (they are all labeled as non-Fourier models) show peak values are higher than the corresponding steady values. As time approaches infinity, there is no difference between the Fourier results and the non-Fourier results. It can be also read that the time delay has great effects on the temperature: the larger the time delay, the smoother the varying curves of temperature, and the wave-like oscillation behaviors become less evident. In other words, the maximum temperature is smaller when the time delay is larger. Moreover, the temperature curves exhibit different forms under different kernel functions: for a given time delay, kernel function 1 t 2 makes the curves smoother than the other two. (a) K t , 1 (b) K t , 1 t (c) K t , 1 t 2 Fig. 3. Effects of the time delay on the temperature with different kernel functions. Fig. 4 illustrates how the temperature distributes within the cracked strip for the time delay examined in Fig. 3(a). The figures are illustrated for temperature values taken at the time when the maximum temperature appears. It is observed that the temperature in the upper half of the strip is higher than that of the lower part, which is owing to the fact: the input, i.e. a higher temperature increase, is applied to the top face of the strip. The thermal insulation of the crack prevents the heat flux from passing through, which explains why the temperature field is no longer uniform along the x-direction at any position in the y-direction. Owing to the existence of the insulated crack, heat flux may be enhanced in a narrow, destroyed path, and result in the higher temperature in the inner region of the strip than that on the boundary, leading to overshooting. An increase in time delay reduces the heat flux, which is of great importance in thermal engineering applications such as safety design of the mechanical devices under severe thermal loadings [43]. (a) CV model (c) 0.5 (b) 0.1 (d) Fourier model Fig. 4. Temperature distribution for different time delays when K t , 1 at the time the highest temperature occurs. From Eqs. (41), it can be observed that the maximum temperature gradients occur at the angle = , which correspond to the upper and the lower crack surfaces around the crack tip. This conclusion is in agreement with the physical intuition that abrupt temperature changes appear around the crack tip. Fig. 5 illustrates the effects of the time delay on the IFTGs under different kernel functions with . For the non-Fourier heat conduction models, the IFTGs fluctuate and increase with time until they reach their peak values, and then they oscillate for some time before stabilizing at the steady values. For the Fourier model, the IFTGs increase monotonically with time until reaching the steady values. The magnitude of the IFTGs for the non-Fourier models is bigger than those of the Fourier model, which shows the effects of the thermal relaxation time on the temperature field. As expected, the larger the time delay, the smaller the peak values of IFTGs. Similarly, the IFTG curves display different forms under different kernel functions: for a given time delay, kernel function 1 t make the curves smoother than the other two. 2 (a) K t , 1 (b) K t , 1 t (c) K t , 1 t 2 Fig. 5. Effects of the time delay on the dynamic IFTGs with different kernel functions. 6.3 Effects of time delay and kernel function on the SIFs When analyzing crack problems, the SIFs are the most important parameters to describe the intensity of singular, elastic stress fields around the crack tip. The effects of different geometric sizes on the SIFs are shown in Fig. 6 with K t , 1 , and h2 = 1. It can be seen that the SIFs increase with time until reaching their peak values. Then they oscillate for some time before stabilizing at the corresponding steady state values. The peak values increase as the size h1 increases from 1 to about 4 and then, they remain almost constant with any further increase in h1. It is also noted that when h1 = 1, K1 vanishes due to the geometric symmetry of the cracked strip. So when the SIFs are calculated in our work, the geometric size is set as h2 = 1 and h1 = 2, and the values of the other parameters are the same as those in subsection 6.2. (a) (b) Fig. 6. Dynamic SIFs of (a) mode-I and (b) mode-II for different h1 values when h2 =1. The effects of the time delay on the transient SIFs are illustrated in Figs. 7-9 under different kernel functions. As expected, a larger value of will result in a lower maximum SIF and a shorter duration to reach this maximum value. For the Fourier model, the SIF increases to the maximum value, and then decreases to the steady state value as time elapses. However, for non-Fourier model, the thermal relaxation time leads to the oscillation behavior of the SIF before it becomes steady. The responses of K1 to kernel function and time delay appear similar to those of K 2 . From Fig. 7, we can see that the maximum values of K1 decrease by 6.70%, 15.91% and 22.16%, respectively, than that predicted by the CV model when the kernel function K t , 1 and time delay is 0.1, 0.3, 0.5 . And the maximum values of K 2 decrease by 10.74%, 23.86% and 31.58%, respectively, than that predicted by the CV model. Similarly, compared with the CV model, the maximum values of K1 decrease by 20.67%, 25.47%, 28.57%, and the maximum values of K 2 decrease by 25.30%, 31.20%, 35.67%, when K t , 1 t and 0.1, 0.3, 0.5 , as shown in Fig. 8; the maximum values of K1 decrease by 27.75%, 30.20%, 31.93%, and the maximum values of K2 decrease by 33.24%, 36.00%, 38.45%, when K t , 1 t and 0.1, 0.3, 0.5 , as shown in Fig. 9. In a word, the 2 greater the time delay, the more significant its effect on the SIFs for any kernel function considered, and when is large enough, for example, ω=100, the current heat conduction model approaches the Fourier model. For all the time delay considered, the kernel function K t , 1 t has the greatest effects on 2 the SIFs. (a) (b) Fig. 7. Effects of the time delay on the dynamic SIFs of (a) mode-I and (b) mode-II with K t, 1 . (a) (b) Fig. 8. Effects of the time delay on the dynamic SIFs of (a) mode-I and (b) mode-II with K t, 1 t . (a) (b) Fig. 9. Effects of the time delay on the dynamic SIFs of (a) mode-I and (b) mode-II with K t , 1 t . 2 In the current model, two factors, namely, the time delay and kernel function can be selected freely. In the above analysis, the results are the effects of different time delays on the responses when the kernel function is determined. Accordingly, the effects of different kernel functions on the responses with given time delay may be presented. It is skipped here and the effects of MDD on the responses are investigated systematically in the following subsection 6.4. 6.4 Effects of MDD on the thermoelastic responses To investigate the effects of MDD on the thermoelastic response systematically, a memory-dependent parameter combining the time delay and kernel function together [31] is introduced: I t t two cases K t , d . Accordingly, it takes different forms for the considered: I case1 t t 1 1 t 1 d 1 2 and 2 1 t 2 d 2 3 . Let I case1 I case 2 , one gets 1 2 = 2 3 . Here t 2 I case 2 1 , 2 t is set as 0.2, 0.3 . It is observed from Fig. 10 that case 2 makes the curves of the variables studied smoother, although the memory-dependent parameters of both cases are identical. Therefore, one may select the flat kernel function to soften the effects of MDD on the dynamic thermoelastic responses. In addition, a better observation of the temperature distribution within the cracked strip is illustrated in Fig. 11 for case 1 and 2, as examined in Fig. 10(a). (a) (b) (c) Fig. 10. Dynamic distribution of (a) temperature, (b) IFTGs, and (c) SIFs (a) case 1 Icase1 Icase 2 . (b) case 2 Fig. 11. Temperature distribution for (a) case 1 and (b) case 2 at the time the highest temperature occurs. It is clear that the stress field around the crack is influenced by the distance between the crack and the boundary characterized by the parameters h1 and h2. From the perspective of engineering application, it is important to know the most unfavorable value h2 (or h1) at which the SIF achieves the maximum values. Fig. 12 shows the influence of the parameter h2 0.02 h2 1.98 on the maximum SIFs for h1 h2 2 with 0.5 , T1 1 , and T2 2 . It is observed that the maximum SIFs are not simply increasing or decreasing monotonically with h2. When the crack is very close to the boundaries, the peak values are relatively large for both mode-I and mode-II. The peak values of mode-II SIF achieve the largest at h2 0.4 . The results can be used as the theoretical basis for the structural safety design. Fig. 12. The variation of peak values of dynamic SIFs versus h2 for case 1 and 2. 7 Conclusions In this paper, the transient thermoelastic problem of a thermally insulated crack parallel to the boundary of a strip under thermal impact loading is investigated using the memory-dependent CV model. Laplace and Fourier transforms are used to reduce the heat conduction problem to a system of singular integral equations. Numerical solutions of the singular integral equations are used to demonstrate the effects of time delay and kernel function on the dynamic temperature field, IFTGs, and SIFs. The results are also compared with those of the Fourier and CV models to build the connections between these models. When the kernel function is set as 1, the memory-dependent CV model can reduce to the CV model or Fourier model as the time delay approaches infinitesimal, or takes a very large value, respectively. For a given kernel function, the larger the time delay, the smoother the curves of the thermoelastic responses. Once the memory-dependent parameter is defined, one may select the flat kernel function to soften the effects of MDD on the dynamic responses. The results also show that the crack location greatly influences the stress field around the crack tip. This finding can be used as the theoretical basis for the structural safety design. The model developed in the present work is especially suitable for analyzing the crack problem in the viscoelastic medium under high rate thermal shocking, or at small temporal and length scales. It better reflects the memory-dependence effect, namely, the instantaneous changing rate depending on the past state. Acknowledgements This study is supported by National Natural Science Foundation of China (11572237, 11732007), the Fundamental Research Funds for the Central Universities and the Natural Sciences and Engineering Research Council of Canada (NSERC), ZX is grateful for the China Scholarship Council (CSC) for providing the support for her visit at University of Alberta. Appendix A G 2b 2a 2 2 2a 2 1 exp p 1 a 2 b exp p 2 2 p p p (A-1) where a and b are constants, (a, b)=(0, 0), (0, 0.5), (1, 1) correspond to 0, 1, 2 , respectively. W * y = T1 exp(qh2 ) T2 exp(qh1 ) exp(qy) T2 exp(qh1 ) T1 exp(qh2 ) exp(qy) 2 p sinh q h1 h2 (A-2) 12 1 exp 2mh2 1 exp 2mh1 (A-3) q p 1 G (A-4) C11 D , 2 m2 C12 D exp 2mh2 2 m2 D exp 2mh1 D C21 12 , C22 122 2 2 m m2 (A-5) J g J g 22 2 M 1 x 21 1 e2 e1 m C21 C22 sin x d 0 g J g J g M 2 x 2 11 1 12 2 e1 C21 C22 cos x d 0 g J 22 f 21 J 21 f 22 K11 x, t 1 sin x t d 0 g 2 J 22 f11 J 21 f12 K12 x, t cos t x d 0 g K 21 x, t 0 2 J11 f 22 J12 f 21 g (A-6) (A-7) cos t x d 3 J12 f11 J11 f12 K 22 x, t 1 sin x t d 0 g J11 1 b11 , J12 b12 , J 21 1 b11 b21 , J 22 1 b12 b22 b11 1 2 h1 exp 2 h1 , b12 2 h12 exp 2 h1 (A-8) b21 2 exp 2 h1 , b22 2 h1 1 exp 2 h1 (A-9) g f11 f 22 f 21 f12 , g1 f 23 f12 f13 f 22 , g2 f13 f 21 f 23 f11 (A-10) f11 2 e21 e41 b21 , f12 2 e22 e42 b22 1 f13 2 e23 e43 e2 f 2 2 m 2 1 exp(2mh2 ) 12 exp(2mh1 ) 1 D (A-11) e31 e11 b11 1 , f 22 2 e32 e12 b12 3 f 23 2 e33 e13 e1 f 3 3 2 m 2 f 21 2 3 3 e1 f1h1 exp h1 , e2 f1 exp h1 e11 e31 (A-12) e f d d J11d12 d13 J d d , e12 12 12 14 , e13 1 2 12 15 d12 d11 d12 d11 d12 d11 d e1 f 2 d11 d13 J11d11 d J d , e32 14 12 11 , e33 15 d12 d11 d12 d11 d12 d11 e41 c14 c11e11 c12e31 c13 , e42 c15 c11e12 c12e32 c13 (A-13) e43 c16 c11e13 c12 e33 c13 e21 J 21 e31 e11 e41 , e22 J 22 e32 e12 e42 e23 e2 e1 f 3 e33 e13 e43 d11 c11c23 c21c13 , d12 c12c23 c22c13 , d13 c14c23 c24c13 d14 c15c23 c25c13 , d15 c16c23 c26c13 (A-14) c11 h2 1 exp h2 , c12 h2 exp h2 exp h2 c13 h2 exp h2 exp h2 , c14 J 21h2 exp h2 c15 J 22 h2 exp h2 , c16 e2 e1 f 3 h2 exp h2 f1 m C21 exp mh1 C22 exp mh1 , f 2 C21 C22 C11 C12 f3 m C21 C22 C11 C12 (A-15) (A-16) References [1] Tzou DY. The generalized lagging response in small-scale and high-rate heating. Int J Heat Mass Tran 1995; 38: 3231–3240. [2] Tzou DY. Macro- to microscale heat transfer: the lagging behavior. Washington, DC: Taylor & Francis; 1997. [3] Cahill DG, Ford WK, Goodson KE, Mahan GD, Majumdar A, Maris HJ, Merlin R, Phillpot SR. Nanoscale thermal transport. J Appl Phys 2003; 93: 793–818. [4] Wang M, Yang N, Guo ZY. Non-Fourier heat conductions in nanomaterials. J Appl Phys 2011; 110: 064310. [5] Wang BL, Han JC. A crack in a finite medium under transient non-Fourier heat conduction. Int J Heat Mass Tran 2012; 55: 4631–4637. [6] Cattaneo C. A form of heat conduction equation which eliminates the paradox of instantaneous propagation. Comp Rend 1958; 247(4): 431–433. [7] Vernotte P. Paradoxes in the continuous theory of the heat conduction. Comp Rend 1958; 246: 3154–3155. [8] Tzou DY. A unified field approach for heat conduction from macro to micro scales. ASME J Heat Tran 1995; 117: 8–16. [9] Jin ZH, Noda N. Transient thermal stress intensity factors for a crack in a semi-infinite plate of a functionally gradient material. Int J Solids Struct 1994; 31(2): 203–218. [10] Sih GC. Heat conduction in the infinite medium with lines of discontinuities. ASME J Heat Tran 1965; 87: 283–298. [11] Yu SW, Qin QH. Damage analysis of thermopiezoelectric properties: part I crack tip singularities. Theor Appl Fract Mech 1996; 25: 263–277. [12] Yu SW, Qin QH. Damage analysis of thermopiezoelectric properties: part II effective crack model. Theor Appl Fract Mech 1996; 25: 279–288. [13] Chang CY, Ma CC. Transient thermal conduction of a rectangular plate with multiple insulated cracks by the alternating method. Int J Heat Mass Tran 2001; 44: 2423–2437. [14] Chang DM, Wang BL. Transient thermal fracture and crack growth behavior in brittle media based on non-Fourier heat conduction. Eng Fract Mech 2012; 94: 29–36. [15] Chen ZT, Hu KQ. Thermo-elastic analysis of a cracked half-plane under a thermal shock impact using the hyperbolic heat conduction theory. J Therm Stress 2012; 33(5): 895–912. [16] Wang BL, Li JE. Hyperbolic heat conduction and associated transient thermal fracture for a piezoelectric material layer. Int J Solids Struct 2013; 50: 1415–1424. [17] Hu KQ, Chen ZT. Thermoelastic analysis of a partially insulated crack in a strip under thermal impact loading using the hyperbolic heat conduction theory. Int J Engng Sci 2012; 51: 144–160. [18] Fu JW, Chen ZT, Qian LF, Xu YD. Non-Fourier thermoelastic behavior of a hollow cylinder with an embedded or edge circumferential crack. Eng Fract Mech 2014; 128: 103–120. [19] Guo SL, Wang BL. Thermal shock fracture of a cylinder with a penny-shaped crack based on hyperbolic heat conduction. Int J Heat Mass Tran 2015; 91: 235–245. [20] Hu KQ, Chen ZT, Transient heat conduction analysis of a cracked half-plane using dual-phase-lag theory. Int J Heat Mass Tran 2013; 62: 445–451. [21] Fu JW, Akbarzadeh AH, Chen ZT, Qian LF, Pasini D. Non-Fourier heat conduction in a sandwich panel with a cracked foam core. Int J Therm Sci 2016; 102: 263–273. [22] Chen LM, Fu JW, Qian LF. On the non-Fourier thermal fracture of an edge-cracked cylindrical bar. Theor Appl Fract Mech 2015; 80: 218–225. [23] Scher H, Montroll EW. Anomalous transit-time dispersion in amorphous solid. Phys Rev B 1975; 12: 2455–2477. [24] Koch DL, Brady JF. Anomalous diffusion in heterogeneous porous media. Phys Rev Fluids 1988; 31: 965–973. [25] Li B, Wang J. Anomalous heat conduction and anomalous diffusion in one-dimensional systems. Phys Rev Lett 2003; 91: 044301–1–4. [26] Sherief HH, El-Sayed AMA, Abd El-Latief AM. Fractional order theory of thermoelasticity. Int J Solids Struct 2010; 47(2): 269–275. [27] Povstenko YZ. Fractional heat conduction equation and associated thermal stress. J Therm Stress 2004; 28(1): 83–102. [28] Youssef HM. Theory of fractional order generalized thermoelasticity. ASME J Heat Tran 2010; 132(6): 061301–1–7. [29] Ezzat MA. Magneto-thermoelasticity with thermoelectric properties and fractional derivative heat transfer. Physica B: Condensed Matter 2011; 406(1): 30–35. [30] Wang JL, Li HF. Surpassing the fractional derivative: concept of the memory-dependent derivative. Comput Math Appl 2011; 62(3): 1562–1567. [31] Yu YJ, Hu W, Tian XG. A novel generalized thermoelasticity model based on memory-dependent derivative. Int J Eng Sci 2014; 81: 123–134. [32] Ezzat MA, El-Karamany AS, El-Bary AA. Generalized thermo-viscoelasticity with memory-dependent derivatives. Int J Mech Sci 2014; 89: 470–475. [33] Ezzat MA, El-Karamany AS, El-Bary AA. Electro-thermoelasticity theory with memory-dependent derivative heat transfer. Int J Engng Sci 2016; 99: 22–38. [34] El-Karamany AS, Ezzat MA. Thermoelastic diffusion with memory-dependent derivative. J Therm Stress 2016; 39(9): 1035–1050. [35] Noda N, Jin ZH. Thermal stress intensity factors for a crack in a strip of a functionally gradient material. Int J Solids Struct 1993; 30(8): 1039–1056. [36] Jin ZH, Noda N. (1993). An internal crack parallel to the boundary of a nonhomogeneous half plane under thermal loading. Int J Engng Sci 1993; 31: 793–806. [37] Erdogan F, Gupta GD, Cook TS. Numerical solution of singular integral equations, in: Sih GC (Ed.), Mechanics of Fracture, Methods of Analysis and Solutions of Crack Problems, vol. 1, Noordhoff, Leyden, Netherlands, 1973; pp. 368–425. [38] Tzou DY. The singular behavior of the temperature gradient in the vicinity of a macrocrack tip. Int J Heat Mass Tran 1990; 33: 2625–2630. [39] Delale F, Erdogan F. Effect of transverse shear and material orthotropy in a cracked spherical cap. Int J Solids Struct 1979; 15: 907–926. [40] Gradshteyn IS, Ryzhic IM. Tables of integrals, series and products. New York: Academic Press; 1965. [41] Theocaris PS, Ioakimidis NI. Numerical integration methods for the solution of singular integral equations. Q Appl Math 1977; 35: 173–183. [42] Miller MK, Guy WT. Numerical inversion of the Laplace transform by use of Jacobi polynomials. SIAM J Numer Anal 1966; 3: 624–635. [43] Xu M, Guo J, Wang L, Cheng L. Thermal wave interference as the origin of the overshooting phenomenon in dual-phase-lagging heat conduction. Int J Therm Sci 2011; 50: 825–830. Highlights Memory-dependent heat conduction model is used for the thermoelastic problem A insulated cracked strip under thermal impact is investigated Integral transform techniques are employed Effects of time delay, kernel function on temperature, IFTGs and SIFs are analyzed

1/--страниц