Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model

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Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model Zhangna Xue , Xiaogeng Tian , Jianlin Liu PII: DOI: Reference:

S0307-904X(19)30694-8 https://doi.org/10.1016/j.apm.2019.11.021 APM 13151

To appear in:

Applied Mathematical Modelling

Received date: Revised date: Accepted date:

10 April 2019 21 October 2019 13 November 2019

Please cite this article as: Zhangna Xue , Xiaogeng Tian , Jianlin Liu , Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model, Applied Mathematical Modelling (2019), doi: https://doi.org/10.1016/j.apm.2019.11.021

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Highlights 

Memory-dependent heat conduction model is used for the thermoelastic problem



A insulated crack in a functionally gradient half-space under thermal impact is investigated



Effects of time delay, kernel function and nonhomogeneity parameters on the temperature and SIFs are analyzed



The present results are compared with those based on Fourier and CV heat conduction model

Thermal shock fracture of a crack in a functionally gradient half-space based on the memory-dependent heat conduction model

Zhangna Xue a, Xiaogeng Tian b, Jianlin Liu a*

a

College of Pipeline and Civil Engineering, China University of Petroleum (East China), Qingdao 266580, China

b

State Key Laboratory for Strength and Vibration of Mechanical Structures, Xi’an Jiaotong University, Xi’an 710049, China *E-mail: [email protected]

Abstract:

In the present study, we consider a thermoelastic half-space made of a

functionally gradient material with an insulated crack, which is subjected to a thermal impact. The memory-dependent heat conduction model is adopted for analysis. By using the Fourier and Laplace transforms, the thermoelastic problem is formulated in terms of singular integral equations which can be solved numerically. Effects of the time delay, kernel function, and nonhomogeneity parameters on the temperature and stress intensity factor are analyzed. Our results are also compared with those based on the Fourier and CV heat conduction models, which can be viewed as two special cases of the present model. In conclusion, the memory-dependent derivative and nonhomogeneity parameters play an essential role in controlling the heat transfer process.

Keywords:

Functionally

gradient

material;

Thermoelastic

analysis;

Memory-dependent derivative; Singular integral equations; Stress intensity factor

1

Introduction Functionally gradient materials (FGMs) are nonhomogeneous composites whose

material properties can change continuously and gradually along desired directions. FGMs have been widely used in many engineering areas, such as heat dissipation structures in advanced space crafts, coating/substrate systems [1] and thermal barrier coatings used in gas-turbine engines or reentry vehicles [2]. When FGMs suffer large thermal gradients or high heat fluxes, they may generate intense thermal stresses inside the material, especially around cracks and other defects. For example, based on the Fourier heat conduction model, Jin and Noda considered the crack problems for a FGM half-space [3] and a FGM strip [4] subjected to a sudden temperature change. Ding and Li [5] investigated the fracture behavior of a functionally graded layered structure with an interface crack under the thermal loading. Chiu et al. [6] studied the problem of an infinite nonhomogeneous plane containing an arbitrarily oriented crack under the action of a uniformly remote heat flux. However, one limitation of the Fourier model is that the speed of the heat propagation is infinite. This means that a thermal disturbance at any position in a body can be felt at any other point instantly, and this capability is ineffective at the very small

length and

time

scales

associated

with small-scale

systems

(nano/micro-scale systems) [7]. Accordingly, the phase-lag concept is introduced to amend the Fourier law. The non-Fourier heat conduction theory was first developed by Cattaneo [8] and Vernottee [9] (CV), who introduced the relaxation concept to approach the wave nature of heat propagation. Since then, considerable effort has

been devoted to the non-Fourier heat conduction problems. For instance, Chang and Wang [10] examined the analytical solutions of the temperature and thermal stress fields for a semi-infinite brittle medium with a surface crack. Hu and Chen [11] obtained the transient temperature and thermal stresses around a crack in a thermoelastic strip under a temperature impact. Eshraghi et al. [12] introduced a weight function method for fracture analysis of a circumferentially cracked functionally graded hollow cylinder subjected to thermal loads. It should be mentioned that, for the Fourier and non-Fourier heat conduction models, the differential and integral terms in the governing equations always appear in the form of integer order. In the past few decades, as a natural extension of the classical differential and integral calculus, the fractional order calculus has been applied to various fields. Mainardi [13] proposed a fractional calculus approach to simulate the processes of some basic physical phenomena, such as diffusion and wave propagation. In some other works [14, 15], the fractional order differential equations are utilized as a powerful approach to describe the anomalous diffusion in complex systems. Inspired by the successful applications of fractional calculus in physics and engineering, various fractional order heat conduction models have been established [16–19] to express the memory-dependence in heat propagation. Povstenko and Kyrylych [20] explored the symmetric stress problem in an infinite plane with a line crack, and the surfaces are exposed to the heat flux loading in the framework of fractional thermoelasticity. Zhang et al. [21] analyzed a thermal shock problem of an elastic strip made of FGMs containing a crack parallel to the free surface on the basis

of the generalized fractional heat conduction theory. However, it is easily seen that the forms of the memory-dependent heat conduction models are different, that is, the fractional order heat conduction models are far from a common agreement, and hence they deserve more investigations. In 2011, Wang and Li [22] proposed a memory-dependent derivative (MDD), which is defined in an integral form of a common derivative with a kernel function on a slipping interval. It behaves better than the fractional one in reflecting the memory effect, i.e. the instantaneous change rate depends on the past state. Inspired by Wang and Li’s work, Yu et al. [23] introduced the MDD instead of the fractional order calculus into the CV model, to denote the memory-dependent property. Subsequently, based on the memory-dependent CV model, the thermoelastic problem of a thermally insulated crack parallel to the boundary of a strip under the thermal impact loading was considered [24]. However, the study mentioned above is limited to homogeneous mediums. When some defects such as cracks exist in the nonhomogeneous medium, it would be interesting to know how the MDD affects the responses. To the best of our knowledge, no information is available on the thermoelastic behavior of FGMs with an internal crack when the MDD is considered in the heat conduction theory. The aim of this work is to apply the heat conduction model with the MDD to analyze the transient thermoelastic responses around a crack in a semi-infinite functionally gradient body under thermal shock. The crack lies parallel to the boundary of the half-space and the crack faces are assumed to be thermally insulated. Fourier and Laplace transforms are employed to reduce the initial boundary problem

to the singular integral equations, which are then solved numerically. The effects of the time delay, kernel function, and nonhomogeneity parameters on the dynamic temperature and stress field are analyzed, and compared to the results based on Fourier and CV heat conduction model.

2

Problem formulation and basic equations

Fig. 1. Geometry and loading of the crack inside a half-space

Let us consider a semi-infinite, thermoelastic plane with a crack of length 2c parallel to the boundary, as shown in Fig. 1. A Cartesian coordinate system O-xy is established, with the x-axis along the crack plane and the material property grading is in the y-direction. For this gradient medium, the Poisson’s ratio  is assumed to be a constant because the effect of its variation on the crack tip stress intensity factor can be negligible [25]. The remaining thermo-mechanical properties are modeled as

follows

E  E0e y , t  t 0e y , k  k0e y ,

(1)

in which E,  t and k are the Young’s modulus, the coefficient of linear thermal expansion and the heat conductivity, respectively. The symbols E0 , t 0 and k0 are the corresponding values along the crack plane y0, and the smpbols ,  and  are the nonhomogeneity parameters with respect to the material property variation. Without loss of generality, the initial temperature of the plane can be postulated as zero. The surface, y−h, is suddenly heated to the temperature T0, where H(t) is the Heaviside function. The crack faces are assumed to be completely insulated, which indicates a temperature drop across the crack surfaces. Throughout this paper, the effects of inertia and thermoelastic coupling are neglected, leading to a quasi-static, and uncoupled problem [26]. Instead of the Fourier and CV heat conduction equations in which the differential and integral terms appear in an integer order, the MDD heat conduction equations are adopted.

2.1 Heat conduction equations The existence of the anomalous diffusion of the material (such as amorphous solid and porous media) leads to the abnormal heat conduction [27, 28], whereas the anomalous diffusion possesses memory and path dependence as well as global correlation. The integer-order derivative cannot describe the history-dependent process, and therefore, the first order MDD is introduced into the CV heat conduction equation [29]

1   D  q  kT .

(2)

Here and hereafter, Eq. (2) is noted as the MDD heat conduction model. The quantity q is the heat flux vector,  is the phase-lag of heat flux,  is the spatial gradient operator, and T is the temperature. The first order memory-dependent derivative D of the function f is defined as [22]

D f  x, t  

1



t

t 

K  t ,   f   x,   d ,

(3)

where the time delay  and the kernel function K(t,) are arbitrarily chosen, and this provides more possibilities to capture the material’s real behaviors. In practice, the kernel function can be understood as the degree of the past effect on the present, so different processes need different kernels to reflect their memory effects. In particular, the kernel function K(t,) can be selected as K t,   1 

2b



t    

a2 t   

2

,

2

(4)

in which (a, b)  (0, 0), (0, 0.5), (1, 1), and the kernel functions correspond to  t    t   K  t ,    1, 1  , 1 .       2

(5)

The energy conservation equation (without any internal heat source) is expressed as

  q =   cET ,

(6)

where  and cE are the mass density and specific heat capacity, respectively. As a consequence, the combination of Eq. (2) with Eq. (6) leads to the memory-dependent heat conduction equation

   k T   1   D   cET .

(7)

2.2 Thermoelastic field equations The equilibrium equations of the planar thermal stress problem for an isotropic, nonhomogeneous, and elastic body (in the absence of body forces) are

 xy  y  x  xy   0,   0, x y x y

(8)

where  x ,  y and  xy are the components of the stress. The strain-displacement relations are listed as

x 

u v 1  u v  ,  y  ,  xy     , x y 2  y x 

(9)

where  x ,  y and  xy are the components of the strain, and u and v are the two components of the displacement. Then we have the compatibility condition 2  2 xy  2 x   y   2 . y 2 x 2 xy

(10)

The constitutive law of the material is

1  x   y   tT , E 1  y   y   x    tT , E

x 

 xy 

2 1     xy . E

(11) (12) (13)

Let U(x, y) be the Airy stress function, then the stresses can be expressed as

x 

 2U  2U  2U ,   ,    . y xy y 2 x 2 xy

(14)

With the aid of Eqs. (11–13), substituting Eq. (14) into Eq. (10), one can get

2   T   y  2 2  U   U  2   U     E0 t 0e    2T  2   2T   0. 2 y y y   2

2

(15)

We then introduce the following non-dimensional quantities

 x , y , h    x, y , h  c ,  u , v    u , v   T c  , T  T T ,   ,  ,      ,  ,   c,     E  T  ,     T  ,  t , ,    k  t , ,     c c  , U  U  E  T c  . t0 0

ij

ij

0

t0 0

0

ij

2

0

ij

t0 0

(16)

2

E

0

t0 0

Then Eqs. (7) and (15) can be rewritten as

T T  1   D  , y t

 2T +

 2 2U  2

(17)

   2U T   y   2U    2 2  e    2T  2   2T   0.  y y y  

(18)

Note that here and hereafter, the bars of dimensionless quantities are omitted for brevity. The initial and thermal boundary conditions have the dimensionless forms

t  0 ,

(19)

T  x,  h   1

 x  ,

(20)

T  x,    0

 x  ,

(21)

T T 0

T  x, 0  y



T  x, 0  y

 V T  x, 0   T  x, 0  

T  x, 0    T  x, 0  

T  x, 0  y



T  x, 0  y

 x  c,

 x  c,

 x  c,

(22) (23)

(24)

where V is the dimensionless thermal conductivity of the crack surface. As the effects of thermal conductivity of the crack surface on the responses have been studied by Hu and Chen [30], they will not be presented in this study. Therefore, here the parameter V is set as 0, that is to say, it indicates the completely insulated crack surface.

The mechanical boundary conditions are

 xy  x,  h    y  x,  h   0

 x   ,

(25)

 xy  x, 0    y  x, 0   0

 x  c,

(26)

 xy  x, 0    xy  x, 0 

 x  c,

(27)

 y  x, 0     y  x, 0  

 x  c,

(28)

u  x, 0    u  x, 0  

 x  c,

(29)

v  x, 0    v  x, 0  

 x  c.

(30)

It should be mentioned that there are some similar studies on the initial boundary problems of the partial differential equations. For example, Marin et al. developed different methods to study the initial boundary problems of micropolar porous materials [31] and dipolar composite materials [32], respectively. Nami and Eskandari [33] used the finite element method for fracture analysis of functionally graded cylinders with semi-elliptical circumferential surface cracks under thermal and mechanical loadings. Here we employ the Fourier and Laplace transforms to deal with the thermal and mechanical problem of a cracked functionally gradient half-space.

3

Temperature field Applying the Laplace transform to both sides of Eq. (17) yields 

T *  x, y, p   L T  x, y, t     T  x, y , t  e  pt dt , 0

T  x, y, t   L1 T *  x, y, p   

1 T *  x, y , p  e pt dp,  Br 2πi

(31)

where Br represents the Bromwich integral path. Considering the zero initial condition in Eq. (19), one obtains

T  T   1  G  pT * , y *

2

*

(32)

2 *   2 * T *     y  2 * 2  U  U    e  T  2    2T *   0,    2 y y y  

 2 2U *  2

(33)

where

  2b 2a 2   2a 2   p   p   2 2    a 2  2b  1  e  1  e .  p   p  p   

G

(34)

The thermal boundary conditions in the Laplace domain read

 x  ,

T * ( x,  h )  1 p

 x  ,

T *  x,    0

T *  x, 0  y



T *  x, 0  y

 V T *  x, 0   T *  x, 0  

T *  x, 0    T *  x, 0  

T *  x, 0  y



(35) (36)

 x  c,

 x  c,

T *  x, 0  y

(37) (38)

 x  c.

(39)

In consideration of the boundary and regularity conditions, the temperature field in the Laplace domain can be expressed as 

T *  x, y    D   e 2 y eix d  W *  y   y  0  , 

2 D     e2h y  e  y  eix d  W *  y   y  0  ,     e 2  h 2 1

T *  x, y   



1

2

(40) (41)

in which D() is an unknown function to be determined, and W *  y  , 1 ,  2 ,  are W *  y =

1 

 2

e q y +h  2 , q  + p 1  G   , p 2 4

  , 2 

 2

 ,  

p 1  G    2 

(42)

2 4

.

(43)

To solve the temperature, we first introduce the density function (x) as follows

  x 

T *  x, 0  x



T *  x, 0  x

.

(44)

From the boundary conditions (38) and (39), one gets

   x dx  0. c

(45)

c

Substituting Eqs. (40) and (41) into (44), and applying the Fourier inverse transform, one gets D   

i  1  2 e2  h  4π

    t  e dt. c

it

c

(46)

Inserting Eqs. (40) and (41) into (37), and using Eq. (46), the singular integral equation for (x) can be derived as



c

c

2πq e qh  1   k  x, t   dt  p t  x 

 x  c .

 t  

(47)

Note that here we have considered the parity of complex functions and used the following relations ei t  x   cos   t  x    i sin   t  x   ,





0

sin   t  x   d 

1 , tx

(48) (49)

where the kernel k(x, t) reads





1 k  x, t    1      2V  2  2 e 2  h  1   sin   x  t   d . 0 

(50)

The singular integral equation (47) under the single value condition of Eq. (45) has the following solution

  x 

  x 1  x2



x  c,

(51)

where (x) is bounded and continuous in the interval c, c. From the single value condition (45), it is seen that (x)(x). Following the numerical techniques in

[34], Eqs. (45) and (47) can be solved at n unknown discrete points as  1  2q e  qh 1  t  k x , t  k   l k    t  x p k 1 n l k  n

 l  1, 2,

, n  1 ,

(52)

π  n  t   0, n

k 1

where tk  cos  2k  1 π  2n   , k  1, 2,

(53)

k

, n; xl  cos  lπ n  , l  1, 2,

, n 1 .

Once the function (x) is obtained, the function D() can be calculated by applying the method of Chebyshev quadrature  1  2 e2  h  n D     i   xi  sin  xi  ,  4π i 1

where xi  cos  2i  1 π  2n   ,  i  cos  π n  , i  1, 2,

(54)

, n.

Bearing in mind the relationship between the exponential and trigonometric function, one may obtain the temperature in the Laplace domain by substituting Eq. (54) into Eqs. (40) and (41) 

T *  x, y, p   2 D   e 2 y cos   x  d  W *  y  0

T *  x, y , p   2 



4

2 D   e   y  e 2  h   y  cos   x  d  W *  y  2  1e2  h  2

0

 y  0 ,

1

(55)

 y  0  . (56)

Thermal stress

4.1 Stress According to Eqs. (40) and (41), the governing equation in terms of the Airy function U can be written as 2 *  2 * 2  U   U  2   U     e    y M *  x, y, p  , 2 y y 2

2

*

(57)

in which

M *  x, y , p   







2 2

  2  22   2  D   e  2 y  ix d

 q    W 2

M *  x, y , p   



*

(58)

 y   y  0 ,

2 D     2   2  21   2    e2  h   y ix  2  h  1 2  1e 1



      22   2 2

2

2

e

 2 y ix

d    q    W 2

*

(59)

 y   y  0.

Considering the regular conditions at infinity, the general solutions of Eq. (57) can be expressed as 



 A1  A2 y  e s y ix d   C11 e     y ix d  y  0 , 

U*  

2

(60)

2



U *    B1  B2 y    B3  B4 y  e2 sy  e s1 y ix d  

   1  y ix   C21  C22 e2  y  e d 

(61)

 y  0 ,

in which Ai (i1, 2) and Bi (i1, 2, 3, 4) are unknowns to be determined, and C11, C21, C22 are defined in Appendix A. The parameters s1, s2, s are s1  

 2

 s, s2  



 s, s    2

2

2 4

(62)

.

With the assistance of Eq. (18), substituting (60) and (61) into (14) leads to (1) When y  0 ,

 x*  





2 A s   A  A y  s  e 2 2

1

2

2 2

 s2 y



 C11      2  e    2  y e ix d , (63) 2



 *y     2  A1  A2 y  e s y  C11 e     y  eix d ,  2





(64)

2



 xy*   i  A2  s2  A1  A2 y  e s y  C11      2  e      y eix d . 

2

2

(65)

(2) When h  y  0 ,

 x*  





2s B 1

2

  B1  B2 y  s12  e  s1 y   2s2 B4   B3  B4 y  s22  e  s2 y 

2  e  ix d   C21      1  e    1  y   2 C22      2  e    2  y  e  ix d , 

 (66)



 *y     2  B1  B2 y    B3  B4 y  e2 sy  e s y ix d  1



   C21  C22 e  2

2  y

 e

    1  y ix

(67)

d ,

 xy*   i  B2  s1  B1  B2 y   e s y   B4  s2  B3  B4 y   e s y  

1

2





   1  y  e  ix d   i C21      1  e  C22      2  e    2  y  e  ix d .

(68)

4.2 Displacement In consideration of Eq. (16), substituting Eq. (9) into (11–13), one gets

The

displacement

 u   u  x, 0    u  x, 0  

u   e  y y  e  y x  e yT , x

(69)

v   e  y x  e  y y  e yT , y

(70)

u v   2  e  y  e  y   xy . y x

(71)

jumps

along

the

line

y0

is

marked

as

and  v   v  x, 0   v  x, 0  . Then from Eqs. (69–71) and

the boundary conditions in Eqs. (25–30), one can obtain  u  x

 2 v x

where

 x 

and

T 

2

  x   T  ,

   x  

  x  y

(72)

  T  ,

(73)

denote the jumps of stress and temperature across y0.

Substituting Eqs. (40), (41), (63–68) into (72) and (73), one gets

 x    2s2  A2  B4   s22  A1  B3    2s1B2  s12 B1  

C21      1    C22  C11      2  2

2

 eix d , 

(74)



T      x  y

2 D   ix e d , 1  2 e2  h

(75)

  3s22  A2  B4   s23  B3  A1    s13 B1  3s12 B2   

(76)

C21      1    C22  C11      2   eix d ,  3

 u 

3

2    2s2  A2  B4   s22  A1  B3    2s1B2  s12 B1   C21      1    x (77) 2 D    ix 2  e d ,   C22  C11      2    1  2e2 h  

 v x









1 i

 2s   3s   A  B    s   s   A  B    2s   3s  B 2 2

2

2

2 2

4

3 2

1

3

1

   s12  s13  B1       1     1  C21       2  2

    2  C22  C11  

2

2 1

2

(78)

2 D    ix  e d . 1  2e2  h 

4.3 Integral equations The substitution of Eqs. (63–68) into the boundary conditions in Eqs. (25–30), the relations for Ai (i1, 2) and Bi (i1, 2, 3, 4) can be obtained as B3   1+2sh  e2 sh B1  2sh 2 e2 sh B2   f1 1  s2 h   f 2 h  e  s2h ,

(79)

B4  2s e 2 sh B1  1  2 sh  e 2 sh B2   f 2  f1s2  e  s2 h ,

(80)

A1  B1  B3  f3 ,

(81)

A2  B2  B4  2sB1  s2 f3  f 4 ,

(82)



lim  i  h11 B1  h12 B2  g3  e

y 0

  y ix





lim   2  h21 B1  h22 B2  g 4  e

y 0



d  0

  y ix

 x  c,

d  0

 x  c .

(83) (84)

By substituting (79–82) into (77) and (78), it yields  u  x







 f11B1  f12 B2  g1  eix d ,

(85)

 v x





1 i

 f 

B  f 22 B2  g 2  eix d ,

(86)

21 1

in which fi , gi  i  1, 2, 3, 4  and hij , f ij  i, j  1, 2  are given in the Appendix A. Similarly, to solve Eqs. (83) and (84), we introduce two dislocation density functions 1  x  and 2  x  as follows:

1  x  

 u  x

, 2  x  

 v y

.

(87)

Applying the inverse Fourier transform and considering the continuity conditions of displacement (29) and (30), it can be obtained from (85) and (86) that B1 

B2  

f 22 g



c

c

1  t  ei t dt 

f12 g



c

c

2  t  ei t dt 

G1 , g

f 21 c f11 c G i t  t e d t  2  t  ei t dt  2 ,   1    c  c g g g

(88)

(89)

in which G1, G2, and g are given in the Appendix A. From Eqs. (29) and (30), one gets



c

c

 j  t dt  0  j  1, 2  .

(90)

The substitution of Eqs. (88) and (89) into Eqs. (83) and (84), the singular integral equations for 1(x) and 2(x) are obtained as 2

  ij



   t  x  K  x, t   t  dt  2πL  x  i  1, 2, c

c

j 1

ij

j

i

x  c,

(91)

where the functions K ij , Li  i, j  1, 2  are given in Appendix A. The solutions of singular integral equations (91) have the following forms

l  x  

l  x  1  x2

 l  1, 2,

x  c,

where  l  x  are continuous and bounded functions in the interval c, c.

(92)

By applying the Lobatto–Chebyshev method as described in the previous works [35, 36], the singular integral equations in (91) and the single value conditions in (90) can be reduced to the following nth linear algebraic equations at n unknown discrete points of   t j  : n



n  1  K11  xr , t j 1  t j     j K12  xr , t j  2  t j   2πL1  xr  , j 1  t j  xr 

j  j 1

   t   0,

(93)

n

j

j 1

1

(94)

j



 1  K 22  xr , t j  2  t j   2πL2  xr  ,  t j  xr 

  j K21  xr , t j 1 t j     j  n

n

j 1

i 1

   t   0,

(95)

n

j 1

j

2

(96)

j

where t j  cos  j  1 π  n  1 

 j  1, 2, , n  , xr  cos  2r  1 π  2  n  1    r  1, 2,  j  π  2  n  1   j  1, n  ,  j  π  n  1  j  2, 3, , n  1 .

, n  1 ,

The stress intensity factor (SIF) is a physical quantity reflecting the strength of elastic stress field at the crack tip. The non-dimensional forms of SIFs under the action of in-plane normal (mode I) and shear (mode II) loadings are

K I*  p  = lim 2π  x  1 *y  x,0, p   

π 2 1, p  , 4

(97)

K II*  p  = lim 2π  x  1 xy*  x,0, p   

π1 1, p  . 4

(98)

x 1

x 1

So far, all the variables in the Laplace domain have been obtained. To achieve the dynamic temperature and SIFs in the time domain, the inverse Laplace transform

must be performed. It is worth noting that no universally accepted method of numerical Laplace inversion is available at present. In order to have confidence in the accuracy of calculations, we adopt the method proposed by Miller and Guy [37]. Functional values of a function f(t) are determined from the values of its Laplace transform function F(p) at discrete points of pi    1  i  ,  i  0, 1, 2

.

So the

evaluation of f(t) may be approximated in terms of the Jacobi polynomials N

f  t    Cn Pn 0,    ,

(99)

n 0

in which   2et 1, Pn

0,  

Pn 0,     

 

 1

n

2n n !

are defined as

1   



dn  n  n 1    1     , n   d

(100)

and Cn can be determined by the system of equations

 i i  i  1 i   n  1  C   F   1  i     n 0  i    1 i    2   i    1  n  n   C0   1   F   1   i  0  .

 i  1, 2,  , (101)

In calculation, the parameter values in the inverse Laplace transform are chosen as N10, 0, and 0.3, which allows the variables to be best described for a period of time.

5

Numerical results and discussion In this section, the effects of the time delay, kernel function, and

nonhomogeneity parameters on the responses will be evaluated. Considering that the non-dimensional crack length is 2c2, the dimensionless distance between the crack

and the boundary h1. It is noted that all of the following variables are non-dimensionalized.

5.1 Validation We first validate the solutions. When the phase-lag of heat flux 0 and nonhomogeneity parameters 0, the current heat conduction model degenerates into the Fourier model in the homogeneous case. In Fig. 2, T  0, 0  and T  0, 0  denote the values of the mid-points on the upper and lower crack faces, respectively. The present results are compared with the previous solutions [38], and excellent agreements are obtained.

Fig. 2. Comparison with Chen’s result

5.2 Effects of the time delay and kernel function on the temperature and SIFs In the following calculations, the phase-lag of heat flux is set as   0.5, nonhomogeneity parameters 1, 0.1, 1, unless otherwise specified. In the MDD

heat conduction model, two factors, namely, the time delay and kernel function may be chosen freely. Fig. 3 shows the effects of the time delay on the transient temperature response at the crack center with kernel function K(t,)1. It should be mentioned that the MDD heat conduction model reduces to the classical Fourier heat conduction model if 0, and to the CV heat conduction model if K(t,)1 and   1010 . For homogeneous materials (0), it is seen from Fig. 3 in Ref. [24] that the temperature on the lower crack face may exceed that on the boundary, which is called the overshooting phenomenon. It is mainly due to the inherent relaxation characteristic and the wave-like oscillation behavior of the current model, in which the phase lag of heat flux  is non-zero. However, the overshooting phenomenon will not occur when the nonhomogeneity parameters are nonzero and Fourier model is used. The introduction of the time delay can reduce the peak value, and the larger the time delay, the smaller the peak value of the temperature. In addition, we can also find that the non-Fourier heat conduction effects vanish, and the temperature based on the current model and Fourier model converges to a steady-state value as time becomes long enough.

Fig. 3. Effects of the time delay on the transient temperature at the crack center

Similar to the transient temperature response, the effect of the time delay on the dynamic SIFs KI and KII around the crack tip with K(t,)1 is shown in Fig. 4. For comparison, the results for the classical Fourier heat conduction are also given. We see that the results for the MDD heat conduction model demonstrate considerable oscillations before converge to the corresponding steady solutions. With the increase of the time delay , the wave-like oscillation behavior becomes weaker, which implies an advantage of the current model, because it not only exhibits the wave-like behavior of the CV model but also captures the characteristic of the classical Fourier heat conduction. In addition, due to the non-local property of the MDD as shown in Eq. (3), the current model is associated with a pronounced memory effect on the evolution of the stress field compared with the CV model. It can be seen that the magnitudes of the SIFs based on the MDD model are lower than its counterparts of the CV model.

(a)

(b) Fig. 4. Effects of the time delay on the dynamic SIFs around the crack tip: (a) KI, (b) KII

In the above analysis, we have investigated the effects of the time delay on the responses once the kernel function is determined. Accordingly, the effect of the kernel function with a given time delay on the responses can also be presented. Figs. 5 and 6 demonstrate the effects of the kernel function on the transient temperature and SIFs with 0.5. Three kernel functions: K  t ,    1, 1   t     , 1   t     

2

are

considered, which will be referred to as Kernel 1, Kernel 2 and Kernel 3, respectively. Clearly, the distributions of the temperature and the SIFs have different forms under different kernel functions. It is noticed that the magnitudes of the temperature and SIFs are the smallest in the case of Kernel 3. The results indicate that different kernel functions reflect different memory effects, therefore one may select Kernel 3 to improve the effects of the memory-dependent derivative.

Fig. 5. Effects of the kernel function on the transient temperature at the crack center

(a)

(b) Fig. 6. Effects of the kernel function on the dynamic SIFs around the crack tip: (a) KI, (b) KII

To better observe the effects of the time delay and kernel function on the temperature distribution within the cracked half-space, the temperature contour distributions when the highest temperature occurs are displayed in Figs. 7 and 8. It is observed that the temperature in the lower half of the half-space is higher than that of the upper part, which may be owed to the fact: the input, i.e. the temperature increase, is applied to the bottom face of the half-space. Due to the assumption of insulation on crack faces, there is a temperature jump across the crack faces. Moreover, from Fig. 7 we can see that an increase in the time delay reduces the heat flux, which is in line with that indicated in Fig. 3. For a given time delay, the heat flux through the medium is the least when Kernel 3 is adopted, as depicted in Fig. 8. The above results are of great importance in thermal engineering applications, such as safety design of the electronic or mechanical devices under severe thermal loadings [39].

(a)

(b)

(c)

(d)

Fig. 7. Temperature distribution for different time delays with K  t ,    1 when the maximum temperature appears: (a)   1010 ; (b)   0.1 ; (c)   0.5 ; (d) Fourier model

(a)

(b)

(c) Fig. 8. Temperature distribution for different kernel functions with   0.5 when the maximum

temperature appears: (a) Kernel 1; (b) Kernel 2; (c) Kernel 3

5.3 Effects of the nonhomogeneity parameters on the temperature and SIFs In fact, when the FGM is used for thermal shielding applications, the lower face of the medium (exposed to thermal loading) should be made of a heat-resistive material such as ceramic and the upper face should be made of a metallic-type material [40]. As a result, in the following discussions, the elastic modulus of the ceramic in FGM should be higher than that of the metal (0). The thermal conductivity and the thermal expansion coefficient of the ceramic should be lower than those of the metal (0, 0). As expected, the nonhomogeneity parameters  and  have no effect on the temperature field, which can be read from Eq. (17). Therefore, here we only discuss the influence of the nonhomogeneity parameter  on the temperature. Fig. 9 presents the transient temperature distribution on the crack faces and the crack extension line y0 for different nonhomogeneity parameters  with 0.5, K(t,)1, 1 and 0.1 for t1.4. It is worth noting that the solid line and dash line stand for the temperature on the lower and upper crack faces, respectively. It can be seen that for the FGM medium, the temperature distribution is not symmetric with respect to the crack faces. This asymmetry can be attributed to two reasons: one is that the crack being insulated is playing the role of a heat barrier, the other is that the thermal shock is applied to the lower surface. With the nonhomogeneity parameter  rising, the temperature of the upper and lower surfaces decreases significantly, and moreover, the temperature jump

across the crack faces is reduced, implying a small temperature gradient.

Fig. 9. Effects of the nonhomogeneity parameter  on the distribution of the temperature field at the plane y0 for t1.4

In order to investigate the effects of the nonhomogeneity parameter  on the temperature of the upper and lower crack surfaces clearly, the temperature variation at the crack center with  for different instants is shown in Fig. 10. Note that ‘Inf’ stands for the long enough time. The solid line represents the temperature of the lower crack center, and the dash line represents the temperature of the upper crack center. One can observe that for short time t0.4, the temperature at the upper crack center is seen not to change since the temperature wave has not yet arrived here. The temperature difference between the upper and lower crack center increases with the time and reaches its peak value at t1.4, and then decreases with the time increasing before it reaches a stable value at tInf. It is also noted that with the increase of , the temperature at the upper and lower crack center decreases, and the temperature at the lower crack center changes greatly, while that at the upper crack center changes little.

Fig. 10.The temperature variation with  for different instants

One of the objectives to introduce FGMs is to relax the thermal stresses in such structures working in the high temperature environment. Therefore, studying the effects of the nonhomogeneity parameters on the SIFs is essential to the design of FGMs. Figs. 11–13 show the variations of the SIFs versus the nonhomogeneity parameters , ,  for different heat conduction models. It appears that the SIF KI is relatively insensitive to either  or , whereas the SIF KII is dramatically affected by . Within the given range, the absolute value of the SIF KI takes the minimum value when 2, and the SIF KII takes the minimum value when 2. This may hold implications in the design of FGMs. Finally, we find that in all cases, the magnitude of the SIFs predicted by the MDD model is significantly lower than that by the CV model. Especially, for the SIF KII, the relative error between the results based on the two heat conduction theories is very distinct. For example, when 1, 0.1, 1, the magnitude of KII under the MDD model decreases 18.90% than the corresponding value from the CV model. Obviously, the memory effect cannot be ignored for small

scale cracks. On the other hand, the SIFs KI predicted by the MDD model and the Fourier model tend to be consistent.

Fig. 11. The variation of SIFs with  for different heat conduction models for 0.5, K(t,)1, 0.1,

1

Fig. 12. The variation of SIFs with  for different heat conduction models for 0.5, K(t,)1, 1,

1

Fig. 13. The variation of SIFs with  for different heat conduction models for 0.5, K(t,)1, 1,

0.1

6

Conclusions In conclusion, we have investigated a transient thermoelastic problem of a

cracked FGM half-space under the thermal shock. The newly proposed memory-dependent heat conduction model has been adopted, and the integral transform technique is employed to reduce the thermal and mechanical problems to a system of singular integral equations. As a consequence, numerical solutions of the singular integral equations are obtained to demonstrate the effects of time delay, kernel function, and nonhomogeneity parameters on the transient temperature and SIFs. The results show that the introduction of the MDD reduces the magnitude of the temperature and SIFs because of the non-local property of the MDD. It also manifests that, the memory effects of MDD can be changed by selecting different kernels and time delays. Moreover, compared with homogeneous materials, FGMs can certainly

reduce the thermal stresses in such structures at small temporal and length scales. These findings may hold implications in many engineering areas. For example, one may select different nonhomogeneity parameters according to the effects of these parameters on the responses when designing FGMs, especially at micro- and nano-scale.

Acknowledgements This study is supported by the National Natural Science Foundation of China (11972375, 11732007), and China Postdoctoral Science Foundation Funded Project (2019TQ0355).

Appendix A 2

C11    +  2   2    2   2  p   2    2  D   2 2 D   C21    +  1    1    2   2  p   2    1  1  2e2  h 2  e2 h D   C22    +  2   2    2   2  p   2    2  2 1  2e2 h

f1   C21 +C22 e 2  h  e    1 h

f 2  C21   +  1  e   1 h  C22   +  2  e   2 h f3  C11   C21  C22  f 4  C11   +  2   C21   +  1   C22   +  2  

f5  C21   +  1    C22  C11   +  2   2

2

2 D   1  2e2  h

f 6  C21   +  1     1    C22  C11   +   2     2   2

2

2 D   1  2 e2  h

g1  f5  2s2 f 4  s22 f3 g 2   s22   2s23  f3   2s2   3s22  f 4  f 6 g3   f 2  f1s2  e s2 h  f 4  s2  f1 1  s2 h   f 2 h  e  s2 h  C11      2  g 4   f1 1  s2 h   f 2 h  e s2 h  f 3  C11

g  f11 f 22  f12 f 21 h11  2se 2 sh  s1  s2 1  2sh  e 2 sh h12  1  1  2 sh  e 2 sh  2 ss2 h 2e 2 sh h21  1  1  2 sh  e 2 sh

h22  2sh2e2 sh

   h12 f 21  h11 f 22   K11   1   sin   x  t   d 0 g  

K12   



K 21   



 2  h12 f11  h11 f12  g

0

 2  h21 f 22  h22 f 21 

0

K 22  



0

cos   t  x  d

g

cos   t  x  d

  3  h22 f11  h21 f12   1   sin   x  t   d g  

G1  g2 f12  g1 f22 G2  g1 f21  g2 f11  h G  h G  L1      11 1 12 2  g3  sin  x  d 0 g    h G  h G  L2    2  21 1 22 2  g 4  cos  x  d 0 g  

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