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j.engfracmech.2018.08.018

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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
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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  kT
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  1T
where
work
1
t   
   1 
 2
t
0
f   d 
(
  
is
the
gamma
function).
 Ezzat
q
[29]
adopted
  
q  kT
 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  kT
(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:
k2T  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
xy
(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
xy
(8)
With the help of Eq. (7), substituting (8) into the Eq. (6) one obtains:
22 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)
22 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)
22 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
12  t K12  x, t  dt  2 M1  x 

1
1
 1


t
K
x
,
t
dt





1
21
1
12  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)
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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
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