Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

55, 3, pp. 765-778, Warsaw 2017
DOI: 10.15632/jtam-pl.55.3.765

EFFECT OF INDUCED TEMPERATURE FIELD ON DEVELOPMENT OF

CURVILINEAR CRACK WITH BONDS BETWEEN THE FACES IN

END ZONES

Vagif M. Mirsalimov

Azerbaijan Technical University and Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku,

Azerbaijan Republic; e-mail: vagif.mirsalimov@imm.az

Azer B. Mustafayev

Institute of Mathematics and Mechanics of NAS of Azerbaijan, Baku, Azerbaijan Republic; e-mail: azer bm@list.ru

Temperature changes near the end of a curvilinear cohesive crack and their influence on
crack growth are investigated. The problem of local temperature changes consists in a delay
or retardationof the cohesive crackgrowth.The bondsbetween the curvilinear crack faces in
the end zones aremodeled by application to the crack surface cohesive forces caused by the
presence of bonds. The boundary value problem of equilibrium of the curvilinear crackwith
interfacial bonds in the end zonesunder actionof external tensile loads, induced temperature
field and tractions in the bonds preventing to its opening, is reduced to a system of singular
integral equations with a Cauchy-type kernel. From the solution of this equation system,
normal and tangential tractions in the bonds are found. Analysis of the limit equilibrium of
the crack using the end zonemodel is performed on the basis of a criterion of bonds limiting
stretching and includes: 1) establishment of tractions depending on opening of the crack
faces; 2) evaluation of the stress state near the curvilinear crack with taking into account
tensile loads, induced temperature field, tractions in the bonds; 3) determination of the
critical external tensile loads.

Keywords: curvilinear crackwith interfacial bonds, thermoelastic stress field, cohesive forces

1. Introduction

Creating reliable emergency response systems is a vital issue, especially when we talk about
unique installations and safety of people. One of the effectivemeans of crack growth retardation
may be temperature and thermoelastic fields (Finkel, 1977; Parton andMorozov, 1985; Potthast
andHerrmann, 2000; Fu et al., 2001;Qin et al., 2007; Liu, 2008, 2011a,b, 2014a,b;Georgantzinos
andAnifantis, 2014; Liu et al., 2015). In fracturemechanics, the healing problem existing in the
crack body is of significant importance (Dimaki et al., 2010). As seen from the results of the
papers (Kadiev and Mirsalimov, 2001; Mirsalimov and Kadiev, 2004; Liu, 2008, 2011a,b; Itou,
2014; Liu, 2014a,b; Mirsalimov and Mustafayev, 2015a,b), the influence of the thermal source
reduces strain of the stretching plane in the direction perpendicular to the crack, and because
of what the stress intensity factor near the crack end lowers. In most of the existing papers,
Griffith’s model of a crack is used. In the present paper, we use a bridged crack model ([4],
Mirsalimov, 2007; Mirsalimov andMustafayev, 2016).
The crack retardation problem is of scientific and significant practical value as its solution

enables one to extend the lifetime and, themain thing, to avoid accidents associatedwith sudden
fracture. Evaluation of efficiency of application of thermal sources in crack growth retardation
in thin-walled structural elements is of interest. Fracture of a constructionmay be prevented by
creating a thermal field in the path of crack growth. Creation of thermal fields was justified by
their ease of preparation andmultilateral nature of influence on the fracture process.



766 V.M.Mirsalimov, A.B. Mustafayev

Technical ease of obtaining in an extended object temperature and thermoelastic field in any
size and distribution creates wide opportunities of change of the direction and retardation of
crack growth. The experiments (Finkel, 1977) show that by heating the crack path to 70-100◦C,
we observe delay and retardation of a crack.
The effect of the temperature field on the retardation of a curvilinear crack with bonds

between the faces still has not been investigated. In this connection, studying the influence of the
reduced thermal stress field on curvilinear crack propagation in a stretchable plane with regard
to the bonds between the faces in the crack end areas is of scientific and practical interest. The
goal of the paper is to develop amathematical model of curvilinear interfacial crack retardation
bymeans of temperature fields.

2. Formulation of problem

Let us consider an unbounded elastic plane weakened with a crack of length 2ℓ = b− a at
the origin of coordinates. In real materials, due to structural and technological factors, crack
surfaces have roughnesses and curvings. A fracturemechanics problem on a curvilinear crack in
a plane assuming that the crack contour has roughnesses (small deviations from a linear form)
is considered. It is assumed that there are areas at which the cohesion forces of the material
continuously distributed at the end area of the crack, ect. It is considered that these areas adjoin
to the crack tips, their sizes are comparable with the crack length [4]. Models of the crack with
end zones were proposed for brittle materials in (Barenblatt, 1961) and for plastic flow state
under constant stress in (Leonov and Panasyuk, 1959; Dugdale, 1960). The end zones of the
curvilinear crackwere simulated by the areas withweakened interparticle bonds in thematerial.
The model of a crack with interfacial bonds at the end zones may be used in different

scales of fracture. Intensive development of crackmodels with explicit account of nonlinear laws
of interaction in conformity to elasto-visco-plastic behavior of materials and various kinds of
loading is connected with this fact. Bibliography on this subject may be found in papers of the
special issue of Engineering Fracture Mechanics (2003).
When the length of the end zone of the crack is not small compared with the crack length,

the methods for evaluating the fracture toughness of the material based on consideration of a
crack with a small end zone are not applicable. In these cases, simulation of the stress state
at the crack end zone should be carried out with regard to deformational characteristics of the
bonds.
The crack faces outside the end zones are free from external loads. At infinity, the strengthe-

ned plane is subjected to uniform tension along the ordinate axis by a stress σ∞y = σ0 (Fig. 1).
For retardation of the crack, on the path of its propagation a zone of compressible stresses is
formed bymeans of heating the domainS by a thermal source to temperatureT0. The following
assumptions are accepted:

a) All thermoelastic characteristics of the plate material are temperature independent.

b) The plate material is homogeneous and isotropic.

It is assumed that at themoment t=0an arbitrary domainS=S1+S2 on the crack growth
path in the plane instantly heats up to a constant temperature T = T0. The remaining part of
the plate at the initial moment has zero temperature.
For many metallic materials (steels, aluminum alloy and so on) at the temperature change

to 300◦C-400◦C, the dependence of thermoelastic characteristics weakly changes according to
temperature. This fact was experimentally established. Thus, for all structural materials there
exists such a temperature range in which the assumption on steadiness of characteristics of the
material is correct. It is established on the basis of temperature dependence of the modulus of
elasticity.



Effect of induced temperature field on development... 767

Fig. 1. Computational diagram of the problem

Let us distinguish a part of the crack of length d1 = a1 − a and d2 = b− b1 (end zones)
adjoining to its tips at which the crack faces interact. The interaction of the crack faces in the
end zones is modeled by introducing between the crack faces the bonds (cohesive forces) having
the given deformation diagram. The physical nature of such bonds and sizes of the end zones in
which the interaction of crack faces is realized, depends on kind of thematerial. A crack existing
in the plane is assumed to be close to a rectilinear form admitting only small deviations of the
crack line from the straight line y=0.The crack line equation is accepted in the form: y= f(x),
a¬x¬ b. Based on the accepted assumption on the form of the crack line, functions f(x) and
f ′(x) are small quantities.
The end zones are small compared with another part of the plane. Therefore, the end zones

may be mentally removed having changed by cuts whose surfaces interact between themselves
by some law corresponding to the action of the removed material.
Under the action of external power and thermal loads on the plane, in the bonds connecting

the crack faces in the end zones, there will arise in the general case, normal qy(x) and tangential
qxy(x) forces. Consequently, to the crack faces in the end zones therewill be applied normal and
tangential stresses equal to qy(x) and qxy(x), respectively. The quantities of these stresses are
not known in advance and they are to be determined.
The boundary conditions of the considered problem have the following form (y= f(x))

σn− iτnt =
{

0 for a1 <x<b1

qy − iqxy for a¬x¬ a1 ∧ b1 ¬x¬ b
(2.1)

where n, t are natural coordinates. The stress state is represented in the form

σx =σx0 +σx1 σy =σy0 +σy1 τxy = τxy0 + τxy1 (2.2)

whereσx0,σy0, τxy0 is the solution to the thermoelasticity problem for a crackless plane;σx1,σy1,
τxy1 is the stress state of the plane with an interfacial crack on the faces on which the stresses
equal in value and opposite in sign, defined by the stress state σx0, σy0, τxy0 for y = 0, are
additionally applied.

3. The method of solution of the boundary-value problem

For determining the stressesσx0,σy0, τxy0, we solve a thermo-elasticity problem for a solid plane.
At first, we find temperature distribution in the plane. For that we solve the following boundary
value problem of heat theory

∂T

∂t
= a∗∆T T =

{

T0 for t=0 ∧ x,y∈S
0 for t=0 ∧ x,y /∈S

(3.1)



768 V.M.Mirsalimov, A.B. Mustafayev

where ∆ is the Laplace operator; a∗ is the coefficient of thermal conductivity of the plane
material.
For a generalized plane stress state, it is assumed that the plate is thermally insulated on

lateral surfaces. When determining the temperature field, for simplifying the problem, we do
not take into account a perturbed temperature field caused by the existence of the crack. Let
the heated by heat source areas S1 and S2 from each vertex are arbitrary and simply connected
regions with centers Ok(Lk,ck) (Fig. 1).
The solution to the thermal conductivity problem has the form

T(x,y,t)=
T0
4πa∗t

∫∫

S1

exp
(

−
R2

4a∗t

)

dξdη+

∫∫

S2

exp
(

−
R2

4a∗t

)

dξdη (3.2)

whereR2 =(x− ξ)2+(y−η)2.
For the thermoelastic potential of displacement, we find

F(x,y,t) =
(1+ν)αT0
4π





t
∫

0

1

τ

∫∫

S

exp
(

− R
2

4a∗t

)

dξdη−2
∫∫

S

ln
1

R
dξdη



 (3.3)

whereS=S1+S2; ν is thePoisson ratio of thematerial;α is the coefficient of linear temperature
expansion.
The components of the stress tensor are expressed through thermoelastic potential of displa-

cement by the known formulas (Parkus, 1959) and have the form

σx0 =−µ(1+ν)αT0

·
{

1+
1

π

∫∫

S

1

R4

[

(x− ξ)2− (y−η)2+2(y−η)2Γ
(

2,
R2

4a∗t

)

−R2exp
(

− R
2

4a∗t

)]

dξdη

}

σy0 =−µ(1+ν)αT0 (3.4)

·
{

1+
1

π

∫∫

S

1

R4

[

−(x− ξ)2+(y−η)2+2(x− ξ)2Γ
(

2,
R2

4a∗t

)

−R2exp
(

−
R2

4a∗t

)]

dξdη

}

τxy0 =−
µ(1+ν)αT0
2π

∫∫

S

4(x− ξ)(y−η)
π

1

R4

[

1−Γ
(

2,
R2

4a∗t

)]

dξdη

Γ(α,x)=

∞
∫

x

e−ttα−1 dt

where µ is the shear modulus of the material.
Boundary conditions (2.1) on the crack with end zones on the basis of (2.2) take the form

(y= f(x))

σn1−iτn1 =
{

−[σy0(x,0)− iτxy0(x,0)] for a1 <x<b1
qy(x,0)− iqxy(x,0)− [σy0(x,0)− iτxy0(x,0)] for a¬x¬ a1 ∧ b1 ¬x¬ b

(3.5)

Let us consider some arbitrary realization of the curved (with small deviations from the
rectilinear form) surface of crack faces.
As the functions f(x) and f ′(x) are small quantities, the function f(x) may be represented

in the form

f(x)= εH(x) a¬x¬ b



Effect of induced temperature field on development... 769

where ε is a small parameter for which we can accept the greatest height of the bulge of irregu-
larity of the upper crack surface related to the crack length.
The stress and the displacement values and tractions in the bonds are sought in the form of

an expansion in the small parameter

σx1 =σ
(0)
x +εσ

(1)
x + . . . σy1 =σ

(0)
y +εσ

(1)
y + . . . τxy1 = τ

(0)
xy +ετ

(1)
xy + . . .

u=u0+εu1+ . . . v= v0+εv1+ . . .

qy = q
0
y +εq

1
y + . . . qxy = q

0
xy+εq

1
xy+ . . .

where the termswith ε of higher orders are neglected for simplification. Here σ
(0)
x , σ

(0)
y , τ

(0)
xy , u0,

v0, q
0
y, q
0
xy and σ

(1)
x , σ

(1)
y , τ

(1)
xy , u1, v1, q

1
y, q
1
xy are the stresses, displacements and tractions in the

bonds of zero and first approximations, respectively.
The values of stresses for y = f(x) can be found by expanding in series the expressions for

stresses in the vicinity of y = 0. Using the procedure of the perturbation method, allowing for
previous formulas, we find boundary conditions for y=0, a¬x¬ b:
— in the zero approximation (y=0)

σ(0)y − iτ(0)xy =
{

−(σy0 − τxy0) for a1 <x<b1
q0y − iq0xy− (σy0 − τxy0) for a0 ¬x¬ a1 ∧ b1 ¬x¬ b0

(3.6)

— in the first approximation (y=0)

σ(1)y − iτ(1)xy =
{

N− iT for a1 ¬x¬ b1
q1y − iq1xy+N− iT for a1 ¬x¬ a1 ∧ b1 ¬x¬ b1

(3.7)

where for y=0

N = τ(0)xy
dH

dx
−H
∂σ
(0)
y

∂y
T =(σ(0)y −σ(0)x )

dH

dx
−H
∂τ
(0)
xy

∂y

a= a0+εa1+ . . . b= b0+εb1+ . . .

(3.8)

Thebasic relations of the statedproblemshouldbecomplementedby theequation connecting
the crack faces opening and tractions in the bonds in the end zones of the crack. Without loss
of generality, we can represent this equation in the form

(v+−v−)− i(u++u−)=Πy(x,σ)qy(x)− iΠx(x,σ)qxy(x) (3.9)

where the functionsΠy(x,σ) andΠx(x,σ) are effective compliances of the bonds dependent on

the tension of the bonds, σ=
√

q2y +q
2
xy is themodulus of the bonds traction vector; (v

+−v−)
are normal and (u+ +u−) are tangential components of the opening of the crack faces in the
end zones.
At constant values of Πy, Πx in (3.9), we have a linear law of deformation. In the general

case, the deformation law is nonlinear and is given.

The stress strain state σ
(0)
x ,σ

(0)
y , τ

(0)
xy andu0, v0 at an infinite plane and conditions of a plane

problem with a cut along the abscissa is described by two piecewise-analytical functions Φ(z)
andΩ(z) (Muskhelishvili, 2010)

σ(0)x +σ
(0)
y =2[Φ0(z)+Φ0(z)] σ

(0)
y − iτ(0)xy =Φ0(z)+Ω0(z)+(z−z)Φ′0(z)

2µ
∂

∂x
(uo+iv0)=κΦ0(z)−Φ0(z)−zΦ′0(z)−Ψ0(z)

Ω(z) =Φ(z)+zΦ′(z)+Ψ(z)

(3.10)

where κ=3−4ν for plane deformation, and κ=(3−ν)/(1+ν) for a plane stress state.



770 V.M.Mirsalimov, A.B. Mustafayev

For determination of the functionsΦ0(z) andΩ0(z) on the basis of boundary conditions and
the superposition principle, we have a linear conjugation problem (Muskhelishvili, 2010)

[Φ0(x)+Ω0(x)]
++[Φ0(x)+Ω0(x)]

− =2p0(x)

[Φ0(x)−Ω0(x)]+− [Φ0(x)−Ω0(x)]− =0
(3.11)

where a¬x¬ b, x is the affix of the points of the contour of the crack with end zones

p0(x)=

{

−(σy0 − τxy0)−σ0 on free faces of the crack
q0y − iq0xy− (σy0 − τxy0)−σ0 on faces of the crack end zones

Since the stresses in the plate are restricted, we should look for the solution to boundary
value problem (3.11) in the class of everywhere bounded functions. The sought for solution to
the boundary value problem is written in the form

Φ0(z) =Ω0(z)=

√

(z−a0)(z− b0)
2πi

b0
∫

a0

p0
√

(t−a0)(t− b0)(t−z)
dt (3.12)

Moreover, all the following solvability conditions of boundary value problem (3.11) should
be fulfilled

b0
∫

a0

p0(t)
√

(t−a0)(b0− t)
dt=0

b0
∫

a0

tp0(t)
√

(t−a0)(b0− t)
gt=0 (3.13)

These relations serve for determination of the unknown parameters a0 and b0.
For the final determination of the complex potentialsΦ0(z) andΩ0(z), it is necessary to find

the tractions in the bonds q0y and q
0
xy. Using the relation

2µ
∂

∂x
(u0+iv0)=κΦ0(z)−Ω0(z)− (z−z)Φ′0(z)

and the boundary values of the functions Φ0(z) and Ω0(z) on the segment a
0 ¬ x¬ b0, we get

the following equality

Φ+0 (x)−Φ
−

0 (x)=
2µ

1+κ

[ ∂

∂x
(u+0 −u

−

0 )+ i
∂

∂x
(v+0 −v

−

0 )
]

(3.14)

Using theSokhotski-Plemelj formulas (Muskhelishvili, 2010) and taking into account formula
(3.12) we find

Φ+0 (x)−Φ
−

0 (x)=−
i
√

(x−a0)(b0−x)
π

b0
∫

a0

p0(t)
√

(t−a0)(b0− t)(t−z)
dt (3.15)

The obtained expression, (3.15), is substituted into the left side of (3.14), and by taking into ac-
count relation (3.9) in the zero approximationwe get the system of nonlinear integro-differential
equations with respect to the unknown functions q0y and q

0
xy

−
√

(x−a0)(b0−x)
π

b0
∫

a0

q0y(t)+fy(t)
√

(t−a0)(b0− t)(t−x)
dt=

2µ

1+κ

d

dx
[Πy(x,σ

0)q0y(x)]

−
√

(x−a0)(b0−x)
π

b0
∫

a0

q0xy(t)+fxy(t)
√

(t−a0)(b0− t)(t−x)
dt=

2µ

1+κ

d

dx
[Πx(x,σ

0)q0xy(x)]

(3.16)

Here fy(t)=−σy0(t)−σ0, fxy(t)=−τxy0(t).



Effect of induced temperature field on development... 771

4. Numerical solution and analysis

As expected, the stated problem in the zero approximation is decomposed into two independent
problems: Eq. (3.16)1 for mode I crack and Eq. (3.16)2 for mode II crack.
Eachof equations (3.16)1 or (3.16)2 is anonlinear integro-differential equationwithaCauchy-

-type integral andmay be solved only numerically. For their solution, one can use a collocation
scheme (Panasyuk et al., 1976;Mirsalimov, 1987;Ladopoulos, 2000, 2013)withanapproximation
of unknown functions.
Now pass to algebraization of integro-differential equations (3.16). At first, in integro-

-differential equations (3.16) all integration segments are reduced to one interval [−1,1]. For
that, we make a change of variables

t=
1

2
(a0+ b0)−

1

2
(a0− b0)τ x=

1

2
(a0+ b0)−

1

2
(a0− b0)η

At such a change of variables, the left side of integro-differential equation (3.16)1 admits the
following form

−1
π

√

1−η2




1
∫

−1

q0y(τ)√
1− τ2(τ −η)

dτ+

1
∫

−1

fy(τ)√
1− τ2(τ −η)

dτ





Respectively, for the left side of equation (3.16)2 we get

−
1

π

√

1−η2




1
∫

−1

q0xy(τ)√
1− τ2(τ −η)

dτ+

1
∫

−1

fxy(τ)√
1− τ2(τ −η)

dτ





Replace the derivative in the right hand side of equation (3.16)1 for an arbitrary internal
i-th node by a finite-difference approximation

d

dx
[Πy(x,σ

0)q0y(x)]i =
Πy
(

xi+1,σ
0(xi+1)

)

q0y(xi+1)−Πy
(

xi−1,σ
0(xi−1)

)

q0y(xi−1)

2∆x

where∆x=(b0−a0)/M.
In the similar way we do it with the right hand side of equation (3.16)2. Therewith we take

into account boundary conditions for η =±1: q0y(a0) = q0y(b0) = 0, q0xy(a0) = q0xy(b0) = 0 (this
corresponds to the conditions v+0 (a

0,0)− v−0 (a0,0) = 0, v
+
0 (b
0,0)− v−0 (b0,0) = 0, u

+
0 (a
0,0)−

u−0 (a
0,0)= 0, u+0 (b

0,0)−u−0 (b0,0)= 0).
Using the quadrature formula

1

2π

1
∫

−1

g(τ)√
1− τ2(τ −η)

dτ =− 1
M sinθ

M
∑

k=1

gk

M−1
∑

m=1

cos(mθk)cos(mθ)

where

τ =cosθ ηm =cosθm θm =
2m−1
2M

π (m=1,2, . . . ,M)

all the integrals in (3.16) are replaced by finite sums, and the derivatives in the right hand sides
of the equations are replaced by finite-difference approximations.
The above formulae enable one to reduce each integro-differential equation to a finite system

of algebraic equationswith respect to approximate values of the sought for function, respectively,
at the nodal points. As a result, we get



772 V.M.Mirsalimov, A.B. Mustafayev

− 2
M

[

M
∑

ν=1

q0y,ν

M−1
∑

k=1

sin(kθm)cos(kθk)+
M
∑

ν=1

fy,ν

M−1
∑

k=1

sin(kθm)cos(kθk)

]

=
M(1+κ)

4µ(b0−a0)
[

Πy
(

xm+1,σ
0(xm+1)

)

q0y,m+1(xm+1)−Πy
(

xm−1,σ
0(xm−1)

)

q0y,m−1(xm−1)
]

− 2
M

[

M
∑

ν=1

q0xy,ν

M−1
∑

k=1

sin(kθm)cos(kθk)+
M
∑

ν=1

fxy,ν

M−1
∑

k=1

sin(kθm)cos(kθk)

]

=
M(1+κ)

4µ(b0−a0)
[

Πx
(

xm+1,σ
0(xm+1)

)

q0xy,m+1(xm+1)−Πx
(

xm−1,σ
0(xm−1)

)

q0xy,m−1(xm−1)
]

(4.1)

where

m=1,2, . . . ,M1 q
0
y,ν = q

0
y(τν) q

0
xy,ν = q

0
xy(τν)

σy0,ν =σy0(τν) τxy0,ν = τxy0(τν) xm+1 =
a0+b0

2
+ b

0
−a0

2
ηm+1

If we take into account the equality

2
M−1
∑

k=0

cos(kθν)sin(kθm)= cot
θm∓θν
2

the systems will take the following forms

M
∑

ν=1

Amν(q
0
y,ν +fy,ν)=

M(1+κ)

4µ(b0−a0)
[

Πy(xm+1,σ
0)q0y,m+1−Πy(xm−1,σ0)q0y,m−1

]

M
∑

ν=1

Amν(q
0
xy,ν +fxy,ν)=

M(1+κ)

4µ(b0−a0)
[

Πx(xm+1,σ
0)q0xy,m+1−Πx(xm−1,σ0)q0xy,m−1

]

(4.2)

where

m=1,2, . . . ,M1 qy,ν = q
0
y(τν) qxy,ν = q

0
xy(τν)

fy,ν = fy(τν) fxy,ν = fxy(τν) Amν =− 1M cot
θm∓θν
2

the upper sign is taken when the number |m−ν| is odd, the lower sign when it is even.
Now pass to algebraization of the solvability conditions of boundary value problem (3.13).

Separating in them the real and imaginary parts, using the change of variables and Gauss’s
quadrature formula, we get the solvability conditions of the problem in the following form

M
∑

ν=1
f∗y(cosθν)= 0

M
∑

ν=1
τνf
∗
y(τν)= 0

M
∑

ν=1
f∗xy(cosθν)= 0

M
∑

ν=1
τνf
∗
xy(τν)= 0

(4.3)

where f∗y = q
0
y +fy, f

∗
xy = q

0
xy+fxy.

As a result of algebraization, instead of each integro-differential equationwith corresponding
additional conditions, we get M1+2 (M1 is the number of nodal points belonging to the crack
end zones) algebraic equations to determine stresses at nodal points and the end zone sizes. Even
in the special case of linear elastic bonds, the obtained system of equations becomes nonlinear
because of theunknownsize of the end zone. In this connection, for solving the obtained systems,
in the case of linear bonds the method of successive approximations has been used.
In the case of a deformation law of nonlinear bonds, to define tractions in the end zones it is

advisable to use an iterative scheme similar to themethod of elastic solutions (Il’yushin, 2003).



Effect of induced temperature field on development... 773

It is accepted that the law of deformation of interparticle bonds (cohesive forces) is linear

for V =
√

(v+0 −v
−

0 )
2+(u+0 −u

−

0 )
2 ¬ V∗. The first step of iterative calculations is solving

the system of equations (4.2) and (4.3) for linear-elastic interparticle bonds. The subsequent
iterations are fulfilled only in the case when the relation V (x)>V∗ holds on a part of the end
zone.For such iterations, the systemof equations is solvedat eachapproximation forquasi-brittle
bonds with effective compliance variable along the crack end zone and dependent on the size of
modulus of the fraction vector obtained in the previous step of calculation. Effective compliance
analysis is conducted through the definition of the secant modulus in elasticity parameters.
The successive approximations process ends as the forces along the end zone, obtained at two
successive iterations, differ a little from each other.
The nonlinear part of the strain curve of interparticle bonds is represented in the form

of a bilinear dependence whose ascending portion corresponds to elastic deformation of the
bonds (0 < V (x) < V∗) with their maximum tension. For V (x) > V∗, the deformation law is
described by a non-linear dependence determined by two points (V∗,σ∗) and (δc,σc). And for
σc ­ σ∗, we have a descending linear dependence (linear hardening corresponding to elastic-
plastic deformation of bonds). Here σ∗ is the maximum elastic stress in the bonds, δc is the
characteristic of fracture toughness of the material determined experimentally. In numerical
calculations, it has been assumed M = 30, which corresponds to partition of the integration
interval into 30 Chebyshev nodal points.
After finding stress components in the zero approximation, we find the functions N and

T from formulas (3.8). The sequence of solution of problem (3.7) in the first approximation is
similar to the solution of the problem in the zero approximation. The solution of the problem
on the definition of piecewise-analytic functionsΦ1(z) andΩ1(z) is of the form

Φ1(z) =Ω1(z)=

√

(z−a1)(z− b1)
2πi

b1
∫

a1

p1(t)
√

(t−a1)(t− b1)(t−z)
dt (4.4)

where

p1(t)=

{

N− iT on free crack faces
q1y − iq1xy+N− iT on faces of the crack end zones

Moreover, all the following solvability conditions of the boundary value problem should be
fulfilled

b1
∫

a1

p1(t)
√

(t−a1)(b1− t)
dt=0

b1
∫

a1

tp1(t)
√

(t−a1)(b1− t)
dt=0 (4.5)

These relations serve for determination of the unknown parameters a1 and b1.
Using the formula and boundary values of the functions Φ1(z), Ω1(z) on the segment

a1 ¬x¬ b1, we find the equality

Φ+1 (x)−Φ
−

1 (x)=
2µ

1+κ

[ ∂

∂x
(u+1 −u

−

1 )+ i
∂

∂x
(v+1 −v

−

1 )
]

(4.6)

Using the Sokhotski-Plemelj (Muskhelishvili, 2010) and taking into account formula (4.4),
we find

Φ+1 (x)−Φ
−

1 (x)=−
i

π

√

(x−a1)(b1−x)
b1
∫

a1

p1(t)
√

(t−a1)(b1− t)(t−x)
dt (4.7)



774 V.M.Mirsalimov, A.B. Mustafayev

Substituting obtained expression (4.7) into the left part of equation (4.6) and taking into
account relation (3.9) for the first approximation, after some transformations we get a system
of nonlinear integro-differential equations with respect to the unknown functions q1y and q

1
xy

−
√

(x−a1)(b1−x)
π

b1
∫

a1

q1y(t)+N(t)
√

(t−a1)(b1− t)(t−x)
dt=

2µ

1+κ

d

dx
[Πy(x,σ

1)q1y(x)]

−
√

(x−a1)(b1−x)
π

b1
∫

a1

q1xy(t)+T(t)
√

(t−a1)(b1− t)(t−x)
dt=

2µ

1+κ

d

dx
[Πx(x,σ

1)q1xy(x)]

(4.8)

where σ1 =
√

(q1y)
2+(q1xy)

2.

Similarly, in the first approximation, in obtaining the algebraic systems all integration in-
tervals are reduced to one interval [−1,1]. Then, using the quadrature formulas of the Gauss
type the integrals are replaced by finite sums. As a result we get the zero approximation while
obtaining algebraic systems all integration intervals are reduced to one interval [−1,1]. Then,
the integrals are replaced by finite sums by means of Gauss-type quadrature formulas. As a
result, we get

M
∑

ν=1

Amν[q
1
y,ν +Nν] =

µM

(1+κ)(b1−a1)
[

Πy(x
1
m+1,σ

1)q1y,m+1−Πy(x1m−1,σ1)q1y,m−1
]

M
∑

ν=1

Amν[q
1
xy,ν +Tν] =

µM

(1+κ)(b1−a1)
[

Πx(x
1
m+1,σ

1)q1xy,m+1−Πx(x1m−1,σ1)q1xy,m−1
]

(4.9)

where

m=1,2, . . . ,M1 x
1
m+1 =

1
2
(a1+ b1)− 1

2
(a1− b1)ηm+1 q1y,ν = q1y(τν)

q1xy,ν = q
1
xy(τν) Nν =N(τν) Tν =T(τν)

As a result of algebraization of the boundary problem solvability conditions (4.5), we obtain

M
∑

ν=1
f∗1y (τν)= 0

M
∑

ν=1
τνf
∗1
y (τν)= 0

M
∑

ν=1
f∗1xy(τν)= 0

M
∑

ν=1
τνf
∗1
xy(τν)= 0

(4.10)

where f∗1y = q
1
y +N, f

∗1
xy = q

1
xy+T .

As a result of algebraization, as in the zero approximation, instead of each integro-differential
equation we get a system ofM1+2 algebraic equations for determining stresses at nodal points
of the crack end zones and end zones sizes.
A solving algorithm for algebraic systems (4.9) and (4.10) is similar to the solution of the

systems for the zeroapproximation.Theopeningof the crack in the endzonesmaybedetermined
from the relations

v+(x,0)−v−(x,0)=Πy(x,σ)qy(x)
u+(x,0)−u−(x,0)=Πx(x,σ)qxy(x)
a¬x¬ a1 b1 ¬x¬ b

(4.11)

Calculations show that for the linear law of deformation of bonds, tractions in the bonds
have always maximal values at the edge of the end zone. The similar picture is observed for



Effect of induced temperature field on development... 775

the quantities of crack openings. Opening of a crack at the edge of end zone has maximum for
linear and nonlinear laws of deformation, and with an increasing compliance of the bonds. To
determine the limit state at which the crack growth occurs, we use the critical condition

|(v+−v−)− i(u++u−)|= δc (4.12)

The joint solution of the obtained equations and condition (4.12) enable determination of
the critical value of external loads, forces in the bonds and the size of the end zone for the limit
equilibrium state under the given characteristics of bonds.
The functionH(x) describing roughness of the crack surface are considered as a determinate

set of the rough surface of the profile contour and also as a stationary random function. In this
case, the random functionH(x) is given by the canonical expansion

H(θ)=
∞
∑

k=−∞

vkε
ikθ

where vk are independent zero random values of the mathematical expectation and dispersions
Dk.
Calculations show that the heated zone promotes flow of plastic deformations in the bonds.
In Fig. 2a, plots of distribution of normal forces qy in the bonds of the crack end zones

are depicted for the following values of free parameters t∗ = 4a∗t/L
2
1 = 10; c1/L1 = −0.2;

c2/L2 = 0.1 = 0.1; ν = 0.3; L1 = L2 = L; M = 30; E = 1.8 · 105MPa; V∗ = 10−6m;
σ∗ =75MPa; σc/σ∗ =2; δc =2.5 ·10−6m; C =2 ·10−7m/MPa (C is the effective compliance
of the bonds). There, curves 1 correspond to the bilinear law of strains of the bonds, and
curves 2 correspond to the linear law of strains. In the computations, we used the dimensionless
coordinates x=(a+ b)/2+(b−a)x′/2.
The compliance of the bonds in the normal and tangential directions is assumed to be equal.
Graphs of the distribution of tangential forces qxy in the bonds of the crack end zones are

shown in Fig. 2b.

Fig. 2. Distribution of normal forces qy/σ0 (a) and tangential forces qxy/σ0 (b) in the bonds of
the crack end zones

Graphs of the length of the crack end zone (b− b1)/(b− a) for the plate against the di-
mensionless value of the tensile stress σ0/σ∗ are shown in Fig. 3 for different crack lengths
ℓ∗ =(b1−a1)/(b−a)= 0.5, 0.7.
The dependence of the critical load σ0/σ∗ on the dimensionless length of the crack (b−a)/R

is shown in Fig. 4. ThereR is the typical linear size of the plate.



776 V.M.Mirsalimov, A.B. Mustafayev

Fig. 3. Dependency of the length of the end zone (b− b1)/(b−a) on the dimensionless value σ0/σ∗ for
different crack lengths

Fig. 4. Distribution of the critical load on the dimensionless length of the crack

Theoretical and experimental investigations show that the created temperature field in the
course of some limited time for the purpose of crack retardation is an insurmountable barrier
(Finkel, 1977) on the path of its propagation. Subsequent removal of the temperature field
(t→∞)will graduallydecrease thevalueof compressive stresses andcrack retardation efficiency.
The crack faces opening at the bottomof the end zone, having attained reduction,will gradually
grow to the size stipulated bymechanical load.

Under action of the temperature field, simultaneouslywith the reduction ofmaximumtensile
stress, there happens its unfolding towards the thermal source. This reduces (Morozov, 1969;
Finkel, 1977) the displacement of the fracture plane observed in the experiment. After removing
the temperaturefield, this circumstancewill help ensuring that for crackpropagation, an increase
of the external tensile load is necessary. Note that the perturbed temperature field amplifies the
inhibitory effect of the induced temperature field of stresses. In conclusion, note that plate-
like elements have found wide application in constructions of different kind transport systems
(aircraft). Based on experimental data and numerical results of this paper, we can recommend
the following schemes of effective retardation of crack propagation:

• On the path of possible fracture of a plate-like construction, it is necessary to create stable
temperature fields. If a crack grows in the direction of temperature increase, then velocity
of its growth will decrease, and sooner or later it will stop.



Effect of induced temperature field on development... 777

• Creating no temperature fields beforehand, heating on the path of crack propagationmay
be conducted impulsively, for example, bymeans of explosive wires. In this case, the crack
tip is found in the site of explosion. As a result of simultaneous action of impact waves
of thermoelastic stress and plastic deformations of the heated material, the crack growth
slows down and the fracture stops.

5. Conclusions

Theoretical investigation of the retardationproblem for a curvilinear crackwith interfacial bonds
by temperature fields has been carried out. An effective calculation scheme of the retardation
of the curvilinear crack with interfacial bonds in a plane under action of external tensile loads
is suggested. Based on the obtained results, we can consider that the temperature field created
in the vicinity of the crack tip is a barrier to its propagation way. Relations for tractions in the
bonds and opening of curvilinear crack faces in the end zone depending on the applied load,
intensity of thermal source, crack length, and geometrical sizes of the heated zone are obtained.
The dependence of the crack length on the applied stretchable load, intensity of the heated zone
and also on physical and geometrical parameters of the plate atmonotone loading is established.

In the case of a crack with bonds in the end zones and temperature stresses induced by heat
sources, the analysis of the limiting equilibrium state of the plane reduces to a parametric study
of the solution of algebraic systems (4.2),(4.3) and (4.9), (4.10) for various laws of deformation
of the bonds, sizes of end zones and thermal and elastic constants of the plane material. The
normal and tangential stresses in the bonds and the crack opening are directly determined by
solving the resulting algebraic systems in each approximation. The crack opening in the end
zones can also be determined from relation (4.11).

References

1. BarenblattG.I., 1961,Themathematical theory for equilibriumcracks formedonbrittle fracture
(in Russian), Journal of Applied Mechanics and Technical Physics, 2, 4, 3-56

2. Dimaki A.V.,Mel’nikov A.G., Pleshanov V.S., Sizova O.V., 2010, Theoretical and experi-
mental study of the healing of surface cracks using induction heating, Inorganic Materials: Applied
Research, 1, 4, 353-358

3. DugdaleD.S., 1960,Yieldingof steel sheets containing slits,Journal of theMechanics andPhysics
of Solids, 8, 100-108

4. Engineering Fracture Mechanics, The special issue: Cohesive Models, 2003, 70, 15, 1741-1987

5. Finkel V.M., 1977,Physical Basis of Fracture Retardation (in Russian), Metallurgiya,Moscow

6. FuY.-M., Bai X.-Z., QiaoG.-Y., HuY.-D., Luan J.-Y., 2001,Technique for producing crack
arrest by electromagnetic heating,Materials Science and Technology, 17, 1653-1656

7. Georgantzinos S.K.,AnifantisN.K., 2014,Crack closure, [In:]Encyclopedia of Thermal Stres-
ses, R.B. Hetnarski (Edit.), Springer Netherlands, ISBN 978-94-007-2738-0, 774-779

8. Il’yushin A.A., 2003,Plasticity (in Russian), Logos,Moscow

9. Itou S., 2014, Thermal stresses around two upper cracks placed symmetrically about a lower
crack in an infinite orthotropic plane under uniform heat flux, Journal of Theoretical and Applied
Mechanics, 52, 617-628

10. KadievR.I.,MirsalimovV.M., 2001, Effect of heat source on the dynamics of crack growth (in
Russian),Vestnik Dagestanskogo Universiteta, 4, 69-73

11. Ladopoulos E.G., 2000, Singular Integral Equations, Linear and Non-linear Theory and its Ap-
plications in Science and Engineering, Springer Verlag, Berlin



778 V.M.Mirsalimov, A.B. Mustafayev

12. Ladopoulos E.G., 2013, Non-linear singular integro-differential equations in Banach spaces by
collocation evaluationmethods,Universal Journal of Integral Equations, 1, 28-38

13. Leonov M.Ya., Panasyuk V.V., 1959, Propagation of fine cracks in solid body (in Russian),
Prikladnaya Mekhanika, 5, 391-401

14. Liu T.J.C., 2008, Thermo-electro-structural coupled analyses of crack arrest by Joule heating,
Theoretical and Applied Fracture Mechanics, 49, 171-184

15. Liu T.J.C., 2011a, Finite element modeling of melting crack tip under thermo-electric Joule he-
ating,Engineering Fracture Mechanics, 78, 666-684

16. Liu T.J.C., 2011b, Fracture mechanics of steel plate under Joule heating analyzed by energy
density criterion,Theoretical and Applied Fracture Mechanics, 56, 154-161

17. Liu T.J.C., 2014a, Compressive stresses near crack tip induced by thermo-electric field, Interna-
tional Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering,
8, 11, 1799-1802

18. LiuT.J.C., 2014b,Crackdetection/arrestwith Joule heating, [In:]Encyclopedia of Thermal Stres-
ses, R.B. Hetnarski (Edit.), Springer Netherlands, ISBN 978-94-007-2738-0, 779-791

19. Liu T.J.C., Tseng J.F., Chen P.H., 2015, Analysis of thermo-electric field in steel strip with
multiple cracks,Proceedings of the 3rd International Conference on Industrial Application Engine-
ering, 408-412

20. Mirsalimov V.M., 1987, Non-one Dimensional Elastoplastic Problems (in Russian), Nauka,
Moscow

21. MirsalimovV.M., 2007,The solutionofaproblemincontact fracturemechanicsonthenucleation
and development of a bridged crack in the hub of a friction pair, Journal of Applied Mathematics
and Mechanics, 71, 120-136

22. Mirsalimov V.M., Kadiev R.I., 2004, Closing of a crack in the sheet element under action of
local thermal field, Journal of Machinery Manufacture and Reliability, 33, 6, 69-75

23. Mirsalimov V.M., Mustafayev A.B., 2015a, A contact problem on partial interaction of faces
of a variable thickness slot under the influence of temperature field,Mechanika, 21, 19-22

24. MirsalimovV.M.,MustafayevA.B., 2015b,Solution of the problemof partial contact between
the faces of a slot of variable width under the action of temperature fields, Materials Science, 51,
96-103

25. Mirsalimov V.M., Mustafayev A.B., 2016, Influence of local temperature field on propaga-
tion of a curvilinear crack with interfacial bonds, ZAMM – Journal of Applied Mathematics and
Mechanics, 96, 1339-1346

26. Morozov E.M., 1969, On the strength analysis by the stage of fracture (in Russian), Deform.
Razrush. Term. Mekh. Vozdeistv, 3, 87-90

27. Muskhelishvili N.I., 2010, Some Basic Problem of Mathematical Theory of Elasticity, Springer,
Amsterdam

28. Panasyuk V.V, Savruk M.P., Datsyshyn A.P., 1976,The Stress Distribution Around Cracks
in Plates and Shells (in Russian), Naukova Dumka, Kiev

29. Parkus H., 1959, Instationäre Wärmespannungen, Springer-Verlag,Wien

30. Parton V.Z., Morozov E.M., 1985, Elastic-Plastic Fracture Mechanics (in Russian), Nauka,
Moscow

31. Potthast B., Herrmann K.P., 2000, Asymptotic analysis for temperature fields induced by
dynamic crack growth in pressure-sensitive materials, International Journal of Fracture, 106,
57-64

32. Qin Z, Librescu L., Hasanyan D., 2007, Joule heating and its implications on crack detec-
tion/arrest in electrically conductive circular cylindrical shells, Journal of Thermal Stresses, 30,
623-637

Manuscript received January 4, 2016; accepted for print January 9, 2017