Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

42, 1, pp. 59-68, Warsaw 2004

THE 3D THERMOELASTIC PROBLEM OF UNIFORM HEAT

FLOW DISTURBED BY AN INTERFACE CRACK IN

A PERIODIC TWO-LAYER COMPOSITE1

Andrzej Kaczyński

Faculty of Mathematics and Information Science, Warsaw University of Technology

e-mail: akacz@alpha.mini.pw.edu.pl

Stanisław J. Matysiak

Faculty of Mechanical Engineering, Białystok University of Technology

e-mail: matysiak@ geo.uw.edu.pl

This paper is concerned with the problem of an interface insulated pla-
ne crack obstructing a uniform heat flux in a two-layer microperiodic
space. An approximate analysis is performed within the framework of
linear stationary thermoelasticitywithmicrolocal parameters.A general
method of solving the resulting boundary-value problem is presented.
It is based on the use of potential functions and an analogy between
the thermal crack problem and the skew-symmetric mechanical loading
problem. The thermal stress singularities are discussed.

Key words: periodic two-layered space, homogenized model, interface
crack, heat flow, thermal stresses

1. Introduction

Due to the rapidly increasing use of composites for engineering structu-
res, considerable attention has been given to the analysis of interface cracks
subjected to mechanical and thermal loading (see the papers included in the
volume edited byRossmanith (1997)). Thewell-known conventional solutions,
in this case, exhibit peculiar oscillatory singularities near crack borders,which
are physically unacceptable.

1
This paper was presented on Symposium Damage Mechanics of Materials Structures,

June 2003, Augustów, Poland



60 A.Kaczyński, S.J.Matysiak

The present work aims at the three-dimensional problem of uniform heat
flow in a bimaterial periodically layered space disturbed by an interface crack.
It is a sequel of our earlier research in the two-dimensional case (cf Kaczyń-
ski and Matysiak, 1989, 1998). The considerations are based on the use of
the homogenized model of a microperiodic two-layer composite, proposed by
Woźniak (1987),Matysiak andWoźniak (1988). This approximate treatment –
employed to interface cracks – fails to predict the oscillatory singular behavior,
and allows us to apply the classical concepts of fracturemechanics in terms of
stress intensity factors.

The investigations are being carried on by the general potential func-
tionmethod devised byKaczyński (1993, 1994). Effective results are obtained
owing to a close resemblance between interface crack problems treated within
the usedhomogenizedmodel and the correspondingproblems in a transversely
isotropic solid. As to the problem under study, a method of constructing its
solution by reducing it to an ordinary problem with shear stresses across the
surfaces of the crack is presented.

The determination of thermal stresses induced by cracks subject to uni-
form heat flow is very important for the study of material failure. There were
few reports on the three-dimensional analysis of thermoelastic crack problems.
The research in this field can be found in the books byKassir and Sih (1975),
Kit and Khay (1989). Related works, pertinent to the present study but con-
cerned with penny-shaped or elliptical cracks in a homogeneous isotropic or
transversely isotropic space, were published by Florence and Goodier (1963),
Kassir (1971), Tsai (1983).

2. Governing relations

Consider a microperiodic-laminated space as shown in Fig.1. A thin re-
peated fundamental layer of the thickness δ = δ1 + δ2 is composed of two
homogeneous sublayers (denoted by l = 1 and l = 2), characterized by the
Lamé constants λl, µl, thermal conductivities kl and coefficients of volume
expansions βl/(λl +2µl/3). Here and in the sequel, the index l = 1,2 refers
to the corresponding sublayers. Referring to the Cartesian coordinate sys-
tem (x1,x2,x3) with the x3-axis normal to the layering, denote at the point
x=(x1,x2,x3) the temperature (strictly, a deviation of the temperature from
the reference state) by θ, the displacement vector by u= [u1,u2,u3] and the
components of stresses by σ11, σ12, σ22, σ13, σ23, σ33.



The 3D thermoelastic problem of uniform heat flow... 61

Fig. 1. A uniform heat flow around an insulated interfacial crack in a two-layer
periodic space

We suppose that this composite is weakened by an interface crack occu-
pying a bounded plane area S with a smooth profile on the interface x3 =0,
and there is a constant heat flow q(∞) = [0,0,−q0] in the direction of the
negative x3-axis (Fig.1). The perfect mechanical bonding and ideal thermal
contact between the layers (excluding the crack region S) are assumed. The
crack surfaces are required to be free from tractions.

Because of the complexity of the composite geometry and boundary con-
ditions, a closed solution to the above crack-thermal stress problem cannot
be obtained. Therefore, a specific homogenization procedure calledmicrolocal
modelling (Woźniak, 1987; Matysiak andWoźniak, 1988; Kaczyński, 1993) le-
ading to a homogenized model of this layered composite will be employed in
order to find an approximate solution. Next, we recall only a brief outline of
governing relations of this model; see the papers cited above for details. The
following notation will be used: latin subscripts always assume values 1,2,3,



62 A.Kaczyński, S.J.Matysiak

and the Greek ones 1,2. The Einstein summation convention holds and sub-
scripts preceded by a comma indicate partial differentiation with respect to
the corresponding coordinates.
The following approximations for the temperature θ, displacements ui,

stresses σ
(l)
ij and fluxes q

(l)
i constitute the foundations of this approach

θ∼=ϑ θ,α ∼=ϑ,α θ
(l)
,3
∼=ϑ,3+h(l)Γ

ui ∼=wi ui,α ∼=wi,α u
(l)
i,3
∼=wi,3+h(l)di

σ
(l)
αβ
∼=µl(wα,β +wβ,α)+ δαβ[λl(wi,i+h(l)d3)−βlϑ]

(2.1)

σ
(l)
α3
∼=µl(wα,3+w3,α+h(l)d3)

σ
(l)
33
∼=(λl+2µl)(w3,3+h(l)d3)+λlwα,α−βlϑ

q(l)α
∼=−klϑ,α q

(l)
3
∼=−kl(ϑ,3+h(l)Γ,3)

Here, δαβ is the Kronecker delta and h
(l) is the derivative of the assumed δ-

periodic sectional shape function that becomes: 1 if l=1 (x∈1st sublayer)
and −η/(1−η) with η = δ1/δ if l = 2 (x∈2nd sublayer). Moreover, ϑ, wi
and di, Γ are unknown functions interpreted as macro-temperature, macro-
displacements andmicrolocal (thermal and elastic) parameters, respectively.
Applying the microlocal procedure to the macro-modelling of this bima-

terial periodically layered composite under stationary conditions, one arrives
at governing equations and constitutive relations of the homogenized model.
They are given (after eliminating allmicrolocal parameters and in the absence
ofbody forces andheat sources) in termsof theunknownmacro-temperature ϑ
andmacro-displacements wi as follows (see Kaczyński, 1994)

ϑ,αα+k
−2
0 ϑ,33 =0 (2.2)

1

2
(c11+ c12)wβ,βα+

1

2
(c11− c12)wα,ββ + c44wα,33+(c13+c44)w3,3α =K1ϑ,α

(2.3)

(c13+ c44)wα,α3+ c44w3,αα+ c33w3,33 =K3ϑ,3

σ
(l)
α3 = c44(wα,3+w3,α)

σ
(l)
33 = c13wα,α+ c33w3,3−K3ϑ

σ
(l)
12 =µl(w1,2+w2,1) (2.4)

σ
(l)
11 = d

(l)
11w1,1+d

(l)
12w2,2+d

(l)
13w3,3−K

(l)
2 ϑ

σ
(l)
22 = d

(l)
12w1,1+d

(l)
11w2,2+d

(l)
13w3,3−K

(l)
2 ϑ



The 3D thermoelastic problem of uniform heat flow... 63

q(l)α =−klϑ,α q
(l)
3 =−Kϑ,3 (2.5)

The positive constants appearing in the above equations, describing material
and geometrical characteristics of the composite constituents, are given in
Appendix. Observe that the condition of the perfect bonding between the
layers is satisfied, and setting µ1 = µ2 ≡ µ, λ1 = λ2 ≡ λ and β1 = β2 ≡ β,
k1 = k2 ≡ kwe get c11 = c33 =λ+2µ, c12 = c13 =λ, c44 =µ, K1 =K3 =β,
K = k, k0 = 1, passing directly to the well-known equations of stationary
thermoelasticity of a homogeneous isotropic body.

3. Mathematical formulation and method of solving to the

boundary-value problem

Within the scope of the above-presented homogenizedmodel, we deal with
the resultingboundary-valueproblem:findfields ϑand wi,σij suitable smooth
on R3−S such that Eqs (2.2)-(2.5) hold, subject to the following boundary
conditions

q3 =−Kϑ,3 =0
σi3 =0

}
∀(x1,x2,x3 =0±)∈S

q3 =−Kϑ,3 =−q0
σi3 =0

}
if
√
x21+x

2
2+x

2
3 →∞

(3.1)

Making use of the superposition principle, we construct the solution as a
sum of two parts – the state of simple flow of uniform heat in the uncracked
medium and the perturbation temperature and stress field due to the crack
that tends to zero at large distances from the origin.
The first 0-problem involves the solution to basic equations (2.2) with

conditions (3.1)2. The results are found to be

ϑ0(x1,x2,x3)=
q0
K
x3

w
0(x1,x2,x3)=

K3q0
K(2c13+ c33)

[
x1x3,x2x3,

1

2
(x23−x21−x22)

]
(3.2)

σ0i3(x1,x2,x3)= 0

Attention will be drawn then on the corrective solution to the perturbed
problem (designated by the tilde). The disturbance due to the crack S re-
sults in amacro-temperature ϑ̃ and induced thermal stresses σ̃ij. The steady-
state temperature field is first determined. In view of the skew-symmetry, the



64 A.Kaczyński, S.J.Matysiak

problem is reduced to that of finding ϑ̃ satisfying so-called quasi-Laplace’s
equation (2.2) in the half space x3 ­ 0, subject to the boundary conditions

ϑ̃,3
∣∣∣
S+
=−
q0
K

ϑ̃
∣∣∣
R2−S+

=0

(3.3)

ϑ̃→ 0 for
√
x21+x

2
2+x

2
3 →∞

It is deducted from the potential theory that the solution of this problem
can be written as

ϑ̃(x1,x2,x3)=
∂

∂z0

∫∫

S

γ(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2+z20

(3.4)

where z0 = k0x3 and γ is an unknown density that satisfies the integro-
differential singular equation of Newton’s potential type

∇2
∫∫

S

γ(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

=
q0
Kk0

(3.5)

in which ∇2 ≡ ∂2/∂x21+∂2/∂x22 stands for the two-dimensional Laplace ope-
rator.
We proceed now to an associated thermoelastic problem. Because of the

anti-symmetry of the stress system, it reduces to that of the half space x3 ­ 0
subjected to the skew-symmetry and stress-free conditions

σ̃33(x1,x2,x3 =0
+)= 0 ∀(x1,x2)∈R2

w̃α(x1,x2,x3 =0
+)= 0 ∀(x1,x2)∈R2−S

σ̃3α(x1,x2,x3 =0
+)= 0 ∀(x1,x2)∈S

(3.6)

An efficient approach is based on the construction of quasi-harmonic po-
tentials that satisfy governing equations (2.3) and are well suited to mixed
boundary conditions (3.6). Utilizing the results derived by Kaczyński (1994),
the perturbed problem reduces to the finding of two harmonic functions g
and h which satisfy the following boundary conditions resulting from Eqs
(3.6)
— for (x1,x2)∈S

g,33+ ν̂(g,22−h,12)= β̂
[∫∫

S

γ(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

]

,1

(3.7)

h,33+ ν̂(h,11−g,12)= β̂
[∫∫

S

γ(ξ1,ξ2) dξ1dξ2√
(x1− ξ1)2+(x2− ξ2)2

]

,2



The 3D thermoelastic problem of uniform heat flow... 65

— for (x1,x2)∈R2−S
g,3 =h,3 =0 (3.8)

with the constants ν̂ and β̂ given in Appendix.
These conditions are seen to be similar to those for shear crack mecha-

nical loadings given by the right-hand sides of equations (3.7), in an iso-
thermal elasticity state (cf Kassir and Sih, 1975). Thus, the thermal crack
problem in hand is reduced to its mechanical skew-symmetrical counterpart
provided we are able to solve Eq. (3.5) and evaluate the corresponding deri-
vatives of potentials appearing in (3.7). Exact results can be obtained for a
crack S in the shape of an ellipse. For illustration, we present them in the
case of a circular interface crack with the radius a, i.e. Sa = {(x1,x2,0) :
r ≡

√
x21+x

2
2 ¬ a}. Then, the solution to Eq. (3.5) is (see Kit and Khay,

1989)

γ(x1,x2)=−
q0
π2Kk0

√
a2−x21−x22 ∀(x1,x2)∈Sa (3.9)

and thederivatives of the correspondingpotentialsmaybe evaluated such that
the perturbed problem described by (3.7) and (3.8) takes the form
— for (x1,x2)∈Sa

g,33+ ν̂(g,22−h,12)=
β̂q0
2Kk0

x1

(3.10)

h,33+ ν̂(h,22−g,12)=
β̂q0
2Kk0

x2

— for (x1,x2)∈R2−Sa
g,3 =h,3 =0 (3.11)

It can be observed that this corresponds to the ordinary boundary space
problem of radial shear in an isotropic case, when the stresses on the crack
surfaces Sa are given as

σ3r =−
β̂q0
2Kk0κ

r x3 =0 r¬ a (3.12)

where κ is a constant defined in Appendix.
Hence, the potential functions g and h leading to the stress distribution in

the laminated medium can be found from the solution to this problem which
is known (see, for example,Kassir andSih, 1975). The singular behavior of the
interface crack-border thermal stresses is similar to that in the homogeneous
case of transversely isotropic bodies. The shearing stress σ3r near the crack



66 A.Kaczyński, S.J.Matysiak

border possesses the inverse square-root type the singularity, and is characte-
rized by the stress intensity factor given by

kII ≡ lim
r→a+

√
2(r−a)σ3r(r,0)=

2a
√
aq0β̂

3πKk0κ
(3.13)

A physically interesting observation is the influence of layering of the con-
sidered body on this factor, expressed by the term β̂/(Kk0κ).

Appendix

Denoting by

Bl =λl+2µl l=1,2

B=(1−η)B1+ηB2 K =(1−η)k1+ηk2

the positive coefficients in governing equations (2.2)-(2.5) are given by the
following formulae

k0 =

√
ηk1K+(1−η)k2K

k1k2
c33 =

B1B2

B

c11 = c33+
4η(1−η)(µ1−µ2)(λ1−λ2+µ1−µ2)

B

c12 =
λ1λ2+2[ηµ2+(1−η)µ1][ηλ1+(1−η)λ2]

B

c13 =
(1−η)λ2B1+ηλ1B2

B
c44 =

µ1µ2
(1−η)µ1+ηµ2

K1 =
ηβ1λ2+(1−η)β2λ1+2[(1−η)µ1+ηµ2][ηβ1+(1−η)β2]

B

K3 =
(1−η)β2B1+ηβ1B2

B
K =

k1k2

K

K
(l)
2 =

2µlβl+λlK3
Bl

d
(l)
11 =

4µl(λl+µl)+λlc13
Bl

d
(l)
12 =

2µlλl+λlc13
Bl

d
(l)
13 =

λlc33
Bl



The 3D thermoelastic problem of uniform heat flow... 67

The constants in Eqs (3.12), (3.7) and (3.8) are derived from Kaczyński
(1994), and are given as follows

κ=
t+c33

c11c33− c213
ν̂ =1−κt3c44

β̂= c1+
K3− c1c13− c2c33k20√

c11c33+ c13
− c44k0κ(c1− c2)

provided

t+ =

√
c11c33−c213−2c13c44

c33c44
+2

√
c11
c33

t3 =

√
ηµ1+(1−η)µ2

c44

c1 =
k20[(c13+ c44)K3− c33K1]+ c44K1

c33c44k
4
0 +(c

2
13+2c13c44− c11c33)k20 +c11c44

c2 =
k0[(c13+ c44)K3−c11K3+k20c44K3]

c33c44k
4
0 +(c

2
13+2c13c44− c11c33)k20 +c11c44

References

1. Florence A.L., Goodier J.N., 1963, The linear thermoelastic problem of
uniform heat flow disturbed by a penny-shaped insulated crack, Int. J. Engng
Sci., 1, 6, 533-540

2. Kaczyński A., 1993, On the three-dimensional interface crack problems in
periodic two-layered composites, Int. J. Fracture, 62, 283-306

3. Kaczyński A., 1994, Three-dimensional thermoelastic problems of interface
cracks in periodic two-layered composites,Engng Fract. Mech., 48, 6, 783-800

4. Kaczyński A., Matysiak S.J., 1989, Thermal stresses in a laminate com-
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5. Kaczyński A., Matysiak S.J., 1998, Thermal stresses in a bimaterial pe-
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inclusions, J. Theor. Appl. Mech., 36, 2, 231-239

6. Kassir M.K., 1971, Thermal crack propagation, Trans. ASME D, 93, 14,
643-648



68 A.Kaczyński, S.J.Matysiak

7. KassirM.K., SihG.C., 1975,Three-Dimensional CrackProblems.Mechanics
of Fracture 2, Noordhooff, Leyden

8. KitH.S.,KhayM.V., 1989,Method ofPotentials inThree-DimensionalTher-
moelastic Problems of Bodies with Cracks (in Russian), Naukova Dumka, Kiev

9. Matysiak S.J., Woźniak C., 1988, On the microlocal modelling of ther-
moelastic periodic composites, J. Tech. Phys., 29, 85-97

10. RossmanithH.P. (Edit.), 1997,Damage and Failure of Interfaces, Balkema,
Rotterdam

11. TsaiY.M., 1983,Transversely isotropic thermoelastic problemof uniformheat
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Trójwymiarowe zagadnienie stałego przepływu ciepła zakłóconego przez

międzywarstwową szczelinę w periodycznym dwuwarstwowym

kompozycie

Streszczenie

Problem stałego przepływu ciepła zakłóconego istnieniem izolowanej termicznie
szczelinymiędzywarstwowejwperiodycznejdwuwarstwowejprzestrzeni jestprzedmio-
tem pracy. Przybliżoną analizę przeprowadzono w ramach liniowej stacjonarnej ter-
mosprężystości z parametramimikrolokalnymi.Podano i zilustrowanometodę rozwią-
zaniawynikającego zagadnienia brzegowego, polegającą na zastosowaniupotencjałów
harmonicznych i ustaleniu analogii z odpowiadającymmechanicznym, niesymetrycz-
nym problemem szczeliny. Zbadano osobliwości naprężeń cieplnych wokół szczeliny.

Manuscript received October 7, 2003; accepted for print November 4, 2003