Acta Mech 224, 1571–1577 (2013)DOI 10.1007/s00707-013-0820-7

Lifeng Ma · Ran He · Jishan Zhang · Brian Shaw

A simple model for the study of the tolerance of interfacialcrack under thermal load

Received: 31 January 2012 / Revised: 23 November 2012 / Published online: 23 February 2013© Springer-Verlag Wien 2013

Abstract Thermal stresses due to a mismatch between material properties may induce failure of a bondedinterface. A simple model for linear dissimilar elastic bonded solids containing an interfacial Zener–Strohcrack under a uniform temperature shift is proposed. Its solution is derived. This result may provide someinformation about the interface defect tolerant size, which is mainly responsible for triggering interface failuresunder thermal load. Thus, it can be used to assess the interface integrity and reliability under thermal load. Onthe other hand, the practical interface crack problem in this study provides another background (mechanism)for the Zener–Stroh crack model.

1 Introduction

Interfaces such as a metal-to-ceramic system have a wide application in both structural materials and micro-electronic packaging in industry. To transfer load, to improve certain surface characteristics (e.g., corrosionand wear resistance), or to implement some special functions, a specific component material is required to bebonded to the other component material. Unfortunately, defects such as micro-voids or cracks in the interfaceare unavoidable during manufacture. The interface will play a vital role since the interface’s reliability anddurability directly influence the mechanical performance and fracture behavior of bi-materials under variousloading conditions [1]. The thermal load on the micro-cracks is an extreme factor which may significantlyinfluence the performance of the interface systems. For example, micro-interface cracks in the thermal barriercoating of blades in a gas turbine may be increased by thermal load cycling [2–6]. Due to a mismatch betweenthe materials’ thermal expansion and elastic properties, the stress concentration at the micro-interface crackleads to crack propagation or eventually causes the structure’s catastrophic failure [7,8]. Therefore, the studyof cracks under given thermal loadings is becoming increasingly important and has attracted considerableattention [2–8]. The objects of these studies are to understand the interface failure mechanism and damageevolution, and therefore to accurately analyze the thermal stresses and strength of bonded interfaces for designand health evaluation (see, e.g., [8,9]).

Considerable efforts have been paid, but there are still many problems which need to be explored. Forexample, in the literature, the interfacial Griffith crack model is frequently used in this topic (e.g., [10,11]).The employed model actually is an interfacial semi-infinite crack model so it hardly reflects the nature ofthe thermal interfacial crack problem with finite length. Also, stress states of the interfacial crack under ther-mal load are evaluated either by finite element method (FEM) (e.g., [12,13]) or by a simplified analytical

L. Ma (B) · R. HeS&V Laboratory, Department of Engineering Mechanics, Xi’an Jiaotong University, 710049 Xi’an, ChinaE-mail: malf@mail.xjtu.edu.cn

J. Zhang · B. ShawDesign Unit, School of Mechanical and Systems Engineering, Newcastle University, NE1 7RU Newcastle Upon Tyne, UK

1572 L. Ma et al.

approach through adopting the constitutive law with the thermal effect term [7,8,14]. FEM could not com-pletely handle some cases where the stress state of interfacial crack problems is singular. The thermal termin the constitutive law is simple and short, but it will make the elasticity governing equation complex andnumerical analysis lengthy. Although recent progress in the generalized finite element method (GFEM) andextended finite element method (XFEM) enables the direct numerical analysis of arbitrary crack or interfaceproblems feasible [15,16], it seems that it still needs great efforts due to its complexity. Alternately, from theresidual stress point of view, the thermal interface crack problems actually are residual stress problems. Themismatch between the material thermal expansions and elastic properties of upper material and lower materialwill spontaneously lead displacement jump between two interface crack surfaces. This real deformation hintsthat the Zener–Stroh crack model is more practical and suitable in comparison with the Griffith crack model.In the literature, the Zener–Stroh crack model is proposed on the basis of metal physical deformation with thedislocation accumulation [17–25].

The aim of the present study is to propose an interfacial Zener–Stroh crack model under a uniform temper-ature shift based on the real deformation, and its analytical solution is to be solved and analyzed. It is expectedthat this model may partially reveal interface failure mechanism and damage evolution and possibly providean easy way for the accurate estimate of the tolerant size of interface crack along bonded interfaces in designand health evaluation. On the other hand, the practical interface crack problem in this study provides anotherbackground (mechanism) for the Zener–Stroh crack model.

2 Formulation

2.1 Kolosov–Muskhelishvili complex potentials for plane elastic problem

In the Kolosov–Muskhelishvili complex formulation of plane elasticity, all components of stress and displace-ment are expressed in terms of two potential functions, �(z) and �(z), as follows [26]:

σ11 + σ22 = 2[�(z)+�(z)

],

σ22 − iσ12 =[�(z)+�(z)+ (z − z)�′ (z)

],

2μ(u1,1 + iu2,1

) = κ� (z)−�(z)− (z − z)�′ (z),

(1)

where i = √−1, z = x1+i x2 = x +iy, � (z) = φ′ (z) , � (z) = [zφ′ (z)+ ψ (z)

]′, μ is the shear modulus,

κ = 3 − 4ν for plane strain, ν is Poisson’s ratio, the comma followed by a subscript i indicates differentiationwith respect to xi , and the bar over a function denotes its complex conjugate.

2.2 Boundary conditions

The interface crack model is shown in Fig. 1, in which the tractions along the surfaces are free and the remoteloading vanishes. Now we suppose that the interface system undergoes a uniform temperature shift T . Themismatch displacement of upper and lower crack surfaces can be obtained by the following virtual operationssimilar to Eshelby’s work [27] : (i) Release the interface and allow the two materials to take free thermal defor-mations; (ii) rejoin the interface to its original shape and size by applying suitable surface tractions except the

x

y

- a a

#1

#2

Fig. 1 An interfacial Zener–Stroh crack formed under a uniform temperature shift. (The thermal expansion coefficients of upperand lower materials are different. #1: Material 1; #2: Material 2)

A simple model for the study of the tolerance 1573

interface crack portion; and (iii) remove the traction and leave two rejoined components in equilibrium state.Then the net initial displacement jump between the upper and lower crack surfaces is as follows:

u1 + iu2 = 2a

a∫

−a

(ε

upperT − εlower

T

)dl =

a∫

−a

(α1T − α2T ) dl = 2a (α1 − α2)T, (2)

where a is the crack half-length, εT is the free thermal expansion strain, α1 and α2 are the thermal expansioncoefficients of the upper material and lower material, respectively.

Thus, with the known jump displacement u1 + iu2 = 2a (α1 − α2)T , the boundary conditions forthe interface crack will be

{σ+

i2 (x) = σ−i2 (x) = 0, (i = 1, 2)

∫ a−a

[(u1 (x)+ iu2 (x))

+,1 − (u1 (x)+ iu2 (x))

−,1

]dx = 2a (α1 − α2)T,

x ∈ (−a, a) , (3)

where the superscript ‘+’ denotes the quantities from the upper crack surface and the superscript ‘-’ denotesthe quantities from the lower one.

It should be pointed out that ifu1+iu2 = 0, Eq. (3) turns to a special Griffith crack boundary condition:⎧⎨

⎩

σi2 (x) = 0, i = 1, 2,a∫

−a

[(u1 (x)+ iu2 (x))

+,1 − (u1 (x)+ iu2 (x))

−,1

]dx = 0, x ∈ (−a, a) . (4)

Here, it is necessary to explain the difference between the Griffith crack problem and the Zener–Stroh crackproblem and their corresponding boundary conditions: For the Griffith crack problem, the crack faces areassumed to be subject to some traction, and the accumulated displacement is zero, while for the Zener–Strohcrack problem, the crack faces are assumed to be traction-free, and the accumulated displacement is a finitevalue [21]. This theoretical description is regardless of the physical mechanism of crack formation. Thus, ifa crack problem can be described with the boundary condition (3), it will be regarded as Zener–Stroh crackproblem whatever its physical evolution mechanism. It should be noted that there is only a jump displacementu1 in Eq. (3) for the thermal problem (u2 is zero). This implies that the two crack tips will be all sharprather than one being sharp and the other blunt in the general Zener–Stroh crack problems.

3 Solution

3.1 Elastic solution for the problem

A detailed review on the interfacial Zener–Stroh crack problem can be found in references such as [18,28].The following is the derivation of the solution for the proposed model in this study.

The potentials in the upper and lower planes are written as

�(z) ={�1 (z) , z ∈ #1�2 (z) , z ∈ #2 �(z) =

{�1 (z) , z ∈ #1�2 (z) , z ∈ #2 (5)

The tractions along the x-axis of the upper and lower planes from (1)2 will be

�1(x+) +�1

(x−) = (σ22 − iσ12)

+ ,�2

(x−) +�2

(x+) = (σ22 − iσ12)

− .(6)

The continuity of (σ22 − iσ12)+ = (σ22 − iσ12)

− along the interface requires that

�1(x+) +�1

(x−) = �2

(x−) +�2

(x+)

, (7)

which leads to

�1 (z) = �2 (z) , z ∈ #1,

�2 (z) = �1 (z) , z ∈ #2.(8)

1574 L. Ma et al.

The derivative of the displacement of jumps across the interface can be written from (1)3 as

(u1 + iu2)+,1 − (u1 + iu2)

−,1 = (c1 + c2)

4

{(1 − β)�1

(x+) − (1 + β)�2

(x−)}

, (9)

where

c1 = (κ1 + 1)

μ1, c2 = (κ2 + 1)

μ2, β = μ1 (κ2 − 1)− μ2 (κ1 − 1)

μ1 (κ2 + 1)+ μ2 (κ1 + 1). (10)

By taking

�1 (z) = (1 + β) f (z) ,

�2 (z) = (1 − β) f (z) ,(11)

Equation (9) can be rewritten as

(u1 + iu2)+,1 − (u1 + iu2)

−,1 = (c1 + c2)

(1 − β2

)

4

[f(x+) − f

(x−)]

. (12)

Then the displacement jump condition (3)2 can be written as

(c1 + c2)(1 − β2

)

4

a∫

−a

[f(x+) − f

(x−)]

dx = 2a (α1 − α2)T . (13)

Next, by using (8) and (11), the traction-free condition of the crack surface can be written as

(1 + β) f(x+) + (1 − β) f

(x−) = (σ22 − iσ12) = 0, (−a < x < a) . (14)

Taking a branch function as

X (z) = 1

(z + a)δ (z − a)(1−δ) (15)

with

δ = 1

2+ iε, ε = − 1

2πln(1 − β)

(1 + β)(16)

and using

X(t+

) = 1

(t + a)δ (a − t)(1−δ) ieεπ,

X(t−

) = 1

(t + a)δ (a − t)(1−δ) (−i) e−επ ,|t | < a, (17)

one may get the solution to (14) as

f (z) = P0 X (z) = P0

(z + a)δ (z − a)(1−δ) . (18)

The constant P0 in (18) can be specified by (13) as

P0 = 4a (α1 − α2)T

(c1 + c2)(1 − β2

)π

i. (19)

Thus, the solution for an interfacial Zener–Stroh crack will be

�(z) ={(1 + β) f (z) , z ∈ #1,(1 − β) f (z) , z ∈ #2,

� (z) ={(1 − β) f (z) , z ∈ #1,(1 + β) f (z) , z ∈ #2.

(20)

A simple model for the study of the tolerance 1575

3.2 Stress intensity factors and crack tip energy release rate

According to the traditional definition of interfacial crack stress intensity factors [29], we define the left andright interfacial crack stress intensity factors (Fig. 1), respectively, as

K L = K LI − i K L

I I = limx→−a−

√2π (− [x + a])δ (σ22 − iσ12) = lim

x→−a− 2√

2π (− [x + a])δ f (x) ,

K R = K RI − i K R

I I = limx→a+

√2π ([x − a])1−δ (σ22 − iσ12) = lim

x→a+ 2√

2π ([x − a])1−δ f (x) .(21)

Inserting (18) into (21) leads to

K R = 8√

2a (α1 − α2)T√π (c1 + c2)

(1 − β2

)(2a)δ

i,

K L = − 8√

2a (α1 − α2)T√π (c1 + c2)

(1 − β2

)(2a)(1−δ) i.

(22)

It can be observed from (22) that K R and K L are antisymmetric. This is the basic property of the Zener–Strohcrack. To validate the result (22), we will consider some special cases as follows:

Case (a): β = 0. When β = 0 in Eq. (10), which is a special combination of the bi-material case, we can getδ = 1/2 from (16), and then from (22)

K R = 8√

a (α1 − α2)T√π (c1 + c2)

i,

K L = −8√

a (α1 − α2)T√π (c1 + c2)

i.

(23)

It can be found from Eq. (23) that the left and right stress intensity factors of the Zener–Stroh crack are theModel II case, and there is no Model I case. In this special case, evidently, both crack tips are sharp rather thanone being sharp and the other blunt in the general Zener–Stroh crack problems [21]. This result is consistentwith the result for the cases of homogenous materials [21].

Case (b): If material #1 and material #2 are identical, Eq. (22) degenerates to

K R = 0, K L = 0 (24)

This means that for a crack embedded in a homogenous material, a uniform temperature shift will not produceany thermal residual stress. This is evident from the thermal elastic mechanics.

The above degenerate cases may partially demonstrate that the result (22) is reliable.It should be pointed out that in the literature the physical mechanism of leading interfacial Zener–Stroh

crack is always believed to be a dislocation pileup (see, e.g., [18,19]), and the two crack tips of a crack are alsoclassified as a blunt one and sharp one. It is also believed that the crack propagation occurs always through theso-called sharp crack tip [21,22]. The thermal interfacial Zener–Stroh crack proposed in this study exhibitssome differences from the previous analysis in the literature: (i) The thermal load can be another resource forforming the Zener–Stroh crack; (ii) the crack tips are all sharp; and thus (iii) the crack can propagate at anycrack tip.

Since the stress intensity factors generally are complex numbers (see Eq. (22)), it is difficult to be used toassess the stability of crack tips. Alternatively, the crack energy release rate G is a real number, which some-times is interpreted as the crack driving force. This advantage suggests that it can be regarded as a practicalparameter. Moreover, we can show that the stress intensity factor K defined above is related to the crack energyrelease rate G as [29]:

G = (c1 + c2) |K |216 cosh2 πε

. (25)

Substituting either K R or K L of (22) into (25), one may find the crack energy release rate G as

G = 4 cosh2 πε

π (c1 + c2)(α1 − α2)

2 (T )2 a. (26)

This is the main result obtained in this study.

1576 L. Ma et al.

4 Discussion

Solution (26) may shed some light on the interface failure mechanism and damage evolution. It may alsoprovide some information on interface design, assessment, and also service life estimation. For the reliabilityand safety of the interface, it must satisfy the following relationship:

G < Gc, (27)

where Gc is the critical interface crack energy release rate, which is just a material parameter. Additionally, itshould be pointed out from Eqs. (26) and (27) that:

(i) In order to reduce the interface crack driving force G, the mismatch between the material thermal expan-sion coefficients of two materials should be chosen to be smaller.

(ii) When the two materials are specified and a uniform temperature shift T is given, the surface tolerantdefect size 2amax will be confined by (26) and (27) as

2amax <π (c1 + c2)Gc

2 cosh2 πε (α1 − α2)2

1

(T )2. (28)

(iii) Once the crack size exceeds the critical size 2amax, it will unstably increase until the structure fails since∂G/∂a > 0 from (26).

The above analysis and solutions are only suitable to problems where the micro-interface crack size isrelatively small compared with the geometric size of the material components. It is expected that this modelcan be adopted for modeling the thermal damage of coating-substrate systems, which conform to the aboveapplication condition.

5 Conclusions

In this study, we proposed an interfacial Zener–Stroh crack model under a uniform temperature shift load. Itsanalytical solution has been derived. In the literature, the Zener–Stroh crack is believed to be only associatedwith physical deformation by dislocation accumulation. This study provided another mechanism for Zener–Stroh crack formation, namely the materials’ thermal coefficient mismatch. The obtained result may be an easyway for the accurate estimation of the thermal stresses of bonded interfaces in design and health assessment.

Acknowledgments L.M. and R.H. would like to thank the financial supports by Ph.D. Programs Foundation of Ministry ofEducation of China and the Fundamental Research Funds for the Central Universities.

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