Published by AMSS Press, Wuhan, ChinaActa Mechanica Solida Sinica, Vol. 24, No. 6, December, 2011 ISSN 0894-9166

THE EFFECT OF BILINEAR BEHAVIOUR ON THEDIRECTION AND MAGNITUDE OF THE PEELING

MOMENT IN A BI-MATERIAL BEAM

Ugur Guven�

(Yildiz Technical University, Department of Mechanical Engineering, 34349 Besiktas, Istanbul, Turkey)

Received 15 June 2010, revision received 14 June 2011

ABSTRACT In this work a bi-material beam exhibiting partly bilinear behaviour under a uniformtemperature change is analyzed. The essence of solution is based on the approach of Timoshenko’smechanics of materials. The main aim of the present analysis is to understand the effect of thebilinear behaviour on the peeling moment. This theoretical mechanics model mentioned here cangive us useful insights to improve the resistance against the delamination.

KEY WORDS peeling moment, layered structures, interface, thermo mechanical, delamination

I. INTRODUCTIONThe bending problem of the bi-material beam under uniformly thermal loading has been an attractive

subject of longstanding considerable scientific and practical interest. Although it has been extensivelyinvestigated, it is still receiving attentions from many researchers because of its importance to structural,mechanical and microelectronic engineering. Existing analytical solutions for the thermal stresses ofbonded structures may be divided into two categories. First one is the elasticity theory approach; thesecond is called as the mechanics of materials approach. Although the second approach is less rigorousthan the first one, the derivation process is easy to understand and the solutions presented are simple andsufficiently accurate. Therefore, the mechanics of materials approaches are frequently used in practicalapplications. However, although the first approach is valid for the whole assembly, the second approachcan be used except the edges of the assembly. The thermo mechanical bending of the bi-material beamwas first examined by Timoshenko in 1925[1]. He made the classical beam theory assumptions, but hisanalysis did not include the interfacial stresses. Although the presence of interfacial stresses was firstnoticed by Timoshenko, he showed no further interest in this issue. After that, the generalized formsof this notable pioneer work[1] have been developed in Refs.[2–6] and [8,9], involving the interfacialstresses. Mirman[7] has reported that the interfacial peeling stress plays a more important role than theinterfacial shear stress in causing interfacial delamination of the cooled bi-material assemblies. Mooreand Jarvis[10] developed a very comprehensive study on the peeling moment and, later on, presentedsome new relationships for the peeling moment[11,12]. Meanwhile, it should be noted a notable pioneerwork in Ref.[13], which discussed the paradox between a peel force and peel angle.

In the present work, a theoretical analysis similar to Ref.[10] is performed for a bi-material beamthat exhibits the partly bilinear elastic behaviour. The analysis is based on Timoshenko’s mechanicsof materials approach as Ref.[10], i.e. the interfacial stresses and boundary conditions are ignored andthe length is taken to be infinite. The presented solution is valid for a generalized plane stress problem.

� Corresponding author. E-mail: uguven@yildiz.edu.tr; Tel: +9002123832823; Fax: +9002122616659

Vol. 24, No. 6 Ugur Guven: Bilinear Behaviour on Direction and Magnitude of Peeling Moment · 507 ·

Furthermore, since the approach based on the evaluation of the peeling moment (Moore and Jarvis)[10]

is more practical than that based on the evaluation of the peeling stress (Suhir and Mirman)[4–6,8,9], weadopt the peeling moment approach in the present paper. The aim of this analysis is to show the effectof the bilinear layer on the peeling moment. The obtained results give useful information to increasethe resistance against the delamination of a bi-material beam.

II. ANALYSESIn the present analysis, the bi-material beam

studied is composed of the upper layer (i.e. layer1) with linear elastic material and the lower layer(i.e. layer 2) with bilinear elastic material (Fig.1).The elasticity modulus of the upper layer is E1,the elasticity modulus of the lower layer in tensionis E2t and in compression is E2c. The thicknessesof the upper and lower layers are t1 and t2, re-spectively and it is assumed that the width of the

Fig. 1 The bi-material beam element far from the ends undera temperature change.

bi-material beam is unit.

Following Ref.[10], one can obtain a general solution for the bending of the bi-material beam underthe thermal mismatch. Here we focus on the peeling moment of the bi-material based on Timoshenko’sapproach[1]. The equilibrium of the internal forces over any cross section of the beam can be writtenas[1]

P1 = −P2 = P (1)

andP (t1 + t2)

2= M1 + M2 (2)

The bending moments M1 and M2 are given as

M1 =E1t

31

12ρand M2 =

E2rt32

12ρ(3)

where E2r =4E2tE2c

(√

E2t +√

E2c)2is defined as reduced elasticity modulus[14]. On the other hand, the

longitudinal strains ε1 and ε2 must satisfy compatibility condition[1] at the interface:

α1ΔT +P

E1t1+

t12ρ

= α2ΔT −P

E2ct2−

t2c

ρ(4)

By using Eqs.(2), (3) and (4), the radius of curvature ρ and P can be obtained as

1

ρ=

(α2 − α1)ΔT

t1 + 2t2c

2+

2(E1I1 + E2rI2)

t1 + t2

(1

E1t1+

1

E2ct2

) (5)

P =2(E1I1 + E2rI2)

(t1 + t2)ρ(6)

where α1 and α2 are the thermal expansion coefficients of the upper and lower layers, respectively, ΔT

is the temperature difference, I1 = t31/12, I2 = t32/12 and t2c =t2√

E2t√E2t +

√E2c

denotes the distance from

the neutral axis of the lower layer to the extreme element in the compression, respectively. It shouldbe noted that t2c, due to a uniform temperature change, is independent of temperature and thermalexpansion coefficient. To realize identical interfacial curvatures, the peeling moment Mp arising at theinterface in a bi-material beam under the uniform temperature change must be taken into account[10].Thus, the interfacial curvature relation is expressed as

M1 −Mp

E1I1=

1

ρ=

M2 + Mp

E2rI2(7)

· 508 · ACTA MECHANICA SOLIDA SINICA 2011

where M1 and M2 are the moments acting to the upper and the lower layer, respectively.Substituting M1 = Pt1/2, M2 = Pt2/2 into Eq.(7) and by using Eq.(6), the peeling moment can

be obtained as

Mp =E1t1t

32

12ρ(t1 + t2)

(4e

1 + k + 2√

k− δ2

)(8)

where k = E2t/E2c, e = E2t/E1 and δ = t2/t1. When E2t > E2c and E2t < E2c, the results correspondto the behaviours which is stronger in tension and compression, respectively.

By using Eqs.(5) and (8), dimensionless peeling moment Mp can be given as

Mp =Mp

E1ΔT t21(α1 − α2)=

δ

(4e

1 + k + 2√

k− δ2

)

2(1 + δ)

⎧⎪⎪⎨⎪⎪⎩3

[δ + 2(1 + δ)

√k

1 +√

k

]+

(1

δ+

k

e

) δ3 +4e

1 + k + 2√

k1 + δ

⎫⎪⎪⎬⎪⎪⎭

(9)

III. NUMERICAL RESULTS AND DISCUSSIONWhen setting k = 1 (which means that the material behaviour of the bi-material beam is the same

as that described by Moore and Jarvis[10–12]) in the present analysis, we can find the same resultsas those obtained by Moore and Jarvis in previous studies. It can be seen from Eq.(4) that whene = δ2(1 +

√k)2/4 the peeling moment becomes zero. This critical value is denoted with ecr in Fig.2.

This important limit case (k = 1) can be realized only if the elasticity modulus of layer 1 E1 is muchgreater than E2 (i.e. the elasticity modulus of the layer 2) and the layers are sufficiently thin. However,in the present study, the value of elasticity modulus E1 can be reduced to a much smaller value (Fig.2)when k > 1. With the increase of k, one can obtain the more realistic values of E1. Furthermore, ingeneral, the magnitude of peeling moment decreases with an increase of the bilinear parameter k (Fig.3).In this numerical example δ is taken to be 0.5.

Fig. 2. Variation of critical value of e(= E2t/E1) with bi-linear parameter k for δ = 0.25 and δ = 0.5. Fig. 3. Variation of dimensionless peeling moment with bi-

linear parameter k for e = 1 and e = 1.5.

IV. CONCLUSIONSThis work considers the problem of a bi-material beam having partly composites properties. Here,

following Moore and Jarvis[10], we extend the peeling moment expression for bilinear material behaviour.The magnitude and sign of the peeling moment can be easily determined with this expression. As statedin Ref.[10], the sign of Mp provides a fundamental design rule for resisting delamination. On the otherhand, according to another view, the difference between the maximum and minimum peeling stresses isa more important indication for resisting delamination. Of course, the peeling stress approach is moresuitable for these calculations. However, a measure of the relative magnitude of the maximum peelingstress may be obtained by the peeling moment approach. In general, although the shearing and the

Vol. 24, No. 6 Ugur Guven: Bilinear Behaviour on Direction and Magnitude of Peeling Moment · 509 ·

peeling stresses are couple, for a simplified engineering analysis, the interfacial shear stresses can bedetermined without taking the role of the peeling stress into account. To obtain the rigorous solutions,the peeling and the shear stresses should be both considered (see e.g. Ref.[15]). However, it has beenreported[17] that when two layers have nearly the same thickness, the approximate methods adequatelypredict the peak values of interfacial stresses.

As is well known, the behaviours of some composites materials can be modelled as bilinear fordifferent industrial applications[16]. Hence, in the framework of the present theoretical analysis, thebehaviour of composite layer is taken into account to be bilinear. In conclusion, it can greatly increasethe resistance against the delamination if the new design mentioned in this work is realized.

Acknowledgement: This work is dedicated to Prof. Dr. I.T.Gurgoze who first inspired my interest insolid mechanics.

References[1] Timoshenko,S.P., Analysis of bimetal thermostats. Journal of Optical Society American, 1925, 11: 233-255.[2] Hess,M.S., The end problem for a laminated elastic strip—I. The general solution. Journal of Composites

Materials, 1969, 3: 262-280.[3] Hess,M.S., The end problem for a laminated elastic strip—II. Differential expansion stresses. Journal of

Composites Materials, 1969, 3: 630-641.[4] Suhir,E., Stresses in bimetal thermostats. ASME Journal of Applied Mechanics, 1986, 53: 657-660.[5] Suhir,E., Interfacial stresses in bimetal thermostats. ASME Journal of Applied Mechanics, 1989, 56: 595-

600.[6] Suhir,E., Predicted thermal stresses in a bi-material assembly adhesively bonded at the ends. Journal of

Applied Physics, 2001, 89: 120-129.[7] Mirman,I.B., Effects of peeling stresses in a bi-material assembly. ASME Journal of Electronic Packaging,

1991, 113: 431-433.[8] Mirman,I.B., Microelectronics and the built-up-bar theory. ASME Journal of Electronic Packaging, 1992,

114: 384-388.[9] Mirman,I.B., Interlaminar stresses in layered beams. ASME Journal of Electronic Packaging, 1992, 114:

389-396.[10] Moore,T.D. and Jarvis,J.L., A simple and fundamental design rule for resisting delamination in bi-material

structures. Microelectronic Reliability, 2003, 43: 487-494.[11] Moore,T.D. and Jarvis,J.L., The peeling moment a key rule for delamination resistance in I.C. assemblies.

ASME Journal Electronic Packaging, 2004, 126: 106-109.[12] Moore,T.D. and Jarvis,J.L., Peeling in bi-material beams: the peeling moment and its relation to the

differential rigidity. ASME Journal of Applied Mechanics, 2004, 71: 290-292.[13] Gent,A.N. and Hamed,G.R., Peel mechanics. Journal of Adhesion, 1975, 7: 91-95.[14] Gere,J.M. and Timoshenko,S.P., Mechanics of Materials. 3rd ed. London: Chapman-Hall, 1991.[15] Yuan,H, Chen,J.F., Teng,J.G. and Lu,X.Z., Interfacial stress analysis of a thin plate bonded to a rigid

substrate and subjected to inclined loading. International Journal of Solids and Structures, 2007, 44: 5247-5271.

[16] Kamiya,N., Thermal stress in a bimodulus thick cylinder. Nuclear Engineering and Design, 1977, 40: 383-391.

[17] Eischen,J.W., Chung,C. and Kim,J.H., Realistic modelling of edge effect stresses in bi-material elements.ASME Journal of Electronic Packaging, 1990, 112: 16-23.