Singular stress fields in interfacial notches ofhybrid metal matrix composites

Anisur Rahmana ,* and A.C.W. Laub

aDepartment of Materials Engineering, Fibrous Materials Research Center, DrexelUniversity, 32nd and Chestnut Streets, Philadelphia, PA 19104, USAbDepartment of Mechanical Engineering and Mechanics, Fibrous Materials ResearchCenter, Drexel University, 32nd and Chestnut Streets, Philadelphia, PA 19104, USA(Received 2 September 1997; accepted 24 May 1998)

This paper presents results from a study on the singular stress fields in metal matrix composites in which dualmatrices exist. The adjoining metallic matrices flanking the interface can deform plastically with powerlaw strainhardening. These matrices may have both different hardening exponents and different yield strengths. Anasymptotic analysis coupled with numerical eigen-analysis solved the spatial structure of the singular stress fieldat radial- and angular-dependent parts:jij ,rlj̃ij (v). The dependence of the strength of the singular stresses on thematrix properties is discussed. The effects of local geometry on the nature of singular stresses are addressed.Highlights for interfacial notches are reported here. The drivers for this study are interfacial notches and free-edges in hybrid metal matrix composites (Figure 1). However, the results can also be applied to other advancedstructures which are composed of two or more distinct components or phases such as bone-implant interfaces andsurface mounts in electronic packages1. q 1998 Elsevier Science Ltd. All rights reserved

INTRODUCTION

In this paper we present a methodology for solving singularstress fields along interfaces in metal matrix compositeswith adjoining dual matrices. When accentuated stresses atgeometric discontinuity along the interface locally exceedsthe yield stress, each of the dual matrices can deformplastically. Specifically, the two metallic matrices candeform with powerlaw strain hardening:t ¼ A1g

N1, andt ¼ A2g

N2, respectively. The dual matrices can havedifferent strain hardening exponents:N1, N2, and differentinitial yield stress as reflected by differentA1 andA2.

The problem is formulated in a general way for bi-metallic notch formed by two arbitrary wedge angles,d anda (Figure 2). By varying the wedge angles, a variety ofgeometries of engineering interest can be analyzed, e.g.,interfacial free-edge, interfacial notches and interfacialcracks (Figure 3). Here, we focus on results for interfacialnotches. Results for the interfacial free-edge or interfacialcracks can be found in2,3. For the interfacial crack geometry,reported results can be found in4–8.

FORMULATION AND GOVERNING EQUATIONS

The problem is modeled as a bi-metallic notch composedof two wedges bonded alongv ¼ 08 (Figure 2). The two

arbitrary wedge angles are denoted asd anda, respectively.By varying these wedge angles, different geometriesof engineering interest can be analyzed. With the subscriptk¼ 1; 2 denoting matrix wedges 1 and 2, respectively,the conditions of equilibrium and compatibility are satisfiedby using stress function and stream function for eachwedge2.

The stress function in wedgek in polar coordinates:

Fk(r, vÞ ¼ rlþ 2fk(v) (1)

The stream function for wedgek:

wk(r, v) ¼ r

l

Nkþ 2

gk(v) (2)

The functionsf andg are related to each other through thematerial law.

The uniaxial stress–strain data of most metals can bemodeled by:

g

g0¼

t

t0þ a

t

t0

� �n

(3)

In this form, the first term on the right hand side representsthe (normalized) elastic strain. The second term is the(normalized) plastic strain, in which the exponentn ( $ 1)describes the work hardening behavior. With high localplastic strains induced by locally high stress levels (suchas those existing at geometric discontinuities), the elastic

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763

* Corresponding author.

portion can be neglected9 and the material law becomes:

g

g0¼a

t

t0

� �n

(4)

Equivalently this can be written as:

t ¼ AgN (5)

WhereN ¼ (1/n) is the strain hardening exponent, andA is amaterial constant reflecting the magnitude of the materialsinitial yield strength.

In a composite with adjoining dual metallic matrices(Figure 2), let the materials in wedges 1 and 2 deformplastically according to:

t ¼ AkgNk (k¼ 1,2) (6)

We use the convention that inFigure 2 the top wedge (v .08, material 1) is taken as the harder material and the bottomwedge (v , 08, material 2) is taken as the softer material,i.e., A1 . A2. A parameterb is defined to quantify thismaterial mismatch:

b¼A1

A2(7)

After generalizing to two-dimensional stress space and aftersome mathematical manipulations, the plane strain govern-ing equation satisfying equilibrium, compatibility andpowerlaw material behavior in each wedge can be expressed

in terms of the angular portion of the stress function,fk(v),and itsv derivatives:

f ivk ¹ l(lþ 2)fk0 þ

1Pk

BkLk

2Nk¹

2Nk ¹ 14N2

kAk(Ak9Bk ¹ NkAkBk9)2

�¹ AkBkBk9 ¹ 1

2 AkMkBk þ 12(Nk þ 1)Ak(Bk9)2

¹l

Nk

l

Nkþ 2

� �AkB

2kg þ 4

l

Nkþ 1

� �3 (l þ 1)

1Pk

fk9Bk

2Nkhk(Ak9Bk ¹ NkAkBk9)

�þ

f 0AkB2k

hk¹

hk9fk9AkB2k

hk

�¼ 0 (k¼ 1,2) ð8Þ

where ( )9, ( )0, ( )- and ( )iv denote the first, second, third

Stress fields in interfacial notches of hybrid metal matrix composites: A. Rahman and A. C. W. Lau

764

Figure 1 Schematic of a hybrid metal matrix composite

Figure 2 Geometry and coordinate system of general bimaterial notch

Figure 3 Bimaterial interface notches

and fourth derivative, respectively, and:

C¼ 32l

Nkþ 1

� �2

lþ 1ð Þ2 (9)

hk ¼ fk0 ¹ l(lþ 2)fk (10)

Ak ¼ h2k þ 4(lþ 1)2(fk9)2 (11)

Bk ¼ 4l

Nkþ 1

� �2

þC(fk9)2

2h2k

(12)

Lk ¼ 2(hk9)2 þ 8(lþ 1)2{ (fk0)2 þ fk-fk9} (13)

Mk¼C

h4k

{ (fk0)2h2

kþfk-fk9h2k¹4fk9fk0hkhk9þ3(hk9fk9)

2} (14)

Pk ¼B2

khk

Nkþ

C

2h3k

(fk9)2AkBk (15)

The nonlinear differential equation contains theeigenvaluel which gives rise to the singular nature of

the dominant local stress field. The boundary conditionsare: traction free along the free surfaces (v ¼ d andv ¼ a),and continuity of displacements and normal and shearstresses along the bond-line (v ¼ 0). Details of theformulation, boundary conditions in terms off and itsderivatives, and the solution method are described inRef. 2.

RESULTS AND DISCUSSIONS

The singular stress field of the formjij ,rlj̃ij (v) the value ofl (l , 0) determines how quickly the stresses will increaseas r → 0. Results indicate thatl is affected by bothgeometry and matrices material mismatch. The angularvariation of the singular stresses,j̃ij (v), show the relativemagnitudes of the stress components at eachv. Of particularinterest is the ratio of the stress component tending to pullthe interface open to the stress component tending to shearthe interface. We use a normalized ratio, theInterfacialMaxity, M2, to measure the relative magnitudes of theinterfacial ‘opening’ stress,jvv, to the interfacial ‘shearing’stress,j rv:

M2 ¼2p

arctanj̃vv

j̃rv

� �v¼ 0

(16)

Absolute values ofj̃vv and j̃rv are used in eqn (16).Results for the interfacial notch problems indicate thatfor a given matrices combination and geometry, the Inter-facial Mixity is fixed for eachl2 (not so for interfacialcrack problems). In theory,M2 can vary between 1 and 0,representing pure interfacial normal stressing and pureinterfacial shear stressing, respectively, in the limitingcases. Results show that at the tip of an interfacialnotch, the stress state is typicallymixed mode, with 1 .M2 . 0.

To demonstrate the effect ofb and geometry, we examinethose results in whichA1 Þ A2 are different while keepingN1 equal toN2. Then results forN1 Þ N2, andA1 Þ A2 arediscussed.

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Stress fields in interfacial notches of hybrid metal matrix composites: A. Rahman and A. C. W. Lau

Figure 4 l–N for symmetric bimaterial notches

Figure 5 j̃ij (v) for symmetric notch,d/a ¼ 1208 (b ¼ 5, N ¼ 0.5)

The effect ofb and notch geometry

KeepingN1 ¼ N2, we can isolate the effect of materialmismatch to only those caused byA1 Þ A2 (i.e., effect ofb).These results should approximate those in actual compositesin which matrices have nearly the same strain hardeningexponents, but rather different initial yield strengths. Afterisolatingb as the only material mismatch parameter, theseresults also show the important effect of geometry.

We considered two groups of interfacial notch geometries:symmetric notch (d ¼ a) and non-symmetric notch (d Þ a),seeFigure 3c andFigure 3d, respectively. For symmetricnotches,Figure 4 shows the variation ofl with the wedgeangled ( ¼ a). In this plot, the wedge angled varies from908 to 1808, thus, modeling geometries from interfacial free-edge through interfacial symmetric notches to interfacialcrack (Figures 1and2). Results of four matrix combinationsare plotted (b ¼ 3, N1 ¼ N2 ¼ N ¼ 0.5), (b ¼

10;N1 ¼ N2 ¼ N ¼ 0:8Þ; ðb¼ 3; N1 ¼ N2 ¼ N ¼ 0.5) and(b ¼ 10,N1 ¼ N2 ¼ N ¼ 0.8). This figure shows the effectsof the geometry (i.e., notch angle¼ 3608 ¹ 2d), the materialmismatch,b, and the magnitude of the common hardeningexponent,N, on the power of the stress singularity,l. Thereis decreasing sensitivity tob asd increases (i.e., notch angledecreases and the bi-matrices notch becomes more and morelike a crack,Figures 2and3). In the limit as the interfacialcrack geometry is approached, the value ofl tends to that ofthe HRR crack9,10 in which l ¼ ¹ N

N þ 1. In that case, thestress-fields become essentially HRR fields and areindependent ofb. Figure 4 also points out thatl of theinterfacial crack geometry (d ¼ 1808). Figure 5depicts theangular variation of stress components,j̃ij (v), for the notchgeometry (d/a) ¼ (1208/1208) for (b ¼ 5, N1 ¼ N2 ¼ N ¼

0.5). The interfacial mixity in this case is 0.83248,indicatingmixed modewith stronger normal stress compo-nent than shear stress component.

Non-symmetric notch results for (d/a) ¼ (608/1808), (908/1808) and (1208/1808) are presented inFigures 6–9. The

Stress fields in interfacial notches of hybrid metal matrix composites: A. Rahman and A. C. W. Lau

766

Figure 6 l versusN for non-symmetric notch ofd/a ¼ 608/1808

Figure 7 Interfacial mixity for non-symmetric notch ofd/a ¼ 608/1808

Figure 8 l versusN for non-symmetric notch ofd/a ¼ 908/1808

effect of geometry is apparent from comparingl, in Figures6, 8 and9. Again as the notch angle decreases,l becomesless sensitive tob. Figure 7portrays the Interfacial Mixitycorresponding to the most dominant singularity (most

negativel) for notch geometry (d/a) ¼ (608/1808). Theseresults cover the entire range ofN and practically the entirerange of meaningfulb.

Results for N1 Þ N2 and A1 Þ A2

When N1 Þ N2, we use the convention thatN1 . N2.WhenN1 Þ N2, as far as the asymptotic singular stress fieldis concerned, the bimatrix composite essentially acts as ifthe less hardening matrix (i.e.,N2, the lower valueN) isattached to a rigid material2. That this is indeed so, for theinterfacial crack geometry, has been reported in Refs.6,7.This simplification decouples the problem and allows us tosolve for l by only focusing on the less strain hardeningmaterial. Results confirm thatl does not depend on theproperties of the more strain hardening (N1) matrix.Consequently,l is independent ofb. An important pointis that theN1 matrix is interpreted as rigid only as far as itsinfluence on the less strain hardening matrix is concerned.For itself, theN1 matrix actually is not rigid and will havedisplacements and stresses. Oncel and the stress distribu-tions on theN2 matrix is determined, the response of themore strain hardeningN1 matrix can be calculated.

Results for symmetric notch geometries are shown inFigures 10–12. Figure 10 depicts l of the dominantsingular stresses as function ofN2. Data ford ranging from508 to 1608 are plotted.Figure 11shows the typical angularvariation of the stress components,j̃ij (v). This plot is for thenotch geometry (d/a) ¼ (1208/1208) and forN2 ¼ 0.1 (withb ¼ 10,N1 ¼ 0.8).Figure 12shows the interfacial mixity ofthe dominant stress singularity for the entire range ofN2 ford from 508 to 1608.

The interfacial notch results presented are thosepertaining to the most dominant singular stresses (corre-sponding to the most negative admissible,l1). For certainwedge angles, there exist a second admissible singularityexponentl2 which is less negative thanl1. This lesspowerful second singular field has its own angularvariations and Interfacial Mixity. Details of the secondstress singularity can be found in2.

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Stress fields in interfacial notches of hybrid metal matrix composites: A. Rahman and A. C. W. Lau

Figure 9 l versusN for non-symmetric notch ofd/a ¼ 1208/1808

Figure 10 l versusN2 for symmetric notch,N2 ¼ 0.5

Figure 11 j̃ij (v) for symmetric notch withN1 fi N2, d/a ¼ 1208/1208,(N2 ¼ 0:1,N1 . N2)

Figure 12 Interfacial mixity for symmetric notchN1 . N2

REFERENCES

1. Rahman, A. and Lau, A. C. W., Stress singularities in surfacemounts allowing for local plastic deformation. InAdvances inElectronic Packaging, EEP-Vol. 4-1, ASME, NY, 1993, pp.29–37.

2. Rahman, A. Singular Stress Fields at Plastically DeformedBimaterials Interfacial Notches and Free-edges. Ph.D. Dissertation.Drexel University, Philadelphia, PA, 1991.

3. Lau, C.W. and Delale, F., Interfacial stress singularities at free edgeof hybrid metal matrix composites.ASME Journal of EngineeringMaterials and Technology, 1988,10, 41–47.

4. Shih, C.F. and Asaro, R.J., Elastic-plastic analysis of cracks onbimaterial interfaces: Part I—Small scale yielding.ASME Journalof Applied Mechanics, 1988,55, 299–316.

5. Shih, C.F. and Asaro, R.J., Elastic-plastic analysis of cracks onbimaterial interfaces: Part II—Small scale yielding fields.ASMEJournal of Applied Mechanics, 1989,56, 763–779.

6. Shih, C.F. and Asaro, R.J., Singular behavior at the end of a tensile crack in ahardening material.International Journal of Fractures, 1990,43, 101–116.

7. Gao, Y. and Lou, Z., Mixed mode interface crack in a pure power-hard-ening bimaterial.International Journal of Fracture, 1990,42, 241–256.

8. Wang, T.C., Elastic-plastic asymptotic fields for cracks on bimaterialinterfaces.Engineering Fracture Mechanics, 1990,3, 527–538.

9. Hutchinson, J.W., Singular behavior at the end of a tensile crack in ahardening material.Journal of the Mechanics and Physics of Solids,1968,16, 13–31.

10. Rice, J.R. and Rosengren, G.F., Plane strain deformation near acrack tip in a power-law hardening material.Journal of theMechanics and Physics of Solids, 1968,16, 1–12.

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