ON THE INTERFACE CRACK MODELS

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ON THE INTERFACE CRACK MODELS

T. Suga, S. Schmauder, G. Elssner

To cite this version:

T. Suga, S. Schmauder, G. Elssner. ON THE INTERFACE CRACK MODELS. Journal de Physique

Colloques, 1988, 49 (C5), pp.C5-539-C5-544. �10.1051/jphyscol:1988565�. �jpa-00228062�

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Colloque C5, supplkment au nolO, Tome 49, octobre

ON THE INTERFACE CRACK MODELS

T. SUGA('), S. SCHMAUDER and G. ELSSNER

Max-Planck-Institut fiir Metallforschung, Institut fiir Werkstoffwissenschaften, 0-7000 Stuttgart, F.R.G.

Resume - Les modeles du continu pour une fissure dans l'interface sont discutes. Les incompatiblites logiques presentes dans le champ de tension du modele conventionnel, cornme par exemple les singularites d'oscillation et les interpenetrations de la surface de fissure, sont dleminees en modifiant les conditions de joints a la pointe de fissure. Un modele de la pointe de fissure est presente et discute par rapport a la dispersion inelastique de l'energie et le manque d'interfaces de materiaux hetero- genes

.

Abstract - The continuum models for an interface crack are reviewed. The logical in- consistencies present in the local stress field of the conventional model, such as the oscillatory singularities and the interpenetration of the crack surfaces, are removed by modifying the boundary conditions at the crack tip. An interpretation of the crack tip models is given and discussed with respect to inelastic energy dissipation during interfacial failure of heterogeneous materials.

I - INTRODUCTION

Semibrittle fracture along interfaces of heterogeneous materials (such as solid state bonded ceramic-metal jqints or composites) can occur when the energy stored in the specimen by external work becomes sufficient to supply the fracture energy needed for creating new surfaces. The term "semibrittle" means that the region in which the energy is consumed by the failure process is limited to the crack tip. The ideal fracture energy of an interface corresponds to the thermodynamic work of adhesion WA which is the difference between the sum of surface energies and the interfacial energy of the materials bonded: WA=7n+7c-7nc. Any failure process is accompanied by irre- versible inelastic energy dissipation. Therefore, denoting its contribution to the fracture energy by W P , the thermodynamic criterion for an interface fracture is written as /I/:

where GC is the critical value of the generalized strain energy release rate C. ^This equation is a rational generalization of the Griffith-Orowan formula for the semibrittle failure of homogeneous materials. If the inelastic deformation of the specimen is limited to a narrow region in the vicinity of the crack, the generalized strain energy release rate corresponds to Irwin's elastic strain energy release rate G which can be calculated by an elastic analysis of the cracked material. If the load-deformation relationship of the specimen is not linear, can also be obtained by modifying the elastic strain energy release rate G,by the experimentally estimated nonlinearity coefficient /2/.

Experimental verification of the energy criterion for interfacial fracture is given in early works by Malyshev and Salganik /3/, Mulville / 4 / , and Saxena 151, followed by the extensive experiments by Suga and Elssner /1.6,7/. The most important result in /1/ is that the dissipated work W p is dependent on the reversible work WA. There-

'')low at: The University of Tokyo, Faculty of Engineering, Department of Precision Engineering, 113 Tokyo, Japan.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1988565

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C5-540 JOURNAL DE PHYSIQUE

fore equation (1) should be written:

Ec

= WA + WP(WA) or, more generally:

In /1/ i t was also shown that the total interfacial fracture energy is strongly dependent on the crystallographic orientation of the bonded materials, and thus is di- rectly affected by the work of adhesion W A , even though the measured value of W A is much smaller than that of WP.

This apparently contradictory result is explainable by Wp being a function of WA. In order to represent the function WP explicitly, an adequate modelling of the energy dissipation process is necessary. The simple model given in /I/ which can estimate the fraction of the reversible fracture energy and the energy dissipated in the vicinity of an in.terfacia1 crack suggests that WP is a monotonic function of WA. There- fore, equation (3) can be written:

where a 1.

A more comprehensive analysis must be done in order to understand the mechanism of interfacial failure related to the nucleation of microcracks in an interface or the interaction of dislocations with an interface crack. Such an analysis requires more detailed knowledge of the stress state in the vicinity of an interface crack, which can be described only by an adequate interface crack model.

In contrast with the well established crack model in single phase materials, interface crack models have not been successfully applied to interfacial fracture phenome- na. In the present paper we review the previous attempts of the modelling of an interface crack. The problems of interface crack models seem to lead to the question of how the interface crack in an actual interface can be simplified and idealized by a model with a sharp and perfect interface within the framework of continuum mechanics.

I1

-

CONVF,NTIONAL INTERFACE CRACK MODEL

The problem of the determination of the stress field in the vicinity of an interface crack in elastically dissimilar materials has been studied by Williams I S / . England /9/. Erdogan

/lo/.

Rice and Sih /1 I/. and Lowengrub and Sneddon /12/ among others. A comprehensive summary of these works is given in /13/. The displacement field u, and ue. and the stress field ug and T~~ in the vicinity of the interface crack tip are

. .

found to possess the following dominant terms:

(ue - iu,)

-

^z^A ^{( 5 )}

(ue

-

" z A-1 (6)

in polar coordinates with z = r exp(i8). The exponent A of the above terms is a complex number of the form 112 + is, where e is the bimaterial constant given by a = 1/27r ln((l+P)/(l-P)) and f3 is one of the Dundurs' composite parameters /I/. For single phase homogeneous materials. E. is reduced to zero. and therefore the stress components are inversely proportional to the square root of the distance r from the crack tip. In contrast with this well-known square root singularity. the non-zero imaginary part of the exponent A of the interface crack leads to undesirable results from the physical point of view: the stress components oscillate and the upper and lower surfaces of the crack wrinkle and overlap in the crack tip region. Indeed. if one defines a complex stress intensity factor K = KI + KI ^I to indicate the strength of the singularity. the stress field at the interface near the crack tip and the opening of the crack surfaces can be written as

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where E' is the effective elastic modulus of the dissimilar material defined in /I/.

As seen in the above equations the term

-ie = cos (slnr)

-

i sin (elnr) ₍₉₎ leads to the oscillatory singularity. If one introduces a characteristic length or a new dimension of length, ro into equation (7):

(erg

- i ~ = (KI ~

-

^iKI1)/-*~ ) (r/ro)-1/2 ~

-

~^ie (10) then the components of the stress intensity factor, K I and K I I , are changed alter- nately due to the term (r,) 1/2 + ie . The period of the oscillation is given by r, = exp (27m/E), and even for the strongest dissimilarity corresponding to

P

= 0.5 and 6 = 0.1749 it takes values at very large intervals:

. . .

6 . 2 ~ 1 0 - ~ ~ , 2.5~10-IS, 1 .O, 4 . 0 ~ 1 0 ~ ~ . 1 . 6 ~ 1 0 ~ ~ . . .

.

etc. It should be emphasized, however, that the sign of the real part of equation (9) is changed at the points of r = 1 . 2 5 ~ 1 0 - ~ and 7.99x103 near the reference length r = 1.

For a central crack at the interface of two elastically dissimilar bonded planes loaded at infinity with a uniform stress field, oo and r,, the stress intensity factor is:

iZ = KI

-

iK1 I = (o,

-

irO)

K.

(21)~" (1

-

218) (11) where 2a is the crack length. It is noted that the term gives an oscillatory character also to the stress intensity factor.

I11 - MODELS FOR THE CRACK TIP REGION

The conventional interface crack model described above has been criticized by many fracture mechanists, because it leads both to the oscillatory behaviour of the stress field and to the overlapping of the crack surfaces in the small region around the crack tip where the interfacial failure will be initiated. Also the stress intensity factor has an ambiguous dependence on an arbitrary length parameter, and therefore i t is not possible to relate the components of the stress intensity factor K I , K I I to the type of loading mode in the vicinity of the crack tip.

These undesirable features result from the traction-free condition assumed on the whole crack surfaces of the conventional interface crack model. The character of the local stress field of the crack tip region obviously depends on the manner in which the conditions change from the interface bond to the crack surfaces. Therefore, by changing the boundary conditions a more realistic character of the local stress field can be obtained. Such attempts have been made by several authors. Comninou /14/ has shown that the stress singularities loose the oscillatory behaviour by assuming that the crack is not completely open and that its surfaces are in contact with each other near the crack tip (contact zone model). An alternative model for the crack tip was introduced by Mak et al. /15/, which assumes that there is no relative slip between the crack surfaces in the crack tip region (interlock model). Although both models eliminate the imaginary part of the stress singularity, an undesirable aspect remains concerning the stress intensity factor, in that one of its components (KI for the contact zone model, K I I for the interlock model) is always eliminated. In ocher words, no normal stress exists in the interface ahead of the crack tip in the contact zone model, regardless of the actual loads applied to the bimaterial. Similarly no shear stress exists in the interlock model.

This dilemma was resolved by the "transition zone model". introduced by Suga and Elssner /16,17/. In this model an interaction between the crack surfaces is assumed.

The transitions acting on the crack surfaces are expressed as:

(ag

-

i ~ ~ = ~ - P E ~ / ~ ) ~ (1-p2)-a(Aur = ~ + ^iAug)/ar ( 12) The stresses in the interface are characterized by both stress intensity factors KI and KII as given in the foIlowing expression:

(ag

-

ⁱ ^~ = (KI ^~- iK1 I ^~) / w e⁾ .-'I2^~ ^~

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This is exactly the same expression as that of homogeneous materials. The crack opening is expressed by:

Au = -8(l-p2)/EX- (KI - iK1 I)/-. r 1/2

(14) The elastic strain energy release rate G is calculated by the following equation:

6

G = -1/2 lim 1/6

S

(og(r) Aug(6-r) + rre(r) bur(&-r))dr

&a

0

(15) giving the relationship

G = (1-p2)/EX* (KI' + K1 12)

The stress functions of this model are constructed by a combination of those of the contact zone model and the interlock model. In Figs. 1 and 2 selected stress distri- butions around the crack tip are compared with those of the conventional crack tip.

Full descriptions for the complex stress functions and the stress and displacement field for the model are given in /17/.

IV

-

INTERPRETATION OF INTERFACE CRACK MODELS

Some fundamen-tal questions are: how is the size of the crack tip region determined, and how can the assumed boundary conditions for the crack tip region be justified?

These questions can be reduced to a more fundamental question: why must an interface crack model be used to characterize the fracture behaviour of an interface?

If one assumes an atomic structure of the interface crack tip and a proper interato- mic potential, the stress field at the crack tip or the ideal cohesion energy of the interface can be calculated under certain boundary conditions such as displacements of the atoms on the crack surfaces or stresses acting on the assembly of atoms. How- ever, how can such displacements or stresses for the small assembly of atoms be determined from macroscopic measurements of the applied force and the deformation of the specimen? In this case a model describing the local stress field of the crack tip is required. The present crack tip model leads, however, to a singular crack tip stress field, which is not physically realistic. The fracture mechanics argument is not that the strength of the stress field itself but rather the critical energy release rate G, should be selected as a physical parameter which represents the atomis- tic cohesive energy independent of the crack model, and that the stress intensity factors derived from the crack model are useful only because of their possibility to characterize the loading mode in the vicinity of the crack tip. Thus the ambiguity relating to the size and the conditions of the crack tip region can be allowed. The selected model should be justified not physically but by fracture toughness tests giving a practical verification of stress intensity factors.

The crack tip zone size has been investigated analytically only for an interface crack with the contact zone or an interlock zone in a dissimilar infinite plane under uniform tensile or shear stress /14,15,18,19,20/. The zone size was determined under the assumption of the existence of compressive stress in the contact zone or a critical crack opening displacement in the interlock zone, respectively. However, from the foregoing viewpoint, a more general method should be developed to determine the stress intensity factors for a given tip zone size and for any configuration of interface cracks, rather than determining the tip zone size.

Other attempts to avoid the oscillatory crack tip stress field are also given in the literature: Sinclair /21/ has proposed a crack tip model with a wedge form. The opening angle of the crack tip is determined such that the stress field has the usual square root sixlgularity. Atkinson /22/ has shown that a crack at the interface region with the elastic moduli continuously varying across the interface can remove the anomalies of the stress singularity. This model may allow treatment of a diffuse interface, although it may be difficult to describe the local stress field explicitly.

Atkinson also demonstrated that a crack in a thin interface layer can likewise remove stress anomalies /22/.

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nal interface crack model (left) and the transition model (right).

Fig.

0 8

2 Normalized shear stress rr0 for K , , = 0 at the crack tips of the conven- tional interface crack model (left) and the transition model (right).

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v

- CONCLUDING REMARKS

The anomalies of the local stress field can be removed by introducing an adequate crack tip model. What remains is the development of a general method to determine the local stress intensity factors. The conventional interface crack model offers no rational local stress field. However, the solution of the model is still valid for the description of the asymptotic behaviour of the stress field far from the crack tip, as verified experimentally /23,24/. With regard to the energy dissipation in the in-

terfacial failure process as mentioned in the introduction, it is still unknown whether dislocations interact with an interface crack in its near-tip local stress field or in the far-from-crack-tip stress field. Further efforts should focus on the micromechanics approach to the interfacial fracture problem from both experimental and analytical points of view.

REFERENCES

/1/ G. Elssner, T. Suga and M. Turwitt, J. de Physique 46 (1985) C4-597 /2/ H. Liebowitz and J. Eftis, Engng. Fract. Mech.

3

(1971) 267

/3/ B.M. Malyshev and R.L. Salganik. Int. J. Fract. Mech.

1

(1965) 114

/4/ D.R. Mulville. P.W. Mast and R.N. Vaishnav. Engng. Fract. Mech. 8 (1976) 555 /5/ A. Saxena. Fibre Sci. Technology 2 (1979) 111

/6/ T. Suga and G. Elssner, 2. Werkstofftech. 16 (1985) 122

/7/ T. Suga, I. Kvernes and G. Elssner. 2 . Werkstofftech.

15

(1984) 371 /8/ M.L. Williams, Bull. Seismol. Soc. Amer. 49 (1959) 199

/9/ A.H. England, J. Appl. Mech. @ (1965) 400 /lo/ F. Erdogan, J. Appl. Mech. (1965) 403

/11/ J.R. Rice and G.C. Sih, J. Appl. Mech. W2 (1965) 418

/12/ M. Lowengrub and I.N. Sneddon. Int. J. Engng. Sci.

11

(1973) 1025 /13/ A. Piva and E. Viola, Engng. Frac. Mech. 13 (1980) 143

/14/ M. Corrminou, J. Appl. Mech. E A (1977) 631

/15/ A.F. Mak. L.M. Keer, S.H. Chen and J.L. Lewis. J. Appl. Mech. 47 (1980) 347 /16/ T. Suga and G. Elssner. J. de Physique 46 (1985) C4-657

/17/ T. Suga, PhD. Thesis, University of Stuttgart (1983) /18/ M. Comninou and D. Schmueser, J. Appl. Mech. 46 (1979) 345 /19/ C. Atkinson, Int. J. Fracture 19 (1982) 131

/20/ A.K. Gautesen and J. Dundurs, J. Appl. Mech. 54 ⁽¹⁹⁸⁷⁾⁹³ /21/ G.B. Sinclair, Int. J. Fracture 16 (1980) 111

/22/ C. Atkinson, Int. J. Fracture 13 (1977) 807

/23/ E.E. Gdoutos and G. Papakaliatakis, Engng. Fract. Mech. 16 (1982) 177 /24/ P.S. Theocaris, Acta Mechanics 24 (1976) 99