DETERMINATION OF THE FRACTURE ENERGY AND THE FRACTURE RESISTANCE OF INTERFACES

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DETERMINATION OF THE FRACTURE ENERGY AND THE FRACTURE RESISTANCE OF

INTERFACES

T. Suga, G. Elssner

To cite this version:

T. Suga, G. Elssner. DETERMINATION OF THE FRACTURE ENERGY AND THE FRACTURE RESISTANCE OF INTERFACES. Journal de Physique Colloques, 1985, 46 (C4), pp.C4-657-C2-663.

�10.1051/jphyscol:1985472�. �jpa-00224729�

JOURNAL DE PHYSIQUE

Colloque C4, supplCment au n04, Tome 46, avril 1985 page 0 - 6 5 7

DETERMINATION OF THE FRACTURE ENERGY AND THE FRACTURE RESISTANCE OF INTERFACES

T. Suga and G. Elssner

Mm-Planck-Institut fiir MetaZZforschung, Institut fiir Werkstoffwissenschaften,

0-7000 Stuttgart I, F.R.G.

A b s t r a c t

-

The basic concept and t h e f r a c t u r e mechanics methods f o r d e t e r -

m i n a t i o n o f t h e f r a c t u r e energy and t h e f r a c t u r e r e s i s t a n c e o f i n t e r f a c e s between s e m i - b r i t t l e m a t e r i a l s are reviewed. The i n t e r f a c i a l f r a c t u r e energy i s d e r i v e d from t h e g l o b a l energy balance concept f o r crack propagations i n an i n t e r f a c e , whereas the f r a c t u r e r e s i s t a n c e parameter i s based on continuum mechanics models of an i n t e r f a c e crack. The method t o determine t h e s t r e n g t h parameters o f an i n t e r f a c e c o n s i s t s o f s t r e s s analyses and mechanical t e s t s on pre-cracked specimens o f a simple geometry such as layered-bonded bend t e s t specimens.

I- INTRODUCTION

Measurements o f the bond s t r e n g t h o f m a t e r i a l j o i n t s are important n o t o n l y from a t e c h n i c a l p o i n t o f view b u t a l s o f o r i n v e s t i g a t i n g t h e r e l a t i o n s h i p o f t h e mechani- c a l p r o p e r t i e s and t h e s t r u c t u r e o f i n t e r f a c e s . For t h i s purpose t h e bond s t r e n g t h data must be represented by a parameter which i s independent o f t h e t e s t i n c proce- dures. The f r a c t u r e mechanics concept provides a method t o determine such parameters, namely, t h e f r a c t u r e energy

GC

and t h e f r a c t u r e r e s i s t a n c e KC. I n Fig. 1 t h e concept i s shown schematically.

The f r a c t u r e energy GC i s d e r i v e d from t h e g l o b a l energy balance p r i n c i p l e f o r the c r e a t i o n o f new surfaces by crack extension. The energy concept can be a p p l i e d t o any type o f f a i l u r e i n t h e i n t e r f a c i a l region, because no d e t a i l e d i n f o r m a t i o n o f t h e f a i l u r e processes i s necessary t o determine t h e f r a c t u r e energy. For a pure i n t e r - f a c i a l f a i l u r e t h e r e v e r s i b l e p a r t o f t h e f r a c t u r e energy corresponds t o t h e thermo- dynamical work o f adhesion which i s determined by t h e chemical c o m p a t i b i l i t y o f the m a t e r i a l s bonded together and the atomic s t r u c t u r e o f the i n t e r f a c e . Therefore a q u a n t i t a t i v e comparison o f t h e f r a c t u r e energy and t h e work o f adhesion can provide an i n s i g h t i n t o t h e c o r r e l a t i o n between the bond-strength and t h e s t r u c t u r e o f t h e i n t e r f a c e and i n t o t h e f a i l u r e mechanism i n t h e i n t e r f a c e region.

The energy d i s s i p a t i o n processes a t f r a c t u r e can be described by means o f microme- chanics models e.g. i n t e r a c t i o n s o f an i n t e r f a c e crack and d i s l o c a t i o n s . Such a modelling r e q u i r e s an a n a l y s i s of t h e s t r e s s around an i n t e r f a c e crack. The e l a s t i c s t r e s s i n t h e v i c i n i t y o f an i n t e r f a c e crack can be-described i n t h e framework o f continuum mechanics by t h e s t r e s s i n t e n s i t y f a c t o r K = KI-iK1l. The s t r e s s i n t e n s i t y f a c t o r K f o r - a s t r a i g h t i n t e r f a c e crack on t h e x-axis i s d e f i n e d by /l/

where y = ( I + B ) / ( l - R ) , R: Dundurs' composite parameter z = x+iy, and X

.

t h e p o s i t i v e eigenvalue w i t h the s m a l l e s t r e a l p a r t o f t h e s t r e s s funcd?ok f o r t h e m a t e r i a l I. Several o t h e r d e f i n i t i o n s f o r t h e s t r e s s i n t e n s i t y f a c t o r have been proposed. The above d e f i n i t i o n , however, i s p r e f e r e d because thereby t h e s t r e s s a c t i n g a t t h e i n t e r f a c e can be described indenendently o f t h e e l a s t i c constants o f t h e bonded m a t e r i a l s by

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985472

JOURNAL DE PHYSIQUE

elostlc analysts of the

carrectlon functlon

(m ^,,l

glcbal energy balance

l

fracture energy

4

G , ^{E* KZ} f mcfure KC resistance

I I

crack models

failure comparison of

different material comb~nat~ons

Fig. 1

-

F r a c t u r e mechanics concept f o r t h e c h a r a c t e r i z a t i o n o f t h e bond s t r e n g t h o f i n t e r f a c e s .

I 1

-

INTERFACE CRACK MODELS

The value of t h e s t r e s s i n t e n s i t y f a c t o r i? depends on t h e choice o f an i n t e r f a c e crack model. Some i n t e r f a c e crack models a r e a v a i l a b l e , which d i f f e r from each o t h e r according t o t h e boundary c o n d i t i o n s assumed f o r t h e crack surfaces. The conventional crack model f o r a s t r e s s - f r e e crack (Fig. 2 ) which has been s u c c e s s f u l l y used f o r s i n g l e homogeneous m a t e r i a l s r e v e a l s p h y s i c a l l y u n r e a l i s t i c f e a t u r e s such as s t r e s s o s c i l l a t i o n and overlapping o f c r a c k surfaces i n t h e v i c i n i t y o f t h e crack t i p due t o t h e non-zero imaginary p a r t E of t h e eigenvalue X /2/. These disadvantages a r e avoided by two h i t h e r t o known i n t e r f a c e crack modil2 w i t h a c o n t a c t zone IFig.3) / 3 , 4 / , i n which one component o f t h e displacements o f t h e crack surfaces i s s e t t o zero. However, one component of t h e s t r e s s i n t e n s i t y f a c t o r i s always zero i n these models. Therefore they cannot be a p p l i e d t o a general d e s c r i p t i o n o f t h e load- i n g mode a t t h e c r a c k t i p . A new i n t e r f a c e crack model / l / combines b o t h c o n t a c t zone models. I n t h i s model stresses a c t on t h e crack surfaces which are r e l a t e d t o t h e g r a d i e n t of t h e displacements o f t h e crack surfaces as shown i n F i g . 4. The s t r e s s f i e l d around t h e c r a c k t i p possesses a f i - s i n g u l a r i t y ; I t i s described by t h e s t r e s s i n t e n s i t y f a c t o r l? = KI

-

^iKII as i n t h e case o f s i n g l e homogeneous m a t e r i a l s . D i f f i c u l t i e s i n t h e i n t e r p r e t a t i o n o f t h e continuum mechanics models f o r an i n t e r f a c e crack a r e due t o t h e l a c k o f i n f o r m a t i o n s concerning the atomic s t r u c t u r e o f an i n t e r f a c e crack t i p . They a r e avoided by a c o n s i d e r a t i o n o f t h e energy released d u r i n g crack extension. A unique r e l a t i o n s h i p holds between t h e ener y r e l e a s e r a t e G and t h e a b s o l u t e value o f t h e s t r e s s i n t e n s i t y f a c t o r K = Kf-T$ independently

of the type o t the i n t e r f a c e crack model:

where D i s Dundurs'composite parameter, and E* the e f f e c t i v e modulus of e l a s t i c i t y . They can be calculated from the e l a s t i c constants of the materials adjacent t o the i n t e r f a c e /5/.

Fig. 2

-

Conventional interface crack model.

Fig. 3

-

Interface crack models with a contact zone. Top: s l i p model, bottom: inter1 ock model.

( K1

-

iKu 1 ,412

( oy

-

itxy)e.o =

.X

^{Fig. 4}

-

New i n t e r f a c e crack model.

A more r e a l i s t i c model f o r the crack t i p in s i n g l e homogeneous materials has been pro- posed by Barenblatt 1 6 1 , i n which no singular s t r e s s a r i s e s a t the crack t i p as ex- pected in real materials. The singular s t r e s s e s produced by the external forces a r e

compensated by a negative s t r e s s singularity which a r i s e s from strong cohesion forces acting between atoms on both sides of the crack surfaces in t h e v i c i n i t y of the crack

C4-660 JOURNAL DE PHYSIQUE

tip. If the distribution of the cohesion forces and the shape of the crack opening re- main constant during the crack extension, the apparent negative stress intensity factor Kg due to cohesion forces, the so-called Barenblatt's cohesion modulus, can be regarded as a material constant which characterizes the fracture resistance of the material.

This idea can be applied also to every tpye of interface crack models. Theanalysis of interface crack models of Barenblatt type lead to the consequence, that the above relationship eq. ( 3 1 , holds also for Barenblatt's cohesion modulus KB / l / . Therefore a fracture resistance parameter KC can be introduced by the equation

without any specification of the interface crack model. Its value can be calculated from the experimentally determined fracture energy GC. Fig. 5 shows the interfacial fracture resistance data KC of some materials joints in comparison to KIC data of several engineering material S .

A1203/A1 plasmasprayed

coating plasmasprayed

A1203. Ti0

glass/epoxy l s i l a n e treated)

Fig. 5

-

Fracture resistance data KIC for some efigineering materials and KC for typical material joints.

111

-

CORRECTION FUNCTIONS

The interfacial fracture energy

E

is determined by the measured fracture load FC of the test specimens which arg pre-cracked in the interface region and by a proper correction function YG which relates the applied force to the energy release rate depending on the geometry of the specimen and the elastic constants of the materials /5/. If the load-displacement curve of the specimen is not linear, the apparent value-of the fracture energy must be multiplied by a

nonlinearity coefficient C which is determined experimentally-from the shape of the load-displacement curve. Details of this procedure are given elsewhere / 7 / . The correction functions YG are calculated by a stress analysis such as the finite element method or determined experimentally by compliance measurements for each specimen configuration and each material combination /l/. Even the calculations are rather tedious but not avoidable in order to obtain accurate data of the fracture energy. Fig. 6 gives a general view of the dependence of the

c o r r e c t i o n f u n c t i o n Y f o r a layer-bonded specimen on the geometrical f a c t o r s , d/W and h/W. The specimen i s composed o f two m a t e r i a l s , 1 and 2, w i t h t h e l a y e r G thickness d. The crack i s s i t u a t e d e i t h e r i n one o f t h e m a t e r i a l s p a r a l l e l t o t h e i n t e r f a c e a t a d i s t a n c e h o r i n t h e i n t e r f a c e a t h = 0. W i s the h e i g h t o f t h e specimen. The c o r r e c t i o n f u n c t i o n YG f o r the crack i n t h e m a t e r i a l 1 o r 2 i s denoted by Y, o r Y2, f o r the i n t e r f a c e crack by YI2, and f o r t h e s p e c i a l case of an i n t e r f a c e crack i n a b i m a t e r i a l , by Ybi. Yiso 1s t h e well-known c o r r e c t i o n f u n c t i o n f o r a s i n g l e homogeneous m a t e r i a l .

interlacial

s ir en a t i2

7

^_lure

t

cohesive la!lure in mnterial 1

a e m e t a s a,,

i

F i g . 6

-

C o r r e c t i o n f u n c t i o n s o f layer-bonded specimens as f u n c t i o n s o f t h e r a t i o s d/W and h/W.

The c o r r e c t i o n f u n c t i o n s f o r an i n t e r f a c e crack Y depending on t h e r a t i o d/W a r e bounded by Y and (I +a) Y A t constant layerl$hickness d and v a r i a b l e r a t i o s h/W t h e c o r r k h t i o n f u n c t i o n i f y ' o r Y changes i t s value d i s c o n t i n u o u s l y from YIZ/(lia) t o Y / ( l t a ) over YIZ a t $he i n t e r f a c e when t h e crack p o s i t i o n s h i f t s from one m a t e r i a l t o t h e other. A t l a r g e r distances o f t h e crack p o s i t i o n from 12 t h e i n t e r f a c e , t h e c o r r e c t i o n f u n c t i o n Y, o r Y2 approach Yiso. The f a c t o r cx i s one o f Dundurs' composite parameters /5/.

An example o f c a l c u l a t e d and e x p e r i m e n t a l l y determined c o r r e c t i o n f u n c t i o n s i s given i n F i g . 7, i n which the normalized compliance C* and t h e c o r r e c t i o n f u n c t i o n YG f o r glass/epoxy/glass f o u r - p o i n t bend t e s t specimens w i t h an i n t e r - face crack a r e p l o t t e d a g a i n s t t h e r a t i o o f t h e crack l e n g t h a t o t h e specimen h e i g h t W. The values o f t h e normalized compliance C* were c a l c u l a t e d by t h e f i n i t e element method u s i n g t h e v i r t u a l crack extension method /l/. The c o r r e c t i o n func- t i o n s a r e evaluated as t h e g r a d i e n t s o f t h e normalized compliance C*:

The c a l c u l a t e d values o f t h e normal i z e d compliance agree w e l l w i t h those obtained experimentally. As expected, t h e curves o f t h e c o r r e c t i o n f u n c t i o n s f o r d i f f e r e n t d/W r a t i o s a r e bounded by t h e curve o f Y,, (d/W = m ) which i s close t o t h e curve o f Y f o r a homogeneous s i n g l e materiav'and by t h e curve o f Y G ^L ( 1 + CX) Yiso withiS0 cc =

-

0.888.

JOURNAL DE PHYSIQUE

Fig. 7

-

Normalized compliance C* and correction function YG for glass-epoxy four-pointed bend test specimens.

IV

-

CONCLUDING REMARKS

With regard to the interpretation of the bond strength data obtained by the fracture mechanics method some precautions must be taken.

First, interfacial failure of most ceramics-/metal joints occurs subcritically rather than abruptly when the loading of the specimens is controlled carefully.

Therefore there might be always a certain influence of kinetic effects on the measured bond strength data, e.g. a dependence of the fracture energy on the crack velocity. The data for the model combinations A1203/Nb and glass/epoxy

/5/ were obtained for a constant crack velocity of 1.7 ^X I O - ~ m/s. The values of the measured fracture energy appeared to increase slightly with increasing crack velocity.

A change in fracture e ergy of smal er than ^10% was observed in a range of crack velocities between 10-l m/s and 10" ./S. A detailed investigation of kinetic effects is the subject for a future study.

Second, large stress concentrations occur at the bonding edges due to differences between the elastic constants of the bonded materials. They add an anti-plane com- ponent to the stresses in the region of the bonding edges. Thereby the deformation and failure mechanism may be changed resulting in an alteration of the fracture energy, especially if thin specimens are used. In fact a curved crack front with a concave form has been observed at the glass-epoxy interface, while a crack in single homogenous materials shows a convex crack front /l/.

Third, the measured values of the interfacial fracture energy may be and are likely larger than the fracture energy value of one material adjacent to the inter- face, as shown for AlpOj/Nbb and glass/epoxy joints /5/. A semi-brittle fracture in the interface region kill take place between those atom~c planes in tne interface or in one adjacent material which are bonded most weakly or in other words where the reversible work of fracture- takes the smallest value. Generally, a crack intends to grow in the direction in which the tensile stress reveals a maximum.

Therefore, for a crack in the interface region the direction of the crack pro- pagation t u r x always into the material with the lower modulus of elasticity. For A1 0 /Nb- and glass/epoxy combinations the crack propagates in the interface, be- ca8sJ the modulus of elasticity of niobium or epoxy is lower than the modulus of elasticity of alumina or glass but the value of the reversible work of fracture of niobium or epoxy is larger than the value of the work of adhesion between alumina

and niobium or glass and epoxy.

REFERENCES

1. T. Suga and G. Elssner, "Bond strength determination of metal-to-ceramic joints by a bend test method", to be published in Z. Werkstofftechnik 2. A. H. England, J. Appl. Mech., 32 (1965) 400

-

402

3. M. Comniou, J. Appl. Mech., 44 (1977) 631

-

636

4. A.F. Mak, L.M. Keer, R.P. Khetan, and S.H. Chen, J. Appl. Mech. 47 (1980) 347

-

350

5. G. Elssner, T. Suga and M. Turwitt, "Fracture of ceramic-to-metal inter- faces", paper presented at this conference.

6. J.R. Willis, J. Mech. Phys. Solids, 15 (1967) 151

-

¹⁶²

7. T. Suga, I. Kvernes, and G. Elssner, "Fracture energy measurements of ceramic thermal barrier coatings", to be published in Z. Werkstofftechnik