INFLUENCE OF ELASTIC ANISOTROPY ON THE EDGE PROBLEM

Transcription

1 INFLUENCE OF ELASTIC ANISOTROPY ON THE EDGE PROBLEM S. Schmauder Max-Planck-Institut für Metallfrschung, Institut für Werkstffwissenschaft, Seestraße 92, D Stuttgart 1, FRG Intrductin Fracture in metaljceramic jints frequently initiates at the interface edge (1, 2) r at a defect situated at the edge regin f the interface. This phenmenn is weil knwn as the edge prblem. It is due t singular stresses at the edge if internal r external lad is applied (3). In the literature, crack nuc1eatin at interface edges in metaljceramic bimaterials with a dme-like crack path in the ceramic has been reprted. Bimaterials may fai! at the interface edge even in the absence f thermal stresses (4). The prblem can be reduced by intrducing interlayers with a prper thermal expansin cefficient (5), r by runding the edges (6). Hwever, such a prcedure is difficult and in many cases impssible. In this paper, the effect f the elastic anistrpy n the edge prblem will be iiluminated fr the case f Al203/Nb. This system is used as a mdel because there is n thermal mismatch. The main idea f the applied prcedure is t estimate the singularity pwer f interface edge stresses fr istrpie metals as well as fr single and plycrystalline Nb bnded t istrpie alumina by examining the energy release rates f very small edge cracks at the interface. General Aspects f Interfaces [1] [2] I 2w CI - r-1/2 crack CI - r-a A = A (lx,ß) ~ 0.41 (8aJY, 1975) FIG. 1. The plane gemetry is cnsidered here. At the interface edge and at the tip f an interface crack different stress singularities prevail.

2 are ften used t describe bimaterials by nly tw elastic cnstants. The range f a and ß is given by a parallelgram in the right half f which all bimaterials can be represented, eventually by changing the numbering f the tw materials s that a ~ O. Many f the technically relevant bimaterial cmbinatins satisfy the relatin (8). /3 ~ a/4. [3] In this scheme different material cmbinatins with the same elastic behaviur are described by the same Dundurs' parameters. The singularity pwer --\f the stresses at the interface edges is the same fr applied and residual stresses (2), and is a functin f a and ß which is nt larger than 0.41 mpared t --\= 0.5 fr the interface crack (Fig. 1). An implicit analytical slutin fr the determinatin f --\frm a and ß fr abimaterial nstituents was given by Bgy (9) as with istrpie elastie D (a, ß; --\) = (cs2 (--\11"f2) - (1---\)2)2 ß2 + 2 (1---\) (s2 (--\11"/2) - (1---\)2) aß + (1 - --\)2 «(1- --\)L 1) a2 + cs2 (--\11"/2) sin2 (--\11"/2) = O. [4] Lines f equal singularity pwer --\in the a-ß-diagram are shwn in Fig. 2. The value fr the bimaterial Al203/Nb is indicated as a crss belnging t a singularity rder f --\~ ß... ~.... ~: ~ cr-r-a1a,ß) , Methd and Mdel FIG. 2. Lines f equal singularity pwer f interface edge stresses. The Numerical methdinvestigatins is described f in the details edgeinstresses the literature, are perfrmed e.g. in ref. by (10). meansinfthe themdel finite the element crack methd is pened(fem). alng the interface and the elastic energy U is calculated fr every crack length a. The strain energy release rate fr a plate f unit thickness is calculated acrding t 1 du G = 2 la-l- [5] The energy release rate is related t the stress intensity factr fr the interface crack by K = (Je r;;. YK(a/h) G =1#K2 [6] [7] where (Je is the external applied stress and YK is a rrectin fr the finite gemetry. Fr an exterir crack in a hmgeneus material YK has the frm (12)

3 hrn YK (ajh) = (ajh) (ajh) (ajh) (ajh)4 [8] ~nd E* is a mean value f the Yung's mduli El and E2 f the invlved materials (8) P=2 1 1 [1 El+E2' 1 J [9] The prcedure t btain the crrectin functin is as fllws: The functin U(ajh) is fitted by a 5th rder plynmial and the strain energy release rate G(a/h) is btained by analytically derivating U(a/h) accrding t eqn. [5]. Then YK (a/h) is calculated thrugh eqns. [6] and [7]. Metal and ceramic are bth treated as linear elastic with the elastic data f the materials given in Table 1. Material Material Data v TABLE E [GPa] 1 A tensile specimen f width h = w = 100 mm and gemetry as shwn in Fig. 1 is mdelied by finite elements (cmpare Fig. 3). The upper part is assumed t cnsist f the stifter phase. The finite element mdel with interface cracks f length a/w = 0.04 and a/w = 0.21 which is externally laded perpendicular t the interface with 1 N/mm2 is shwn in Fig. 3 fr Al203/Nb. In rder t calculate the energies f very small interface cracks the mdel is refined at the interface edge as shwn in Fig. 4. FIG.3. Defrmatin (magnified 25 OOOx)f an Al203/Nb bimaterial with interface cracks f length (a) a/w = 0.04 and (b) a/w = FIG. 4. Clse-up view f the mesh at interface edge t calculate elastic energies fr shrt interface cracks.

4 Results and Discussin The strain energy U is an increasing functin f the crack length fr bth, cracks in hmgeneus materials as weil as in the bimaterial, with slightly higher values far the latter ne. The gemetry functin f the crack in the hmgeneus material (eqn. (8]) and in Al203jNb is shwn in Fig. 5..., GO " z ~ <>:0 ~~j\-. ~~ ~ GO '" \ REL. CRACK LENGTH A/H FIG. 5. Crrectin functin YK fr a crack in a hmgeneus material (triangles ) and in A1203jNb (circles) where H = w. The influence f the stress singularity at the edge is seen frm the increase f the curve fr the cmpsite in the range ajw ~ 0.1, while the influence f the dissimilarity f the materials is present as lng as the cnditin ajh ~ 0.25 is fulfil1ed. Fr larger crack lengths the crrectin functin is n lnger dminated by the dissimilarity f the materials. The rati R f the crrectin functins f the bimaterial and the hmgeneus material (Fig. 6) at a shrt crack length is used as a measure, e.g. R(ajh = 0.01). The rati fr A1203jNb is given by (cmpare Fig. 6) RNbj A1203 = 1.4. [10]. ~ REL. CRACK LENGTH A/H FIG.6. Rati f crrectin functins YK (A1203/Nb)/YK (A1203). The material cmpsite is characterized A1203/Nbby beys (R, aa) singularity = (1, 0). Apwer simple f linear A = interplatin accrdingthrugh t eqn. these [4] while tw apairs hmgeneus f values leads t the equatin A = (R-I). [11]

5 A cmparisn f analytieaily determined singularity rders frm eqn. [4] with values frm eqn. [11] and R :'rm the FEM -ealculatins is given in Table 2 fr a wide range f different material prperties. The discernibility between the bimaterials A1203/Nb and Al203/Si demnstrates the sensibility f the methd. ß \ A [2] [11] [4] Cl: [1] f the Stress Singularity Pwers TABLE 2 The rati f the crrectin funetins fr tw Al203/Nb bimaterials where a (100) and a (110) type Nb plane are assumed t be bnded t elastieally istrpie Al203 as weil as the crrectin funetin fr hmgeneus materials are shwn in Fig. 7 tgether with the crrespnding rati f the crrectin functins fr the bimaterial with istrpie cnstituents. In bth f these anistrpie calculatins the crack prpagates in a [001] directin. The interesting result is the fact that A is a functin f the Nb rientatin and with respect t the interface is distinctively higher r lwer cmpared t the case f istrpie cnstituents. The systematlgy f this dramatic change in A remains a tpie fr further studies. Hwever, this result f the anistrpie calculatins implies a higher prbability fr crack nucleatin at the edge fr different Nb rientatins. In that sense the Al203/Nb(110) empsite shuld be mst sensible fr interface edge crack nucleatin. yinh k /yhm k yhm k a/w Al2 3/ 1110lNb Nb 110.0lNb A ,04 Cnclusins FIG.7. Rati f crrectin functins fr elastically anistrpie behaviur f Nb. Estimated singularity pwers f stresses at the interface edge are tabulated (fr details see text). Aside frm the quantitative infrmatin given in Table 2 and in the Figures, the results f the paper may be summarized as fllws: A new methd is prpsed t determine the singularity pwer f interface edge stresses. The methd is successful in the case f istrpie cnstituents. The methd is applied t Al203(istrpic)/Nb(anistrpic) cmpsites with a (100) and a (110) type plane f Nb bnded t Al203 and crack prpagatin in a [100] directin. The sin~larity pwer f the stresses at the interface edge shws a strng rientatin dependence. A lwer singulanty pwer f A = 0.04 fr the (100) Nb rientatin and a higher singularity pwer f A = 0.17 fr the (110) Nb rientatin cmpared t the crrespnding bimaterial with istrpie cnstituents is fund. The stress singularity pwer fr bimaterials f istrpie cnstituents is knwn frm eqn. [4]. Thus, slving eqn. [11]fr R immediately leads t the crack driving frce fr these bimaterials R = A. [12]

6 References 1. RD. Adams and J. Cppendale, J. Adh. 10, 49 (1979) T. Suga and G. Elssner, Prceedings f the Japanese MRS (1988), t be published RS. Alwar and Y.R Nagaraja, J. Adh. 7, 279 (1976) 4. A. Piva and E. Vila, Engng. Fract. Mech. 13, 143 (1980) 5. K. Suganuma, T. Okamt, M. Kizumi and M. Shimada, J. Mat. Sei. Let. 4, 648 (1985) J. M. Dundurs, Okajima, J. Ph.D. Appl. Thesis, Mech. Carnegie-Melln 36, 650 (1985) University, Pittsburgh (1985) 8. S. Schmauder, Cer. Frum Int. 2, p. 101 (1986) D.B. Bgy, J. Appl. Mech. 38, 911 (1971) O.c. Zienkiewicz, Th<:Finite Element Methd, McGraw Hill, Lndn (1975) T. Suga, Ph.D. Thesis, University f Stuttgart (1983) K. -H. Schwalbe, Fracture Mechanics f Metallic Materials, Carl Hanser Verlag, Munich (1980)