Nabil Kazi Tani 1, Tawfik Tamine 1, Guy Pluvinage 2

Transcription

1 The paper was presented at the Eighth Meeting New Trends in Fatigue and Fracture (NTF8) Ankaran, Slovenia, 3 4 Octoer, 008 Nail Kazi Tani 1, Tawfik Tamine 1, Gu Pluvinage NUMERICAL EVALUATION OF THE STRAIN ENERGY RELEASE RATE WITH ORIENTATION AND POSITION OF A CRACK NEAR A BIMATERIAL INTERFACE NUMERIČKA OCENA BRZINE OSLOBAĐANJA ENERGIJE DEFORMACIJE SA ORIJENTACIJOM I POLOŽAJEM PRSLINE BLIZU INTERFEJSA BIMATERIJALA Originalni naučni rad / Original scientific paper UDK /UDC: 539.4: Rad primljen / Paper received: Kewords i-material interfacial crack mixed mode fracture stress intensit factor energ release rate finite element Astract The finite element method and fracture mechanics concept were used to stud the interfacial fracture of imaterial structure. Effects of mechanical materials properties, position and orientation etween crack and imaterial interface are presented. Numerical results show that the energ release rate of the interfacial crack is influenced consideral parameters of these effects. For several examples studied, a good agreement is otained with different authors in term of normalised stress intensit factor and energ release rate. INTRODUCTION Man authors have studied cracks in complex media as nonhomogeneous materials. Recentl, some researchers have een interested in the prolems of the interfacial cracks in imaterial plates. P.R. Marur and H.V. Tippur, /1/, have solved interfacial crack prolems in imaterial plates analticall the development of stress field formulations into analtical series. Using the oundar element method (BEM), Yijun J. Liu and Nan Xu, //, have determined stress intensit factors for curved interfacial cracks. The same parameters have een expressed analticall J.P Shi, /3/, using variational approach. In their paper, C. Bjerken and C. Persson, /4/, have used the crack closure integral method to determine complex formulation of stress intensit factors for cracks at the interface in imaterial plates. For cracks perpendicular to the interface, T.C.Wang and P. Stahle, /5/, have determined the stress field near the crack tip using Muskhelishivili s potentials. Yilan and Adresa autora / Author's address: 1) Laorator LCGE, Facult of Mechanical Engineering, Universit of Sciences and Technolog of Oran, BP 1505 EL M NAUER Oran, Algeria, Kazitani_nail@ahoo.fr ) Laoratoire de Fiailité Mécanique ENIM, Metz, France pluvina@univ-metz.fr Ključne reči imaterijal prslina na interfejsu mešoviti olik loma faktor intenziteta napona rzina osloađanja energije konačni element Izvod Metoda konačnih elemenata i koncept mehanike loma su korišćeni za proučavanje loma interfejsa strukture od imaterijala. Izraženi su uticaji mehaničkih osoina materijala, položaja i orijentacije prsline u odnosu na interfejs imaterijala. Numerički rezultati pokazuju da rzina osloađanja energije prsline na interfejsu u značajnoj meri zavisi od parametara ovih uticaja. Za više proučavanih primera doijena je dora saglasnost rezultata različitih autora prikazanih preko normalizovanog faktora intenziteta napona i rzine osloađanja energije. Hua, /6/, have calculated stress intensit factors from displacement equations near the crack tip for an edge crack perpendicular to the interface, where the position of crack tip is at the interface. Using photoelasticit experimental techniques, A. Cirello and B. Zuccarello, /7/, have studied the effect of crack propagation perpendicular to the interface in nonhomogenous media which contains two dissimilar materials. The aim of present stud is to descrie the effects of position and crack orientation with the presence of interfaces in imaterial plates on the fracture parameters, especiall the stress intensit factor K and energ release rate G. The influence of crack length a and fracture mixed modes have een developed in this paper for cases of cracks parallel and perpendicular, and also inclined to the interface. The results otained numericall have een compared to recent research works. Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

2 BASIC FORMULAE Stress field for crack parallel to imaterial interface Figure 1. Crack geometr and coordinate sstem in imaterial plate. Slika 1. Geometrija prsline i koordinatni sistem imaterijalne ploče The complex form of the stress field near the crack tip for a crack at the interface can e expressed as follows, /8/: iε iε Re Kr Im Kr I II σαβ αβ( θε, ) Σ + Σ αβ( θε, ) (1) πr πr where (r, θ) are polar coordinates of the sstem and (α, β) are material indices 1 and. The parameter, ε, that characterizes imaterial specimen is defined the equation: 1 1 β ln D ε π 1 + βd () with D, the Dundur s parameter, /9/, given as follows: µ 1/ µ ( k1 1) ( k 1) βd µ / µ ( k + 1) + ( k + 1) 1 1 with k j 3 4ν j in plane strain; k j (3) 3 ν j in plane stress. 1 + ν The complex stress intensit factor K is defined as: 1 i K K + ik K e κ (4) where κ is the phase angle that depends on the imaterial constant ε. I II Functions Σ αβ( θ, ε ) and Σ αβ( θ, ε ) equal unit along the interface ahead of the crack tip for θ 0. For the case θ 0, the angular functions can e found in /10/. For ε 0, Eq. (1) is reduced to the expression for the stress field close to crack tip in a homogeneous material. For θ 0, stresses σ and σ 1 at the interface directl ahead of the tip are given : K1+ ik i σ σ + iσ1 r ε (5) π r Associated crack surface displacements, δ 1 and δ 1, at a distance r ehind the tip, θ π, are given, /4/: with iε 8( K1+ ik) r r i 1 * δ δ + δ (1+ iε) cosh( πε) π E j (6) * + (7) E E1 E j j j E E /(1 ν ) in plane strain; Ej E j in plane stress; and j 1;. When the possiilit of crack advance is considered, the energ release rate is often used as a measure of driving force. For an interfacial crack, Malshev and Salganik, /11/, showed that the energ release rate, G, in terms of complex stress intensit factor, K, is: K G cosh ( πε ) E Stress field for the crack perpendicular to imaterial interface Consider the plane elastic prolem as shown in Fig., when a finite crack is perpendicular to the imaterial interface. A Cartesian coordinate sstem Ox is attached on the interface. The x axis is along the interface and the axis is normal to the interface and coincides with the crack elongation direction, /5/. Both materials are isotropic and homogeneous. Material I is occupied the upper half plane S 1 and material II is occupied the lower half plane S. Stress and displacement in an elastic solid can e represented two Muskhelishivili potentials: σ + σ 4 Re( Φ( z)) x σ iτ Φ ( z) +Ω ( z) + ( z z) Φ'( z) x µ ( u + iu ) κφ( z) Ω( z) ( z z) Φ( z) x Here Φ(z) and Ω(z) are complex potentials. Figure. A finite crack perpendicular to a imaterial interface. Slika. Konačna prslina upravna na interfejs imaterijala The complex potentials for an edge dislocation at Z S, in an infinite elastic solid can e expressed as follows: B B s s Φ ( z ) ; Ω ( z) + B z s z s ( z s) µ B ( x + i) πi( κ + 1) 0 0 * (8) (9) (10) where x and are x and components of dislocation, κ 3 4ν for plane strain, ν is Poisson s ratio, and µ is the shear modulus. If the edge dislocation is emedded in material II, the complex potentials are (see Suo, /1/): Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

3 (1 +Λ1) Φ 0( z) z S1 Φ () z Φ 0() z +Λ Ω 0() z z S Φ 0() z +Λ1Ω 0() z z S1 Φ () z (1 +Λ ) Φ 0( z) z S αd + βd αd β Λ D 1 Λ 1 β 1+ β D where α D and β D are two Dundur s, /9/, parameters: ( κ + 1)/ µ ( κ1+ 1)/ µ 1 αd ( κ + 1)/ µ + ( κ1+ 1)/ µ 1 ( κ 1)/ µ ( κ1 1)/ µ 1 βd ( κ + 1)/ µ + ( κ + 1)/ µ 1 D 1 (11) (1) (13) (14) The crack can e considered as a continuous distriution of dislocations. Hence for our prolem, we have: µ ( x + i) Φ 0() z dt (15) πi(1 + κ ) z+ it ( x i) t( x i) 0 ( z) µ + dt µ Ω + dt πi(1 + κ) a z+ it π(1 + κ) a z+ it a ( ) where a and are distances from crack tips A and B to the interface, respectivel. Let us introduce a new complex variale z * and a new function I(z * ), where z * iz, * 1 x + i Ι ( z ) dt (16) * π a z 1 The function I(z * ) is a holomorphic function in the whole complex plane z * outside the cut (, a). Using the following variale transformations: * a + a, a + a z + ζ t + ξ (17) the function I(z * ) can e represented as: x + i I( z ) dξ (18) π 1 ς ξ Let us assume that dislocation densit can e expanded as a series of the first Cheshev polnomial 1 x + i αmtm( ξ) (19) 1 ξ m 0 where T m (ξ) is the first polnomial of Cheshev T ( ξ ) cos( m θ ), ξ cosθ. m Components x and of the dislocation densit have the square-root singularit r 1/ at crack tips. The first term of the right-hand side of Eq. (19) characterizes the desired singularit at the crack tips. In order to get a closed analtical solution of Eq. (18), the series of the first Cheshev polnomial is introduced in Eq. (19). The opening displacement on the crack surface can e expressed /5/ as: t ξ 1 δ x + δ ( x + i) dt αmt ( ξ) dξ. a0 1 1 ξ m 0 (0) sin( mθ ) a0α0( π θ) + a0 αm m m 1 where a 0 is crack length, c is distance from centre of crack to the interface (Fig. ), we it is here: a 0 (a )/; c (a + )/; ξ cos(θ) (t c)a 0. Opening displacement at point A should e zero, what is possile when t a; ξ 1; θ 0; producing a 0 0. NUMERICAL SIMULATION OF CRACKS IN BIMATE- RIAL PLATES Interfacial central crack etween two dissimilar homogenous materials To stud the evolution of energ restitution rate for the case of interfacial crack in imaterial plates, a specimen of ceramic/metal joint etween Si 3 N 4 (silicon nitride) and S45C steel was used. The silver ased razing allo was the onding etween Si 3 N 4 ceramic and S45C steel, //, Fig. 3. The FEM is applied for the resolution of the interface prolem using 8 nodes quadratic finite element meshing. The procedure G-théta availale in code CASTEM, /13/, is used for evaluation of parameter G. To compare our results to those otained Yijun J. Liu and Nan Xu, //, we have evaluated G parameter using the following equations: where 1 G (1 β ) + 1 ν1 1 ν K E1 E (1) µ 1 ν µ ν 1 1 (1 ) (1 ) β µ (1 ν ) + µ (1 ν ) 1 1 I II () K K + K (3) where m 1, m represent shear modulus of the materials, and K I, K II are stress intensit factors relative to fracture modes I and II, respectivel. Figure 3. Bimaterial plate containing a crack at the interface sujected to an uniform tensile load. Slika 3. Bimaterijalna ploča sa prslinom na interfejsu izložena ravnomernom zateznom opterećenju Due to the vertical smmetr of specimens, onl half of the cracked plate is considered in the modelization 8 nodes quadratic finite elements, as shown in Fig. 4. Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

4 Figure 4. Finite elements meshing of the interfacial crack tip in imaterial plate. Slika 4. Mreža konačnih elemenata oko vrha prsline na interfejsu imaterijalne ploče The values of released energ rate are given in Fig. 5. Numerical results are in a good agreement with the results of /14/ and //, maximum difference is less than 5%. Cracks normal to the interface in imaterial plates In this application, imaterial specimens of aluminium and PSM-1, of elastic constants given in Tale 1, with a crack (W L 15 mm, t 5.5 mm), are considered (Fig. 6) for different crack lengths corresponding to crack ratios (a/w 0.1, 0., 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.99), /7/. Tale 1. Elastic constants of the materials. Taela 1. Elastične konstante materijala Young modulus E (MPa) Poisson ratio ν Aluminium PSM Specimens have een otained joining two aluminium thin plates and a plate made of PSM-1 polcaronate with liquid Araldite D adhesive, /7/. The specimens are sujected to tensile force F 6600 N, that produced a stress field at the upper and lower edges of the plate as shown in Fig. 7. Figure 6. Geometr of the cracked specimen sujected to tensile loading, /7/. Slika 6. Geometrija epruvete sa prslinom izložene zateznom opterećenju, /7/ Due to the smmetr according to oth axes x, of reference, onl the quarter of plate is considered in the modelization of the specimen as shown in Fig. 7. Applied finite elements meshing near the crack with 8 nodes quadratic finite elements is shown in Fig. 8. Figure 7. Modelization of imaterial plate. Slika 7. Model imaterijalne ploče Figure 5. Numerical results FEM calculation of G parameters vs. crack ratio a/w in an approximate evaluation from /14/ and //. Slika 5. Numerički rezultati proračuna MKE zavisnosti G parametra od odnosa prsline a/w u priližnoj oceni iz Ref. /14/ i // Figure 8. FEM meshing of imaterial plate near the crack tip. Slika 8. Mreža KE za imaterijalnu ploču u lizini vrha prsline Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

5 Figure 9 shows results of stress intensit factors calculated FEM and those reported in Ref. /7/ using Red Green Blue (RGB photoelasticit) experiments, /15/, and a semi-analtic relationship defined as: K λ ( α β λ ) π a 1 tan λ ( α β λ ) w 1 D D 1 I 1 D+ D 1 α D α D+ βdλ1 (4) where α D and β D are two Dundur s parameters and λ 1 is the order of singularit. Values of normalised stress intensit factor otained numericall FEM are in a good agreement with the ones evaluated in /7/ Red, Green, Blue (RGB) experimental photoelasticit and Eq. (4). In Fig. 9, the presence of the crack in the less stiffened material and the effect of interface materials mismatch produced a closing ridging stress intensit factor that decreases the stress intensit factor resulting from external load with the crack length, i.e. when the crack tip is close to the interface. Figure 9. Evaluation FEM of normalised stress intensit factor ( K K / σ πa) vs crack length ratio (a/w). I I o Experimental method /7/ FEM evaluation /14/ Numerical resolution /7/ Slika 9. Ocena pomoću MKE zavisnosti normalizovanog faktora intenziteta napon ( K K / σ πa) i odnosa dužine prsline (a/w) I I The numerical simulation in Fig. 10 shows the decreased tendenc of normalised stress intensit factors when crack 1.40E E E-07 propagates in the domain of material I (PSM-1, Tale 1). The evaluation has een considered also when the crack propagates in the more stiffened domain of the imaterial plate (Aluminium, Tale 1). In this case the normalised stress intensit factor increases with ratio a/w. Figure 10. Evaluation of the normalised stress intensit factor with the crack propagation and material properties Slika 10. Zavisnost normalizovanog faktora intenziteta napona od rasta prsline i osoina materijala It is important to note that the ratio of evolution of parameter in Fig. 10 will e more significant when the geometric crack ratio is greater than a/w 0.4; it means that when the crack tip is close to the interface. Cracks parallel to the interface In the next simulation, an attempt is made to show the effect of lateral position of crack on the release energ rate G when the crack is located at a distance h from the interface. The evolution of parameter G is plotted in Fig. 11. Results presented in Fig. 11 show that G parameter increases when the crack is too far from the interface. The evolution depends on the material elastic constants in the domain containing the crack and crack ratio a/w. For the same geometr crack ratio a/w, greater G values are noted when the crack is positioned in the material of the minor value of Young modulus (E 06 GPa, ν 0.30). 1.40E E E-07 K I G 8.00E E E-08.00E E+00 a/w h 0,5w h w h 1,5w G 8.00E E E-08.00E E+00 a/w a) Crack in aluminium ) Crack in PSM-1 material a) Prslina u aluminijumu ) Prslina u PSM-1 materijalu Figure 11. Evolution of the released energ rate G with the lateral position h of the crack Slika 11. Zavisnost od rzine osloađanja energije G od očnog položaja h prsline h 0,5w h w h 1,5w Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

6 Interface cracks in mixed mode fracture This section concerns the effect of mixed mode and mechanical properties of the imaterial on fracture parameters, especiall on the released energ rate G. The specimen of imaterial plate contained an inclined edge crack of various ratio (a/w 0.1, 0., 0.3, 0.4, 0.5, 0.6) and inclination angles. The cracked plate is sujected to a tensile loading stress σ g 1 MPa, Fig. 1. The prolem has een solved in plane stress condition using the finite element method for specimen modelization (Fig. 13). After numerical resolution in term of G eq, according to eq I II G G + G (5) Released energ rate values are plotted in Fig. 14 for several crack orientations in aluminium and the PSM 1 material. Otained results show that the released energ rate G increases with crack length a for cracks positioned in oth materials. Figure 1. Inclined edge crack in imaterial plate. Slika 1. Nagnuta prslina u imaterijalnoj ploči Figure 13. Finite element modelling of interface crack near the tip. Slika 13. Model konačnih elemenata prsline na interfejsu oko vrha 1.48E-0 1.8E E-0 Graphs of G eq in Fig. 14 show that this parameter increases with crack length. It is important to note that when the crack propagation is located in the domain of more stiff material of specimen (aluminium, Tale 1), the values of released energ rate G eq decreased (Fig. 14) for all studied inclinations of crack angle. CONCLUSIONS In this paper, the stress field near the crack tip has een studied for several orientations and positions of crack means of numerical analsis. The singularit of the stress and its complexit for cracks along the interface was considered evaluating energ release rate G. The results for G parameter for a central interfacial crack shown in Fig. 5 are in a good agreement with those otained, presented in Ref. /14/. This fracture parameter increases with crack length a. This tendenc of G has the same evolution as in the case when the crack is in a homogenous media. The elastic mismatch has a significant effect on the values of the released energ rate G. When the crack is perpendicular to the interface, the decreased curve of normalised stress intensit factor K I otained A. Cirello and B. Zuccarello in Ref. /7/ is confirmed for all geometric crack ratios a/w (Fig. 9). This tendenc of stress intensit factor decrease is explained in /7/ the significant presence of ridging effect. The graphs plotted in Fig. 10 show that the influence of the ridging effect is negligile when the crack propagates in the aluminium domain, i.e. the normalised stress intensit factor increases with crack length. For the inclined edge crack emanating from the interface, results in Fig. 14 show that G parameter increases with crack inclination angle; this tendenc is more significant for ratio a/w 0.4 and for inclination angle θ 60. The elastic mismatch effect on the energ release rate is more important when the crack tip orientation is closed to the interface. As mentioned aove, the elastic constants of the domain containing the crack have the same effect on the G parameter. 1.48E-0 1.8E E-0 G eq. 8.81E E E-03.81E E E a/w G eq. 8.81E E E-03.81E E E a) Crack in aluminium ) Crack in PSM 1 material a) Prslina u aluminijumu ) Prslina u PSM 1 materijalu Figure 14. Evolution of released energ rate G eq with crack inclination angle and ratio a/w. Slika 14. Zavisnost rzine osloađanja energije G eq od ugla nagia i odnosa a/w Vol. 9, r. 1 (009), str Vol. 9, No 1 (009), pp. 15 1

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