UDK 539.42 Original scientific article/Izvirni znanstveni članek ISSN 1580-2949 MTAEC9, 40(4)157(2006) ESTIMATION OF THE FATIGUE THRESHOLD VALUES FOR A CRACK PROPAGATING THROUGH A BI-MATERIAL INTERFACE TAKING INTO ACCOUNT RESIDUAL STRESSES OCENA UTRUJENOSTNEGA PRAGA ZA RAZPOKO, KI NAPREDUJE SKOZI VMESNO PLOSKEV MED DVEMA MATERIALOMA Z UPOŠTEVANJEM REZIDUALNIH NAPETOSTI Lubos Náhlík Institute of Physics of Materials, Academy of Sciences of the Czech Republic, Zizkova 22, 616 62 Brno, Czech Republic nahlikŽipm.cz Prejem rokopisa — received: 2006-05-17; sprejem za objavo - accepted for publication: 2006-07-18 This article deals with the behaviour of a fatigue crack propagating across a bi-material interface. A stability criterion for the crack touching the bi-material interface, taking into account the residual stresses closing the crack faces, is formulated. Linear elastic fracture mechanics are assumed to apply and the finite-element method is used in the calculations. The criterion proposed is applied to determine the fatigue threshold stress for crack propagation across the interface. Two different geometries are considered. It is shown that the threshold values for crack propagation are influenced by the residual stresses closing the crack and by the specific combination of the elastic constants of the materials used. The results contribute to a better understanding of the failure of structures with bi-material interfaces (protective layers, composite materials, etc.). Key words: bi-material interface, residual stress, plasticity-induced crack closure, critical stress Članek obravnava utrujenostno razpoko, ki napreduje skozi ploskev med dvema materialoma. Formulirano je merilo za razpoko, kise dotakne vmesne ploskve, kiupošteva rezidualno napetost, kizapira ustnicirazpoke. Za izračune sta uporabljenilinearna mehanika loma in metoda končnih elementov. Merilo je uporabljeno za določitev praga napetosti za prehod razpoke preko vmesne ploskve. Upoštevani sta dve geometriji. Znano je, da na napetost na pragu vplivajo rezidualne napetosti, ki zapirajo razpoko, in specifična kombinacija elastičnih konstant obeh materialov. Rezultati omogočajo boljše razumevanje zloma struktur z vmesnimiploskvami(varovalnisloji, kompozitnimaterialiitd). Ključne besede: vmesna ploskev, rezidualne napetosti, plastično zaprtje razpoke, kritična napetost 1 INTRODUCTION The interface between two dissimilar media represents a weak point for many applications of structures composed of different materials. The presence of regions with different mechanical properties and the existence of an interface between them have a Figure 1: The plastic zone created by a crack with its tip at the interface. Material properties are described by the Young’s modulus E, the Poisson’s ratio v and the yield stress <70 of each material. Slika 1: Plastična zona zaradi razpoke z vrhom na vmesni površini. Lastnosti materiala opisujejo Youngov modul E, Poissonovo število v in meja plastičnosti <70 za vsak material pronounced influence on the stress distribution of composite bodies. The characteristics of fracture in the vicinity of, and through, the interface are influenced strongly by the properties of the interface and of the materials on either side of the interface. Fracture usually starts at a defect in the interface, especially at an interface microcrack, or at the edge of the interface. Another important factor is the influence of the interface on a crack penetrating that interface from one material into the second in a bi-material body. 2 A CRACK TERMINATING PERPENDICULAR TO THE INTERFACE The stress distribution around the crack tip (in the case of a crack perpendicular to the interface) can be expressed in its general form, as follows (e.g., 1,2): Hi -/,(A«,/?) (1) 2nrp where fij(p,aß) is a known function of the bi-material parameters a and ß, as defined in 1, and 0 < p = p (a,ß) < 1 is the stress singularity exponent. For given materials and loading conditions the stress MATERIALI IN TEHNOLOGIJE 40 (2006) 4 157 L. NÁHLÍK: ESTIMATION OF THE FATIGUE THRESHOLD VALUES FOR A CRACK PROPAGATING ... distribution around the crack tip is determined by the value of the generalised stress-intensity factor Hi. The value of HI is proportional to the applied load and has to be estimated numerically. For homogeneous materials, Hi = Ki is the stress-intensity factor, and p = ˝. The stress distribution around a crack with its tip at the interface of two different elastic materials as given by Eq. (1) represents a general singular stress concentrator. The fact that the stress singularity exponent differs from ˝ means that the linear elastic fracture mechanics procedures and criteria cannot be used 3. In a homogeneous body (we consider for simplicity that the corresponding value of the load ratio R = 0, i.e., AK, = K,) and under the conditions corresponding to a high cycle fatigue, the fatigue threshold condition has the form KI(oT) = Kth (2) The fatigue crack will not propagate if the value of the applied load, expressed in terms of the stress intensity factor range KI, is less than the corresponding threshold value of the material Kth. Similarly, for a crack with its tip at the interface, the stability condition can be written in the form HI(°th ) = Hth (Kth) (3) In other words, the fatigue crack stays arrested at the interface if the value of the applied load, expressed in terms of the generalised stress-intensity factor range HI, is less than the corresponding generalised fatigue threshold value of the material Hth. The generalised threshold value Hth is a function of the fatigue threshold value Kth of the material M2 and, moreover, it depends on the elastic mismatch of the materials M1 and M2, as expressed by the bi-material parameters a,ß, i.e., Hth = Hth (Kth,aß), see 4. Instead of the values Kth and Hth the fatigue threshold stress can be used as the quantity describing the behaviour of the crack. The threshold stress oth is the value of the externally applied tensile stress aappl at which the crack will start to grow. The fatigue threshold condition for a propagating crack is: Pappl < O-th (4) A method introduced in previous publications (see 4 for details) was used to determine the generalised threshold values Hth and the threshold stress 0th. The method, which is based on an assessment of the plastic zone size (see Figure 1), yields the following expression for the critical value of the generalised stress-intensity factor: Hth(Kth ) = Kt2hpo(01-2 p) J hom ( ) f(a,ß,v)_ where p is the stress singularity exponent, Kth is the threshold value of the stress-intensity factor determined for the material M2 into which the crack is to grow, tf0 is the yield stress of the material M2, and the expression within the square brackets represents the ratio of the areas of the plastic zones in front of the crack tip in a homogeneous material (only the material M2 is considered, see Figure 1), and in the real bi-material case. v = V2 is the Poisson’s ratio of the material M2. The threshold stress is then given by: th appl #zappl ) (6) where Hth is the threshold value determined from equation (5) and HI(oappl) is the generalised stress-intensity factor determined numerically (using the FEM) for an applied tensile stress aappl. The procedure mentioned above and the results obtained from the criterion (6) neglect the existence of a reversed plastic zone and the closure of the fatigue crack during propagation. The phenomenon of fatigue-crack closure was investigated and described by Elber 5 as so-called "plasticity induced crack closure". Numerical elasto-plastic FEM calculations can be used to estimate the level of the applied external loading in the sense of Newman’s calculations of opening stresses, e.g., 6. For simplicity, a pulsating (sinusoidal) loading is assumed. The effective value of the threshold stress is then: :oth+oop (7) where of is the effective threshold stress taking into account the plasticity-induced crack closure, Oth is the threshold stress given by Eq. (5), and aop is the CTnl a) Ml Ď Ml 1 1 m: T i 1, -------L- ---------------------------1 C„vi (5) b) J M2 T 1 J Ml Ml ] it Ml T 1 1, 1 ------------------------------k Figure 2: The bi-material body with "edge" (a) and "central" (b) crack under tensile loading considered in the numerical example. T = 25 mm, t = 12.5 mm, L = 75 mm. Slika 2: Telo iz dvojnega materiala z robno (a) in centralno (b) razpoko prinatezniobremenitvi, upoštevaniprinumeričnem primeru: T = 25 mm, t = 12.5 mm, L = 75 mm 158 MATERIALI IN TEHNOLOGIJE 40 (2006) 4 L. NÁHLÍK: ESTIMATION OF THE FATIGUE THRESHOLD VALUES FOR A CRACK PROPAGATING • ¦» u ¦: ¦ ¦ I" ft» «0 •!- 1* LH e.e Figure 3: a) Dependence of the normalised effective threshold stress on Young’s modulus ratio; b) normalised (aopcl /trappl) values of the opening/closing stresses as a function of the Young’s modulus ratio for a bi-material body with an "edge" crack. Slika 3: a) Odvisnost normaliziranega dejanskega praga napetosti od razmerja Young modulov; b) normalizirane (ffop,cl/Cappl) vrednostiza napetosti odprtja in zaprtja v odvisnosti od Young modula za telo iz dveh materialov z robno razpoko V ö Ou,) Op ¦ t t '¦ y - Otf 02 «t ¦:¦ un i» Figure 4: a) Dependence of the normalised effective threshold stress on Young’s modulus ratio; b) normalised (oop,cl /crappl) values of the opening/closing stresses as a function of the Young’s modulus ratio for a bi-material body with a "central" crack. Slika 4: a) Odvisnost normaliziranega dejanskega praga napetosti od razmerja Young modulov; b) normalizirane (ffop,cl/Cappl) vrednostiza napetostiodprtja/zaprtja v odvisnostiod razmerja Young modulov za telo iz dveh materialov s centralno razpoko computed opening stress 7. Eq. (7) can also take into account various loading ratios, R. 3 NUMERICAL EXAMPLE The proposed procedure is applied to the estimation of the threshold stress oth corresponding to the threshold level in the case of cracked bi-material bodies (see Figure 2). First, the procedure published in 4 and Eq. (6) were used to determine oth. Then the model was subject to cyclic loading with a loading ratio of R = 0. The crack tip was located 0.3 mm in front of the interface at the beginning of the calculation and directly at the interface at the end of the calculation. Six load cycles were modelled, with the crack length increasing by 0.05 mm at each step. From the numerical calculation the opening (and closing) stress oop (and ocl), were determined, see Figure 3a and 4a, and from Eq.7 the value of E2), the fatigue threshold value decreases in comparison to the homogeneous material, with elastic properties of material M2. The results for the case of "edge" and "central" crack are more or less similar, but it is clear that the influence of Young’s modulus ratio is stronger for case of the "edge" crack. Figures 3b and 4b show the ratio between the maximum of the applied stress and the opening (closing) stresses. The differences between the individual cases are caused by different geometries and boundary conditions. The results obtained contribute to a better understanding of the damage that such cracks may cause in composite materials. Acknowledgements This research was supported by the grant No. 106/04/P084 of the Czech Science Foundation. 5 REFERENCES 1 K. Y. Lin, J. W. Mar: Int. Journal of Fracture, 12 (1976), 52 2 S. A. Meguid, M. Tan, Z. H. Zhu: Int. Journal of Fracture, 73 (1995), 1 3Z. Knésl: Int. Journal of Fracture, 48 (1991), R79 4Z. Knésl, L. Náhlík, J. C. Radon: Comput. Mater. Science 28 (2003), 620 5 W. Elber: Engng. Fracture Mechanics, 2 (1970), 37 6K. Solanki, S.R. Daniewicz, J.C. Newman, Jr.: Engng. Fracture Mechanics, 71 (2004), 165 7 L. Náhlík, P. Hutaf, Z. Knésl: Key Engng. Materials (2006)-accepted for publication 160 MATERIALI IN TEHNOLOGIJE 40 (2006) 4