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Crack Tip Plasticity Influence on Cracks Approaching Cu-Si Interface
Available online at www.sciencedirect.com
ScienceDirect Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia
Procedia Structural Integrity 23 (2019) 239–244
9th International Conference on Materials Structure and Micromechanics of Fracture
Crack Tip Plasticity Influence on Cracks Approaching Cu-Si Interface Stanislav Žáka,* and Reinhard Pippana a Erich
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstraβe 12, 8700 Leoben, Austria
Abstract Thin copper (Cu) films are widely used in numerous electronic applications where such films are frequently applied on silicon (Si) substrate. These devices can be exposed to severe loading conditions during their lifetime, induced by thermal, static and/or fatigue loading, which may lead to critical failure of such components due to cracking. Such failure may lead to a shortcut or disconnection of the electrical circuit which, in majority of cases, renders the whole device useless. Experimental studies showed that when short cracks propagate through Cu layer towards the Si interface, the crack tip plasticity tends to influence the crack driving force more when it approaches the Cu-Si interface, where the crack propagation speed is decreasing. This fact leads to possible decrease of the crack driving force (shielding effect) due to the Cu-Si interface. The change of the crack driving force in such systems was investigated in this study by the means of finite element calculations, whereas the cases of elastic-elastic and plastic-elastic transitions at the Cu-Si interface were considered and compared. The impact of the crack tip plastic region and its characteristic dimension (in comparison with the thickness of Cu film) on the crack driving force was quantified and related to the changes in crack driving force magnitude. Several magnitudes of loading (for models with the same yield stress) were considered to simulate different stages of evolution of the crack tip plastic zone. The results then divided these stages into several groups from no plasticityinduced influence on fully plastic Cu film. These findings can lead to better understanding of the crack propagation through the thin plastic films on elastic substrates and permit a better lifetime prediction of electronical or electro-mechanical devices. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review underresponsibility responsibility of scientific the scientific committee of the IC organizers MSMF organizers. Peer-review under of the committee of the ICMSMF Keywords: crack driving force; crack tip plasticity; Cu thin films; Cu-Si interface
* Corresponding author. Tel.: +420 603799811. E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the IC MSMF organizers.
2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the ICMSMF organizers 10.1016/j.prostr.2020.01.093
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Stanislav Žák et al. / Procedia Structural Integrity 23 (2019) 239–244 Stanislav Žák et al. / Structural Integrity Procedia 00 (2019) 000–000
1. Introduction Nowadays, in electronic industry the composite materials are mostly used for creation of circuit boards and other electronical or electro-mechanical devices. Such a composite, which is commonly used, consists of the combination of Si substrate and thin Cu films. Despite the low probability of ordinary electronical components to undergo severe mechanical stresses, a significant fatigue loading can occur e.g. due to thermal loading. Hence, the crack driving force investigation for cases of such composite materials plays significant role in the design of electronical components. It has been proved that the material properties mismatch at such interfaces influences the crack propagation. Related research on cracks in bodies consisting of two linear elastic materials (e.g. interfacial cracks (Rice, 1988; Williams, 1959), cracks perpendicular to the interface (Cook and Erdogan, 1972; Pan and Amadei, 1999; Romeo and Ballarini, 1997; Zak and Williams, 1963) or crack in elastic bi-materials with special composition (Bleeck et al., 1998; Erdogan, 1995; Rousseau and Tippur, 2000)) showed in general that the local stress intensity factors (SIFs) or crack driving force G or J-integral at the crack tip approaching the interface from stiffer to softer material (from area with higher to area with lower Young’s modulus E) increases and for the opposite case (crack propagation from lower to higher E) the local SIFs, G or J-integral values should decrease. When elastic-plastic material properties are taken into account, the size and evolution of the crack tip plastic zone strongly influences the G. Early works aimed at small scale yielding (Delfin et al., 1995; Romeo and Ballarini, 1997; Shih, 1991) and numerical solutions (Kim et al., 1997; Sugimura et al., 1995) of elastic-plastic interface crack problems showed the differences between the applied far-field crack driving force and the local tip J-integral. This deviation was closely related with the plastic zone reaching the interface. More recent works (Kolednik et al., 2010; Pippan et al., 2000; Pippan and Riemelmoser, 1998; Simha et al., 2003) showed the development of the crack plastic zone throughout the bi-material interface from computational and experimental point of view. The decrease of both crack driving force and crack tip opening displacement (CTOD) was observed for soft to hard material transitions and vice versa for the hard to soft material transition. In comparison to simple elastic solution, these results suggest some sort of transition between G values for both materials for cracks approaching the interface and the same applies for CTOD. However, the mentioned works took into account only small-scale yielding conditions or a relatively small crack tip plastic zones in comparison to the crack length or the thickness of the respective material layer. Contrary to them, this research is aimed at the evaluation of crack driving force (in terms of the J-integral) in a relatively thin film layer in elastic-plastic Cu material on top of the purely elastic Si substrate. The material combination simulates real electronical devices (such as printed circuit boards) subjected to external loading. Used numerical procedures enables to evaluate even extreme cases when the Cu layer is completely plastically deformed with the Si substrate still under elastic conditions. 2. Model and methods
Fig. 1. Used FE model: (a) geometry model with highlighted main parameters (orange color represents the thin Cu film on top of the grey Si substrate); (b) detail of FE mesh around the crack tip (area denoted by blue rectangle on the left, tip element dimensions ~ 5e-4·t)
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To evaluate the J-integral as a function of the crack length, a 2D parametric model was created using the Abaqus finite element (FE) code. The model was constructed to simulate a thin Cu layer on top of the Si substrate, where the Cu layer thickness t and the crack length a were the main parameters (see Fig. 1) dividing the geometry model into several sections. In the modelling procedure, the symmetry conditions were used and only one crack flank has been modelled. The whole model was created with general dimension (width and height) by two orders of magnitude larger than the Cu film thickness t and on the far sides of the model (for xmax and ymax coordinates) the infinite boundary conditions were applied to avoid influencing the results. The model was loaded by the force F applied perpendicular to the crack advance direction (to model mode I loading) and it was used to control the nominal stress σyy in the Cu film. Used FE mesh consisted of quadratic 2D elements (CPE8) (Dassault-Systemes, 2015) under the plane strain conditions. The mesh was well-refined in the vicinity of the crack tip (see Fig. 1 b) by a circular alignment of the mesh elements. This elements distribution is suitable for the evaluation of the J-integral by the contour (I-integral) integration method and also for a good description of the crack tip plastic zone. The integration method and used I-integral approach were well-described in (Walters et al., 2005) or for use with Abaqus code in (Dassault-Systemes, 2015). The Si material model was defined by the Young’s modulus ESi = 165 000 MPa and Poisson’s ratio μSi = 0.22 (Dolbow and Gosz, 1996) to model elastic behavior of Si. The Cu thin film material model was defined by its elastic properties ECu = 130 000 MPa and μCu = 0.34 (Freund and Suresh, 2003), its yield stress σyield = 100 MPa (Zhang et al., 2011) and hardening modulus ECu, pl = 600 MPa which is low (bringing used model close to the elastic-ideally plastic material) but still in conjunction with experiments in (Zhang et al., 2011). The simulation was executed only with the use of elastic material properties for the first time to get a reference results and then with the elastic-plastic behavior of Cu 7-times in succession with increasing loading force F. For each loading level the crack length was changed from a/t = 0.06 to a/t = 0.94 (0.994 for the fully elastic model). In all simulations the J-integral and the crack tip plastic zone radius rp (length in the crack extension direction, see Fig. 2) were evaluated. 3. Results The main observed results throughout the simulations were the shape and the characteristic dimension rp of the crack tip plastic zone as well as the J-integral as functions of the relative crack length and loading level represented by the ratio of nominal stress σyy in the Cu thin film and its yield stress σyield.
Fig. 2. Comparison of the crack tip plastic zone for crack with a/t = 0.56 (on the left) and a/t = 0.8 (on the right) for loading ratio σyy/σyield = 0.75, red line represents the interface between Cu and Si; on the left figure an indication of the major model dimensions (a and t), material composition and where the crack tip plastic zone radius rp was evaluated is denoted
From Fig. 2 it is clearly visible that the presence of the interface close to the crack tip has some influence on the crack tip plastic zone which changes the shape and seems to be attracted by the interface. Moreover, this change in crack tip plasticity and the proximity of the interface changes the actual value of the mode I J-integral (JI).
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Fig. 3. J-integral for mode I loading (normalized to Cu material characteristics ECu and μCu with use of plane strain conditions, specimen cross-section area S, loading force F and crack length a) and crack tip plastic zone radii (normalized to the thickness of remaining Cu film) as functions of crack length normalized to the entire Cu thin film thickness
From Fig. 3 one can clearly see that the introduction of significant plastic behavior of the Cu material to the FE model changes the results significantly. The linear solution (dotted black line) shows constant value of the normalized JI (linearly increasing absolute value of JI with increasing crack length a) for almost the whole range of the crack length. Only for a/t above 0.95, hence for the crack close to the interface, resulting elastic JI starts to decrease, which is in a good agreement with other authors (Cook and Erdogan, 1972; Pan and Amadei, 1999; Romeo and Ballarini, 1997; Zak and Williams, 1963), since in our case the transition on the bi-material interface is from compliant material to the stiffer one. Results from elastic-plastic FE simulation (separated according to the plasticity level described by the ratio σyy/σyield) show increase in JI for cases with a short crack and decrease of JI when the crack is approaching the interface. This effect is more pronounced for higher plasticity levels. Moreover, observed crack tip plastic zone radius rp shows expected increase with increasing a until the plasticity reaches the interface where the growth of the plastic zone stops due to elastic conditions behind the interface (purely elastic model of Si). 4. Discussion The resulting dependencies of JI on the crack length for cracks in thin films (when plasticity is taken into account) can be divided into three groups according to the plasticity (or loading) levels and the direct comparison with the linear-elastic solution can show the influence of the crack tip plasticity. The first elastic-plastic solution with a low plasticity ratio of 0.43 shows minimal to no difference in comparison with the elastic solution. Increase of the JI with increasing a is linear. Furthermore, the crack tip plastic zone is small for this case and thus the plasticity has no impact on the crack driving force. Some differences in comparison to the linear solution start to be noticeable for plasticity ratio between 0.5 and 1 (whereas the ratio 1 means fully plasticized Cu thin film). For these cases a small, rather insignificant, increase in normalized JI for short cracks is visible. Moreover, for long cracks (relatively to the Cu film thickness) some shielding effects are starting to play a significant role. This is noticeable by the sudden decrease of normalized JI and
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its deflection from the elastic solution. When comparing the JI and rp charts in Fig. 3, one can see that this change (deviation) corresponds to crack length when the crack tip plastic zone reaches the interface (i.e. when rp/(t-a) = 1). When the plastic zone ahead of the crack tip reaches the interface, the stress-strain field starts to be deformed by the discontinuity in both elastic and plastic material properties of used material models. However, simple difference between Young’s moduli of the mentioned material models cannot explain such a difference – the crack tip is much further away from the interface than the difference between ESi and ECu needs to manifest (for elastic case it was not visible until a was approximately 95% of t). Therefore, the crack tip plasticity is the governing element. Moreover, this corresponds well with other author’s research for lower levels of loading (Kolednik et al., 2010; Pippan et al., 2000; Pippan and Riemelmoser, 1998; Simha et al., 2003) - the mismatch between yield stresses at the interface causes the JI (or GI) to increase or decrease for higher to lower or lower to higher yield stress transitions respectively (in presented case the yield stress of Si substrate equals , i.e. pure elastic behavior - modeled transition is from lower to higher yield stress). The loading levels with plasticity ratio above 1 (full thickness of Cu film is plastically deformed) show completely different JI behavior. The normalized crack driving force is rapidly decreasing with increase of the crack length and there is barely any connection between the JI and rp. The full thickness of the Cu film is plastically deformed and only small area along the crack flanks remains under elastic deformation, therefore the exact crack tip plastic region could not be obtained for the two highest plasticity ratios. However, for the case of σyy/σyield = 1.07 the exact crack tip plastic region was still noticeable, but it reached the interface very soon in the simulations. Moreover, it should be mentioned here that the normalization function used in Fig. 3 a is based solely on the elastic solution of JI and thus the highest loading levels are not represented well, but the trend of results should be independent on the used normalization function. 5. Conclusions The presented case-study of thin elastic-plastic Cu film on the purely elastic Si substrate with a crack approaching the Cu-Si interface showed interesting behavior of the crack driving force (described by J-integral) when the Cu plasticity and high loading levels are considered. Performed FE simulations of the problem revealed that the crack behavior in such a case can be divided into three groups according to the loading/plasticity levels: • In the first group for σyy/σyield < 0.5 the crack tip plastic zone is relatively small in comparison with the crack length a and the Cu film thickness t. There is no influence of the Cu plasticity and the elastic-plastic solution merges with purely elastic one. Hence, no plasticity induced shielding is present. • In the second group with nominal stress level as follows: 0.5 < σyy/σyield < 1, the plasticity starts to play significant role. A decrease in evaluated JI corresponding to the point when the crack tip plasticity reaches the interface was observed. When this occurs, the crack tip plastic zone starts to stick to the interface and its shape deforms according to the interface, hence a strong plasticity induced shielding was observed for such cases. • In the third group for σyy/σyield > 1 the full Cu film thickness is plastically deformed, and the main load-bearing part of the modeled body is the Si substrate. The plasticity induced shielding is strong for such high loading levels. The crack tip plastic zone merges with the overall plasticity of the Cu film (or for plasticity ratios close to 1, the crack tip plasticity is suddenly intensified) and only the increase of the plastic deformation (around the crack tip) above the nominal values in the Cu film could be observed (not the precise rp-value). Therefore, no direct connection between the JI progression and the point when the crack tip plasticity reaches the interface could be observed. The revealed behavior of the crack driving force in the thin elastic-plastic Cu films on the elastic Si substrate can help with understanding of the influence of the elastic-plastic interface on the crack driving force and the crack tip plasticity. Although the loading levels for nominal stress below the Cu yield stress were described and the connection between the plastic zone radius and the JI was revealed, the higher loading above Cu yielding point need more in-depth research and slightly modified approach.
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References Bleeck, O., Munz, D., Schaller, W., Yang, Y., 1998. Effect of a graded interlayer on the stress intensity factor of cracks in a joint under thermal loading. Eng. Fract. Mech. 60, 615–623. Doi: 10.1016/S0013-7944(98)00044-7 Cook, T.S., Erdogan, F., 1972. Stresses in bonded materials with a crack perpendicular to the interface. Int. J. Eng. Sci. 10, 677–697. Doi: 10.1016/0020-7225(72)90063-8 Dassault-Systemes, 2015. Abaqus/CAE 2016 (users manual). Delfin, P., Gunnars, J., Stahle, P., 1995. Effect of elastic mismatch on the growth of a crack initially terminated at an interface in elastic plastic bimaterials. Fatigue Fract. Eng. Mater. Struct. 18, 1201–1212. Doi: 10.1111/j.1460-2695.1995.tb00848.x Dolbow, J., Gosz, M., 1996. Effect of out-of-plane properties of a polyimide film on the stress fields in microelectronic structures. Mech. Mater. 23, 311–321. Doi: 10.1016/0167-6636(96)00021-X Erdogan, F., 1995. Fracture mechanics of functionally graded materials. Compos. Eng. 5, 753–770. Doi: 10.1016/0961-9526(95)00029-M Freund, L.B., Suresh, S., 2003. Thin film materials : stress, defect formation, and surface evolution. Cambridge University Press, Cambridge, UK Kim, A.S., Suresh, S., Shih, C.F., 1997. Plasticity effects on fracture normal to interfaces with homogeneous and graded compositions. Int. J. Solids Struct. 34, 3415–3432. Doi: 10.1016/S0020-7683(96)00225-9 Kolednik, O., Predan, J., Fischer, F.D., 2010. Cracks in inhomogeneous materials: Comprehensive assessment using the configurational forces concept. Eng. Fract. Mech. 77, 3611–3624. Doi: 10.1016/j.engfracmech.2010.10.010 Pan, E., Amadei, B., 1999. Boundary element analysis of fracture mechanics in anisotropic bimaterials. Eng. Anal. Bound. Elem. 23, 683–691. Doi: 10.1016/S0955-7997(99)00018-1 Pippan, R., Flechsig, K., Riemelmoser, F.O., 2000. Fatigue crack propagation behavior in the vicinity of an interface between materials with different yield stresses. Mater. Sci. Eng. A 283, 225–233. Doi: 10.1016/S0921-5093(00)00703-6 Pippan, R., Riemelmoser, F.O., 1998. Fatigue of bimaterials. Investigation of the plastic mismatch in case of cracks perpendicular to the interface. Comput. Mater. Sci. 13, 108–116. Doi: 10.1016/s0927-0256(98)00051-2 Rice, J.R., 1988. Elastic Fracture Mechanics Concepts for Interfacial Cracks. J. Appl. Mech. 55, 98. Doi: 10.1115/1.3173668 Romeo, A., Ballarini, R., 1997. A cohesive zone model for cracks terminating at a bimaterial interface. Int. J. Solids Struct. 34, 1307–1326. Doi: 10.1016/S0020-7683(96)00144-8 Rousseau, C.-E., Tippur, H.V., 2000. Compositionally graded materials with cracks normal to the elastic gradient. Acta Mater. 48, 4021–4033. Doi: 10.1016/S1359-6454(00)00202-0 Shih, C.F., 1991. Cracks on bimaterial interfaces: elasticity and plasticity aspects. Mater. Sci. Eng. A 143, 77–90. Doi: 10.1016/0921-5093(91)90727-5 Simha, N.K., Fischer, F.D., Kolednik, O., Chen, C.R., 2003. Inhomogeneity effects on the crack driving force in elastic and elastic-plastic materials. J. Mech. Phys. Solids 51, 209–240. Doi: 10.1016/S0022-5096(02)00025-X Sugimura, Y., Lim, P.G., Shih, C.F., Suresh, S., 1995. Fracture normal to a bimaterial interface: Effects of plasticity on crack-tip shielding and amplification. Acta Metall. Mater. 43, 1157–1169. Doi: 10.1016/0956-7151(94)00295-S Walters, M.C., Paulino, G.H., Dodds, R.H., 2005. Interaction integral procedures for 3-D curved cracks including surface tractions. Eng. Fract. Mech. 72, 1635–1663. Doi: 10.1016/j.engfracmech.2005.01.002 Williams, M.L., 1959. The Stresses Around a Fault or Crack in Dissimilar Media. Bull. Seismol. Soc. Am. 49, 199–204. Zak, A.R., Williams, M.L., 1963. Crack Point Stress Singularities at a Bi-Material Interface. J. Appl. Mech. 30, 142. Doi: 10.1115/1.3630064 Zhang, S., Sakane, M., Nagasawa, T., Kobayashi, K., 2011. Mechanical Properties of Copper Thin Films Used in Electronic Devices. Procedia Eng. 10, 1497–1502. Doi: 10.1016/J.PROENG.2011.04.250
ScienceDirect Available online at www.sciencedirect.com Structural Integrity Procedia 00 (2019) 000–000
ScienceDirect
www.elsevier.com/locate/procedia
Procedia Structural Integrity 23 (2019) 239–244
9th International Conference on Materials Structure and Micromechanics of Fracture
Crack Tip Plasticity Influence on Cracks Approaching Cu-Si Interface Stanislav Žáka,* and Reinhard Pippana a Erich
Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstraβe 12, 8700 Leoben, Austria
Abstract Thin copper (Cu) films are widely used in numerous electronic applications where such films are frequently applied on silicon (Si) substrate. These devices can be exposed to severe loading conditions during their lifetime, induced by thermal, static and/or fatigue loading, which may lead to critical failure of such components due to cracking. Such failure may lead to a shortcut or disconnection of the electrical circuit which, in majority of cases, renders the whole device useless. Experimental studies showed that when short cracks propagate through Cu layer towards the Si interface, the crack tip plasticity tends to influence the crack driving force more when it approaches the Cu-Si interface, where the crack propagation speed is decreasing. This fact leads to possible decrease of the crack driving force (shielding effect) due to the Cu-Si interface. The change of the crack driving force in such systems was investigated in this study by the means of finite element calculations, whereas the cases of elastic-elastic and plastic-elastic transitions at the Cu-Si interface were considered and compared. The impact of the crack tip plastic region and its characteristic dimension (in comparison with the thickness of Cu film) on the crack driving force was quantified and related to the changes in crack driving force magnitude. Several magnitudes of loading (for models with the same yield stress) were considered to simulate different stages of evolution of the crack tip plastic zone. The results then divided these stages into several groups from no plasticityinduced influence on fully plastic Cu film. These findings can lead to better understanding of the crack propagation through the thin plastic films on elastic substrates and permit a better lifetime prediction of electronical or electro-mechanical devices. © 2019 The Authors. Published by Elsevier B.V. © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review underresponsibility responsibility of scientific the scientific committee of the IC organizers MSMF organizers. Peer-review under of the committee of the ICMSMF Keywords: crack driving force; crack tip plasticity; Cu thin films; Cu-Si interface
* Corresponding author. Tel.: +420 603799811. E-mail address: [email protected] 2452-3216 © 2019 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the scientific committee of the IC MSMF organizers.
2452-3216 © 2019 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the ICMSMF organizers 10.1016/j.prostr.2020.01.093
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Stanislav Žák et al. / Procedia Structural Integrity 23 (2019) 239–244 Stanislav Žák et al. / Structural Integrity Procedia 00 (2019) 000–000
1. Introduction Nowadays, in electronic industry the composite materials are mostly used for creation of circuit boards and other electronical or electro-mechanical devices. Such a composite, which is commonly used, consists of the combination of Si substrate and thin Cu films. Despite the low probability of ordinary electronical components to undergo severe mechanical stresses, a significant fatigue loading can occur e.g. due to thermal loading. Hence, the crack driving force investigation for cases of such composite materials plays significant role in the design of electronical components. It has been proved that the material properties mismatch at such interfaces influences the crack propagation. Related research on cracks in bodies consisting of two linear elastic materials (e.g. interfacial cracks (Rice, 1988; Williams, 1959), cracks perpendicular to the interface (Cook and Erdogan, 1972; Pan and Amadei, 1999; Romeo and Ballarini, 1997; Zak and Williams, 1963) or crack in elastic bi-materials with special composition (Bleeck et al., 1998; Erdogan, 1995; Rousseau and Tippur, 2000)) showed in general that the local stress intensity factors (SIFs) or crack driving force G or J-integral at the crack tip approaching the interface from stiffer to softer material (from area with higher to area with lower Young’s modulus E) increases and for the opposite case (crack propagation from lower to higher E) the local SIFs, G or J-integral values should decrease. When elastic-plastic material properties are taken into account, the size and evolution of the crack tip plastic zone strongly influences the G. Early works aimed at small scale yielding (Delfin et al., 1995; Romeo and Ballarini, 1997; Shih, 1991) and numerical solutions (Kim et al., 1997; Sugimura et al., 1995) of elastic-plastic interface crack problems showed the differences between the applied far-field crack driving force and the local tip J-integral. This deviation was closely related with the plastic zone reaching the interface. More recent works (Kolednik et al., 2010; Pippan et al., 2000; Pippan and Riemelmoser, 1998; Simha et al., 2003) showed the development of the crack plastic zone throughout the bi-material interface from computational and experimental point of view. The decrease of both crack driving force and crack tip opening displacement (CTOD) was observed for soft to hard material transitions and vice versa for the hard to soft material transition. In comparison to simple elastic solution, these results suggest some sort of transition between G values for both materials for cracks approaching the interface and the same applies for CTOD. However, the mentioned works took into account only small-scale yielding conditions or a relatively small crack tip plastic zones in comparison to the crack length or the thickness of the respective material layer. Contrary to them, this research is aimed at the evaluation of crack driving force (in terms of the J-integral) in a relatively thin film layer in elastic-plastic Cu material on top of the purely elastic Si substrate. The material combination simulates real electronical devices (such as printed circuit boards) subjected to external loading. Used numerical procedures enables to evaluate even extreme cases when the Cu layer is completely plastically deformed with the Si substrate still under elastic conditions. 2. Model and methods
Fig. 1. Used FE model: (a) geometry model with highlighted main parameters (orange color represents the thin Cu film on top of the grey Si substrate); (b) detail of FE mesh around the crack tip (area denoted by blue rectangle on the left, tip element dimensions ~ 5e-4·t)
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To evaluate the J-integral as a function of the crack length, a 2D parametric model was created using the Abaqus finite element (FE) code. The model was constructed to simulate a thin Cu layer on top of the Si substrate, where the Cu layer thickness t and the crack length a were the main parameters (see Fig. 1) dividing the geometry model into several sections. In the modelling procedure, the symmetry conditions were used and only one crack flank has been modelled. The whole model was created with general dimension (width and height) by two orders of magnitude larger than the Cu film thickness t and on the far sides of the model (for xmax and ymax coordinates) the infinite boundary conditions were applied to avoid influencing the results. The model was loaded by the force F applied perpendicular to the crack advance direction (to model mode I loading) and it was used to control the nominal stress σyy in the Cu film. Used FE mesh consisted of quadratic 2D elements (CPE8) (Dassault-Systemes, 2015) under the plane strain conditions. The mesh was well-refined in the vicinity of the crack tip (see Fig. 1 b) by a circular alignment of the mesh elements. This elements distribution is suitable for the evaluation of the J-integral by the contour (I-integral) integration method and also for a good description of the crack tip plastic zone. The integration method and used I-integral approach were well-described in (Walters et al., 2005) or for use with Abaqus code in (Dassault-Systemes, 2015). The Si material model was defined by the Young’s modulus ESi = 165 000 MPa and Poisson’s ratio μSi = 0.22 (Dolbow and Gosz, 1996) to model elastic behavior of Si. The Cu thin film material model was defined by its elastic properties ECu = 130 000 MPa and μCu = 0.34 (Freund and Suresh, 2003), its yield stress σyield = 100 MPa (Zhang et al., 2011) and hardening modulus ECu, pl = 600 MPa which is low (bringing used model close to the elastic-ideally plastic material) but still in conjunction with experiments in (Zhang et al., 2011). The simulation was executed only with the use of elastic material properties for the first time to get a reference results and then with the elastic-plastic behavior of Cu 7-times in succession with increasing loading force F. For each loading level the crack length was changed from a/t = 0.06 to a/t = 0.94 (0.994 for the fully elastic model). In all simulations the J-integral and the crack tip plastic zone radius rp (length in the crack extension direction, see Fig. 2) were evaluated. 3. Results The main observed results throughout the simulations were the shape and the characteristic dimension rp of the crack tip plastic zone as well as the J-integral as functions of the relative crack length and loading level represented by the ratio of nominal stress σyy in the Cu thin film and its yield stress σyield.
Fig. 2. Comparison of the crack tip plastic zone for crack with a/t = 0.56 (on the left) and a/t = 0.8 (on the right) for loading ratio σyy/σyield = 0.75, red line represents the interface between Cu and Si; on the left figure an indication of the major model dimensions (a and t), material composition and where the crack tip plastic zone radius rp was evaluated is denoted
From Fig. 2 it is clearly visible that the presence of the interface close to the crack tip has some influence on the crack tip plastic zone which changes the shape and seems to be attracted by the interface. Moreover, this change in crack tip plasticity and the proximity of the interface changes the actual value of the mode I J-integral (JI).
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Fig. 3. J-integral for mode I loading (normalized to Cu material characteristics ECu and μCu with use of plane strain conditions, specimen cross-section area S, loading force F and crack length a) and crack tip plastic zone radii (normalized to the thickness of remaining Cu film) as functions of crack length normalized to the entire Cu thin film thickness
From Fig. 3 one can clearly see that the introduction of significant plastic behavior of the Cu material to the FE model changes the results significantly. The linear solution (dotted black line) shows constant value of the normalized JI (linearly increasing absolute value of JI with increasing crack length a) for almost the whole range of the crack length. Only for a/t above 0.95, hence for the crack close to the interface, resulting elastic JI starts to decrease, which is in a good agreement with other authors (Cook and Erdogan, 1972; Pan and Amadei, 1999; Romeo and Ballarini, 1997; Zak and Williams, 1963), since in our case the transition on the bi-material interface is from compliant material to the stiffer one. Results from elastic-plastic FE simulation (separated according to the plasticity level described by the ratio σyy/σyield) show increase in JI for cases with a short crack and decrease of JI when the crack is approaching the interface. This effect is more pronounced for higher plasticity levels. Moreover, observed crack tip plastic zone radius rp shows expected increase with increasing a until the plasticity reaches the interface where the growth of the plastic zone stops due to elastic conditions behind the interface (purely elastic model of Si). 4. Discussion The resulting dependencies of JI on the crack length for cracks in thin films (when plasticity is taken into account) can be divided into three groups according to the plasticity (or loading) levels and the direct comparison with the linear-elastic solution can show the influence of the crack tip plasticity. The first elastic-plastic solution with a low plasticity ratio of 0.43 shows minimal to no difference in comparison with the elastic solution. Increase of the JI with increasing a is linear. Furthermore, the crack tip plastic zone is small for this case and thus the plasticity has no impact on the crack driving force. Some differences in comparison to the linear solution start to be noticeable for plasticity ratio between 0.5 and 1 (whereas the ratio 1 means fully plasticized Cu thin film). For these cases a small, rather insignificant, increase in normalized JI for short cracks is visible. Moreover, for long cracks (relatively to the Cu film thickness) some shielding effects are starting to play a significant role. This is noticeable by the sudden decrease of normalized JI and
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its deflection from the elastic solution. When comparing the JI and rp charts in Fig. 3, one can see that this change (deviation) corresponds to crack length when the crack tip plastic zone reaches the interface (i.e. when rp/(t-a) = 1). When the plastic zone ahead of the crack tip reaches the interface, the stress-strain field starts to be deformed by the discontinuity in both elastic and plastic material properties of used material models. However, simple difference between Young’s moduli of the mentioned material models cannot explain such a difference – the crack tip is much further away from the interface than the difference between ESi and ECu needs to manifest (for elastic case it was not visible until a was approximately 95% of t). Therefore, the crack tip plasticity is the governing element. Moreover, this corresponds well with other author’s research for lower levels of loading (Kolednik et al., 2010; Pippan et al., 2000; Pippan and Riemelmoser, 1998; Simha et al., 2003) - the mismatch between yield stresses at the interface causes the JI (or GI) to increase or decrease for higher to lower or lower to higher yield stress transitions respectively (in presented case the yield stress of Si substrate equals , i.e. pure elastic behavior - modeled transition is from lower to higher yield stress). The loading levels with plasticity ratio above 1 (full thickness of Cu film is plastically deformed) show completely different JI behavior. The normalized crack driving force is rapidly decreasing with increase of the crack length and there is barely any connection between the JI and rp. The full thickness of the Cu film is plastically deformed and only small area along the crack flanks remains under elastic deformation, therefore the exact crack tip plastic region could not be obtained for the two highest plasticity ratios. However, for the case of σyy/σyield = 1.07 the exact crack tip plastic region was still noticeable, but it reached the interface very soon in the simulations. Moreover, it should be mentioned here that the normalization function used in Fig. 3 a is based solely on the elastic solution of JI and thus the highest loading levels are not represented well, but the trend of results should be independent on the used normalization function. 5. Conclusions The presented case-study of thin elastic-plastic Cu film on the purely elastic Si substrate with a crack approaching the Cu-Si interface showed interesting behavior of the crack driving force (described by J-integral) when the Cu plasticity and high loading levels are considered. Performed FE simulations of the problem revealed that the crack behavior in such a case can be divided into three groups according to the loading/plasticity levels: • In the first group for σyy/σyield < 0.5 the crack tip plastic zone is relatively small in comparison with the crack length a and the Cu film thickness t. There is no influence of the Cu plasticity and the elastic-plastic solution merges with purely elastic one. Hence, no plasticity induced shielding is present. • In the second group with nominal stress level as follows: 0.5 < σyy/σyield < 1, the plasticity starts to play significant role. A decrease in evaluated JI corresponding to the point when the crack tip plasticity reaches the interface was observed. When this occurs, the crack tip plastic zone starts to stick to the interface and its shape deforms according to the interface, hence a strong plasticity induced shielding was observed for such cases. • In the third group for σyy/σyield > 1 the full Cu film thickness is plastically deformed, and the main load-bearing part of the modeled body is the Si substrate. The plasticity induced shielding is strong for such high loading levels. The crack tip plastic zone merges with the overall plasticity of the Cu film (or for plasticity ratios close to 1, the crack tip plasticity is suddenly intensified) and only the increase of the plastic deformation (around the crack tip) above the nominal values in the Cu film could be observed (not the precise rp-value). Therefore, no direct connection between the JI progression and the point when the crack tip plasticity reaches the interface could be observed. The revealed behavior of the crack driving force in the thin elastic-plastic Cu films on the elastic Si substrate can help with understanding of the influence of the elastic-plastic interface on the crack driving force and the crack tip plasticity. Although the loading levels for nominal stress below the Cu yield stress were described and the connection between the plastic zone radius and the JI was revealed, the higher loading above Cu yielding point need more in-depth research and slightly modified approach.
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