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Numerical modeling and simulation of intergranular fracture due to dynamic embrittlement for a CuNiSi alloy
Mechanics Research Communications 75 (2016) 81–88
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Numerical modeling and simulation of intergranular fracture due to dynamic embrittlement for a Cu Ni Si alloy Zhidan Sun a,∗ , Lahouari Benabou b , Hongqian Xue c a b c
ICD, P2 MN, LASMIS, University of Technology of Troyes, UMR 6281, CNRS, Troyes, France LISV, Université de Versailles-Saint-Quentin-en-Yvelines, 45 Avenue des Etats-Unis, 78000 Versailles, France School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 13 December 2015 Received in revised form 21 April 2016 Accepted 8 June 2016 Available online 9 June 2016 Keywords: Dynamic embrittlement Intergranular cracking Crystal plasticity Cohesive zone method Homogenization
a b s t r a c t The involvement of the dynamic embrittlement phenomenon during tensile tests of a Cu Ni Si alloy leads to decohesion of grain boundaries, which is dependent on the energy decrease at grain boundaries caused by sulfur segregation. The severity of dynamic embrittlement increases with the increasing of temperature and/or the decreasing of straining rate. A crystal plasticity approach associated with the cohesive zone method is used in order to analyze the fracture caused by dynamic embrittlement. A cohesive zone model is used to describe the grain boundary cracking and one of its main parameters is identified using a homogenization scheme for the nonlinear response of a polycrystal in presence of degradable grain boundaries. Numerical simulations are carried out in order to simulate the mechanical behavior related to dynamic embrittlement under different conditions of strain rate and temperature. The obtained results such as tensile stress-strain curves and intergranular crack distribution are compared with experimental observations. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Materials for various electrical applications such as lead frame or squirrel cage of rotors in large electrical motors are required to have good mechanical strength and electrical conductivity as well. Some copper alloys of the precipitation-strengthened type have been used for such applications. However, some alloys among them, such as Cu Cr [1], and Cu Ni Si [2] have been found to suffer from dynamic embrittlement in the temperature range 400–800 K. The dynamic embrittlement is a time-dependent form of brittle intergranular fracture that occurs in many alloys when stress and elevated temperatures are applied [3,4]. This phenomenon is due to the reduction in intergranular cohesive strength resulting from stress-driven diffusion of embrittling element into grain boundaries. It is well known that sufficient quantities of embrittling element present at grain boundaries by diffusive segregation can change the fracture mode of copper alloys from ductile rupture to intergranular mode [5]. This is because the fracture energy is decreased by the presence of the segregated embrittling element. Some attempts have been made to model the dynamic embrtittlement mainly in the macro-scale [6–8]. In these analyses, the
∗ Corresponding author. E-mail address: [email protected] (Z. Sun). http://dx.doi.org/10.1016/j.mechrescom.2016.06.003 0093-6413/© 2016 Elsevier Ltd. All rights reserved.
microstructure features were generally not introduced in modeling. The use of the finite element method has enabled the integration of the grain morphology into the description of the mechanical behavior of a polycrystalline material. In this study, the investigation is aimed at computational procedure for numerical modeling of intergranular embrittlement process coupled with crystal plasticity approach. Such procedure will be useful in analyzing, for example, the sensitivity of crystallographic orientation/texture to dynamic embrittlement. Furthermore, by optimizing the texture, the severity of dynamic embrittlement can be reduced and the performance of components may be increased. With this specific goal of developing a numerical approach suitable for the study of dynamic embrittlement, the crystal plasticity formulation coupled to the cohesive zone method was implemented in the finite element code ABAQUS. The crystal plasticity model is used to describe the mechanical behavior of each grain as function of its crystallographic orientation and slip systems, while the cohesive zone model is used to simulate the intergranular cracking at grain boundaries. The parameters of the cohesive zone model, which are dependent on experimental conditions such as temperature and strain rate, are identified using a self-consistent homogenization scheme in which conditions of imperfect and damageable interface have been added [9]. The results of the numerical
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Fig. 3. Schematic representation of the contribution of various constituents (grains and grain boundaries) in the polycrystal and their combined effect on the overall response.
Fig. 1. Typical stress-strain curves obtained at RT and 473 K with high strain rate (ε˙ = 2 × 10−3 s−1 denoted by fast) and low strain rate (ε˙ = 5 × 10−6 s−1 denoted by slow).
simulations are compared with the experimental data to validate the numerical modeling approach. 2. Highlighting of dynamic embrittlement in Cu Ni Si alloy A commercial Cu Ni Si alloy was investigated, with the following chemical composition (in wt%): 2.16 Ni, 0.72 Si, 0.23 Zr, 0.02 Fe. The material was prepared in the form of bars according to industrial conditions. The tensile tests were carried out with dumbbell shape specimens under displacement control. The straining rates were in the range 5 × 10−6 s−1 to 2 × 10−3 s−1 . The tests were performed in air at room temperature (RT) and up to 473 K. Typical experimental stress-strain curves are presented in Fig. 1. It can be seen that the ductility, represented by elongation to rupture R , decreases when the temperature is increased and/or the ·
strain rate is decreased. For the high straining rate = 2 × 10−3 s−1 , the curve at RT exhibits a ductile behavior with R , and at 473 K the ductility is slightly reduced with Fig. 1 also demonstrates that decreasing the straining rate enhances strongly the reduction of ductility. εR is ε˙ = 5 × 10−6 s−1 reduced down to 3.9% for the low straining rate at All the fracture surfaces were examined using scanning electron microscopy (SEM) and optical microscopy in order to analyze cracking mechanisms. Under the condition of 473 K with ε˙ = 5 × 10−6 s−1 where the ductility is strongly reduced according to the stress-strain curve, the fracture mode appears clearly to be of intergranular type (Fig. 2a). This investigation was confirmed by
observing the longitudinal cut section. As shown in Fig. 2b, the damage seems appear in the form of intergranular cracking. Additional observation using MEB on cut section confirmed that the cracks shown in Fig. 2b are effectively of intergranular type. The experimental results shown above highlight that the investigated Cu Ni Si alloy exhibits an intergranular embrittlement which is typically time-dependent. This intergranular embrittlement is induced by stress-assisted diffusion of an embrittling element during the tensile test itself. This process is usually called “dynamic embrittlement”. The embrittlement process is more pronounced with increasing temperature and decreasing strain rate, which promotes the segregation of embrittling species and consequently the reduction of the cohesive strength of grain boundaries. Regarding the identity of the embrittling element, according to different tests and measures carried out in [10], the embrittling element in the studied Cu Ni Si alloy is very likely the sulfur. In the literature, sulfur has been identified as embrittling element in other metals both experimentally and theoretically. For example, the sulfur-induced embrittlement of nickel is clearly associated with sulfur segregation to grain boundaries and the transition from ductile to brittle behavior requires a critical intergranular concentration of sulfur [11]. 3. Modeling of brittle intergranular cracking As described in Section 2, the brittle fracture of the investigated Cu Ni Si alloy is caused by intergranular cracking due to the cohesive strength reduction of grain boundaries. In this case, the fracture process of grain boundaries can be simulated using the cohesive zone method (CZM). As a matter of fact, the decreasing of grain boundary strength is controlled by the fracture energy release rate, which is one of the important parameters of a cohesive zone model. In this work, an analytical approach is used to model the
Fig. 2. (a) Fracture surface, and (b) longitudinal cut section observations, after tensile test at 473 K with ε˙ = 5 × 10−6 s−1 .
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state variable ˛s , respectively. The resolved shear stress s is used as a critical variable to calculate the viscoplastic slip rate: ␥˙ s =
| s − s | − s n sign s − s k
(2)
where k and n are material constants characterizing the viscosity. The hardening variables, as well as the evolution of the state variables, are given by: s = 0 + bQ
·s
r
·s
hsr r with = 1 − bs
(3)
·s
s = c˛s with˛˙ s = ˙ s − d˛s Fig. 4. Self-consistent scheme applied to a polycrystalline aggregate with imperfect grain boundaries.
(4) ·s
where 0 represents the initial critical shear stress, = | ˙ s |, the parameters c and d characterize the kinematic hardening, while b, Q and the matrix hsr define the isotropic hardening. The plasfrom the contribution of all slip tic strain rate in grain g, resulting pg systems, is calculated with ε˙ =
˙ s ms . s 3.2. Cohesive zone model
Fig. 5. Comparison between the numerical simulation and experimental tensile stress-strain curves with different strain rates (from very slow to fast) at 473 K.
overall polycrystal behavior, while taking account of both crystal plasticity inside the grains and interface decohesion at the grain boundaries (Fig. 3). This approach allows identifying the parameters of the cohesive zone model. Accordingly, in this section, the crystal plasticity model and the cohesive zone model used in this work will be outlined and the parameter identification for these models will be briefly presented. It is important to mention that the current developed approach coupling crystal plasticity and cohesive zone method are suitable for simulating intergranular fracture. In the case where the fracture is not dominated by intergranular decohesion, the prediction capacity of this approach is limited. 3.1. Crystal plasticity model Single crystal plasticity is defined at the grain scale where slip occurs by dislocation motion along a slip system. For FCC polycrys12 octahedral slip systems are considered, based on 4 slip talline, planes 111 and 3 slip directions 110. According to the Schmid’s law, the resolved shear stress s for the system s in the grain g can be obtained by: s = g : ms withms =
1s l ⊗ ns + ns ⊗ ls 2
(1)
where ns and ls are respectively the normal to the plane of the slip system s and the slip direction in this plane. In this work, a viscoplastic model developed by Méric and Cailletaud [12] is used. In this phenomenological model, isotropic hardening s and kinematic hardening s are introduced, which are associated with the isotropic hardening state variable s and the kinematic hardening
The cracking process at grain boundary is modeled by means of the cohesive zone method, which was motivated by the observed intergranular cracking, as shown in Section 2. Cohesive elements are placed at each grain boundary in the synthetic polycrystal microstructure. The cohesive elements link two crystal grains on both sides, and are characterized by a constitutive law where the cohesive force is a function of the relative displacement between the two crystal surfaces. The fracture process is lumped into the crack line and the constitutive law, also called traction-separation law, which relates tractions T and displacement jumps across the cohesive surface. The traction-separation behavior can be taken in different forms (bilinear [13] or exponential [14], for example) to represent the grain boundary response. In what follows, an exponential Smith-Ferrante law, approximating well the separation of atomic planes due to cleavage, is employed. The cohesive energy density, represented by the area beneath the traction-separation curve, is the most important parameter since it measures the amount of energy needed to completely separate the bounded surfaces to create free surfaces. The cohesive zone model begins from a traction potential from which the coupled cohesive traction T is derived:
= eTc ıc 1 − 1 + ı/ıc e−ı/ıc T=
∂ ı = eTc e−ı/ıc ıc ∂ı
, (5)
where e is the e-number, Tc is the maximum cohesive traction, ıc is a characteristic opening displacement, and ı is the effective open
u2n + 2 u2t . The parameter ing displacement given by ı = represents the ratio of shear to normal opening relative displacement. The normal and shear tractions, Tn and Tt respectively, can be obtained in the same way by differentiating the potential by each component of ı. Hence, only three parameters are involved in the cohesive law, namely Tc , ıc , and. In our study, will be taken as 1, which means a same contribution of displacement jumps in mode I and mode II. The cohesive energy of the grain boundary, also called the critical energy release rate, can be obtained from these parameters as Gc = eTc ıc . The calculations of Van der Ven and Ceder [15] suggest that ıc is insensitive to impurity coverage and thus, can be considered, to a first approximation, as constant. It is taken as 8 × 10−5 mm for this study. In this case, Tc (consequently Gc ) is the only parameter to be identified for this cohesive zone model.
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Fig. 6. Crack initiation and propagation process during monotonic loading with low strain rate (ε˙ = 5 × 10−6 s−1 ).
3.3. Parameters identification The approach developed in this work couples a crystal plasticity model (responsible for the constitutive behavior) and a cohesive zone model (responsible for the fracture behavior). The determination of these two groups of material parameters was performed using a mean filed homogenization scheme. From the curves to rupture experimentally obtained, it is possible to fit the values of these two groups of parameters, respectively using a model with per-
fect interfaces (without fracture), and a model with an imperfect interface (with fracture). For the crystal plasticity model, the values of the parameters should be identified using all the experimental testing conditions, as some parameters are sensitive to strain rate which ranges from ε˙ = 5 × 10−6 s−1 to ε˙ = 2 × 10−3 s−1 , at 473 K. To do this, a homogenization scheme is used to obtain the effective behavior of a Representative Elementary Volume (REV) made of 40 grains, as presented in [9]. In the polycrystal, each grain is treated as an ellipsoidal inclusion embedded in an infinite medium made of
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Fig. 7. Von Mises stress distribution obtained under different loading levels corresponding to different crack lengths. Notice that once a crack is formed, on its two sides, the stress tends to be released.
all the remaining grains, also called the homogeneous equivalent medium (HEM). Perfect interface condition is first considered for the grain boundaries. For such a class of material, the self-consistent method can thus be used as a technique for determining the effective properties. The identification of these parameters was realized using the secant moduli method of Berveiller-Zaoui [16]. In this as homogenization model, the local stress in each grain is given function of the macroscopic fields of stress and plastic strain, and E p , by the following equation: g =
+ ˇ E p − εpg
(6)
The function ˇ, involving a secant plastic modulus, evolves with the total plastic deformation during loading.
Regarding the identification of the interface parameter Tc , it can be easily performed since the imperfect interface conditions were included in the mean field homogenization scheme, and thus will have an effect on the overall behavior which is fitted. To present briefly the insertion of these conditions, it should be recalled that failure of grain boundaries is reflected by discontinuity of the displacement field across the interface. A recent equivalent inclusion solution, developed by Othmani et al. [17], was incorporated in the polycrystal transition rule to describe debonding at grain boundaries. We use here the same approach that we adapt for a self-consistent scheme: an aggregate of Ng grains with imperfect interfaces between the grains is considered. As loading is applied, the grains may be weakened at their interfaces, which may affect the overall stress-strain behavior of the polycrystal. Accounting for
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Fig. 8. Equivalent plastic strain evolution during the tensile test of the REV. Macro shear bands are formed with about 45 ◦ as respect to the loading direction.
imperfect interface conditions, Eq. (6) thus becomes: g =
g −1 −1 eff eff + I + L : S −1 − I : C : L :
g S −1 − I : Ep − εpg
(7) eff
In this new transition rule, L
is the effective tangent moduli of
because there is always a stress concentration. If we consider a spherical inclusion embedded in a homogeneous equivalent medium which has the same property as the inclusion (Fig. 4), a √ stress concentration factor of 3 should be applied to obtain the maximum cohesive traction Tc in a full field description (Table 1). The values of these material parameters identified from experiments were then used in full field finite element simulations, with appropriate boundary and loading conditions.
g
S is the modthe polycrystal with non-perfectly bonded grains, C is the elastic stiffness tensor of the ified Eshelby tensor, and damaged polycrystal. Detailed description of the derivation of this homogenization scheme is given in [9]. The maximum cohesive traction Tc , identified with mean field homogenization scheme, has a value averaged along the entire grain boundary, as assumed for the sake of simplicity in Ref. [9]. However, as a matter of fact, in a deformed polycrystal microstructure, the stress field along a grain boundary is not homogeneous
4. Results and discussion A 2D polycrystalline microstructure was generated using Voronoi tessellation. The microstructure was then extruded with a small thickness. Boundary conditions are applied so that the computation is performed under plane stress conditions. In this work, an REV of 44 grains is used. According to literature, this number of grains could be enough to faithfully describe the mechanical
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Table 1 √ Identified values of the maximum cohesive traction Tc in both mean field [9], and full field description (the latter with a stress concentration factor of 3 applied).
Tc mean field (MPa) Tc full field (MPa)
Fast (ε˙ = 2 × 10−3 s−1 ) at 200 ◦ C
Slow (ε˙ = 5 × 10−6 s−1 ) at 200 ◦ C
Very slow (ε˙ = 5 × 10−7 s−1 ) at 200 ◦ C
471 816
414 717
387 670
behavior of the material. The orientation of each grain is randomly assigned, i.e., there is no texture. No explicit cracks are introduced because the cohesive elements inserted along grain boundaries allow the boundaries to fail to form cracks. It is important to notice that the type of grain boundaries is not taken into account, as the same cohesive zone model was applied to all the grain boundaries. Numerical simulations are performed to simulate the intergranular cracking process. In this work, only the temperature of 473 K has been considered in association with different strain rates (Table 1), because under these conditions the fracture is dominated by intergranular cracking, as demonstrated by Fig. 2a and b. The results are discussed in order to analyze the occurrence at mesoscale of intergranular cracking due to dynamic embrittlement which was discussed in Section 2. 4.1. Tensile stress-strain curves The stress-strain curves obtained from the numerical simulations are presented in Fig. 5. To make the comparison easier, the curves obtained experimentally are also included. Basically, the numerical simulation curves are in good agreement with the experimental ones. The elongations to rupture of the simulation curves are quite consistent with the experimental results. This means that the numerical tools coupling the crystal plasticity model (which describes the behavior of the grains) and the cohesive zone model (which describes the fracture behavior of the grains boundaries) has a good capacity of prediction in terms of fracture. The stress levels reached for simulation curves are slightly lower than the stress levels of the experimental curves. One possible reason for this difference could come from the small number of grains (about 40 grains) in the REV or from the fact that the grains are not real tridimentional grains but only extruded ones. Regarding the final rupture behavior, the simulation curves are different from the experimental ones. As can be seen in Fig. 5, the final rupture is less brutal in the case of the simulated curves. Typically, after a stress drop of small amplitude, the stress decrease slows down progressively, while in the case of the experimental curves, the rupture is very brutal until the specimen is completely broken. This means that in the case of simulation, the crack propagation is not sufficiently rapid to reproduce the real final rupture which occurs in practice. This could be due to the fact that the diffusion effect of embrittling species is not accounted for in the present developed approach. However in reality for a specimen under tensile load, some grains boundaries can be weakened by the intergranular segregation of embrittling species, and this weakening phenomenon is reinforced by the stress concentration which occurs at crack tip. According to the literature, the diffusion process is also driven by the hydrostatic stress gradient, which means that the presence of stress concentration is able to accelerate the diffusion process and promotes very rapid crack propagation. Thus a final brutal rupture was observed in experiment. From this point of view, the stress-induced diffusion process of embrittling species should be incorporated in the numerical tool in order to consistently reproduce the final brutal rupture. 4.2. Intergranular cracking process Microscopically, stress concentration can be observed at grains boundaries and triple point junction of grain boundaries, which is
related to the heterogeneity of mechanical behavior as the grain orientation changes from one grain to another. Under the effect of intergranular embrittlement, crack initiates at grain boundaries or triple point junctions, and propagates along grain boundaries because the strength of the grain boundaries is weakened due to the presence of the embrittling elements such as impurities. A typical intergranular cracking process is presented in Fig. 6, which was obtained with a low strain rate. As shown in Fig. 6a, the crack initiation occurred at a triple point junction of grain boundaries. This is a phenomenon frequently observed because a triple point junction plays a role as a stress raiser. Under further loading, the first formed crack did not propagate, whereas a second crack initiated, as indicated in Fig. 6b. Then these two micro-cracks joined together to form a larger one (Fig. 6c). As the traction continued, the crack propagated in both two directions (Fig. 6d). Under further loading, only one side of the crack continued to propagate, while on the other side the crack ceased to propagate. This pause of crack propagation is due to the fact that the deviation angle is larger on this side than on the other side (Fig. 6e). After the crack reached one border, and under the effect of further loading, the crack propagated continuously until failure (Fig. 6f). Globally, the crack propagated along the grain boundaries which are more or less perpendicular to the loading direction. As a matter of fact, the grain boundaries perpendicular to the loading direction are more loaded than other grain boundaries in terms of opening. Consequently the damage developed more easily at these grain boundaries and promoted crack propagation. This observation is consistent with the experimental results revealed by cross sectional observation using optical microscopy, as shown in Fig. 2b. In the case of other strain rates, the cracking process including crack initiation and propagation seems to be the same as in the case of low strain rate. 4.3. Stress and plastic strain fields Crystal plasticity model is based on the shear stress induced plastic slip theory, and the cohesive zone model used in this work accounts for mixed-mode loading (normal and tangential, respectively). In this case, the presentation of shear stress field can be more interesting. However, it is not easy to show the local shear stress field in a global coordinate, as each grain has a different orientation and is associated to its proper local coordinate. Hence, in this paper, Von Mises stress field and equivalent plastic strain field are used to give a qualitative estimation of stress distribution. Fig. 7 shows the evolution of the Von Mises stress distribution in the REV and Fig. 8 shows the plastic strain distribution. In Fig. 7a, the stress rises rapidly not only in the grain boundary but also inside some of the crystals with “harder” elastic orientations. The stress is heterogeneously distributed in the entire area of the REV as seen in Fig. 7a. With further traction, some crystal lattices more and less tend to reorient, and the high stress area begin to coalesce. Gradually, macro shear bands with about 45 ◦ to the loading direction are formed, as show in Fig. 8a. This plastic shear flow penetrates through the grain interiors, which is due to the effect of the shear stress during the tensile loading. Some grain boundaries are subjected to severe normal and shear loading, resulting in the decrease of the strength and consequently the separation of the grain boundaries. Once a grain boundary is separated, on its two sides the stress is released, as indicated in Fig. 7b and c. Under further loading, the
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crack propagates and the region of stress release becomes more and more large (Fig. 7d and e). Finally, in more than half of the REV, the stress becomes very low, which is due to the formation of the main intergranular crack (Fig. 7f). As for the plastic strain fields, under further loading, the shear bands continue to develop (Fig. 8b–e). The plastic strain is more and more concentrated in several shear bands. Other complementary shear bands also appear when the loading level is high and it seems that the plastic strain tends to be generalized in the entire REV (Fig. 8f). 5. Concluding remarks and future work In this work, a numerical approach suitable for the study of intergranular cracking has been developed. The approach presented in this paper incorporates the following elements: (i) a crystal plasticity model to describe the mechanical behavior of each grain; (ii) the cohesive zone method to simulate the intergranular cracking at grain boundaries; (iii) a self-consistent homogenization scheme to identify the parameter from the experimental results. The results of numerical simulation have been presented in form of stressstrain curves and of stress and plastic strain fields. The intergranular cracking process has also been illustrated. These results of simulation are consistent with the experimental investigation, which suggests that this approach is efficient to simulate the mechanical behavior of the material and its intergranular cracking caused by dynamic embrittlement. The present approach offers good development perspectives for modeling intergranular fracture of polycrystals. For example, in order to better simulate the final brutal rupture observed in experiment, future work aims to incorporate the process of intergranular cracking using the cohesive zone model accounting for the sulfur concentration at grain boundaries. This final coupled micromechanics-diffusion-damage model should help to analyze and understand the role of the different parameters which are related to both experimental conditions and material features such as grain size, texture, etc.
References [1] H. Nathani, R.D.K. Misra, Characteristics of intermediate temperature dynamic embrittlement of age hardenable copper-chromium alloys, Mater. Sci. Technol. 20 (2004) 546–549. [2] Z. Sun, C. Laitem, A. Vincent, Dynamic embrittlement at intermediate temperature in a Cu–Ni–Si alloy, Mater. Sci. Eng. A 477 (2008) 145–152. [3] U. Krupp, C.J. McMahon Jr., Dynamic embrittlement-time-dependent brittle fracture, J. Alloy Compd. 378 (2004) 79–84. [4] U. Krupp, Dynamic embrittlement – time dependent quasi-brittle intergranular fracture at high temperatures, Int. Mater. Rev. 50 (2005) 83–97. [5] L. Priester, Joints de grains et plasticité crystalline, Lavoisier, Paris, 2011. [6] D. Bika, C.J. Mcmahon Jr., A kinetic model for dynamic embrittlement, Phys. Status Solidi A 131 (1992) 639–649. [7] R.C. Muthiah, J.A. Pfaendtner, S. Ishikawa, C.J. Mcmahon Jr., Tin-induced dynamic embrittlement of Cu–7%Sn bicrystals with a 5 boundary, Acta Mater. 47 (1999) 2797–2808. [8] T.D. Xu, A model for intergranular segregation/dilution induced by applied stress, J. Mater. Sci. 35 (2000) 5621–5628. [9] L. Benabou, Z. Sun, Homogenization scheme for brittle intergranular decohesion in polycrystalline aggregates, Mech. Res. Commun. 55 (2014) 114–119. [10] Z. Sun, Contribution à l’étude des propriétés mécaniques des alliages de cuivre à durcissement structural Cu–Ni–Si et Cu–Cr–Zr: influence de la microstructure et des conditions d’utilisaiton. PhD thesis, INSA de Lyon, 2008. [11] M. Yamaguchi, M. Shiga, H. Kaburaki, Grain boundary decohesion by impurity segregation in a nickel-sulfur system, Science 307 (2005) 393–397. [12] L. Méric, G. Cailletaud, M. Gaspérini, F.E. calculations of copper bicrystal specimens submitted to tension–compression tests, Acta Metall. 42 (1994) 921–935. [13] V. Tvergaard, J.W. Hutchinson, The relation between crack growth resistance and fracture process parameters in elastic-plastic solids, J. Mech. Phys. Solids 40 (1992) 1377–1397. [14] J.H. Rose, J. Ferrante, J.R. Smith, Universal binding energy curves for metals and bimetallic interfaces, Phys. Rev. Lett. 47 (1981) 675–678. [15] A. Van der Ven, G. Ceder, Impurity induced van der Waals transition during decohesion, Phys. Rev. B 67 (2003) 060101. [16] M. Berveiller, A. Zaoui, An extension of the self-consistent scheme to plastically flowing polycrystal, J. Mech. Phys. Solids 26 (1978) 325–344. [17] Y. Othmani, L. Delannay, I. Doghri, Equivalent inclusion solution adapted to particle debonding with a non-linear cohesive law, Int. J. Solids Struct. 48 (2011) 3326–3335.
Contents lists available at ScienceDirect
Mechanics Research Communications journal homepage: www.elsevier.com/locate/mechrescom
Numerical modeling and simulation of intergranular fracture due to dynamic embrittlement for a Cu Ni Si alloy Zhidan Sun a,∗ , Lahouari Benabou b , Hongqian Xue c a b c
ICD, P2 MN, LASMIS, University of Technology of Troyes, UMR 6281, CNRS, Troyes, France LISV, Université de Versailles-Saint-Quentin-en-Yvelines, 45 Avenue des Etats-Unis, 78000 Versailles, France School of Mechanical Engineering, Northwestern Polytechnical University, Xi’an 710072, China
a r t i c l e
i n f o
Article history: Received 13 December 2015 Received in revised form 21 April 2016 Accepted 8 June 2016 Available online 9 June 2016 Keywords: Dynamic embrittlement Intergranular cracking Crystal plasticity Cohesive zone method Homogenization
a b s t r a c t The involvement of the dynamic embrittlement phenomenon during tensile tests of a Cu Ni Si alloy leads to decohesion of grain boundaries, which is dependent on the energy decrease at grain boundaries caused by sulfur segregation. The severity of dynamic embrittlement increases with the increasing of temperature and/or the decreasing of straining rate. A crystal plasticity approach associated with the cohesive zone method is used in order to analyze the fracture caused by dynamic embrittlement. A cohesive zone model is used to describe the grain boundary cracking and one of its main parameters is identified using a homogenization scheme for the nonlinear response of a polycrystal in presence of degradable grain boundaries. Numerical simulations are carried out in order to simulate the mechanical behavior related to dynamic embrittlement under different conditions of strain rate and temperature. The obtained results such as tensile stress-strain curves and intergranular crack distribution are compared with experimental observations. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Materials for various electrical applications such as lead frame or squirrel cage of rotors in large electrical motors are required to have good mechanical strength and electrical conductivity as well. Some copper alloys of the precipitation-strengthened type have been used for such applications. However, some alloys among them, such as Cu Cr [1], and Cu Ni Si [2] have been found to suffer from dynamic embrittlement in the temperature range 400–800 K. The dynamic embrittlement is a time-dependent form of brittle intergranular fracture that occurs in many alloys when stress and elevated temperatures are applied [3,4]. This phenomenon is due to the reduction in intergranular cohesive strength resulting from stress-driven diffusion of embrittling element into grain boundaries. It is well known that sufficient quantities of embrittling element present at grain boundaries by diffusive segregation can change the fracture mode of copper alloys from ductile rupture to intergranular mode [5]. This is because the fracture energy is decreased by the presence of the segregated embrittling element. Some attempts have been made to model the dynamic embrtittlement mainly in the macro-scale [6–8]. In these analyses, the
∗ Corresponding author. E-mail address: [email protected] (Z. Sun). http://dx.doi.org/10.1016/j.mechrescom.2016.06.003 0093-6413/© 2016 Elsevier Ltd. All rights reserved.
microstructure features were generally not introduced in modeling. The use of the finite element method has enabled the integration of the grain morphology into the description of the mechanical behavior of a polycrystalline material. In this study, the investigation is aimed at computational procedure for numerical modeling of intergranular embrittlement process coupled with crystal plasticity approach. Such procedure will be useful in analyzing, for example, the sensitivity of crystallographic orientation/texture to dynamic embrittlement. Furthermore, by optimizing the texture, the severity of dynamic embrittlement can be reduced and the performance of components may be increased. With this specific goal of developing a numerical approach suitable for the study of dynamic embrittlement, the crystal plasticity formulation coupled to the cohesive zone method was implemented in the finite element code ABAQUS. The crystal plasticity model is used to describe the mechanical behavior of each grain as function of its crystallographic orientation and slip systems, while the cohesive zone model is used to simulate the intergranular cracking at grain boundaries. The parameters of the cohesive zone model, which are dependent on experimental conditions such as temperature and strain rate, are identified using a self-consistent homogenization scheme in which conditions of imperfect and damageable interface have been added [9]. The results of the numerical
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Fig. 3. Schematic representation of the contribution of various constituents (grains and grain boundaries) in the polycrystal and their combined effect on the overall response.
Fig. 1. Typical stress-strain curves obtained at RT and 473 K with high strain rate (ε˙ = 2 × 10−3 s−1 denoted by fast) and low strain rate (ε˙ = 5 × 10−6 s−1 denoted by slow).
simulations are compared with the experimental data to validate the numerical modeling approach. 2. Highlighting of dynamic embrittlement in Cu Ni Si alloy A commercial Cu Ni Si alloy was investigated, with the following chemical composition (in wt%): 2.16 Ni, 0.72 Si, 0.23 Zr, 0.02 Fe. The material was prepared in the form of bars according to industrial conditions. The tensile tests were carried out with dumbbell shape specimens under displacement control. The straining rates were in the range 5 × 10−6 s−1 to 2 × 10−3 s−1 . The tests were performed in air at room temperature (RT) and up to 473 K. Typical experimental stress-strain curves are presented in Fig. 1. It can be seen that the ductility, represented by elongation to rupture R , decreases when the temperature is increased and/or the ·
strain rate is decreased. For the high straining rate = 2 × 10−3 s−1 , the curve at RT exhibits a ductile behavior with R , and at 473 K the ductility is slightly reduced with Fig. 1 also demonstrates that decreasing the straining rate enhances strongly the reduction of ductility. εR is ε˙ = 5 × 10−6 s−1 reduced down to 3.9% for the low straining rate at All the fracture surfaces were examined using scanning electron microscopy (SEM) and optical microscopy in order to analyze cracking mechanisms. Under the condition of 473 K with ε˙ = 5 × 10−6 s−1 where the ductility is strongly reduced according to the stress-strain curve, the fracture mode appears clearly to be of intergranular type (Fig. 2a). This investigation was confirmed by
observing the longitudinal cut section. As shown in Fig. 2b, the damage seems appear in the form of intergranular cracking. Additional observation using MEB on cut section confirmed that the cracks shown in Fig. 2b are effectively of intergranular type. The experimental results shown above highlight that the investigated Cu Ni Si alloy exhibits an intergranular embrittlement which is typically time-dependent. This intergranular embrittlement is induced by stress-assisted diffusion of an embrittling element during the tensile test itself. This process is usually called “dynamic embrittlement”. The embrittlement process is more pronounced with increasing temperature and decreasing strain rate, which promotes the segregation of embrittling species and consequently the reduction of the cohesive strength of grain boundaries. Regarding the identity of the embrittling element, according to different tests and measures carried out in [10], the embrittling element in the studied Cu Ni Si alloy is very likely the sulfur. In the literature, sulfur has been identified as embrittling element in other metals both experimentally and theoretically. For example, the sulfur-induced embrittlement of nickel is clearly associated with sulfur segregation to grain boundaries and the transition from ductile to brittle behavior requires a critical intergranular concentration of sulfur [11]. 3. Modeling of brittle intergranular cracking As described in Section 2, the brittle fracture of the investigated Cu Ni Si alloy is caused by intergranular cracking due to the cohesive strength reduction of grain boundaries. In this case, the fracture process of grain boundaries can be simulated using the cohesive zone method (CZM). As a matter of fact, the decreasing of grain boundary strength is controlled by the fracture energy release rate, which is one of the important parameters of a cohesive zone model. In this work, an analytical approach is used to model the
Fig. 2. (a) Fracture surface, and (b) longitudinal cut section observations, after tensile test at 473 K with ε˙ = 5 × 10−6 s−1 .
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state variable ˛s , respectively. The resolved shear stress s is used as a critical variable to calculate the viscoplastic slip rate: ␥˙ s =
| s − s | − s n sign s − s k
(2)
where k and n are material constants characterizing the viscosity. The hardening variables, as well as the evolution of the state variables, are given by: s = 0 + bQ
·s
r
·s
hsr r with = 1 − bs
(3)
·s
s = c˛s with˛˙ s = ˙ s − d˛s Fig. 4. Self-consistent scheme applied to a polycrystalline aggregate with imperfect grain boundaries.
(4) ·s
where 0 represents the initial critical shear stress, = | ˙ s |, the parameters c and d characterize the kinematic hardening, while b, Q and the matrix hsr define the isotropic hardening. The plasfrom the contribution of all slip tic strain rate in grain g, resulting pg systems, is calculated with ε˙ =
˙ s ms . s 3.2. Cohesive zone model
Fig. 5. Comparison between the numerical simulation and experimental tensile stress-strain curves with different strain rates (from very slow to fast) at 473 K.
overall polycrystal behavior, while taking account of both crystal plasticity inside the grains and interface decohesion at the grain boundaries (Fig. 3). This approach allows identifying the parameters of the cohesive zone model. Accordingly, in this section, the crystal plasticity model and the cohesive zone model used in this work will be outlined and the parameter identification for these models will be briefly presented. It is important to mention that the current developed approach coupling crystal plasticity and cohesive zone method are suitable for simulating intergranular fracture. In the case where the fracture is not dominated by intergranular decohesion, the prediction capacity of this approach is limited. 3.1. Crystal plasticity model Single crystal plasticity is defined at the grain scale where slip occurs by dislocation motion along a slip system. For FCC polycrys12 octahedral slip systems are considered, based on 4 slip talline, planes 111 and 3 slip directions 110. According to the Schmid’s law, the resolved shear stress s for the system s in the grain g can be obtained by: s = g : ms withms =
1s l ⊗ ns + ns ⊗ ls 2
(1)
where ns and ls are respectively the normal to the plane of the slip system s and the slip direction in this plane. In this work, a viscoplastic model developed by Méric and Cailletaud [12] is used. In this phenomenological model, isotropic hardening s and kinematic hardening s are introduced, which are associated with the isotropic hardening state variable s and the kinematic hardening
The cracking process at grain boundary is modeled by means of the cohesive zone method, which was motivated by the observed intergranular cracking, as shown in Section 2. Cohesive elements are placed at each grain boundary in the synthetic polycrystal microstructure. The cohesive elements link two crystal grains on both sides, and are characterized by a constitutive law where the cohesive force is a function of the relative displacement between the two crystal surfaces. The fracture process is lumped into the crack line and the constitutive law, also called traction-separation law, which relates tractions T and displacement jumps across the cohesive surface. The traction-separation behavior can be taken in different forms (bilinear [13] or exponential [14], for example) to represent the grain boundary response. In what follows, an exponential Smith-Ferrante law, approximating well the separation of atomic planes due to cleavage, is employed. The cohesive energy density, represented by the area beneath the traction-separation curve, is the most important parameter since it measures the amount of energy needed to completely separate the bounded surfaces to create free surfaces. The cohesive zone model begins from a traction potential from which the coupled cohesive traction T is derived:
= eTc ıc 1 − 1 + ı/ıc e−ı/ıc T=
∂ ı = eTc e−ı/ıc ıc ∂ı
, (5)
where e is the e-number, Tc is the maximum cohesive traction, ıc is a characteristic opening displacement, and ı is the effective open
u2n + 2 u2t . The parameter ing displacement given by ı = represents the ratio of shear to normal opening relative displacement. The normal and shear tractions, Tn and Tt respectively, can be obtained in the same way by differentiating the potential by each component of ı. Hence, only three parameters are involved in the cohesive law, namely Tc , ıc , and. In our study, will be taken as 1, which means a same contribution of displacement jumps in mode I and mode II. The cohesive energy of the grain boundary, also called the critical energy release rate, can be obtained from these parameters as Gc = eTc ıc . The calculations of Van der Ven and Ceder [15] suggest that ıc is insensitive to impurity coverage and thus, can be considered, to a first approximation, as constant. It is taken as 8 × 10−5 mm for this study. In this case, Tc (consequently Gc ) is the only parameter to be identified for this cohesive zone model.
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Fig. 6. Crack initiation and propagation process during monotonic loading with low strain rate (ε˙ = 5 × 10−6 s−1 ).
3.3. Parameters identification The approach developed in this work couples a crystal plasticity model (responsible for the constitutive behavior) and a cohesive zone model (responsible for the fracture behavior). The determination of these two groups of material parameters was performed using a mean filed homogenization scheme. From the curves to rupture experimentally obtained, it is possible to fit the values of these two groups of parameters, respectively using a model with per-
fect interfaces (without fracture), and a model with an imperfect interface (with fracture). For the crystal plasticity model, the values of the parameters should be identified using all the experimental testing conditions, as some parameters are sensitive to strain rate which ranges from ε˙ = 5 × 10−6 s−1 to ε˙ = 2 × 10−3 s−1 , at 473 K. To do this, a homogenization scheme is used to obtain the effective behavior of a Representative Elementary Volume (REV) made of 40 grains, as presented in [9]. In the polycrystal, each grain is treated as an ellipsoidal inclusion embedded in an infinite medium made of
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Fig. 7. Von Mises stress distribution obtained under different loading levels corresponding to different crack lengths. Notice that once a crack is formed, on its two sides, the stress tends to be released.
all the remaining grains, also called the homogeneous equivalent medium (HEM). Perfect interface condition is first considered for the grain boundaries. For such a class of material, the self-consistent method can thus be used as a technique for determining the effective properties. The identification of these parameters was realized using the secant moduli method of Berveiller-Zaoui [16]. In this as homogenization model, the local stress in each grain is given function of the macroscopic fields of stress and plastic strain, and E p , by the following equation: g =
+ ˇ E p − εpg
(6)
The function ˇ, involving a secant plastic modulus, evolves with the total plastic deformation during loading.
Regarding the identification of the interface parameter Tc , it can be easily performed since the imperfect interface conditions were included in the mean field homogenization scheme, and thus will have an effect on the overall behavior which is fitted. To present briefly the insertion of these conditions, it should be recalled that failure of grain boundaries is reflected by discontinuity of the displacement field across the interface. A recent equivalent inclusion solution, developed by Othmani et al. [17], was incorporated in the polycrystal transition rule to describe debonding at grain boundaries. We use here the same approach that we adapt for a self-consistent scheme: an aggregate of Ng grains with imperfect interfaces between the grains is considered. As loading is applied, the grains may be weakened at their interfaces, which may affect the overall stress-strain behavior of the polycrystal. Accounting for
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Fig. 8. Equivalent plastic strain evolution during the tensile test of the REV. Macro shear bands are formed with about 45 ◦ as respect to the loading direction.
imperfect interface conditions, Eq. (6) thus becomes: g =
g −1 −1 eff eff + I + L : S −1 − I : C : L :
g S −1 − I : Ep − εpg
(7) eff
In this new transition rule, L
is the effective tangent moduli of
because there is always a stress concentration. If we consider a spherical inclusion embedded in a homogeneous equivalent medium which has the same property as the inclusion (Fig. 4), a √ stress concentration factor of 3 should be applied to obtain the maximum cohesive traction Tc in a full field description (Table 1). The values of these material parameters identified from experiments were then used in full field finite element simulations, with appropriate boundary and loading conditions.
g
S is the modthe polycrystal with non-perfectly bonded grains, C is the elastic stiffness tensor of the ified Eshelby tensor, and damaged polycrystal. Detailed description of the derivation of this homogenization scheme is given in [9]. The maximum cohesive traction Tc , identified with mean field homogenization scheme, has a value averaged along the entire grain boundary, as assumed for the sake of simplicity in Ref. [9]. However, as a matter of fact, in a deformed polycrystal microstructure, the stress field along a grain boundary is not homogeneous
4. Results and discussion A 2D polycrystalline microstructure was generated using Voronoi tessellation. The microstructure was then extruded with a small thickness. Boundary conditions are applied so that the computation is performed under plane stress conditions. In this work, an REV of 44 grains is used. According to literature, this number of grains could be enough to faithfully describe the mechanical
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Table 1 √ Identified values of the maximum cohesive traction Tc in both mean field [9], and full field description (the latter with a stress concentration factor of 3 applied).
Tc mean field (MPa) Tc full field (MPa)
Fast (ε˙ = 2 × 10−3 s−1 ) at 200 ◦ C
Slow (ε˙ = 5 × 10−6 s−1 ) at 200 ◦ C
Very slow (ε˙ = 5 × 10−7 s−1 ) at 200 ◦ C
471 816
414 717
387 670
behavior of the material. The orientation of each grain is randomly assigned, i.e., there is no texture. No explicit cracks are introduced because the cohesive elements inserted along grain boundaries allow the boundaries to fail to form cracks. It is important to notice that the type of grain boundaries is not taken into account, as the same cohesive zone model was applied to all the grain boundaries. Numerical simulations are performed to simulate the intergranular cracking process. In this work, only the temperature of 473 K has been considered in association with different strain rates (Table 1), because under these conditions the fracture is dominated by intergranular cracking, as demonstrated by Fig. 2a and b. The results are discussed in order to analyze the occurrence at mesoscale of intergranular cracking due to dynamic embrittlement which was discussed in Section 2. 4.1. Tensile stress-strain curves The stress-strain curves obtained from the numerical simulations are presented in Fig. 5. To make the comparison easier, the curves obtained experimentally are also included. Basically, the numerical simulation curves are in good agreement with the experimental ones. The elongations to rupture of the simulation curves are quite consistent with the experimental results. This means that the numerical tools coupling the crystal plasticity model (which describes the behavior of the grains) and the cohesive zone model (which describes the fracture behavior of the grains boundaries) has a good capacity of prediction in terms of fracture. The stress levels reached for simulation curves are slightly lower than the stress levels of the experimental curves. One possible reason for this difference could come from the small number of grains (about 40 grains) in the REV or from the fact that the grains are not real tridimentional grains but only extruded ones. Regarding the final rupture behavior, the simulation curves are different from the experimental ones. As can be seen in Fig. 5, the final rupture is less brutal in the case of the simulated curves. Typically, after a stress drop of small amplitude, the stress decrease slows down progressively, while in the case of the experimental curves, the rupture is very brutal until the specimen is completely broken. This means that in the case of simulation, the crack propagation is not sufficiently rapid to reproduce the real final rupture which occurs in practice. This could be due to the fact that the diffusion effect of embrittling species is not accounted for in the present developed approach. However in reality for a specimen under tensile load, some grains boundaries can be weakened by the intergranular segregation of embrittling species, and this weakening phenomenon is reinforced by the stress concentration which occurs at crack tip. According to the literature, the diffusion process is also driven by the hydrostatic stress gradient, which means that the presence of stress concentration is able to accelerate the diffusion process and promotes very rapid crack propagation. Thus a final brutal rupture was observed in experiment. From this point of view, the stress-induced diffusion process of embrittling species should be incorporated in the numerical tool in order to consistently reproduce the final brutal rupture. 4.2. Intergranular cracking process Microscopically, stress concentration can be observed at grains boundaries and triple point junction of grain boundaries, which is
related to the heterogeneity of mechanical behavior as the grain orientation changes from one grain to another. Under the effect of intergranular embrittlement, crack initiates at grain boundaries or triple point junctions, and propagates along grain boundaries because the strength of the grain boundaries is weakened due to the presence of the embrittling elements such as impurities. A typical intergranular cracking process is presented in Fig. 6, which was obtained with a low strain rate. As shown in Fig. 6a, the crack initiation occurred at a triple point junction of grain boundaries. This is a phenomenon frequently observed because a triple point junction plays a role as a stress raiser. Under further loading, the first formed crack did not propagate, whereas a second crack initiated, as indicated in Fig. 6b. Then these two micro-cracks joined together to form a larger one (Fig. 6c). As the traction continued, the crack propagated in both two directions (Fig. 6d). Under further loading, only one side of the crack continued to propagate, while on the other side the crack ceased to propagate. This pause of crack propagation is due to the fact that the deviation angle is larger on this side than on the other side (Fig. 6e). After the crack reached one border, and under the effect of further loading, the crack propagated continuously until failure (Fig. 6f). Globally, the crack propagated along the grain boundaries which are more or less perpendicular to the loading direction. As a matter of fact, the grain boundaries perpendicular to the loading direction are more loaded than other grain boundaries in terms of opening. Consequently the damage developed more easily at these grain boundaries and promoted crack propagation. This observation is consistent with the experimental results revealed by cross sectional observation using optical microscopy, as shown in Fig. 2b. In the case of other strain rates, the cracking process including crack initiation and propagation seems to be the same as in the case of low strain rate. 4.3. Stress and plastic strain fields Crystal plasticity model is based on the shear stress induced plastic slip theory, and the cohesive zone model used in this work accounts for mixed-mode loading (normal and tangential, respectively). In this case, the presentation of shear stress field can be more interesting. However, it is not easy to show the local shear stress field in a global coordinate, as each grain has a different orientation and is associated to its proper local coordinate. Hence, in this paper, Von Mises stress field and equivalent plastic strain field are used to give a qualitative estimation of stress distribution. Fig. 7 shows the evolution of the Von Mises stress distribution in the REV and Fig. 8 shows the plastic strain distribution. In Fig. 7a, the stress rises rapidly not only in the grain boundary but also inside some of the crystals with “harder” elastic orientations. The stress is heterogeneously distributed in the entire area of the REV as seen in Fig. 7a. With further traction, some crystal lattices more and less tend to reorient, and the high stress area begin to coalesce. Gradually, macro shear bands with about 45 ◦ to the loading direction are formed, as show in Fig. 8a. This plastic shear flow penetrates through the grain interiors, which is due to the effect of the shear stress during the tensile loading. Some grain boundaries are subjected to severe normal and shear loading, resulting in the decrease of the strength and consequently the separation of the grain boundaries. Once a grain boundary is separated, on its two sides the stress is released, as indicated in Fig. 7b and c. Under further loading, the
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crack propagates and the region of stress release becomes more and more large (Fig. 7d and e). Finally, in more than half of the REV, the stress becomes very low, which is due to the formation of the main intergranular crack (Fig. 7f). As for the plastic strain fields, under further loading, the shear bands continue to develop (Fig. 8b–e). The plastic strain is more and more concentrated in several shear bands. Other complementary shear bands also appear when the loading level is high and it seems that the plastic strain tends to be generalized in the entire REV (Fig. 8f). 5. Concluding remarks and future work In this work, a numerical approach suitable for the study of intergranular cracking has been developed. The approach presented in this paper incorporates the following elements: (i) a crystal plasticity model to describe the mechanical behavior of each grain; (ii) the cohesive zone method to simulate the intergranular cracking at grain boundaries; (iii) a self-consistent homogenization scheme to identify the parameter from the experimental results. The results of numerical simulation have been presented in form of stressstrain curves and of stress and plastic strain fields. The intergranular cracking process has also been illustrated. These results of simulation are consistent with the experimental investigation, which suggests that this approach is efficient to simulate the mechanical behavior of the material and its intergranular cracking caused by dynamic embrittlement. The present approach offers good development perspectives for modeling intergranular fracture of polycrystals. For example, in order to better simulate the final brutal rupture observed in experiment, future work aims to incorporate the process of intergranular cracking using the cohesive zone model accounting for the sulfur concentration at grain boundaries. This final coupled micromechanics-diffusion-damage model should help to analyze and understand the role of the different parameters which are related to both experimental conditions and material features such as grain size, texture, etc.
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