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Micromechanical simulation of fracture behavior of bimodal nanostructured metals
Materials Science & Engineering A 618 (2014) 479–489
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Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Micromechanical simulation of fracture behavior of bimodal nanostructured metals X. Guo a,b,n, R. Ji a, G.J. Weng c, L.L. Zhu d, J. Lu e a
School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China c Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA d Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, Zhejiang, China e Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2014 Received in revised form 31 August 2014 Accepted 5 September 2014 Available online 16 September 2014
Nanostructured (NS) metals with bimodal grain size distribution that consist of coarse grained (CG) and nano-grained (NG) regions have proved to have both high strength and good ductility. In this paper a numerical investigation, using the combination of a mechanism-based strain gradient plasticity theory, a micromechanics composite model, and the Johnson–Cook failure model, is conducted to investigate the effects of the distribution of the CG inclusions and their shape on fracture behavior of a bimodal NS copper. Load–response relations are employed to evaluate the loading history stability, while apparent crack length and strain energy history are used to analyze the fracture resistance. This study shows that both crack bridging in the CG inclusions and crack deflection in the NG matrix can significantly toughen the bimodal NS Cu. Our simulations also show that there exists a critical volume fraction of CG inclusions for some microstructures at which the fracture resistance of the bimodal NS Cu is at its minimal state and thus it should be avoided in material design. & 2014 Elsevier B.V. All rights reserved.
Keywords: Bimodal grain size distribution Mechanism-based strain gradient plasticity Micromechanics composite model Johnson–Cook failure model Fracture resistance
1. Introduction Nanocrystalline metals are known to have high yield stress but they generally lack good ductility. In recent years nanostructured (NS) metals with bimodal grain size distribution, by mixing up the length scales with nano-grained (NG) or ultrafine-grained (UFG) regions as the matrix phase and coarse grained (CG) regions as toughening inclusions, have shown high yield stress and large gain in work hardening and failure strain. The bimodal concept is now a viable way for new material processing and design aimed at both high strength and high ductility. Many experimental efforts have been devoted to study fracture behavior and toughening mechanisms of the bimodal NS metals. Compared with the strongest commercialized Al alloy, bimodal NS Al alloy had a 30% increase in strength and no loss in ductility due to the presence of the CG zone [1]. A bimodal NS Cu exhibited a failure strain of about 65% and yield stress that is several times of its CG counterpart [2]. As volume fraction of the CG inclusions increased, the ductility of bimodal NS Al alloy was found to increase due to crack bridging and the debonding between NG
n Corresponding author at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China. Tel.: þ 86 22 2740 4934; fax: þ 86 22 8740 1979. E-mail address: [email protected] (X. Guo).
http://dx.doi.org/10.1016/j.msea.2014.09.036 0921-5093/& 2014 Elsevier B.V. All rights reserved.
and CG regions [3]. In general bimodal grain size distribution can be produced by consolidating a simple mixture of powders of different grain sizes and, depending on the microstructural variations, a wide range of property improvement could be achieved [4–6]. Equally intriguing, NS Al sub-micrometer thin films were found to become ductile due to the bimodal grain distribution [7]. In bimodal UFG Al–Mg alloy the CG region was also found to contribute to their overall ductility [8]. The CG could bridge surfaces of cracks originating in the NG regions so that bimodal Ti samples tended to exhibit lower strength but enhanced ductility as compared to NG and UFG Ti [9,10]. By modifying grain size distribution in a multimodal grain structure in Ti, a combination of high yield stress and large uniform elongation has also been reported [11]. The CG bands could also arrest cracks by local blunting, resist crack propagation by bridging, and impede crack propagation by crack deflection, branching, and debonding [12]. The same mechanisms also worked in bimodal UFG Al–Mg alloy and it was found that increasing the volume fraction of the CG regions could increase the ductility with only a slightly-reduced strength [13]. Large localized plastic strain within the CG in the bimodal UFG Ni alloy could also lead to strain localization and debonding [14]. Even with these promising observations, it must be recognized that direct experimental measurement still has its intrinsic limitations. Specifically the exact grain size distribution, grain shape, and
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arrangement, which depend on the fabrication processes, are difficult to reproduce, and the overall mechanical responses hence become tricky to predict [4]. There is very limited theoretical modeling and numerical investigation on fracture behavior of bimodal NS metals. Among these limited studies, we have found that a unit-cell model was used in [15] to study the failure process in a bimodal NS Al alloy, but in the study only one set of yield stress and tensile strength of the NG was extracted from experiments. There was a lack of reliable constitutive relations for the NG phase. As a consequence systematic quantification of the important factors in the fracture behavior of the bimodal NS metals could not be carried out. In another study it was reported that there was initiation of nanocracks at the boundaries between the CG and NG regions, and indicated that good ductility came from the CGs in suppressing the crack growth and providing strain hardening [16]. A micromechanical model of dual-phase composite showed that smaller ductile particles favored not only uniform elongation but also the overall strength, resulting from the higher strain hardening ability of the CG inclusions [17]. But numerical investigations which could address the effects of inclusion distribution and its shape on the fracture behavior of bimodal NS metals still remain unavailable in the literature. This observation has motivated us to undertake this study. In this paper, the mechanism-based strain gradient plasticity theory will be adopted to describe the constitutive relation of the NG phase and the Johnson–Cook plasticity model will be employed to describe the constitutive relation of each phase at high strain rate. In addition, a micromechanics composite model will be incorporated into this bimodal framework. The dependence of fracture resistance on a series of important factors, including the NG grain size, and microstructure and volume fraction of the CG phase, will be fully analyzed. Our computational investigation will focus on how the toughening mechanisms work in such a diverse microstructure environment so that the key factors which could serve to improve its strength and ductility could be identified.
2. Problem statement and numerical framework 2.1. Specimen configuration and idealized microstructures A center-cracked tension bimodal NS Cu specimen is studied. It has a width 2.0 mm, a height 0.6 mm, and a pre-crack with a length 0.44 mm. Only one-half of the specimen is used due to symmetry, as shown in Fig. 1. The microstructure in a small region (60 300 μm2) in front of the pre-crack tip is taken into consideration. Fig. 1 also illustrates finite element discretization and a sampled idealized microstructure of the bimodal NS Cu, with the NG phase in green and the CG phase in red. The material outside the microstructure is assumed homogeneous, whose properties are determined by a micromechanics composite model in Section 2.3. Fine meshes are used to resolve an intense stress field and a detailed fracture process. Specifically, linear triangular elements [18] with each size in the order of 1 μm are used in the microstructure region. The number of nodes is about 235 k and that of triangular elements about 468 k. Note that the mesh size cannot resolve individual grain, grain boundary, and orientation of both the NG and the CG phases. Therefore, these two phases will be represented by homogenized constitutive relations, as described in Section 2.2 below. The morphological parameters of the actual bimodal NS Cu are influenced by the microstructures. To study the influences of phase attributes, six idealized microstructures are considered, as shown in Fig. 2. The volume fraction of the CG Cu inclusions is 24.5% on average, while the arrangement and shape vary. Microstructures A and C (Fig. 2a and c) consist of regular arrays of uniform regions
Fig. 1. Finite element discretization and a sampled microstructure of bimodal NS Cu. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
with square unit cells. Microstructures B and D (Fig. 2b and d) have a staggered region arrangement. The circular regions in microstructures A and B have a radius of 5 μm. The square regions in microstructures C and D have the same area with the circular ones in microstructures A and B, respectively. It should be recognized that square CG inclusions embedded in an NG phase are difficult to achieve experimentally, but they could serve as approximations to the irregular or faceted morphology of the CG phase. But particles with almost square shape have been reported for Bi2WO6 products [19]. Microstructures E and F have randomly-distributed unidirectional elliptical regions with half-major and half-minor axes 10 and 2.5 μm, respectively, resulting in an aspect ratio of 4. In microstructure E (Fig. 2e), the direction of the pre-crack path is aligned along the major axis, while in microstructure F it is perpendicular. It may be noted that random arrangement of the elliptical CG inclusions with aspect ratio 4 is consistent with experimental observations of some typical bimodal microstructures such as as-extruded Al–Mg alloy with CG 30% [12]. Elliptical inclusions have also been reported in other types of material systems [20]. Here, the specimen is initially stress free and at room temperature. Tensile load is applied by imposing symmetric velocity boundary conditions along the upper and lower edges of the specimen. The upper boundary velocity is 1 m/s, corresponding to the overall strain rate 3.3 103 s–1. Symmetric boundary condition is applied to the left edge since only half of the center-cracked specimen is used in the simulation, and the right edge is traction free. Condition of plain strain is assumed to prevail. 2.2. Constitutive relation of the NG phase The volume fraction of grain boundary increases significantly in the NG phase and its influence on the deformation needs to be
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Fig. 2. Six idealized microstructures.
considered. With the grain size decreasing, the number of geometrically necessary dislocations (GNDs) increases. GNDs are mainly stored in the regions near the grain boundaries and related to the strain gradient there. To account for this effect, the mechanismbased strain gradient plasticity [21] is adopted to describe the constitutive relation of the NG Cu. Note that a portion of GNDs is also stored in the boundaries between the NGs and CGs; this effect will not be considered in this paper. The strain rate tensor ε_ can be decomposed into elastic and plastic parts: ε_ ¼ ε_ e þ ε_ p
ð1Þ
The elastic strain rate ε_ e is obtained from the stress rate in a linear elastic fashion as _ ε_ e ¼ M : σ;
ð2Þ
where M is the elastic compliance tensor. According to the J2-plasticity flow rule, the plastic strain rate ε_ p is proportional to the deviatoric stress σ0 : ε_ p ¼
3_εpe 0 σ: 2σ e
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, σ 0ij ¼ σ ij σ kk δij =3 ði; j ¼ 1; 2; 3Þ, σ e ¼ 3σ 0ij σ 0ij =2 is von Mises' effective stress, and ε_ pe , the equivalent plastic strain rate, is determined by the power law σ e m0 ; ð4Þ ε_ pe ¼ ε_ e σ flow qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ε_ e ¼ 2_ε0ij ε_ 0ij =3 is the equivalent strain rate, ε_ 0ij ¼ ε_ ij ε_ kk δij =3, m0 ð 4 20Þ is the strain rate-sensitivity parameter, and σ flow the flow stress of the NG phase. Here the dislocation density in the grain boundary dislocation pile-up zones (GBDPZ), ρGB , is adopted to analyze the contribution of grain boundaries on the flow stress based on Taylor's model. It can be expressed by [22–25] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ σ flow ¼ σ 0 þ Mαμb ρI þ ρGB : Here, α, μ, and M are the Taylor constant, shear modulus, and Taylor factor, respectively, σ 0 is the lattice friction stress, ρI the dislocation density in the crystal interior of the NG, and b is the Burgers constant. ρI depends on the balance between accumulation and annihilation of dislocations by dynamic recovery [26–29]; its growth can be expressed as ∂ρI k pffiffiffiffi ¼ M þ k ρ ρ k ð6Þ 1 2 I ; I dG ∂εpe where εpe is the equivalent plastic strain, k ¼ 1=b, dG the NG grain 0 0 size, k1 ¼ ψ=b, ψ a proportionality factor, k2 ¼ k2 ð_εpe =_ε00 Þ 1=n0 , k2 a 0 constant, ε_ 0 a reference strain rate, and n0 inversely proportional
to temperature. Here, the first and second terms on the right of Eq. (6) are affiliated with the athermal storage of dislocations, and the third one is associated with the annihilation of dislocations during the dynamic process. It will become evident later that the NG grain size, dG , plays a very prominent role in the toughening and strengthening of bimodal NS metals. On the other hand ρGB is described by ρGB ¼ k
GB η
GB
b
;
ð7Þ
where k ¼ 6dGBDPZ = ϕGB dG , dGBDPZ is a thickness of the GBDPZ, GB ϕ a geometric factor, and ηGB the local strain gradient due to dislocations near the grain boundaries. We can obtain ρI by solving Eq. (6) numerically, and then substitute it and ρGB in Eq. (7) into Eq. (5) to obtain the flow stress of the NG phase. Early research showed that Young's modulus of NG crystals was lower than that of the corresponding CG crystals by 15–50% and it was attributed to the voids [30]. There are multiple theories to explain this decrease from view points of existence of voids, inaccurate measurement, and low sample density [31]. The experimental investigations in the past decade showed that Young's modulus of NG Cu was almost equal to or slightly lower than that of the corresponding CG Cu [32,33]. Our focus in this paper is not on elasticity so Young's modulus of the NG Cu is taken to be the same as that of the CG Cu, at 124 GPa. Poisson's ratio is taken to be 0.34. From the above two values, its shear modulus can be calculated as 46.3 GPa. By comparing the predictions of the above mechanism-based strain gradient plasticity theory with the experimental results of the NG with grain sizes 23, 74, and 200 nm [2,34], we obtained the strain gradient in the GBDPZ to be 0.04 nm–1, which is used throughout this paper. Descriptions, symbols, and values of the parameters in the strain gradient plasticity theory are tabulated in Table 1. The intrinsic length parameter is taken to be ðμ=σ y Þ2 b; where σ y is the yield stress [22,35]. In this paper, it is 1.23–3.76 μm for the NG Cu with different grain sizes. The constitutive relation of the CG phase can be directly obtained [18] and the constitutive parameters will be listed in Table 2 in Section 2.4. GB
2.3. Micromechanics composite model In the composite models of nanocrystalline metals, grain interiors and grain boundaries are treated as two separate phases [36]. Micromechanics-based composite models have been developed to investigate the grain-size hardening, the breakdown of Hall–Petch relation, as well as the competition between grain size and porosity, for both rate-independent and rate-dependent behaviors of nanocrystalline metals [29,37–42]. In this paper, the
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Table 1 Descriptions, symbols, and values of parameters in the mechanism-based strain gradient plasticity theory.
modulus and Poisson's ratio of the ith phase, respectively. The corresponding secant bulk and shear moduli of the ith phase satisfy the isotropic relations S
ki ¼
ESi ; 3ð1 2υSi Þ
μSi ¼
ESi : 2ð1 þυSi Þ
ð9Þ
Suppose that the composite is subject to a boundary displacement giving rise to a uniform strain ε. Based on the composite model developed by Weng [44], the relations between the hydrostatic and deviatoric strains of the constituent phases and those of the composite follow: ð0Þ εkk ¼ ðA0 þ A2 Þεkk ; ð1Þ εkk ¼ A0 εkk ;
εijð0Þ0 ¼ B2 ε0ij c1 B1 εpð1Þ ij ;
ð10Þ
pð1Þ 0 εð1Þ0 ij ¼ B0 εij þ c0 B1 εij ;
ð11Þ
and the mean stress components of the matrix phase and inclusions are given by h i pð1Þ S 0 ¼ 3κ 0 ðA0 þA2 Þεkk ; σ ð0Þ0 ; ð12Þ σ ð0Þ ij ¼ 2μ0 ðB0 þB2 Þεij c1 B2 εij kk ¼ 3κ 0 A1 εkk ; σ ð1Þ kk
σ ð1Þ0 ij ¼
2μS0 βS0
h i ; B2 ε0ij ð1 c0 βS0 Þεpð1Þ ij
ð13Þ
in which A0 ¼
κ0 ; c0 αS0 ðκ1 κ 0 Þ þ κ0
A1 ¼
κ1 ; c0 αS0 ðκ 1 κ 0 Þ þ κ 0
A2 ¼
αS0 ðκ 1 κ 0 Þ ; c0 αS0 ðκ 1 κ0 Þ þ κ 0
ð14Þ B0 ¼
μS0 ; S S c0 β0 ðμ1 μS0 Þ þ μS0
B1 ¼
βS0 μS1 ; S S c0 β0 ðμ1 μS0 Þ þ μS0
B2 ¼
βS0 ðμS1 μS0 Þ c0 βS0 ðμS1 μS0 Þ þ μS0
:
ð15Þ Here, ci is the volume fraction of the ith phase, and ki and μi are the bulk and shear moduli of the ith phase, respectively. αS0 and βS0 are the components of Eshelby tensor for spherical inclusions following SS0 ¼ ðαS0 ; βS0 Þ, in which αS0 ¼ ð1 þ υS0 Þ=3ð1 υS0 Þ and βS0 ¼ 2ð4 5υS0 Þ=15ð1 υS0 Þ. Therefore, the dilatational and the deviatoric stresses and strains of the composite are connected by ( ! ) c 1 B2 c 1 B1 σ kk ¼ 3κ 0 ½1 þ c1 ðA1 A0 Þεkk ; σ 0ij ¼ 2μS0 1 þ S ε0ij S εpð1Þ : β0 β0 ij
Table 2 Constitutive parameters of each phase in the bimodal NS Cu.
NG (23 nm)
NG (74 nm)
NG (200 nm)
CG
A (MPa)
669
527
382
90
B (MPa)
912
561
206
292
n
0.28
0.263
0.15
0.31
d1
0.13
0.0575
0.0319
0.54
ð16Þ
composite model based on the modified Mori–Tanaka mean field approach [24,43,44] is employed to estimate the constitutive relation of the homogenized phase. The composite theory with the secant modulus is suggested to describe the nonlinear behavior of individual phase. The flow stress obtained in Section 2.2 is used in the following derivation of the composite behavior. With the secant moduli approach, the secant Young's modulus and secant Poisson's ratio of the ith phase can be written as ESi ¼
σ ðiÞ εeðiÞ þ εpðiÞ
¼
Ei ; ðiÞ ðiÞ m0 1 1 þ ðEi εðiÞ =σ ðiÞ Þ½ðσ 11 =σ flow Þ flow
S 1 1 E υSi ¼ υi i : Ei 2 2
ð8Þ Here, NG or UFG phase is referred to as phase 0, the matrix phase, and CG as phase 1, the inclusions. Ei and υi denote Young's
As the applied load increases, the bimodal metal experiences three stages of deformation. In the first stage both phases are elastic, and thus, the superscript s can be deleted in every term. In our material system, during the second stage the CG phase yields first, while the NG phase still remains elastic. Finally, during the third stage both phases are in plastic state. With Eq. (16) and this observation the entire stress–strain curve of the composite, i.e., the homogenized phase, can be obtained. 2.4. The Johnson–Cook plasticity and failure models The Johnson–Cook plasticity model [45], a phenomenological model, is widely employed to describe the constitutive relation of metals and alloys at high strain rates. It describes a competition among thermal softening, strain hardening, and strain rate hardening as follows [45,46]: p
ε_ T Tr m σ e ¼ A þBðεpe Þn 1 þ C ln e 1 ðT r r T rT m Þ; ε_ 0 Tm Tr ð17Þ where A, B, C(¼ 0.0004 for NG and¼ 0.025 for CG), m(¼1.09), and n are model parameters, ε_ 0 a reference strain rate (taken as 1 s 1),
X. Guo et al. / Materials Science & Engineering A 618 (2014) 479–489
T the temperature, T r the room temperature (25 1C), and T m the melting temperature of Cu (1083 1C). The Johnson–Cook failure model [47] takes the effect of stress triaxiality, strain rate, and temperature on the failure strain into account. It establishes a linear incremental relation between the damage parameter D and the equivalent plastic strain increment dεpe as Z 1 d εpe ; ð18Þ D¼ εf where εf is the failure strain. Eq. (18) states that damage occurs when plastic deformation occurs and accumulates linearly with respect to d εpe . An element is taken to fail when D reaches 1. In Eq. (18), εf is defined by [47] p h i p ε_ T Tr 1 þ d5 ; ð19Þ εf ¼ d1 þ d2 ed3 σe 1 þ d4 ln e ε_ 0 Tm Tr where d1 to d5 are material constants, and p is the hydrostatic pressure, i.e., opposite of the trace of the stress tensor. In this paper, failed elements will be directly removed in both simulation and post-processing. With the failed elements removing, the precrack propagates in the microstructure. If loading state does not change too abruptly during deformation, d2 ¼ d3 ¼ 0 is considered to be a good approximation [48,49]. Here, d4 is taken as 0.014 and d5 as 1.12. Due to the high strain rate, the deformation is assumed adiabatic. The temperature increase ΔT due to the plastic dissipation can be written as Z χ ΔT ¼ σ d εpij ; ð20Þ ρcv ij where ρ ¼ 8900 kg=m3 is the density, cv ¼ 383 J=ðkg 1CÞ the specific heat [50], and χ ¼ 0:9 the ratio of plastic dissipation converted to heat [51–53]. Good agreement was found between predictions of the Johnson–Cook failure model and experimental results of metals under dynamic loading [51–53]. By fitting the constitutive relation of the NG phase obtained from the mechanism-based strain gradient plasticity theory and the constitutive relation of the CG inclusion in Section 2.2, we can obtain constitutive parameters of each phase in the bimodal NS Cu. They are listed in Table 2. Similarly, by fitting the constitutive relation of the composite obtained from the micromechanics composite model in Section 2.3, we can extract the constitutive parameters of the composite and use them for the homogenized phase. These are listed in Table 3. From Tables 2 and 3 Table 3 Constitutive parameters of the homogenized phase of the bimodal NS Cu with CG volume fraction 24.5% but different NG grain sizes.
Homogenized
Homogenized
Homogenized
phase with NG
phase with NG
phase with NG
grain size 23 nm
grain size 74 nm
grain size 200 nm
A (MPa)
467.8
398.2
350.4
B (MPa)
771.6
315.14
195.22
n
0.2705
0.2404
0.4011
d1
0.195
0.08625
0.0319
483
it can be found that both A (yield stress) and B decrease monotonically as NG grain size increases.
3. Results and discussion This micromechanics-based numerical scheme, which combines the mechanism-based strain gradient plasticity theory, the micromechanics composite model, and the Johnson–Cook failure model, will now be applied to study the fracture behavior of the bimodal NS Cu. In this process the six idealized microstructures in Fig. 2 and the constitutive parameters listed in Tables 2 and 3 are utilized. Crack path depends on material heterogeneity, i.e., the particular microstructural phase distribution. The crack stops at an interface of CG inclusion and NG matrix. Since CG phase has lower yield stress and larger failure strain, it can resist rapid propagation of the crack. The CG left at the wake plays a bridging role, consistent with the numerical investigation via the unit-cell model [15]. When there are favorably-oriented interfaces between the CG and NG regions, debonding occurs and may also contribute to fracture resistance [3,12]. Debonding and severe plastic deformation of CG inclusions can cause significant energy loss [3]. When debonding was perpendicular to the fracture plane, it might contribute to the ductility of bimodal specimen [12]. The crack usually alters propagation direction and deflects. It increases the real crack length, and thus, the crack takes a longer time to go through the entire microstructure. The experimental investigation on bimodal UFG Al–Mg alloy also showed that CG inclusions impeded crack propagation by crack deflection and debonding [12]. Stress gradient near the tip of a curved crack is smaller than that of a straight crack, the crack-tip region can accumulate more strain energy, and thus, a larger boundary load is required for the crack propagation [54]. For the bimodal NS Cu, both bridging of CG inclusions and crack deflection in NG matrix are responsible for enhancing fracture resistance. In most bridging cases, CG inclusions deform locally at stress concentrations and are in severe plastic deformation. During crack deflection, CG inclusions are not in severe plastic deformation since they remain almost in their original shape in the wake. 3.1. Effects of the microstructure and the NG grain size, dG Fig. 3 illustrates the boundary load versus the upper boundary displacement, with the NG grain sizes 23, 74, and 200 nm. For comparison, results of the same computational configuration with unimodal NG Cu are also illustrated. It can be found that with the NG size increasing, peak load decreases, which originates from the fact that strength of the NG and thus strength of the homogenized phase decrease. In all cases, the peak load of the bimodal NS Cu is lower than that of the unimodal NG Cu. After the peak, the load decreasing rate of the bimodal NS Cu is significantly smaller than that of the unimodal NG Cu, which implies that loading history of the former is more stable than that of the latter. Fig. 4 shows apparent crack length versus the upper boundary displacement. In the cases of 23 and 200 nm grain sizes, the crack in microstructure C has the highest propagation speed. In the case of 74 nm size, the crack in microstructure C has one of the highest propagation speeds. In all cases, the crack either in microstructure D or F has the lowest propagation speed. The lower the propagation speed is, the larger the fracture resistance is. Compared with other microstructures, bridging is dominant in microstructure F and crack deflection is dominant in microstructure D. Therefore, the two mechanisms can significantly improve fracture resistance. The propagation speed in the bimodal NS Cu is essentially lower
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Fig. 3. Load versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
than that in the unimodal NG Cu, which indicates that the CG phase in the bimodal NS Cu increases fracture resistance. Damage dissipation is important in quantifying fracture resistance [25,55–57]. Fig. 5 illustrates the damage dissipation versus the upper boundary displacement. It can be found that microstructure B has the largest damage dissipation rate (i.e., the damage dissipation over time) and that microstructure D the smallest one. The peak damage dissipation decreases substantially. Fracture process is closely related to the failure strain of NG phase. In the above three NG sizes, the descending of the failure strain coincides with that of the peak damage dissipation in Fig. 5. Fig. 6 shows strain energy versus the upper boundary displacement. When CG inclusions are in large deformation, more elastic energy will be accumulated. In all three cases (23, 74, and 200 nm), strain energy either in microstructure D or in microstructure F is the largest among six microstructures. This coincides with the facts that microstructure D or F is associated with the largest boundary load after the load peak (as illustrated in Fig. 3) and that the crack in microstructure D or F has the lowest propagation speed (as shown in Fig. 4). Therefore, similar with the apparent crack length, the strain energy is also an index to evaluate fracture resistance. Both external work and plastic dissipation can also be the output for further analysis. In each case of 23, 74, and 200 nm, the external work or the plastic dissipation for six microstructures almost coincides, which indicates that both of them are insensitive to the crack path, and thus, they are loosely related to the microstructures in the condition of the same volume fraction of CG inclusions. Both of their peak values decrease. The external work is converted into the plastic dissipation, damage dissipation,
and strain energy. The damage dissipation is much smaller than the plastic dissipation, i.e., most of the external work is consumed into plastic dissipation. 3.2. Group comparisons To compare fracture resistance of diverse microstructures, we make five groups: (i) A and B, (ii) C and D, (iii) E and F, (iv) A and C, and (v) B and D. In the first group, microstructure A has regular arrays of circular CG inclusions, while microstructure B has staggered arrangement of circular CG inclusions. From Fig. 4, it can be found that when the NG grain size is 23 nm, microstructure A has larger fracture resistance than microstructure B, while when the NG grain size is 74 and 200 nm, the former has smaller fracture resistance than the latter. Note that the volume fraction of the CG regions in microstructure B is slightly larger than that in microstructure A. When the NG has a smaller failure strain (e.g., 0.0575 and 0.0319 for the NG size 74 and 200 nm, respectively), volume fraction of the CG regions plays an important role in enhancing fracture resistance. Specifically, the microstructure with larger volume fraction of CG regions (i.e., microstructure B) behaves better than that with smaller volume fraction of CG regions (i.e., microstructure A). In the second group, microstructure C consists of regular arrays of square inclusions with square unit cells, while microstructure D has staggered square inclusions. Bridging occurs more often in microstructure C, while crack deflection more often in microstructure D, and thus, the CG inclusions in microstructure D deform moderately. It seems that fracture resistance of microstructure C is
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485
Fig. 4. Apparent crack length versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
larger than that of microstructure D. However, Fig. 4 shows that it is not the case whenever the NG grain sizes are 23, 74, and 200 nm. This can be explained from a view point of the real crack length. On observation of the fracture pattern, it can be found that the crack in microstructure C deflects with an angle about 451 to connect the centers of the CG inclusions, while the crack in microstructure D deflects with an angle larger than 451 to connect the vertices of the CG inclusions. Therefore, the real crack length in microstructure D is larger than that in microstructure C, i.e., staggered arrangement of square CG inclusions has larger fracture resistance than arrayed arrangement. In the third group, microstructure E consists of random unidirectional elliptical inclusions with their major axes aligned in the direction of the pre-crack path, while microstructure F perpendicular to it. Since elliptical inclusions in microstructure F have larger total projection length perpendicular to the direction of the pre-crack path than those in microstructure E, bridging occurs more often in microstructure F, while debonding more often in microstructure E. In Fig. 4 it can be found that fracture resistance of microstructure F is always much larger than that of microstructure E. It can also be observed that fracture resistance of microstructure E is even lower than that of the unimodal NG Cu when the NG grain size is 23 nm, and that it can moderately enhance fracture resistance when the NG grain sizes are 74 and 200 nm. Therefore, bimodal structure with microstructure E should not be used to enhance NG metals with moderate failure strain. In some particular situations, one can estimate potential microcrack nucleation sites and dominant propagation direction, based on which major axis direction of the elliptical inclusions can be chosen properly.
In the fourth group, it is important to investigate the loadcarrying capacity of circular inclusions in microstructure A and square inclusions in microstructure C. Fig. 7 illustrates fracture processes of CG inclusions in microstructures A and C. It can be found that the circular CG inclusions in microstructure A have severe plastic deformation before fracture and that the NG matrix also has large deformation. It can also be observed that the square CG inclusions in microstructure C only have larger deformation around their vertices, while the NG matrix has much smaller deformation. Large proportion of fracture in the circular CG inclusions is in mode-I, while that in the square CG inclusions is in mode-II. Therefore, when both of them are in bridging mode, load-carrying capacity of the circular inclusions is higher than that of square ones. It is preferable to use microstructure with circular inclusions to improve fracture resistance. In the fifth group, crack in microstructure B deflects together with the CG bridging. After the crack in microstructure D penetrates several CG inclusions, it deflects between two rows of CG inclusions, and thus, the CG inclusions play a less important role in bridging. Compared with microstructure B, the crack path in microstructure D is more tortuous. Specifically, the crack in microstructure B follows a zigzag path to connect the centers of CG inclusions, while that in microstructure D follows a trapezoidal path to connect vertices of the square CG inclusions. This is the intrinsic reason that crack in microstructure D has a longer real crack length, and thus, it has a larger fracture resistance. Note that the above comparison is only valid when crack deflection is more dominant than bridging. In the opposite case, i.e., when bridging plays a more important role, the microstructure with square inclusions will have smaller fracture resistance, as shown in the comparison in the fourth group.
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Fig. 5. Damage dissipation versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
3.3. Effect of volume fraction of the CG inclusions The dependence of fracture resistance on volume fraction of CG inclusions is an important issue for bimodal NS metals as it provides important information for toughening. To investigate this issue, we use microstructures having the same shape but different characteristic lengths of CG inclusions. Specifically, five radii, namely 2, 3, 4, 5, and 6 μm, are chosen for circular CG inclusions in microstructure B. First, the NG grain size is chosen as 23 nm. With the radius of CG inclusions increasing, their volume fraction increases and the constitutive relation of the corresponding homogenized phase also changes. Table 4 lists constitutive parameters of the homogenized phase, obtained by the micromechanics composite model with consideration of different volume fractions of CG inclusions in Section 2.3. With the radius increasing, A (yield stress) decreases monotonically (as shown in Table 4), and this in turn leads to the decrease of boundary load, external work, plastic dissipation, and strain energy versus the upper boundary displacement. Figs. 8 and 9 illustrate apparent crack length and damage dissipation versus the upper boundary displacement for the cases of five radii. With the radius increasing at each boundary displacement, both the apparent crack length and the damage dissipation first increase slowly and then decrease quickly, implying minimum fracture resistance and maximum damage dissipation. The critical radius is 4 μm for microstructure B and the corresponding critical volume fraction of CG inclusions is 15.68%. Similar phenomenon also occurs in the cases of 74 and 200 nm of the NG size, while the critical radius changes to 3 μm. This can be explained as follows. On observation of fracture pattern, when the
radius is smaller than the critical one, bridging does not work, the crack bypasses the CG inclusions with ease, and only crack deflection is effective. Therefore, the real crack length and fracture resistance decrease as the radius increases. When the radius is larger than the critical one, bridging plays an important role, and thus, fracture resistance increases with increasing radius. Note that if the main toughening mechanism does not change when the characteristic length of CG inclusions varies, the critical characteristic length and volume fraction will not exist – this is the case, for instance, in microstructure D.
4. Conclusions In this paper a numerical investigation via the mechanismbased strain gradient plasticity theory, the micromechanics composite model, and the Johnson–Cook failure model has been carried out to study fracture behavior and toughening mechanisms of bimodal NS metals. The boundary load, the apparent crack length, the damage dissipation, and the strain energy have been analyzed to evaluate the stability of loading history and the fracture resistance. It is found that both bridging in CG inclusions and crack deflection in NG matrix can have strong effects in enhancing fracture resistance of bimodal NS metals. Main conclusions can be drawn as follows: 1. Loading history of the bimodal NS metal is more stable than that of the unimodal NG metal since the load decreasing rate of the bimodal NS metal is significantly slower than that of the unimodal NG metal.
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Fig. 6. Strain energy versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
Fig. 7. Fracture processes of CG regions in (a) microstructure A and (b) microstructure C.
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Table 4 Constitutive parameters of the homogenized phase of the bimodal NS Cu with NG grain size of 23 nm but different CG radii.
metal at its minimal state. Therefore this critical volume fraction should be avoided in material design. 4. The staggered arrangement of square CG inclusions has larger fracture resistance than the arrayed arrangement. 5. In some particular situations, one needs to estimate potential microcrack nucleation sites and dominant propagation direction, based on which major axis direction of elliptical inclusions can be chosen properly for improved toughness. This study provides the needed insights into the design of bimodal microstructure for superior fracture resistance in highstrength NS metals.
Acknowledgments Fig. 8. Apparent crack length versus boundary displacement for the cases of five radii.
We wish to thank an anonymous reviewer for his/her helpful comments. X. Guo acknowledges the support from the National Key Basic Research Program (Grant 2012CB932203), the National Natural Science Foundation of China (Project nos. 11102128 and 11372214), and the Elite Scholar Program of Tianjin University. G.J. Weng thanks the support of NSF Mechanics of Materials Program under CMMI-1162431. L.L. Zhu acknowledges the support from National Natural Science Foundation of China (Project nos. 11302189 and 11472243), Doctoral Fund of Ministry of Education of China (Grant no. 20130101120175), Zhejiang Provincial Qianjiang Talent Program (Grant no. QJD1202012), and Educational Commission of Zhejiang Province of China (Grant no. Y201223476). J. Lu acknowledges the supports of GRF/CityU519110 and the Croucher Foundation CityU9500006. References
Fig. 9. Damage dissipation versus boundary displacement for the cases of five radii.
2. When bridging plays an important role, the microstructure with circular inclusions has more severe plastic deformation. This offers larger fracture resistance than the square ones. When crack deflection is dominant, the microstructure with square inclusions has a longer real crack length and thus larger fracture resistance as compared to the circular counterpart. 3. When dominant toughening mechanism changes from crack deflection to bridging, a critical volume fraction of CG inclusions exists that makes fracture resistance of the bimodal NS
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Contents lists available at ScienceDirect
Materials Science & Engineering A journal homepage: www.elsevier.com/locate/msea
Micromechanical simulation of fracture behavior of bimodal nanostructured metals X. Guo a,b,n, R. Ji a, G.J. Weng c, L.L. Zhu d, J. Lu e a
School of Mechanical Engineering, Tianjin University, Tianjin 300072, China Tianjin Key Laboratory of Nonlinear Dynamics and Chaos Control, Tianjin 300072, China c Department of Mechanical and Aerospace Engineering, Rutgers University, New Brunswick, NJ 08903, USA d Department of Engineering Mechanics, School of Aeronautics and Astronautics, Zhejiang University, Hangzhou 310027, Zhejiang, China e Department of Mechanical and Biomedical Engineering, City University of Hong Kong, Hong Kong b
art ic l e i nf o
a b s t r a c t
Article history: Received 9 April 2014 Received in revised form 31 August 2014 Accepted 5 September 2014 Available online 16 September 2014
Nanostructured (NS) metals with bimodal grain size distribution that consist of coarse grained (CG) and nano-grained (NG) regions have proved to have both high strength and good ductility. In this paper a numerical investigation, using the combination of a mechanism-based strain gradient plasticity theory, a micromechanics composite model, and the Johnson–Cook failure model, is conducted to investigate the effects of the distribution of the CG inclusions and their shape on fracture behavior of a bimodal NS copper. Load–response relations are employed to evaluate the loading history stability, while apparent crack length and strain energy history are used to analyze the fracture resistance. This study shows that both crack bridging in the CG inclusions and crack deflection in the NG matrix can significantly toughen the bimodal NS Cu. Our simulations also show that there exists a critical volume fraction of CG inclusions for some microstructures at which the fracture resistance of the bimodal NS Cu is at its minimal state and thus it should be avoided in material design. & 2014 Elsevier B.V. All rights reserved.
Keywords: Bimodal grain size distribution Mechanism-based strain gradient plasticity Micromechanics composite model Johnson–Cook failure model Fracture resistance
1. Introduction Nanocrystalline metals are known to have high yield stress but they generally lack good ductility. In recent years nanostructured (NS) metals with bimodal grain size distribution, by mixing up the length scales with nano-grained (NG) or ultrafine-grained (UFG) regions as the matrix phase and coarse grained (CG) regions as toughening inclusions, have shown high yield stress and large gain in work hardening and failure strain. The bimodal concept is now a viable way for new material processing and design aimed at both high strength and high ductility. Many experimental efforts have been devoted to study fracture behavior and toughening mechanisms of the bimodal NS metals. Compared with the strongest commercialized Al alloy, bimodal NS Al alloy had a 30% increase in strength and no loss in ductility due to the presence of the CG zone [1]. A bimodal NS Cu exhibited a failure strain of about 65% and yield stress that is several times of its CG counterpart [2]. As volume fraction of the CG inclusions increased, the ductility of bimodal NS Al alloy was found to increase due to crack bridging and the debonding between NG
n Corresponding author at: School of Mechanical Engineering, Tianjin University, Tianjin 300072, China. Tel.: þ 86 22 2740 4934; fax: þ 86 22 8740 1979. E-mail address: [email protected] (X. Guo).
http://dx.doi.org/10.1016/j.msea.2014.09.036 0921-5093/& 2014 Elsevier B.V. All rights reserved.
and CG regions [3]. In general bimodal grain size distribution can be produced by consolidating a simple mixture of powders of different grain sizes and, depending on the microstructural variations, a wide range of property improvement could be achieved [4–6]. Equally intriguing, NS Al sub-micrometer thin films were found to become ductile due to the bimodal grain distribution [7]. In bimodal UFG Al–Mg alloy the CG region was also found to contribute to their overall ductility [8]. The CG could bridge surfaces of cracks originating in the NG regions so that bimodal Ti samples tended to exhibit lower strength but enhanced ductility as compared to NG and UFG Ti [9,10]. By modifying grain size distribution in a multimodal grain structure in Ti, a combination of high yield stress and large uniform elongation has also been reported [11]. The CG bands could also arrest cracks by local blunting, resist crack propagation by bridging, and impede crack propagation by crack deflection, branching, and debonding [12]. The same mechanisms also worked in bimodal UFG Al–Mg alloy and it was found that increasing the volume fraction of the CG regions could increase the ductility with only a slightly-reduced strength [13]. Large localized plastic strain within the CG in the bimodal UFG Ni alloy could also lead to strain localization and debonding [14]. Even with these promising observations, it must be recognized that direct experimental measurement still has its intrinsic limitations. Specifically the exact grain size distribution, grain shape, and
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arrangement, which depend on the fabrication processes, are difficult to reproduce, and the overall mechanical responses hence become tricky to predict [4]. There is very limited theoretical modeling and numerical investigation on fracture behavior of bimodal NS metals. Among these limited studies, we have found that a unit-cell model was used in [15] to study the failure process in a bimodal NS Al alloy, but in the study only one set of yield stress and tensile strength of the NG was extracted from experiments. There was a lack of reliable constitutive relations for the NG phase. As a consequence systematic quantification of the important factors in the fracture behavior of the bimodal NS metals could not be carried out. In another study it was reported that there was initiation of nanocracks at the boundaries between the CG and NG regions, and indicated that good ductility came from the CGs in suppressing the crack growth and providing strain hardening [16]. A micromechanical model of dual-phase composite showed that smaller ductile particles favored not only uniform elongation but also the overall strength, resulting from the higher strain hardening ability of the CG inclusions [17]. But numerical investigations which could address the effects of inclusion distribution and its shape on the fracture behavior of bimodal NS metals still remain unavailable in the literature. This observation has motivated us to undertake this study. In this paper, the mechanism-based strain gradient plasticity theory will be adopted to describe the constitutive relation of the NG phase and the Johnson–Cook plasticity model will be employed to describe the constitutive relation of each phase at high strain rate. In addition, a micromechanics composite model will be incorporated into this bimodal framework. The dependence of fracture resistance on a series of important factors, including the NG grain size, and microstructure and volume fraction of the CG phase, will be fully analyzed. Our computational investigation will focus on how the toughening mechanisms work in such a diverse microstructure environment so that the key factors which could serve to improve its strength and ductility could be identified.
2. Problem statement and numerical framework 2.1. Specimen configuration and idealized microstructures A center-cracked tension bimodal NS Cu specimen is studied. It has a width 2.0 mm, a height 0.6 mm, and a pre-crack with a length 0.44 mm. Only one-half of the specimen is used due to symmetry, as shown in Fig. 1. The microstructure in a small region (60 300 μm2) in front of the pre-crack tip is taken into consideration. Fig. 1 also illustrates finite element discretization and a sampled idealized microstructure of the bimodal NS Cu, with the NG phase in green and the CG phase in red. The material outside the microstructure is assumed homogeneous, whose properties are determined by a micromechanics composite model in Section 2.3. Fine meshes are used to resolve an intense stress field and a detailed fracture process. Specifically, linear triangular elements [18] with each size in the order of 1 μm are used in the microstructure region. The number of nodes is about 235 k and that of triangular elements about 468 k. Note that the mesh size cannot resolve individual grain, grain boundary, and orientation of both the NG and the CG phases. Therefore, these two phases will be represented by homogenized constitutive relations, as described in Section 2.2 below. The morphological parameters of the actual bimodal NS Cu are influenced by the microstructures. To study the influences of phase attributes, six idealized microstructures are considered, as shown in Fig. 2. The volume fraction of the CG Cu inclusions is 24.5% on average, while the arrangement and shape vary. Microstructures A and C (Fig. 2a and c) consist of regular arrays of uniform regions
Fig. 1. Finite element discretization and a sampled microstructure of bimodal NS Cu. (For interpretation of the references to colour in this figure, the reader is referred to the web version of this article.)
with square unit cells. Microstructures B and D (Fig. 2b and d) have a staggered region arrangement. The circular regions in microstructures A and B have a radius of 5 μm. The square regions in microstructures C and D have the same area with the circular ones in microstructures A and B, respectively. It should be recognized that square CG inclusions embedded in an NG phase are difficult to achieve experimentally, but they could serve as approximations to the irregular or faceted morphology of the CG phase. But particles with almost square shape have been reported for Bi2WO6 products [19]. Microstructures E and F have randomly-distributed unidirectional elliptical regions with half-major and half-minor axes 10 and 2.5 μm, respectively, resulting in an aspect ratio of 4. In microstructure E (Fig. 2e), the direction of the pre-crack path is aligned along the major axis, while in microstructure F it is perpendicular. It may be noted that random arrangement of the elliptical CG inclusions with aspect ratio 4 is consistent with experimental observations of some typical bimodal microstructures such as as-extruded Al–Mg alloy with CG 30% [12]. Elliptical inclusions have also been reported in other types of material systems [20]. Here, the specimen is initially stress free and at room temperature. Tensile load is applied by imposing symmetric velocity boundary conditions along the upper and lower edges of the specimen. The upper boundary velocity is 1 m/s, corresponding to the overall strain rate 3.3 103 s–1. Symmetric boundary condition is applied to the left edge since only half of the center-cracked specimen is used in the simulation, and the right edge is traction free. Condition of plain strain is assumed to prevail. 2.2. Constitutive relation of the NG phase The volume fraction of grain boundary increases significantly in the NG phase and its influence on the deformation needs to be
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481
Fig. 2. Six idealized microstructures.
considered. With the grain size decreasing, the number of geometrically necessary dislocations (GNDs) increases. GNDs are mainly stored in the regions near the grain boundaries and related to the strain gradient there. To account for this effect, the mechanismbased strain gradient plasticity [21] is adopted to describe the constitutive relation of the NG Cu. Note that a portion of GNDs is also stored in the boundaries between the NGs and CGs; this effect will not be considered in this paper. The strain rate tensor ε_ can be decomposed into elastic and plastic parts: ε_ ¼ ε_ e þ ε_ p
ð1Þ
The elastic strain rate ε_ e is obtained from the stress rate in a linear elastic fashion as _ ε_ e ¼ M : σ;
ð2Þ
where M is the elastic compliance tensor. According to the J2-plasticity flow rule, the plastic strain rate ε_ p is proportional to the deviatoric stress σ0 : ε_ p ¼
3_εpe 0 σ: 2σ e
ð3Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Here, σ 0ij ¼ σ ij σ kk δij =3 ði; j ¼ 1; 2; 3Þ, σ e ¼ 3σ 0ij σ 0ij =2 is von Mises' effective stress, and ε_ pe , the equivalent plastic strain rate, is determined by the power law σ e m0 ; ð4Þ ε_ pe ¼ ε_ e σ flow qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where ε_ e ¼ 2_ε0ij ε_ 0ij =3 is the equivalent strain rate, ε_ 0ij ¼ ε_ ij ε_ kk δij =3, m0 ð 4 20Þ is the strain rate-sensitivity parameter, and σ flow the flow stress of the NG phase. Here the dislocation density in the grain boundary dislocation pile-up zones (GBDPZ), ρGB , is adopted to analyze the contribution of grain boundaries on the flow stress based on Taylor's model. It can be expressed by [22–25] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð5Þ σ flow ¼ σ 0 þ Mαμb ρI þ ρGB : Here, α, μ, and M are the Taylor constant, shear modulus, and Taylor factor, respectively, σ 0 is the lattice friction stress, ρI the dislocation density in the crystal interior of the NG, and b is the Burgers constant. ρI depends on the balance between accumulation and annihilation of dislocations by dynamic recovery [26–29]; its growth can be expressed as ∂ρI k pffiffiffiffi ¼ M þ k ρ ρ k ð6Þ 1 2 I ; I dG ∂εpe where εpe is the equivalent plastic strain, k ¼ 1=b, dG the NG grain 0 0 size, k1 ¼ ψ=b, ψ a proportionality factor, k2 ¼ k2 ð_εpe =_ε00 Þ 1=n0 , k2 a 0 constant, ε_ 0 a reference strain rate, and n0 inversely proportional
to temperature. Here, the first and second terms on the right of Eq. (6) are affiliated with the athermal storage of dislocations, and the third one is associated with the annihilation of dislocations during the dynamic process. It will become evident later that the NG grain size, dG , plays a very prominent role in the toughening and strengthening of bimodal NS metals. On the other hand ρGB is described by ρGB ¼ k
GB η
GB
b
;
ð7Þ
where k ¼ 6dGBDPZ = ϕGB dG , dGBDPZ is a thickness of the GBDPZ, GB ϕ a geometric factor, and ηGB the local strain gradient due to dislocations near the grain boundaries. We can obtain ρI by solving Eq. (6) numerically, and then substitute it and ρGB in Eq. (7) into Eq. (5) to obtain the flow stress of the NG phase. Early research showed that Young's modulus of NG crystals was lower than that of the corresponding CG crystals by 15–50% and it was attributed to the voids [30]. There are multiple theories to explain this decrease from view points of existence of voids, inaccurate measurement, and low sample density [31]. The experimental investigations in the past decade showed that Young's modulus of NG Cu was almost equal to or slightly lower than that of the corresponding CG Cu [32,33]. Our focus in this paper is not on elasticity so Young's modulus of the NG Cu is taken to be the same as that of the CG Cu, at 124 GPa. Poisson's ratio is taken to be 0.34. From the above two values, its shear modulus can be calculated as 46.3 GPa. By comparing the predictions of the above mechanism-based strain gradient plasticity theory with the experimental results of the NG with grain sizes 23, 74, and 200 nm [2,34], we obtained the strain gradient in the GBDPZ to be 0.04 nm–1, which is used throughout this paper. Descriptions, symbols, and values of the parameters in the strain gradient plasticity theory are tabulated in Table 1. The intrinsic length parameter is taken to be ðμ=σ y Þ2 b; where σ y is the yield stress [22,35]. In this paper, it is 1.23–3.76 μm for the NG Cu with different grain sizes. The constitutive relation of the CG phase can be directly obtained [18] and the constitutive parameters will be listed in Table 2 in Section 2.4. GB
2.3. Micromechanics composite model In the composite models of nanocrystalline metals, grain interiors and grain boundaries are treated as two separate phases [36]. Micromechanics-based composite models have been developed to investigate the grain-size hardening, the breakdown of Hall–Petch relation, as well as the competition between grain size and porosity, for both rate-independent and rate-dependent behaviors of nanocrystalline metals [29,37–42]. In this paper, the
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Table 1 Descriptions, symbols, and values of parameters in the mechanism-based strain gradient plasticity theory.
modulus and Poisson's ratio of the ith phase, respectively. The corresponding secant bulk and shear moduli of the ith phase satisfy the isotropic relations S
ki ¼
ESi ; 3ð1 2υSi Þ
μSi ¼
ESi : 2ð1 þυSi Þ
ð9Þ
Suppose that the composite is subject to a boundary displacement giving rise to a uniform strain ε. Based on the composite model developed by Weng [44], the relations between the hydrostatic and deviatoric strains of the constituent phases and those of the composite follow: ð0Þ εkk ¼ ðA0 þ A2 Þεkk ; ð1Þ εkk ¼ A0 εkk ;
εijð0Þ0 ¼ B2 ε0ij c1 B1 εpð1Þ ij ;
ð10Þ
pð1Þ 0 εð1Þ0 ij ¼ B0 εij þ c0 B1 εij ;
ð11Þ
and the mean stress components of the matrix phase and inclusions are given by h i pð1Þ S 0 ¼ 3κ 0 ðA0 þA2 Þεkk ; σ ð0Þ0 ; ð12Þ σ ð0Þ ij ¼ 2μ0 ðB0 þB2 Þεij c1 B2 εij kk ¼ 3κ 0 A1 εkk ; σ ð1Þ kk
σ ð1Þ0 ij ¼
2μS0 βS0
h i ; B2 ε0ij ð1 c0 βS0 Þεpð1Þ ij
ð13Þ
in which A0 ¼
κ0 ; c0 αS0 ðκ1 κ 0 Þ þ κ0
A1 ¼
κ1 ; c0 αS0 ðκ 1 κ 0 Þ þ κ 0
A2 ¼
αS0 ðκ 1 κ 0 Þ ; c0 αS0 ðκ 1 κ0 Þ þ κ 0
ð14Þ B0 ¼
μS0 ; S S c0 β0 ðμ1 μS0 Þ þ μS0
B1 ¼
βS0 μS1 ; S S c0 β0 ðμ1 μS0 Þ þ μS0
B2 ¼
βS0 ðμS1 μS0 Þ c0 βS0 ðμS1 μS0 Þ þ μS0
:
ð15Þ Here, ci is the volume fraction of the ith phase, and ki and μi are the bulk and shear moduli of the ith phase, respectively. αS0 and βS0 are the components of Eshelby tensor for spherical inclusions following SS0 ¼ ðαS0 ; βS0 Þ, in which αS0 ¼ ð1 þ υS0 Þ=3ð1 υS0 Þ and βS0 ¼ 2ð4 5υS0 Þ=15ð1 υS0 Þ. Therefore, the dilatational and the deviatoric stresses and strains of the composite are connected by ( ! ) c 1 B2 c 1 B1 σ kk ¼ 3κ 0 ½1 þ c1 ðA1 A0 Þεkk ; σ 0ij ¼ 2μS0 1 þ S ε0ij S εpð1Þ : β0 β0 ij
Table 2 Constitutive parameters of each phase in the bimodal NS Cu.
NG (23 nm)
NG (74 nm)
NG (200 nm)
CG
A (MPa)
669
527
382
90
B (MPa)
912
561
206
292
n
0.28
0.263
0.15
0.31
d1
0.13
0.0575
0.0319
0.54
ð16Þ
composite model based on the modified Mori–Tanaka mean field approach [24,43,44] is employed to estimate the constitutive relation of the homogenized phase. The composite theory with the secant modulus is suggested to describe the nonlinear behavior of individual phase. The flow stress obtained in Section 2.2 is used in the following derivation of the composite behavior. With the secant moduli approach, the secant Young's modulus and secant Poisson's ratio of the ith phase can be written as ESi ¼
σ ðiÞ εeðiÞ þ εpðiÞ
¼
Ei ; ðiÞ ðiÞ m0 1 1 þ ðEi εðiÞ =σ ðiÞ Þ½ðσ 11 =σ flow Þ flow
S 1 1 E υSi ¼ υi i : Ei 2 2
ð8Þ Here, NG or UFG phase is referred to as phase 0, the matrix phase, and CG as phase 1, the inclusions. Ei and υi denote Young's
As the applied load increases, the bimodal metal experiences three stages of deformation. In the first stage both phases are elastic, and thus, the superscript s can be deleted in every term. In our material system, during the second stage the CG phase yields first, while the NG phase still remains elastic. Finally, during the third stage both phases are in plastic state. With Eq. (16) and this observation the entire stress–strain curve of the composite, i.e., the homogenized phase, can be obtained. 2.4. The Johnson–Cook plasticity and failure models The Johnson–Cook plasticity model [45], a phenomenological model, is widely employed to describe the constitutive relation of metals and alloys at high strain rates. It describes a competition among thermal softening, strain hardening, and strain rate hardening as follows [45,46]: p
ε_ T Tr m σ e ¼ A þBðεpe Þn 1 þ C ln e 1 ðT r r T rT m Þ; ε_ 0 Tm Tr ð17Þ where A, B, C(¼ 0.0004 for NG and¼ 0.025 for CG), m(¼1.09), and n are model parameters, ε_ 0 a reference strain rate (taken as 1 s 1),
X. Guo et al. / Materials Science & Engineering A 618 (2014) 479–489
T the temperature, T r the room temperature (25 1C), and T m the melting temperature of Cu (1083 1C). The Johnson–Cook failure model [47] takes the effect of stress triaxiality, strain rate, and temperature on the failure strain into account. It establishes a linear incremental relation between the damage parameter D and the equivalent plastic strain increment dεpe as Z 1 d εpe ; ð18Þ D¼ εf where εf is the failure strain. Eq. (18) states that damage occurs when plastic deformation occurs and accumulates linearly with respect to d εpe . An element is taken to fail when D reaches 1. In Eq. (18), εf is defined by [47] p h i p ε_ T Tr 1 þ d5 ; ð19Þ εf ¼ d1 þ d2 ed3 σe 1 þ d4 ln e ε_ 0 Tm Tr where d1 to d5 are material constants, and p is the hydrostatic pressure, i.e., opposite of the trace of the stress tensor. In this paper, failed elements will be directly removed in both simulation and post-processing. With the failed elements removing, the precrack propagates in the microstructure. If loading state does not change too abruptly during deformation, d2 ¼ d3 ¼ 0 is considered to be a good approximation [48,49]. Here, d4 is taken as 0.014 and d5 as 1.12. Due to the high strain rate, the deformation is assumed adiabatic. The temperature increase ΔT due to the plastic dissipation can be written as Z χ ΔT ¼ σ d εpij ; ð20Þ ρcv ij where ρ ¼ 8900 kg=m3 is the density, cv ¼ 383 J=ðkg 1CÞ the specific heat [50], and χ ¼ 0:9 the ratio of plastic dissipation converted to heat [51–53]. Good agreement was found between predictions of the Johnson–Cook failure model and experimental results of metals under dynamic loading [51–53]. By fitting the constitutive relation of the NG phase obtained from the mechanism-based strain gradient plasticity theory and the constitutive relation of the CG inclusion in Section 2.2, we can obtain constitutive parameters of each phase in the bimodal NS Cu. They are listed in Table 2. Similarly, by fitting the constitutive relation of the composite obtained from the micromechanics composite model in Section 2.3, we can extract the constitutive parameters of the composite and use them for the homogenized phase. These are listed in Table 3. From Tables 2 and 3 Table 3 Constitutive parameters of the homogenized phase of the bimodal NS Cu with CG volume fraction 24.5% but different NG grain sizes.
Homogenized
Homogenized
Homogenized
phase with NG
phase with NG
phase with NG
grain size 23 nm
grain size 74 nm
grain size 200 nm
A (MPa)
467.8
398.2
350.4
B (MPa)
771.6
315.14
195.22
n
0.2705
0.2404
0.4011
d1
0.195
0.08625
0.0319
483
it can be found that both A (yield stress) and B decrease monotonically as NG grain size increases.
3. Results and discussion This micromechanics-based numerical scheme, which combines the mechanism-based strain gradient plasticity theory, the micromechanics composite model, and the Johnson–Cook failure model, will now be applied to study the fracture behavior of the bimodal NS Cu. In this process the six idealized microstructures in Fig. 2 and the constitutive parameters listed in Tables 2 and 3 are utilized. Crack path depends on material heterogeneity, i.e., the particular microstructural phase distribution. The crack stops at an interface of CG inclusion and NG matrix. Since CG phase has lower yield stress and larger failure strain, it can resist rapid propagation of the crack. The CG left at the wake plays a bridging role, consistent with the numerical investigation via the unit-cell model [15]. When there are favorably-oriented interfaces between the CG and NG regions, debonding occurs and may also contribute to fracture resistance [3,12]. Debonding and severe plastic deformation of CG inclusions can cause significant energy loss [3]. When debonding was perpendicular to the fracture plane, it might contribute to the ductility of bimodal specimen [12]. The crack usually alters propagation direction and deflects. It increases the real crack length, and thus, the crack takes a longer time to go through the entire microstructure. The experimental investigation on bimodal UFG Al–Mg alloy also showed that CG inclusions impeded crack propagation by crack deflection and debonding [12]. Stress gradient near the tip of a curved crack is smaller than that of a straight crack, the crack-tip region can accumulate more strain energy, and thus, a larger boundary load is required for the crack propagation [54]. For the bimodal NS Cu, both bridging of CG inclusions and crack deflection in NG matrix are responsible for enhancing fracture resistance. In most bridging cases, CG inclusions deform locally at stress concentrations and are in severe plastic deformation. During crack deflection, CG inclusions are not in severe plastic deformation since they remain almost in their original shape in the wake. 3.1. Effects of the microstructure and the NG grain size, dG Fig. 3 illustrates the boundary load versus the upper boundary displacement, with the NG grain sizes 23, 74, and 200 nm. For comparison, results of the same computational configuration with unimodal NG Cu are also illustrated. It can be found that with the NG size increasing, peak load decreases, which originates from the fact that strength of the NG and thus strength of the homogenized phase decrease. In all cases, the peak load of the bimodal NS Cu is lower than that of the unimodal NG Cu. After the peak, the load decreasing rate of the bimodal NS Cu is significantly smaller than that of the unimodal NG Cu, which implies that loading history of the former is more stable than that of the latter. Fig. 4 shows apparent crack length versus the upper boundary displacement. In the cases of 23 and 200 nm grain sizes, the crack in microstructure C has the highest propagation speed. In the case of 74 nm size, the crack in microstructure C has one of the highest propagation speeds. In all cases, the crack either in microstructure D or F has the lowest propagation speed. The lower the propagation speed is, the larger the fracture resistance is. Compared with other microstructures, bridging is dominant in microstructure F and crack deflection is dominant in microstructure D. Therefore, the two mechanisms can significantly improve fracture resistance. The propagation speed in the bimodal NS Cu is essentially lower
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Fig. 3. Load versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
than that in the unimodal NG Cu, which indicates that the CG phase in the bimodal NS Cu increases fracture resistance. Damage dissipation is important in quantifying fracture resistance [25,55–57]. Fig. 5 illustrates the damage dissipation versus the upper boundary displacement. It can be found that microstructure B has the largest damage dissipation rate (i.e., the damage dissipation over time) and that microstructure D the smallest one. The peak damage dissipation decreases substantially. Fracture process is closely related to the failure strain of NG phase. In the above three NG sizes, the descending of the failure strain coincides with that of the peak damage dissipation in Fig. 5. Fig. 6 shows strain energy versus the upper boundary displacement. When CG inclusions are in large deformation, more elastic energy will be accumulated. In all three cases (23, 74, and 200 nm), strain energy either in microstructure D or in microstructure F is the largest among six microstructures. This coincides with the facts that microstructure D or F is associated with the largest boundary load after the load peak (as illustrated in Fig. 3) and that the crack in microstructure D or F has the lowest propagation speed (as shown in Fig. 4). Therefore, similar with the apparent crack length, the strain energy is also an index to evaluate fracture resistance. Both external work and plastic dissipation can also be the output for further analysis. In each case of 23, 74, and 200 nm, the external work or the plastic dissipation for six microstructures almost coincides, which indicates that both of them are insensitive to the crack path, and thus, they are loosely related to the microstructures in the condition of the same volume fraction of CG inclusions. Both of their peak values decrease. The external work is converted into the plastic dissipation, damage dissipation,
and strain energy. The damage dissipation is much smaller than the plastic dissipation, i.e., most of the external work is consumed into plastic dissipation. 3.2. Group comparisons To compare fracture resistance of diverse microstructures, we make five groups: (i) A and B, (ii) C and D, (iii) E and F, (iv) A and C, and (v) B and D. In the first group, microstructure A has regular arrays of circular CG inclusions, while microstructure B has staggered arrangement of circular CG inclusions. From Fig. 4, it can be found that when the NG grain size is 23 nm, microstructure A has larger fracture resistance than microstructure B, while when the NG grain size is 74 and 200 nm, the former has smaller fracture resistance than the latter. Note that the volume fraction of the CG regions in microstructure B is slightly larger than that in microstructure A. When the NG has a smaller failure strain (e.g., 0.0575 and 0.0319 for the NG size 74 and 200 nm, respectively), volume fraction of the CG regions plays an important role in enhancing fracture resistance. Specifically, the microstructure with larger volume fraction of CG regions (i.e., microstructure B) behaves better than that with smaller volume fraction of CG regions (i.e., microstructure A). In the second group, microstructure C consists of regular arrays of square inclusions with square unit cells, while microstructure D has staggered square inclusions. Bridging occurs more often in microstructure C, while crack deflection more often in microstructure D, and thus, the CG inclusions in microstructure D deform moderately. It seems that fracture resistance of microstructure C is
X. Guo et al. / Materials Science & Engineering A 618 (2014) 479–489
485
Fig. 4. Apparent crack length versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
larger than that of microstructure D. However, Fig. 4 shows that it is not the case whenever the NG grain sizes are 23, 74, and 200 nm. This can be explained from a view point of the real crack length. On observation of the fracture pattern, it can be found that the crack in microstructure C deflects with an angle about 451 to connect the centers of the CG inclusions, while the crack in microstructure D deflects with an angle larger than 451 to connect the vertices of the CG inclusions. Therefore, the real crack length in microstructure D is larger than that in microstructure C, i.e., staggered arrangement of square CG inclusions has larger fracture resistance than arrayed arrangement. In the third group, microstructure E consists of random unidirectional elliptical inclusions with their major axes aligned in the direction of the pre-crack path, while microstructure F perpendicular to it. Since elliptical inclusions in microstructure F have larger total projection length perpendicular to the direction of the pre-crack path than those in microstructure E, bridging occurs more often in microstructure F, while debonding more often in microstructure E. In Fig. 4 it can be found that fracture resistance of microstructure F is always much larger than that of microstructure E. It can also be observed that fracture resistance of microstructure E is even lower than that of the unimodal NG Cu when the NG grain size is 23 nm, and that it can moderately enhance fracture resistance when the NG grain sizes are 74 and 200 nm. Therefore, bimodal structure with microstructure E should not be used to enhance NG metals with moderate failure strain. In some particular situations, one can estimate potential microcrack nucleation sites and dominant propagation direction, based on which major axis direction of the elliptical inclusions can be chosen properly.
In the fourth group, it is important to investigate the loadcarrying capacity of circular inclusions in microstructure A and square inclusions in microstructure C. Fig. 7 illustrates fracture processes of CG inclusions in microstructures A and C. It can be found that the circular CG inclusions in microstructure A have severe plastic deformation before fracture and that the NG matrix also has large deformation. It can also be observed that the square CG inclusions in microstructure C only have larger deformation around their vertices, while the NG matrix has much smaller deformation. Large proportion of fracture in the circular CG inclusions is in mode-I, while that in the square CG inclusions is in mode-II. Therefore, when both of them are in bridging mode, load-carrying capacity of the circular inclusions is higher than that of square ones. It is preferable to use microstructure with circular inclusions to improve fracture resistance. In the fifth group, crack in microstructure B deflects together with the CG bridging. After the crack in microstructure D penetrates several CG inclusions, it deflects between two rows of CG inclusions, and thus, the CG inclusions play a less important role in bridging. Compared with microstructure B, the crack path in microstructure D is more tortuous. Specifically, the crack in microstructure B follows a zigzag path to connect the centers of CG inclusions, while that in microstructure D follows a trapezoidal path to connect vertices of the square CG inclusions. This is the intrinsic reason that crack in microstructure D has a longer real crack length, and thus, it has a larger fracture resistance. Note that the above comparison is only valid when crack deflection is more dominant than bridging. In the opposite case, i.e., when bridging plays a more important role, the microstructure with square inclusions will have smaller fracture resistance, as shown in the comparison in the fourth group.
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Fig. 5. Damage dissipation versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
3.3. Effect of volume fraction of the CG inclusions The dependence of fracture resistance on volume fraction of CG inclusions is an important issue for bimodal NS metals as it provides important information for toughening. To investigate this issue, we use microstructures having the same shape but different characteristic lengths of CG inclusions. Specifically, five radii, namely 2, 3, 4, 5, and 6 μm, are chosen for circular CG inclusions in microstructure B. First, the NG grain size is chosen as 23 nm. With the radius of CG inclusions increasing, their volume fraction increases and the constitutive relation of the corresponding homogenized phase also changes. Table 4 lists constitutive parameters of the homogenized phase, obtained by the micromechanics composite model with consideration of different volume fractions of CG inclusions in Section 2.3. With the radius increasing, A (yield stress) decreases monotonically (as shown in Table 4), and this in turn leads to the decrease of boundary load, external work, plastic dissipation, and strain energy versus the upper boundary displacement. Figs. 8 and 9 illustrate apparent crack length and damage dissipation versus the upper boundary displacement for the cases of five radii. With the radius increasing at each boundary displacement, both the apparent crack length and the damage dissipation first increase slowly and then decrease quickly, implying minimum fracture resistance and maximum damage dissipation. The critical radius is 4 μm for microstructure B and the corresponding critical volume fraction of CG inclusions is 15.68%. Similar phenomenon also occurs in the cases of 74 and 200 nm of the NG size, while the critical radius changes to 3 μm. This can be explained as follows. On observation of fracture pattern, when the
radius is smaller than the critical one, bridging does not work, the crack bypasses the CG inclusions with ease, and only crack deflection is effective. Therefore, the real crack length and fracture resistance decrease as the radius increases. When the radius is larger than the critical one, bridging plays an important role, and thus, fracture resistance increases with increasing radius. Note that if the main toughening mechanism does not change when the characteristic length of CG inclusions varies, the critical characteristic length and volume fraction will not exist – this is the case, for instance, in microstructure D.
4. Conclusions In this paper a numerical investigation via the mechanismbased strain gradient plasticity theory, the micromechanics composite model, and the Johnson–Cook failure model has been carried out to study fracture behavior and toughening mechanisms of bimodal NS metals. The boundary load, the apparent crack length, the damage dissipation, and the strain energy have been analyzed to evaluate the stability of loading history and the fracture resistance. It is found that both bridging in CG inclusions and crack deflection in NG matrix can have strong effects in enhancing fracture resistance of bimodal NS metals. Main conclusions can be drawn as follows: 1. Loading history of the bimodal NS metal is more stable than that of the unimodal NG metal since the load decreasing rate of the bimodal NS metal is significantly slower than that of the unimodal NG metal.
X. Guo et al. / Materials Science & Engineering A 618 (2014) 479–489
Fig. 6. Strain energy versus boundary displacement. The NG grain sizes are (a) 23 nm, (b) 74 nm, and (c) 200 nm.
Fig. 7. Fracture processes of CG regions in (a) microstructure A and (b) microstructure C.
487
488
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Table 4 Constitutive parameters of the homogenized phase of the bimodal NS Cu with NG grain size of 23 nm but different CG radii.
metal at its minimal state. Therefore this critical volume fraction should be avoided in material design. 4. The staggered arrangement of square CG inclusions has larger fracture resistance than the arrayed arrangement. 5. In some particular situations, one needs to estimate potential microcrack nucleation sites and dominant propagation direction, based on which major axis direction of elliptical inclusions can be chosen properly for improved toughness. This study provides the needed insights into the design of bimodal microstructure for superior fracture resistance in highstrength NS metals.
Acknowledgments Fig. 8. Apparent crack length versus boundary displacement for the cases of five radii.
We wish to thank an anonymous reviewer for his/her helpful comments. X. Guo acknowledges the support from the National Key Basic Research Program (Grant 2012CB932203), the National Natural Science Foundation of China (Project nos. 11102128 and 11372214), and the Elite Scholar Program of Tianjin University. G.J. Weng thanks the support of NSF Mechanics of Materials Program under CMMI-1162431. L.L. Zhu acknowledges the support from National Natural Science Foundation of China (Project nos. 11302189 and 11472243), Doctoral Fund of Ministry of Education of China (Grant no. 20130101120175), Zhejiang Provincial Qianjiang Talent Program (Grant no. QJD1202012), and Educational Commission of Zhejiang Province of China (Grant no. Y201223476). J. Lu acknowledges the supports of GRF/CityU519110 and the Croucher Foundation CityU9500006. References
Fig. 9. Damage dissipation versus boundary displacement for the cases of five radii.
2. When bridging plays an important role, the microstructure with circular inclusions has more severe plastic deformation. This offers larger fracture resistance than the square ones. When crack deflection is dominant, the microstructure with square inclusions has a longer real crack length and thus larger fracture resistance as compared to the circular counterpart. 3. When dominant toughening mechanism changes from crack deflection to bridging, a critical volume fraction of CG inclusions exists that makes fracture resistance of the bimodal NS
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