- Email: [email protected]
Application of phase-field modeling to deformation of metallic glasses
Current Opinion in Solid State and Materials Science 15 (2011) 116–124
Contents lists available at ScienceDirect
Current Opinion in Solid State and Materials Science journal homepage: www.elsevier.com/locate/cossms
Application of phase-field modeling to deformation of metallic glasses G.P. Zheng ⇑ Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
a r t i c l e
i n f o
Article history: Available online 2 February 2011 Keywords: Phase-field modeling Metallic glasses Defects Plastic deformation Fracture
a b s t r a c t Metallic glass or amorphous alloy is a new class of metallic materials with unique mechanical deformation mechanisms. In the past decade, the encouraging progress has been made on revealing the deformation mechanisms and mechanical behaviors of metallic glasses using various theoretical and computational methods. In this review, the application of the recently developed phase-field method to deformation of metallic glasses is emphasized. We first briefly introduce the features of deformation of metallic glasses, following by a review on the modeling methods that are based on kinetic theory or thermodynamics of the deformation defects. Then we focus our attentions on the phase-field modeling which typically treats the localized plastic flow or shear banding as the collective behaviors of structural transformation of the deformation defects. Application of the phase-field modeling to shear banding, local heating, shear band instability in metallic glasses and metallic-glass–matrix composites are discussed. We also highlight the research directions and challenges of phase-field modeling on the deformation of metallic glasses. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Metallic glasses are currently emerging as a novel class of metallic materials with many attractive physical, chemical and mechanical properties for structural and functional applications. Recent advance in the determination of alloy systems with excellent glass forming ability has led to the synthesis of bulk metallic glasses (BMGs) with typical dimensions of inches or more. One of the areas that benefits greatly from this advancement is the understanding of mechanical properties, since BMGs can be made sufficiently large to allow standardized mechanical testing. In the past decade, many well controlled tests have been conducted, such as tension, compression, torsion, fracture toughness tests, fatigue test and indentation test. Compared with conventional crystalline metallic materials, metallic glasses have many attractive mechanical properties [1,2], such as high fracture strength, distinctly higher tensile strength, much lower Young’s modulus, outstanding super-plasticity at their supercooled regions. Unlike crystalline alloys, metallic glasses do not have dislocation defects during the deformation process and the deformation is localized in nature. Therefore, metallic glasses have attracted significant scientific interests besides the technological attentions. Deformation and fracture behaviors of glassy materials (e.g., metallic glasses, silicate glasses, polymers and granular materials), or more generally, disordered or heterogeneous materials, are quite different from those of crystalline solids. Perhaps the best ⇑ Tel.: +852 27666660; fax: +852 23654703. E-mail address: [email protected] 1359-0286/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cossms.2011.01.004
way to show such differences is to compare the deformation behaviors in metallic glasses and the crystalline metals. In crystalline metal under large deformation, dislocations whose geometries are characterized by well-defined crystallography quantities such as sense vectors and Burgers vectors would proliferate, resulting in significant yielding and plastic deformation. In metallic glasses, dislocations are unlikely to be responsible for the deformation due to the lack of long range translational symmetry in their structure. Instead metallic glasses deform in two distinct modes [3], depending on applied stress, temperature and strain rate. Below the glass transition temperature Tg and above 0.7Tg, the metallic glass deforms homogeneously in that every portion of the solid participates in the flow process. At low temperature (typically T < 0.7Tg) and high strain rates the deformation is heterogeneous, i.e., the plastic deformation is localized or is restricted to occur in shear band with sizes ranging from tens of nm to sub-microns, away from which the deformation is elastic. Although large plastic deformation are developed when metallic glasses are subjected to constrained deformation such as compression, bending or drawing, no appreciable amount of macroscopic plasticity is observed when they are deformed under tension or torsion. The metallic glasses are typical elastic-perfectly plastic materials showing no apparent work hardening. When they are subjected to tensile loading at the yield stress, metallic glasses may have catastrophic failure which occurs along one of the shear bands where the deformation becomes so large that it physically separates the system. The fracture of metallic glass is well characterized by experiments and is found to be much more sophisticate than the brittle fracture in crystalline solid. One of the most interesting phenom-
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
ena in the fracture of metallic glass is the vein patterns presented at the fractured surface [2]. Another important characteristic observed at the fractured surface is the wavy surface [4] with roughness and wavelength of tens of nanometers. These features have played an important role in promoting the theory for the deformation of metallic glasses. For example, the vein pattern prompted the idea of low viscosity and liquid-like disordered structure inside the shear bands. The wavy cracking phenomenon suggested the Griffith theory for brittle fracture could be incorporated with the idea of shear banding instability to describe the fracture of metallic glasses. During the fracture processes, it was reported that temperature near the deformed region could rise up to hundreds or even several thousands Kelvin, and recrystalization and atomistic structure transformation were found inside the shear band [5]. Although they are still in debate, the crystallization and atomistic structure changes in the shear bands seem to support the fact that there is local heating caused by rapid deformation localized in shear band [6]. Because shear banding is the only deformation mechanism operating at temperatures below 0.7Tg, the defects that are responsible for the shear banding must be atomistic and collective. The habitual shear bands generated at the same locations under repeated deformation show the existence of region with higher degrees (lower density) of chemical or structural disorders compared with dense disordered structures [7] which ideally are considered as random close packed (RCP) structure or icosahedral structures in glassy alloys [8]. Naturally, the free-volume defects which are atoms surrounding by excess atomic volume and always exist in metallic glasses [9], can be considered to be the deformation defects that are responsible for the shear banding. However, evidences from experiments reveal that deformation defects or flow defects in metallic glasses are much more complicated than free-volume defects [10]. Especially the structural details and strain/stress states around the deformation defects are still not clear. Nonetheless, the deformation defects should exist most likely in the free-volume regions. With promising structural and functional applications in many areas, the metallic glasses have advanced rapidly in the past decade. It is an urgent need to connect the activities of microscopic deformation defects and the macroscopic deformation behaviors in metallic glasses. In crystalline metals such connection between mechanical properties and dislocation activities is well established and has contributed significantly to our understanding of their deformation behaviors, which plays a key role in the development of these materials for industrial applications. If the mechanical properties of metallic glasses can be directly related to the activities of their deformation defects, such relation can provide references for the design and use of metallic glasses with improved strength, toughness and ductility. Because of the complexity of deformation mechanisms as indicated in the above-mentioned features of deformation in metallic glasses, despite considerable effort has been devoted to these materials, theoretical descriptions of the macroscopic deformation behaviors comprehensively based on the activities of atomistic deformation defects are still not well developed. Firstly, it is because there are a number of puzzles remain to be resolved for the structural characteristics of deformation defects. Secondly, it is caused by the lack of computational model capable of relating the deformation defects with shear banding. The shear bands are typically tens of nanometers in size, it is either too large for the molecular dynamics (MD) simulation to simulate their branching and interaction processes, or too small for the finite element method (FEM) to capture its atomistic scale deformation details. Although the modeling methods [11–17] based on kinetic theory and thermodynamics of the deformation defects successfully describe some features of the deformation of metallic glass, they rely
117
on many structural details and parameters of the deformation defects which turn out to be still unclear. Considering the fact that the atomistic details of the deformation defects are still not well characterized in metallic glasses, phenomenological model could provide us with useful tool to describe the collective activities of the deformation defects during plastic deformation. Phase-field modeling is essentially based on the phenomenological model which is formulized by coarse-grained free-energy functional of the system containing deformation defects, and has been successfully applied to connect the activities of dislocations and macroscopic deformation behaviors in crystalline solids [18,19]. When it is employed to model the deformation behaviors of metallic glasses, the phase-field modeling could show some advantages. First, the free-energy functional of the metallic glass can be constructed to describe the energetics of the deformation defects with the help of sufficient experimental data or atomistic simulation results. Second, phase-field modeling is capable of looking into the mechanisms of shear band formation, propagation and branching since such behaviors occur in the length scales from tens of nanometers to microns. Third, although the length scale of this simulation method is typically mesoscopic, the phenomenological model is based on theoretical and experimental understanding of the atomistic deformation defects of the metallic glasses. Hence it essentially bridges the scientific gap to correlate the atomistic deformation defects and the macroscopic mechanical properties of the metallic glasses. In this paper, we first introduce the recent advance in the theories of plastic flow of metallic glasses in Section 2. Then we focus our attentions on the phase-field modeling which typically treats the localized plastic flow or shear banding as the operation of structural transformation of the deformation defects in Section 3. In Section 4, we discuss some examples of the application of the phase-field modeling to shear banding, adiabatic heating, shear band instability in metallic glasses and metallic-glass–matrix composites. We comment on several research directions and challenges of this method in Section 5.
2. Theories of plastic flow in metallic glass Deformation mechanisms of metallic glasses have been extensively investigated for decades. Several theoretical models have been proposed for the deformation mechanisms in metallic glasses. These theories include free volume model [20,21] and atomic level stress fluctuation [22,23]. As pointed out by Spaepen [20] and Argon [21] in 1970s, deformation of metallic glasses is mainly through the activation of free-volume defects, i.e., either the diffusion or transformation of free volumes when the metallic glasses are deformed. Structural details in the free-volume regions were given by Argon [21] who predicted the deformation defects to be in the free-volume regions with a typical size of 5 atom diameters and are in spherical shape at high temperatures or disk shape at low temperatures. This concept was further developed by Falk and Langer [24] who described the region consisting of activated free volumes as shear transformation zone (STZ). Based on the kinetic theory for free-volume coalescence and annihilation, Spaepen [20] derived the explicit expression of the inhomogeneous flow rate, which is recently used by Huang et al. [25] to analyze the shear localization in one dimension, and Dai et al. [26] to analyze the adiabatic shear banding. Langer and his coworkers [12,13] have incorporated thermodynamics into conventional STZ model [24] and describe accurately the macroscopic deformation behaviors of metallic glasses. In their mean-field continuum model [12], an effective temperature, as an important macroscopic state variable, governs the density of the shear transformation zones. Under shear stress, each STZ deforms
118
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
to sustain the plastic strain, and the STZ can flip in the direction of the applied strain, or revert to their original configuration under the shear stress in the opposite direction. Each STZ interacts with other STZs through continuum fields, and so the rearrangement occurs at a rate R(s)/s0, depending on the stress and a characteristic attempt frequency. STZ can flip at most once in the direction of applied stress, and can be created or annihilated to accommodate plastic flow. The differential equation describing the number density of STZs is given by [13]:
n
s0 n_ ¼ RðsÞn RðsÞn þ C
1
2
e1=v n ;
ð1Þ
where R(±s)/s0 is the rate of switching per STZ as a function of stress s, C is the rate at which energy is dissipated per STZ, and n1e1/v is the steady state density of the STZs in equilibrium. v is defined as the dimensionless effective temperature. Based on the equations derived above, in a simple shear geometry with periodic Lees–Edwards boundary conditions, shear localization is found by numerical integration of the STZ equations. The macroscopic stress–strain curves also show consistency with the experiments and MD simulations. By approximating these nonlinear dynamics, a localization criterion has been obtained from the initial conditions for the effective temperature [13]. Thamburaja and Ekambaram [14] have developed a finitedeformation-based, thermo-mechanically-coupled constitutive model for metallic glasses, which describes the plastic flow of metallic glasses at various temperatures. Based on the principles of the thermodynamics and the concept of micro-force balance, the theory has been formulated and a kinetic equation for the free volume concentration n which considers free volume diffusion and structural relaxation has also been derived as
c_ 0 sn2 c_ 0 p n_ ¼ Dðr2 nÞ þ fc_ ðn nT Þ: sn3 s n3
the free volume vf of an atom is defined by vf = vi v0 and vi is the volume of Voronoi polyhedron of the ith atom; v0 = 4(R0)3/3 and R0 is the radius of the atom. vm is the maximum of vi when complete decohesion occurs. Therefore Pij is the scaled density tensor. v⁄TrPij = 0 denotes no defect or dense random packing which we treat as the reference state. v⁄TrPij = qc represents a structural transformation of the deformation defects and v⁄TrPij qc denotes stable fractured atomistic structure. a = a0 (Tg T)/Tg. Combined Eq. (3) with Eq. (4), the free energy of stress-free metallic glass can be written as
ð2Þ
DF ¼
# Z " @ qij 2 1 dV; aq2ij þ bq3ij þ cq4ij þ j 2 @xj
ð5Þ
where j is the surface energy of the interface between the deformation defect and defect-free region. To write down the free-energy functional (Eq. (5)) for deformation defects, it has been assumed that there is no any long range interaction among sites containing deformation defects. This is true for metallic glass. On the other hand, the concept of deformation defect mentioned above may be slightly different with STZ proposed by Argon [21] and Langer [14]. 3. Phase-field model for fracture in metallic glasses Considering the brittle fracture of metallic glass under tensile deformation, the formation of vein patterns or nano-scale roughness at the fracture surface could result from the collective activities of deformation defects. We may introduce the displacement field u(r) = u(x, y)z as an order parameter to describe the formation of fractured surface, where u(x, y)Z0 represents the displacement of atom from its equilibrium position along the stress direction, and Z0 = (3vm/4)1/3. As shown in Fig. 1, the fractured surface can be characterized by its height hðx; yÞ ¼ hðx; yÞ þ uðx; yÞ.
Being calibrated by fitting the constitutive model to compression experiments, the model can reproduce the stress–strain curves for both the jump-in strain-rate experiment and the simple compression experiment conducted at different temperatures. Furukawa et al. [15] considered the plastic flow in metallic glasses as the viscoelastic deformation which is described by the rheological dynamics and a constitutive model for stress fluctuation. This model is similar to the dynamic equation for the concentration in viscoelastic fluid mixtures and is particular successful in explaining the strain-rate controlled plastic flow in metallic glasses. Zheng [27], and Zheng and Li [28] developed a phenomenological model for deformation defects in metallic glasses. Because the deformation defects consist of sites with free volumes that are more vulnerable to internal rearrangement than their dense random packing counterpart, they can be thermally or mechanically activated. When the accumulation of these activated free-volume sites reaches a critical value, the deformation defect transforms into a structure which is more stable than dense random packing structure. The nucleation of such transformed deformation defect results in shear band. This model simply treats the formation of shear band as a consequence of the structural transformation of deformation defects. The free energy change of a metallic glass due to the introduction of deformation defects is written as
DF ¼
Z
fs dV ¼
Z
½f0 þ K c dV;
ð3Þ
where fs is the free energy density of the defects. Kc is the curvature of the defect surface; f0 is given by:
f0 ¼ aq2ij þ bq3ij þ cq4ij rij qij ;
ð4Þ
where rij is the applied stress tensor; qij = v Pij is the defect density tensor. v⁄ is a coarse-grained average of [vi(r) v0]/(vm v0), where ⁄
Fig. 1. The schematic of model for the simulation on fracture of metallic glass and the definition of crack-opening field h(x, y). The displacement field u near the fractured surface is defined by hðx; yÞ ¼ hðx; yÞ þ uðx; yÞ.
119
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
Here hðx; yÞ is the average of the height of fracture surface and it describes in macroscopic scales the profile of the fracture surface. For example, in most bulk metallic glasses the fracture surface from macroscopic view is a plane inclining with an angle of 40– 45° to the loading direction and can be described by a definite function hðx; yÞ. Another order parameter is the deformation defect density which is now simplified as a scalar q(r) = v⁄TrPij. Taking the effect of deformation defect into account, we can write the local elastic energy as
fe ¼
E 2 E0 ð1 qÞ 2 u ¼ u ¼ eð1 qÞu2 ; 2 2
ð6Þ
where E is the Young’s modulus of metallic glass and E0 is the reference Young’s modulus assumed to be that of its crystalline counterpart. The binding energy between atom cluster can be written as 2
fb ¼ d½1 eu :
ð7Þ
Combine Eqs. (6) and (7), we are able to construct the free-energy functional with two order parameters u(x, y) and q(x, y) for the deformed metallic glass under uniaxial tensile stress rzz:
Fðq; uÞ ¼ DF u þ DF q þ DF int :
ð8Þ 2
Here DF u ¼ eu2 þ d½1 eu þ ujruj2 rzz u;
ð9Þ
1 DF q ¼ aq2 þ bq3 þ cq4 þ jjrqj2 ; 2
ð10Þ
and DF int ¼ equ2 wqjruj2 :
ð11Þ
The |ru|2 term in Eq. (9) is the elastic energy contribution from stretched atomic cluster surface. This surface can form macroscopic crack surface. We write in Eq. (11) the interaction term between displacement field u and deformation defect field as q|ru|2 with a positive coupling strength w. This is because that the crack formation tends to occur in the zone (shear band) that consists of deformation defects. The plastic deformation of system described by Eq. (8) can be studied qualitatively. We are interested in the spatial extension of the plastic deformation on the fractured surface. Shear band formation in plastic deformation is directly connected to the unstable mode in model system described by Eq. (8)–(11). Prior to fracture the instability of atomic displacement field u(x, y) can be represented by the fluctuating field w(x, y): u(x, y, t) = us(x, y) + w(x, y,t). Here w(x, y) is small and large w(x, y) represents microcrack. us(x, y) is the displacement field prior to fracture and is assumed to be a constant u0 in brittle fracture. u0 is equivalent to the strain in the elastic limit. The corresponding fracture strength rF is given by
dF ¼ 0; duu¼u0
1 uðr; tÞeikr dr 2p Z 1 qk ðtÞ ¼ qðr; tÞeikr dr 2p
uk ðtÞ ¼
2
rF ¼ 2½eð1 qÞ þ deu0 u0 :
ð12Þ
The defect density q at rzz < rF is small. We can describe the dynamics of atomic cluster under rF by using kinetic equation,
@u dF ¼ Cu du @t 1 u2 ¼ 2Cu eð1 qÞu þ de u ur2 u wrq ru rF : 2 ð13Þ
Cu is the mobility constant. Neglecting higher order terms than O(q2), O(w2) and O(u2) and using the Fourier transform
ð14Þ
we get the equation for w(t) by substituting Eq. (12) in Eq. (13)
h i @ Wk 2 2 ¼ 2Cu ðe þ dÞ uk þ ðe þ wk Þqk Wk : @t
ð15Þ
The unstable modes to this deformed system exist when
qk ¼
ðe þ dÞ þ uk 2
e þ wk
2
ð16Þ
:
The inverse Fourier transform of Eq. (16) leads to the nontrivial description of deformation defect density inside the deformed zone:
h
pffiffiffiffiffiffiffiffiffiffiffiffiffii
qð~rÞ ¼ C 1 dð~rÞ þ C 2 exp j~rj ðe=wÞ :
ð17Þ
Eq. (17) shows that the structural transformation of deformation defects driven by unstable displacement field is localized. This leads to shear band formation during plastic deformation. The fluctuation of deformation defect density near the shear band can be written as:
rffiffiffiffi e ~ ~ dqðrÞ / exp jrj exp ½j~ rj=r 0 ; w
ð18Þ
pffiffiffiffiffiffiffiffiffi where r 0 ¼ w=e is the correlation length of shear bands. r0 measures the localization of shear bands and is controlled by parameter w. If w = 0, r0 = 0, which means the defect simply do not form shear band at the fractured surface. With increasing interaction strength w, shear band width increases. These analyses could explain the vein pattern observed at the fracture surface.
4. Phase-field modeling of shear banding in metallic glasses As we mentioned in Section 2, deformation of metallic glasses is through shear banding at low temperatures. To describe the shear banding behaviors under different thermal and stress conditions, the effects of temperature and stress–strain relation around the deformation defect should be considered in Eq. (5). The phenomenological model has considered the temperature by setting a = a0 (Tg T)/Tg. Based on the characteristics of the shear banding that the regions other than the shear bands are deformed elastically, the stress–strain relation around the deformation defects is governed by the theory of elasticity [29], and the local strain energy can be written as
1 C ijkl ekl eij 2
¼ l dik djl þ dil djk þ kdik djl ;
e½eij ¼ C ijkl
or
Z
ð19Þ
where eij is the strain tensor defined by the displacement field u through the relation eij = ( o uj/ o xi + o ui/ o xj)/2 and i, j = 1, 2, 3; l is the shear modulus and k is related to the bulk modulus B by B = k + 2 l/3. The response of deformation defects to the local shear is reflected in the defect free energy (Eq. (5)) through expanding the coefficients b and c with the plastic work De = e[eij] e0, where e0 is the strain energy of metallic glass at the elastic limit. The free-energy functional of the deformed metallic glass is thus given explicitly as
Z h i2
j q0 ~_ ~ qj2 dV; F¼ ð20Þ u þ f q; eij þ jr 2 2
a0 b0 c0 a1 2 b1 3
q þ q e½eij e0 ; ð21Þ f q; eij ¼ e eij þ q2 þ q3 þ q4 þ 2 3 4 2 3
120
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
where q0 is the mass density of metallic glass; a0 = a0 b0 T/Tg and a0 , b0 , a0, b0 and c0, a1, b1 are constants. The equations of motions for ~ u and q are described respectively by
sq
@q dF ¼ dq @t
¼ jr2 q a0 q þ b0 q2 þ c0 q3
qða1 þ b1 qÞ e½eij e0 ;
ð22Þ
and
q0
@ 2~ dF a1 b1 u ¼ lr 1 þ q2 u ; ¼ r þ q r~ 2 deij 2 3 @t
ð23Þ
where sq is the characteristic time for defect activation. When the coefficients of Eq. (21) are given by a0 = 4(2 T/Tg) DG, b0 = 32DG, c0 = 16DG, it can be clearly seen that there is a metastable state of the deformation defect (q = qc). The stable defect-free state (q = 0) can be activated either by applied stress or temperature elevation. The activation energy is DG. Usually DG is determined by the strain-rate dependent flow stresses at various temperatures [30]. The surface energy j of the interface between the deformation defect and defect-free region can be estimated from the surface energy of the metallic glasses and the characteristic length of the vein pattern at the fracture surface. Shear banding, shear band branching and local heating in Vitrelloy-1 BMG (Zr41Ti14Ni10Cu12.5Be22.5) are simulated by the phasefield model [28]. Fig. 2c–e shows shear banding and shear band branching under different loading stresses [28]. When the heat conduction equation is included in the modeling, the temperatures near the shear bands are calculated as shown in Fig. 2b. A crack resulting from the shear banding would become unstable when its tip velocity reaches a critical value VC. The phase-field modeling reveals that the maximum velocity that a crack can achieve in Zrbased BMG is 0.63VR, where VR is the Rayleigh wave speed.
In Fe–Si–B metallic glass, the relation between shear band thickness and the local shear stress is obtained by the phase-field modeling [31]. The relation facilitates the determination of macroscopic mechanical properties such as shear strength and fracture toughness of metallic glass from the shear band characteristics. The advantage of phase-field modeling for shear banding can be seen in Fig. 3 which shows the mechanisms of shear band interaction and multiplication. In the sub-micron scales it can be found that the deflection and attraction of two meeting shear bands occur depending on the signs of shear stresses near the approaching parts of these shear bands. Moreover from the simulation study it is found that the integration of two attracting shear bands will result in the generation of secondary shear bands [32]. One of the most useful applications of phase-field modeling is in the simulation of mechanical behaviors of metallic-glass–matrix composites. In the tungsten fibers reinforced Zr-based BMG, the results from phase-field modeling reproduce all the important features of shear banding observed in experiments. Fig. 4 shows the effect of reinforcing fibers on shear banding in the BMG composites [33]. It is found the tensile residual stress at the interface between BMG matrix and the reinforcement is preferred for the purpose of enhancing the plasticity of the BMG composite, while the compressive residual stress would improve its fracture toughness. Furthermore, the residual stress g and atomic bonding condition represented by the defect density q = w0 at the interface can be used to determine the deformation modes in the BMG–matrix composite, as shown in Fig. 5 [41].
5. Future research directions In the previous sections, we have mentioned that there are several theoretical and phenomenological models for the description of shear banding. For metallic glasses, however, the essential features of plastic deformation accommodated by the shear banding
Fig. 2. (a) The model system of a BMG plate. (b) Temperature distribution around the shear band with plastic work to heat conversion ratio b = 0.5. (c–e) Shear bands and crack propagations under stress intensity factors KI = 25 MPa m1/2, 35.4 MPa m1/2 and 100.2 MPa m1/2 respectively. The gray scales correspond to the values of 1 q. The color scale represents temperatures (K). The length bars in (b)–(e) represent 2 lm [28].
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
121
banding, shear band branching and shear band interaction and multiplication, there are still some unsolved issues such as sample size effect on shear localization, brittle-to-ductile transition, mechanisms of fatigue, shear band interaction with microstructures, micromechanics of shear bands leading to fractures. We briefly summarize the challenges and directions of research on the modeling and simulation of deformation of metallic glasses using phase-field method as follows. 5.1. Combination of first-principles calculation with phase-field modeling There are only two coefficients (DG and j) of the phenomenological model for deformation defect which paves a way for the phase-field modeling of deformation in metallic glasses. These coefficients in particular j are difficult to be determined from experiments, while the ab inito simulation [34] could be used to determine DG and j consistent with those defined by the phenomenological model. In Zr-based BMG [28] and Fe–Si–B [31] metallic glasses ribbons, DG determined from first-principles calculation are comparable with those measured from experiments. It has been shown that the phase-field modeling with DG and j determined from first-principles calculations successfully predict the deformation behaviors of Zr-based BMG [28] and Fe–Si–B [31] metallic glasses ribbons. Considering the evidence that the activation volume of the deformation defects in metallic glasses is typically less than 1 nm3, although ab initio simulation usually handles systems with less than one thousand atoms, the structural details, activation energy, surface energy, elastic constants relating to the deformation defects could be readily obtained. Combined with first-principles calculation in constructing the energy landscape of the metallic glass, the phase-field modeling could link the activities of atomistic defects with macroscopic deformation behaviors. Fig. 3. Shear band multiplication and interaction. (a) Formation of one secondary shear band marked by a dash circle; (b) Formation of two secondary shear bands marked by dash circles. The gray scales correspond to the values of 1 q. The color bars ( 870 MPa) are for the contour plots of stresses [32].
have not been fully captured by these models. Although the phasefield modeling has shown its advantages in the simulation of shear
5.2. Implementation of phase-field modeling by considering the shape and size of deformation defects In the free-energy functional given in Sections 3 and 4, the density of deformation defects are simplified as a scalar order parameter q(r) = v⁄TrPij. Because of the existence of structural and chemical inhomogeneity in metallic glasses, the density of deformation defects has to be described by a tensor order parameter
Fig. 4. Shear banding in fiber-reinforced BMG. (a)–(d) Fibers with different lengths L. The length bar represents 0.4 lm. (e)–(h) Fibers with different orientation angles. The length bar represents 0.8 lm [33].
122
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
Fig. 5. Different deformation modes for shear banding and crack propagation in the BMG composite: (a)–(b) shear banding induced cracking inside the reinforcement after the shear band interacts with reinforcement. g = 0, w0 = 0; KI = 24.2 MPa m1/2 (a), and 27.5 MPa m1/2; (b) and (c) propagation of shear band along the reinforcement surface with compressive residual stress g = 435 MPa; w0 = 0; (d) shear banding induced cracking inside the reinforcement with tensile residual stress g = 175 MPa at its surface; w0 = 0; (e) branching of shear band along the reinforcement surface; w0 = 0.8; g = 0. (f) Shear banding at opposite sides of the reinforcement surface; w0 = 1; g = 0. KI = 27.5 MPa m1/2 in (c)–(f). u is the crack field of the tungsten fiber [41].
qij. Although the geometry of deformation defect is still not clear, the coupling between the defect density tensor with local displacement field u may be quantitatively determined by the first-principles calculation. Implementation of phase-field modeling by
considering the shape and size of deformation defects will facilitate the precise connection between the microscopic activities of defects with macroscopic deformation behaviors of metallic glasses under complicate loading conditions
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
5.3. Simulation of the length scale impact on the deformation of metallic glasses Recently, the effects of sample sizes on the mechanical strength and ductility of metallic glasses are of interests. In experiments the length scale impacts on the deformation of metallic glasses are found to occur in a range of tens of nanometers to hundreds of microns [35,36]. Phase-field modeling is the best methods to investigate such impacts and could provide a fundamental insight into such impacts. 5.4. Simulation of crystallization and deformation in in situ formed BMG composites The lack of tensile ductility is one of the major obstacles to the structural application of bulk metallic glasses (BMGs). Recently various experimental investigations have shown the feasibility of making BMG–matrix composites with excellent combination of mechanical properties from some glassy alloy systems [37,38]. Among those BMG–matrix composites some in situ-formed BMGs containing crystalline particles, which are precipitation of a crystalline phase during solidification with rest of the melt forming a glassy–alloy matrix, possess the most desirable mechanical properties, e.g., high fracture strength, impressive fracture toughness and ductility more than 10%. For example Zr-based BMG–matrix composites with an in situ dendritic b phase are found to be among the toughest bulk materials known to date [38]. The crystallization of the dendritic phase is determined by a lot of factors due to the multi-component nature of the BMGs. Phase-field modeling has been proved to be very robust in the simulation of solidification and the formation of dendritic phases [39,40]. When combined with that for dendritic phase modeling, the phase-field modeling for deformation of BMG–matrix composites will provide us with useful simulation approaches to predict and understand their mechanical behaviors, avoiding expensive trail-and-error materials preparation and mechanical tests. 6. Concluding remarks Although the length scale of phase-field modeling is typically mesoscopic, the phenomenological model for the modeling is based on theoretical and experimental understanding of the atomistic deformation defects of the metallic glasses. Hence it essentially bridges the scientific gap to correlate the atomistic deformation defects and the macroscopic mechanical properties of the metallic glasses. Application of the phase-field modeling to shear banding, local heating, shear band instability in metallic glasses and metallic-glass–matrix composites are discussed. Despite many challenges in the characterization of the deformation defects in metallic glasses, the energy landscape of metallic glasses consisting of the deformation defects could be quantitatively determined, facilitating the formulation of the free-energy functional of the system for phase-field modeling. The phase-field modeling could be very useful for the simulation of deformation behaviors of metallic glasses, with examples and research directions given or proposed in this paper. Acknowledgment The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 7196/06E). The author is grateful for the supports provided by the Research Funds of Hong Kong Polytechnic University (Project No. A-SA29).
123
References [1] Ashby MF, Greer AL. Metallic glasses as structural materials. Scripta Mater 2006;54:321–6. [2] Yavari AR, Lewandowski JJ, Eckert J. Mechanical properties of bulk metallic glasses. MRS Bull 2007;32:635–8. [3] Berry BS. Metallic glasses. Metals Park: ASM; 1978. [4] Zhang ZF, Wu FF, Gao W, Tan J, Wang ZG, Stoica M, et al. Wavy cleavage fracture of bulk metallic glass. Appl Phys Lett 2006;89:251917. [5] Lewandowski JJ, Greer AL. Temperature rise at shear bands in metallic glasses. Nat Mater 2006;5:15–8. [6] Zhang H, Subhash G, Maiti S. Local heating and viscosity drop during shear band evolution in bulk metallic glasses under quasistatic loading. J Appl Phys 2007;102:043519. [7] Zhu A, Shiflet GJ, Poon SJ. Unified approach to atomic transport phenomena in metallic glasses from the bond deficiency perspective. Phys Rev B 2010;81:224209. [8] Zachariason WH. The atomic arrangement in glass. J Am Chem Soc 1932;54:3841–51. [9] Spaepen F, Turnbull D. A mechanism for flow and fracture of metallic glasses. Scripta Metall 1974;8:563. [10] Lekka CE, Ibenskas A, Yavari AR, Evangelakis GA. Tensile deformation accommodation in microscopic metallic glasses via subnanocluster reconstructions. Appl Phys Lett 2007;91:214103. [11] Daub EG, Carlson JM. Stick-slip instabilities and shear strain localization in amorphous materials. Phys Rev E 2009;80:066113. [12] Langer JS. Shear-transformation-zone theory of deformation in metallic glasses. Scripta Mater 2006;54:375–9. [13] Manning ML, Langer JS, Carlson JM. Strain localization in a shear transformation zone model for amorphous solids. Phys Rev E 2007;76:056106. [14] Thamburaja P, Ekambaram R. Coupled thermo-mechanical modelling of bulkmetallic glasses: theory, finite-element simulations and experimental verification. J Mech Phys Solids 2007;55:1236–73. [15] Furukawa A, Tanaka H. Inhomogeneous flow and fracture of glassy materials. Nat Mater 2009;8:601–9. [16] Homer ER, Schuh CA. Mesoscale modeling of amorphous metals by shear transformation zone dynamics. Acta Mater 2009;57:2823–33. [17] Demetriou MD, Johnson WL, Samwer K. Coarse-grained description of localized inelastic deformation in amorphous metals. Appl Phys Lett 2009:94. [18] Onuki A. Nonlinear strain theory of plastic flow in solids. J Phys Condens Matter 2003;15:S891–901. [19] Ghoniem NM. Modeling the dynamics of dislocation ensembles. In: Yip S, editor. Handbook of materials modeling. New York: Springer; 2005. p. 2269–86. [20] Spaepen F. A macroscopic mechanism for steady state inhomogeneous flow in metallic glasses. Acta Metall 1977;25:407415. [21] Argon AS. Plastic deformation in metallic glasses. Acta Metall 1979;27:47–58. [22] Egami T, Levashov VA, Morris JR, Haruyama O. Statistical mechanics of metallic glasses and liquids. Metall Mater Trans 2010;41A:1628–33. [23] Egami T. Formation and deformation of metallic glasses: atomistic theory. Intermetallics 2006;14:882–7. [24] Falk ML, Langer JS, Pechenik L. Thermal effects in the shear-transformationzone theory of amorphous plasticity comparisons to metallic glass data. Phys Rev E 2004;70:011507. [25] Huang R, Suo Z, Prevost J, Nix W. Inhomogeneous deformation in metallic glasses. J Mech Phys Solids 2002;50:1011–27. [26] Dai LH, Yan M, Liu LF, Bai YL. Adiabatic shear banding instability in bulk metallic glasses. Appl Phys Lett 2005;87:141916. [27] Zheng GP. PhD thesis. The Johns Hopkins University; 2001. [28] Zheng GP, Li M. Mesoscopic theory of shear banding and crack propagation in metallic glasses. Phys Rev B 2009;80:104201. [29] Landau LD, Lifshitz EM. Theory of elasticity. 3rd ed. Oxford: Butterworth; 1986. p. 11–8. [30] Lu J, Ravichandran G, Johnson WL. Deformation behavior of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures. Acta Mater 2003;51:3429–43. [31] Zheng GP, Shen Y. Multi-scale modeling of shear banding in iron-based metallic glasses. J Alloy Compd 2010;504S:56–9. [32] Shen Y, Zheng GP. Modeling of shear band multiplication and interaction in metallic glass matrix composites. Scripta Mater 2010;63:181–4. [33] Zheng GP, Shen Y. Simulation of crack propagation in fiber reinforced bulk metallic glasses. Int J Solids Struct 2009;47:320–9. [34] Ganesh P, Widom M. Ab initio simulations of geometrical frustration in supercooled liquid Fe and Fe-based metallic glass. Phys Rev B 2008;77:014205. [35] Conner RD, Johnson WL, Paton NE, Nix WD. Shear bands and cracking of metallic glass plates in bending. J Appl Phys 2003;94:904–11. [36] Delogu F. Molecular dynamics study of size effects in the compression of metallic glass nanowires. Phys Rev B 2009;79:184109. [37] Launey ME, Hofmann DC, Johnson WL, Ritchie RO. Solution to the problem of the poor cyclic fatigue resistance of bulk metallic glasses. PNAS 2009;106:4986–91. [38] Hofmann DC, Suh JY, Wiest A, Duan G, Lind ML, Demetriou MD, et al. Designing metallic glass matrix composites with high toughness and tensile ductility. Nature 2008;451:1085-U3.
124
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
[39] Gurevich S, Karma A, Plapp M, Trivedi R. Phase-field study of threedimensional steady-state growth shapes in directional solidification. Phys Rev E 2010;81:011603. [40] Galenko PK, Reutzel S, Herlach DM, Fries SG, Steinbach I, Apel M. Dendritic solidification in undercooled Ni–Zr–Al melts: experiments and modeling. Acta Mater 2009;57:6166–75.
[41] Zheng GP, Shen Y. Simulation of shear banding and crack propagation in bulk metallic glass matrix composites. Contributed paper in the 17th International Symposium on Metastable, Amorphous and Nanostructured Materials, Zurich, Switzerland; 2010.
Contents lists available at ScienceDirect
Current Opinion in Solid State and Materials Science journal homepage: www.elsevier.com/locate/cossms
Application of phase-field modeling to deformation of metallic glasses G.P. Zheng ⇑ Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong, China
a r t i c l e
i n f o
Article history: Available online 2 February 2011 Keywords: Phase-field modeling Metallic glasses Defects Plastic deformation Fracture
a b s t r a c t Metallic glass or amorphous alloy is a new class of metallic materials with unique mechanical deformation mechanisms. In the past decade, the encouraging progress has been made on revealing the deformation mechanisms and mechanical behaviors of metallic glasses using various theoretical and computational methods. In this review, the application of the recently developed phase-field method to deformation of metallic glasses is emphasized. We first briefly introduce the features of deformation of metallic glasses, following by a review on the modeling methods that are based on kinetic theory or thermodynamics of the deformation defects. Then we focus our attentions on the phase-field modeling which typically treats the localized plastic flow or shear banding as the collective behaviors of structural transformation of the deformation defects. Application of the phase-field modeling to shear banding, local heating, shear band instability in metallic glasses and metallic-glass–matrix composites are discussed. We also highlight the research directions and challenges of phase-field modeling on the deformation of metallic glasses. Ó 2011 Elsevier Ltd. All rights reserved.
1. Introduction Metallic glasses are currently emerging as a novel class of metallic materials with many attractive physical, chemical and mechanical properties for structural and functional applications. Recent advance in the determination of alloy systems with excellent glass forming ability has led to the synthesis of bulk metallic glasses (BMGs) with typical dimensions of inches or more. One of the areas that benefits greatly from this advancement is the understanding of mechanical properties, since BMGs can be made sufficiently large to allow standardized mechanical testing. In the past decade, many well controlled tests have been conducted, such as tension, compression, torsion, fracture toughness tests, fatigue test and indentation test. Compared with conventional crystalline metallic materials, metallic glasses have many attractive mechanical properties [1,2], such as high fracture strength, distinctly higher tensile strength, much lower Young’s modulus, outstanding super-plasticity at their supercooled regions. Unlike crystalline alloys, metallic glasses do not have dislocation defects during the deformation process and the deformation is localized in nature. Therefore, metallic glasses have attracted significant scientific interests besides the technological attentions. Deformation and fracture behaviors of glassy materials (e.g., metallic glasses, silicate glasses, polymers and granular materials), or more generally, disordered or heterogeneous materials, are quite different from those of crystalline solids. Perhaps the best ⇑ Tel.: +852 27666660; fax: +852 23654703. E-mail address: [email protected] 1359-0286/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.cossms.2011.01.004
way to show such differences is to compare the deformation behaviors in metallic glasses and the crystalline metals. In crystalline metal under large deformation, dislocations whose geometries are characterized by well-defined crystallography quantities such as sense vectors and Burgers vectors would proliferate, resulting in significant yielding and plastic deformation. In metallic glasses, dislocations are unlikely to be responsible for the deformation due to the lack of long range translational symmetry in their structure. Instead metallic glasses deform in two distinct modes [3], depending on applied stress, temperature and strain rate. Below the glass transition temperature Tg and above 0.7Tg, the metallic glass deforms homogeneously in that every portion of the solid participates in the flow process. At low temperature (typically T < 0.7Tg) and high strain rates the deformation is heterogeneous, i.e., the plastic deformation is localized or is restricted to occur in shear band with sizes ranging from tens of nm to sub-microns, away from which the deformation is elastic. Although large plastic deformation are developed when metallic glasses are subjected to constrained deformation such as compression, bending or drawing, no appreciable amount of macroscopic plasticity is observed when they are deformed under tension or torsion. The metallic glasses are typical elastic-perfectly plastic materials showing no apparent work hardening. When they are subjected to tensile loading at the yield stress, metallic glasses may have catastrophic failure which occurs along one of the shear bands where the deformation becomes so large that it physically separates the system. The fracture of metallic glass is well characterized by experiments and is found to be much more sophisticate than the brittle fracture in crystalline solid. One of the most interesting phenom-
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
ena in the fracture of metallic glass is the vein patterns presented at the fractured surface [2]. Another important characteristic observed at the fractured surface is the wavy surface [4] with roughness and wavelength of tens of nanometers. These features have played an important role in promoting the theory for the deformation of metallic glasses. For example, the vein pattern prompted the idea of low viscosity and liquid-like disordered structure inside the shear bands. The wavy cracking phenomenon suggested the Griffith theory for brittle fracture could be incorporated with the idea of shear banding instability to describe the fracture of metallic glasses. During the fracture processes, it was reported that temperature near the deformed region could rise up to hundreds or even several thousands Kelvin, and recrystalization and atomistic structure transformation were found inside the shear band [5]. Although they are still in debate, the crystallization and atomistic structure changes in the shear bands seem to support the fact that there is local heating caused by rapid deformation localized in shear band [6]. Because shear banding is the only deformation mechanism operating at temperatures below 0.7Tg, the defects that are responsible for the shear banding must be atomistic and collective. The habitual shear bands generated at the same locations under repeated deformation show the existence of region with higher degrees (lower density) of chemical or structural disorders compared with dense disordered structures [7] which ideally are considered as random close packed (RCP) structure or icosahedral structures in glassy alloys [8]. Naturally, the free-volume defects which are atoms surrounding by excess atomic volume and always exist in metallic glasses [9], can be considered to be the deformation defects that are responsible for the shear banding. However, evidences from experiments reveal that deformation defects or flow defects in metallic glasses are much more complicated than free-volume defects [10]. Especially the structural details and strain/stress states around the deformation defects are still not clear. Nonetheless, the deformation defects should exist most likely in the free-volume regions. With promising structural and functional applications in many areas, the metallic glasses have advanced rapidly in the past decade. It is an urgent need to connect the activities of microscopic deformation defects and the macroscopic deformation behaviors in metallic glasses. In crystalline metals such connection between mechanical properties and dislocation activities is well established and has contributed significantly to our understanding of their deformation behaviors, which plays a key role in the development of these materials for industrial applications. If the mechanical properties of metallic glasses can be directly related to the activities of their deformation defects, such relation can provide references for the design and use of metallic glasses with improved strength, toughness and ductility. Because of the complexity of deformation mechanisms as indicated in the above-mentioned features of deformation in metallic glasses, despite considerable effort has been devoted to these materials, theoretical descriptions of the macroscopic deformation behaviors comprehensively based on the activities of atomistic deformation defects are still not well developed. Firstly, it is because there are a number of puzzles remain to be resolved for the structural characteristics of deformation defects. Secondly, it is caused by the lack of computational model capable of relating the deformation defects with shear banding. The shear bands are typically tens of nanometers in size, it is either too large for the molecular dynamics (MD) simulation to simulate their branching and interaction processes, or too small for the finite element method (FEM) to capture its atomistic scale deformation details. Although the modeling methods [11–17] based on kinetic theory and thermodynamics of the deformation defects successfully describe some features of the deformation of metallic glass, they rely
117
on many structural details and parameters of the deformation defects which turn out to be still unclear. Considering the fact that the atomistic details of the deformation defects are still not well characterized in metallic glasses, phenomenological model could provide us with useful tool to describe the collective activities of the deformation defects during plastic deformation. Phase-field modeling is essentially based on the phenomenological model which is formulized by coarse-grained free-energy functional of the system containing deformation defects, and has been successfully applied to connect the activities of dislocations and macroscopic deformation behaviors in crystalline solids [18,19]. When it is employed to model the deformation behaviors of metallic glasses, the phase-field modeling could show some advantages. First, the free-energy functional of the metallic glass can be constructed to describe the energetics of the deformation defects with the help of sufficient experimental data or atomistic simulation results. Second, phase-field modeling is capable of looking into the mechanisms of shear band formation, propagation and branching since such behaviors occur in the length scales from tens of nanometers to microns. Third, although the length scale of this simulation method is typically mesoscopic, the phenomenological model is based on theoretical and experimental understanding of the atomistic deformation defects of the metallic glasses. Hence it essentially bridges the scientific gap to correlate the atomistic deformation defects and the macroscopic mechanical properties of the metallic glasses. In this paper, we first introduce the recent advance in the theories of plastic flow of metallic glasses in Section 2. Then we focus our attentions on the phase-field modeling which typically treats the localized plastic flow or shear banding as the operation of structural transformation of the deformation defects in Section 3. In Section 4, we discuss some examples of the application of the phase-field modeling to shear banding, adiabatic heating, shear band instability in metallic glasses and metallic-glass–matrix composites. We comment on several research directions and challenges of this method in Section 5.
2. Theories of plastic flow in metallic glass Deformation mechanisms of metallic glasses have been extensively investigated for decades. Several theoretical models have been proposed for the deformation mechanisms in metallic glasses. These theories include free volume model [20,21] and atomic level stress fluctuation [22,23]. As pointed out by Spaepen [20] and Argon [21] in 1970s, deformation of metallic glasses is mainly through the activation of free-volume defects, i.e., either the diffusion or transformation of free volumes when the metallic glasses are deformed. Structural details in the free-volume regions were given by Argon [21] who predicted the deformation defects to be in the free-volume regions with a typical size of 5 atom diameters and are in spherical shape at high temperatures or disk shape at low temperatures. This concept was further developed by Falk and Langer [24] who described the region consisting of activated free volumes as shear transformation zone (STZ). Based on the kinetic theory for free-volume coalescence and annihilation, Spaepen [20] derived the explicit expression of the inhomogeneous flow rate, which is recently used by Huang et al. [25] to analyze the shear localization in one dimension, and Dai et al. [26] to analyze the adiabatic shear banding. Langer and his coworkers [12,13] have incorporated thermodynamics into conventional STZ model [24] and describe accurately the macroscopic deformation behaviors of metallic glasses. In their mean-field continuum model [12], an effective temperature, as an important macroscopic state variable, governs the density of the shear transformation zones. Under shear stress, each STZ deforms
118
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
to sustain the plastic strain, and the STZ can flip in the direction of the applied strain, or revert to their original configuration under the shear stress in the opposite direction. Each STZ interacts with other STZs through continuum fields, and so the rearrangement occurs at a rate R(s)/s0, depending on the stress and a characteristic attempt frequency. STZ can flip at most once in the direction of applied stress, and can be created or annihilated to accommodate plastic flow. The differential equation describing the number density of STZs is given by [13]:
n
s0 n_ ¼ RðsÞn RðsÞn þ C
1
2
e1=v n ;
ð1Þ
where R(±s)/s0 is the rate of switching per STZ as a function of stress s, C is the rate at which energy is dissipated per STZ, and n1e1/v is the steady state density of the STZs in equilibrium. v is defined as the dimensionless effective temperature. Based on the equations derived above, in a simple shear geometry with periodic Lees–Edwards boundary conditions, shear localization is found by numerical integration of the STZ equations. The macroscopic stress–strain curves also show consistency with the experiments and MD simulations. By approximating these nonlinear dynamics, a localization criterion has been obtained from the initial conditions for the effective temperature [13]. Thamburaja and Ekambaram [14] have developed a finitedeformation-based, thermo-mechanically-coupled constitutive model for metallic glasses, which describes the plastic flow of metallic glasses at various temperatures. Based on the principles of the thermodynamics and the concept of micro-force balance, the theory has been formulated and a kinetic equation for the free volume concentration n which considers free volume diffusion and structural relaxation has also been derived as
c_ 0 sn2 c_ 0 p n_ ¼ Dðr2 nÞ þ fc_ ðn nT Þ: sn3 s n3
the free volume vf of an atom is defined by vf = vi v0 and vi is the volume of Voronoi polyhedron of the ith atom; v0 = 4(R0)3/3 and R0 is the radius of the atom. vm is the maximum of vi when complete decohesion occurs. Therefore Pij is the scaled density tensor. v⁄TrPij = 0 denotes no defect or dense random packing which we treat as the reference state. v⁄TrPij = qc represents a structural transformation of the deformation defects and v⁄TrPij qc denotes stable fractured atomistic structure. a = a0 (Tg T)/Tg. Combined Eq. (3) with Eq. (4), the free energy of stress-free metallic glass can be written as
ð2Þ
DF ¼
# Z " @ qij 2 1 dV; aq2ij þ bq3ij þ cq4ij þ j 2 @xj
ð5Þ
where j is the surface energy of the interface between the deformation defect and defect-free region. To write down the free-energy functional (Eq. (5)) for deformation defects, it has been assumed that there is no any long range interaction among sites containing deformation defects. This is true for metallic glass. On the other hand, the concept of deformation defect mentioned above may be slightly different with STZ proposed by Argon [21] and Langer [14]. 3. Phase-field model for fracture in metallic glasses Considering the brittle fracture of metallic glass under tensile deformation, the formation of vein patterns or nano-scale roughness at the fracture surface could result from the collective activities of deformation defects. We may introduce the displacement field u(r) = u(x, y)z as an order parameter to describe the formation of fractured surface, where u(x, y)Z0 represents the displacement of atom from its equilibrium position along the stress direction, and Z0 = (3vm/4)1/3. As shown in Fig. 1, the fractured surface can be characterized by its height hðx; yÞ ¼ hðx; yÞ þ uðx; yÞ.
Being calibrated by fitting the constitutive model to compression experiments, the model can reproduce the stress–strain curves for both the jump-in strain-rate experiment and the simple compression experiment conducted at different temperatures. Furukawa et al. [15] considered the plastic flow in metallic glasses as the viscoelastic deformation which is described by the rheological dynamics and a constitutive model for stress fluctuation. This model is similar to the dynamic equation for the concentration in viscoelastic fluid mixtures and is particular successful in explaining the strain-rate controlled plastic flow in metallic glasses. Zheng [27], and Zheng and Li [28] developed a phenomenological model for deformation defects in metallic glasses. Because the deformation defects consist of sites with free volumes that are more vulnerable to internal rearrangement than their dense random packing counterpart, they can be thermally or mechanically activated. When the accumulation of these activated free-volume sites reaches a critical value, the deformation defect transforms into a structure which is more stable than dense random packing structure. The nucleation of such transformed deformation defect results in shear band. This model simply treats the formation of shear band as a consequence of the structural transformation of deformation defects. The free energy change of a metallic glass due to the introduction of deformation defects is written as
DF ¼
Z
fs dV ¼
Z
½f0 þ K c dV;
ð3Þ
where fs is the free energy density of the defects. Kc is the curvature of the defect surface; f0 is given by:
f0 ¼ aq2ij þ bq3ij þ cq4ij rij qij ;
ð4Þ
where rij is the applied stress tensor; qij = v Pij is the defect density tensor. v⁄ is a coarse-grained average of [vi(r) v0]/(vm v0), where ⁄
Fig. 1. The schematic of model for the simulation on fracture of metallic glass and the definition of crack-opening field h(x, y). The displacement field u near the fractured surface is defined by hðx; yÞ ¼ hðx; yÞ þ uðx; yÞ.
119
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
Here hðx; yÞ is the average of the height of fracture surface and it describes in macroscopic scales the profile of the fracture surface. For example, in most bulk metallic glasses the fracture surface from macroscopic view is a plane inclining with an angle of 40– 45° to the loading direction and can be described by a definite function hðx; yÞ. Another order parameter is the deformation defect density which is now simplified as a scalar q(r) = v⁄TrPij. Taking the effect of deformation defect into account, we can write the local elastic energy as
fe ¼
E 2 E0 ð1 qÞ 2 u ¼ u ¼ eð1 qÞu2 ; 2 2
ð6Þ
where E is the Young’s modulus of metallic glass and E0 is the reference Young’s modulus assumed to be that of its crystalline counterpart. The binding energy between atom cluster can be written as 2
fb ¼ d½1 eu :
ð7Þ
Combine Eqs. (6) and (7), we are able to construct the free-energy functional with two order parameters u(x, y) and q(x, y) for the deformed metallic glass under uniaxial tensile stress rzz:
Fðq; uÞ ¼ DF u þ DF q þ DF int :
ð8Þ 2
Here DF u ¼ eu2 þ d½1 eu þ ujruj2 rzz u;
ð9Þ
1 DF q ¼ aq2 þ bq3 þ cq4 þ jjrqj2 ; 2
ð10Þ
and DF int ¼ equ2 wqjruj2 :
ð11Þ
The |ru|2 term in Eq. (9) is the elastic energy contribution from stretched atomic cluster surface. This surface can form macroscopic crack surface. We write in Eq. (11) the interaction term between displacement field u and deformation defect field as q|ru|2 with a positive coupling strength w. This is because that the crack formation tends to occur in the zone (shear band) that consists of deformation defects. The plastic deformation of system described by Eq. (8) can be studied qualitatively. We are interested in the spatial extension of the plastic deformation on the fractured surface. Shear band formation in plastic deformation is directly connected to the unstable mode in model system described by Eq. (8)–(11). Prior to fracture the instability of atomic displacement field u(x, y) can be represented by the fluctuating field w(x, y): u(x, y, t) = us(x, y) + w(x, y,t). Here w(x, y) is small and large w(x, y) represents microcrack. us(x, y) is the displacement field prior to fracture and is assumed to be a constant u0 in brittle fracture. u0 is equivalent to the strain in the elastic limit. The corresponding fracture strength rF is given by
dF ¼ 0; duu¼u0
1 uðr; tÞeikr dr 2p Z 1 qk ðtÞ ¼ qðr; tÞeikr dr 2p
uk ðtÞ ¼
2
rF ¼ 2½eð1 qÞ þ deu0 u0 :
ð12Þ
The defect density q at rzz < rF is small. We can describe the dynamics of atomic cluster under rF by using kinetic equation,
@u dF ¼ Cu du @t 1 u2 ¼ 2Cu eð1 qÞu þ de u ur2 u wrq ru rF : 2 ð13Þ
Cu is the mobility constant. Neglecting higher order terms than O(q2), O(w2) and O(u2) and using the Fourier transform
ð14Þ
we get the equation for w(t) by substituting Eq. (12) in Eq. (13)
h i @ Wk 2 2 ¼ 2Cu ðe þ dÞ uk þ ðe þ wk Þqk Wk : @t
ð15Þ
The unstable modes to this deformed system exist when
qk ¼
ðe þ dÞ þ uk 2
e þ wk
2
ð16Þ
:
The inverse Fourier transform of Eq. (16) leads to the nontrivial description of deformation defect density inside the deformed zone:
h
pffiffiffiffiffiffiffiffiffiffiffiffiffii
qð~rÞ ¼ C 1 dð~rÞ þ C 2 exp j~rj ðe=wÞ :
ð17Þ
Eq. (17) shows that the structural transformation of deformation defects driven by unstable displacement field is localized. This leads to shear band formation during plastic deformation. The fluctuation of deformation defect density near the shear band can be written as:
rffiffiffiffi e ~ ~ dqðrÞ / exp jrj exp ½j~ rj=r 0 ; w
ð18Þ
pffiffiffiffiffiffiffiffiffi where r 0 ¼ w=e is the correlation length of shear bands. r0 measures the localization of shear bands and is controlled by parameter w. If w = 0, r0 = 0, which means the defect simply do not form shear band at the fractured surface. With increasing interaction strength w, shear band width increases. These analyses could explain the vein pattern observed at the fracture surface.
4. Phase-field modeling of shear banding in metallic glasses As we mentioned in Section 2, deformation of metallic glasses is through shear banding at low temperatures. To describe the shear banding behaviors under different thermal and stress conditions, the effects of temperature and stress–strain relation around the deformation defect should be considered in Eq. (5). The phenomenological model has considered the temperature by setting a = a0 (Tg T)/Tg. Based on the characteristics of the shear banding that the regions other than the shear bands are deformed elastically, the stress–strain relation around the deformation defects is governed by the theory of elasticity [29], and the local strain energy can be written as
1 C ijkl ekl eij 2
¼ l dik djl þ dil djk þ kdik djl ;
e½eij ¼ C ijkl
or
Z
ð19Þ
where eij is the strain tensor defined by the displacement field u through the relation eij = ( o uj/ o xi + o ui/ o xj)/2 and i, j = 1, 2, 3; l is the shear modulus and k is related to the bulk modulus B by B = k + 2 l/3. The response of deformation defects to the local shear is reflected in the defect free energy (Eq. (5)) through expanding the coefficients b and c with the plastic work De = e[eij] e0, where e0 is the strain energy of metallic glass at the elastic limit. The free-energy functional of the deformed metallic glass is thus given explicitly as
Z h i2
j q0 ~_ ~ qj2 dV; F¼ ð20Þ u þ f q; eij þ jr 2 2
a0 b0 c0 a1 2 b1 3
q þ q e½eij e0 ; ð21Þ f q; eij ¼ e eij þ q2 þ q3 þ q4 þ 2 3 4 2 3
120
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
where q0 is the mass density of metallic glass; a0 = a0 b0 T/Tg and a0 , b0 , a0, b0 and c0, a1, b1 are constants. The equations of motions for ~ u and q are described respectively by
sq
@q dF ¼ dq @t
¼ jr2 q a0 q þ b0 q2 þ c0 q3
qða1 þ b1 qÞ e½eij e0 ;
ð22Þ
and
q0
@ 2~ dF a1 b1 u ¼ lr 1 þ q2 u ; ¼ r þ q r~ 2 deij 2 3 @t
ð23Þ
where sq is the characteristic time for defect activation. When the coefficients of Eq. (21) are given by a0 = 4(2 T/Tg) DG, b0 = 32DG, c0 = 16DG, it can be clearly seen that there is a metastable state of the deformation defect (q = qc). The stable defect-free state (q = 0) can be activated either by applied stress or temperature elevation. The activation energy is DG. Usually DG is determined by the strain-rate dependent flow stresses at various temperatures [30]. The surface energy j of the interface between the deformation defect and defect-free region can be estimated from the surface energy of the metallic glasses and the characteristic length of the vein pattern at the fracture surface. Shear banding, shear band branching and local heating in Vitrelloy-1 BMG (Zr41Ti14Ni10Cu12.5Be22.5) are simulated by the phasefield model [28]. Fig. 2c–e shows shear banding and shear band branching under different loading stresses [28]. When the heat conduction equation is included in the modeling, the temperatures near the shear bands are calculated as shown in Fig. 2b. A crack resulting from the shear banding would become unstable when its tip velocity reaches a critical value VC. The phase-field modeling reveals that the maximum velocity that a crack can achieve in Zrbased BMG is 0.63VR, where VR is the Rayleigh wave speed.
In Fe–Si–B metallic glass, the relation between shear band thickness and the local shear stress is obtained by the phase-field modeling [31]. The relation facilitates the determination of macroscopic mechanical properties such as shear strength and fracture toughness of metallic glass from the shear band characteristics. The advantage of phase-field modeling for shear banding can be seen in Fig. 3 which shows the mechanisms of shear band interaction and multiplication. In the sub-micron scales it can be found that the deflection and attraction of two meeting shear bands occur depending on the signs of shear stresses near the approaching parts of these shear bands. Moreover from the simulation study it is found that the integration of two attracting shear bands will result in the generation of secondary shear bands [32]. One of the most useful applications of phase-field modeling is in the simulation of mechanical behaviors of metallic-glass–matrix composites. In the tungsten fibers reinforced Zr-based BMG, the results from phase-field modeling reproduce all the important features of shear banding observed in experiments. Fig. 4 shows the effect of reinforcing fibers on shear banding in the BMG composites [33]. It is found the tensile residual stress at the interface between BMG matrix and the reinforcement is preferred for the purpose of enhancing the plasticity of the BMG composite, while the compressive residual stress would improve its fracture toughness. Furthermore, the residual stress g and atomic bonding condition represented by the defect density q = w0 at the interface can be used to determine the deformation modes in the BMG–matrix composite, as shown in Fig. 5 [41].
5. Future research directions In the previous sections, we have mentioned that there are several theoretical and phenomenological models for the description of shear banding. For metallic glasses, however, the essential features of plastic deformation accommodated by the shear banding
Fig. 2. (a) The model system of a BMG plate. (b) Temperature distribution around the shear band with plastic work to heat conversion ratio b = 0.5. (c–e) Shear bands and crack propagations under stress intensity factors KI = 25 MPa m1/2, 35.4 MPa m1/2 and 100.2 MPa m1/2 respectively. The gray scales correspond to the values of 1 q. The color scale represents temperatures (K). The length bars in (b)–(e) represent 2 lm [28].
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
121
banding, shear band branching and shear band interaction and multiplication, there are still some unsolved issues such as sample size effect on shear localization, brittle-to-ductile transition, mechanisms of fatigue, shear band interaction with microstructures, micromechanics of shear bands leading to fractures. We briefly summarize the challenges and directions of research on the modeling and simulation of deformation of metallic glasses using phase-field method as follows. 5.1. Combination of first-principles calculation with phase-field modeling There are only two coefficients (DG and j) of the phenomenological model for deformation defect which paves a way for the phase-field modeling of deformation in metallic glasses. These coefficients in particular j are difficult to be determined from experiments, while the ab inito simulation [34] could be used to determine DG and j consistent with those defined by the phenomenological model. In Zr-based BMG [28] and Fe–Si–B [31] metallic glasses ribbons, DG determined from first-principles calculation are comparable with those measured from experiments. It has been shown that the phase-field modeling with DG and j determined from first-principles calculations successfully predict the deformation behaviors of Zr-based BMG [28] and Fe–Si–B [31] metallic glasses ribbons. Considering the evidence that the activation volume of the deformation defects in metallic glasses is typically less than 1 nm3, although ab initio simulation usually handles systems with less than one thousand atoms, the structural details, activation energy, surface energy, elastic constants relating to the deformation defects could be readily obtained. Combined with first-principles calculation in constructing the energy landscape of the metallic glass, the phase-field modeling could link the activities of atomistic defects with macroscopic deformation behaviors. Fig. 3. Shear band multiplication and interaction. (a) Formation of one secondary shear band marked by a dash circle; (b) Formation of two secondary shear bands marked by dash circles. The gray scales correspond to the values of 1 q. The color bars ( 870 MPa) are for the contour plots of stresses [32].
have not been fully captured by these models. Although the phasefield modeling has shown its advantages in the simulation of shear
5.2. Implementation of phase-field modeling by considering the shape and size of deformation defects In the free-energy functional given in Sections 3 and 4, the density of deformation defects are simplified as a scalar order parameter q(r) = v⁄TrPij. Because of the existence of structural and chemical inhomogeneity in metallic glasses, the density of deformation defects has to be described by a tensor order parameter
Fig. 4. Shear banding in fiber-reinforced BMG. (a)–(d) Fibers with different lengths L. The length bar represents 0.4 lm. (e)–(h) Fibers with different orientation angles. The length bar represents 0.8 lm [33].
122
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
Fig. 5. Different deformation modes for shear banding and crack propagation in the BMG composite: (a)–(b) shear banding induced cracking inside the reinforcement after the shear band interacts with reinforcement. g = 0, w0 = 0; KI = 24.2 MPa m1/2 (a), and 27.5 MPa m1/2; (b) and (c) propagation of shear band along the reinforcement surface with compressive residual stress g = 435 MPa; w0 = 0; (d) shear banding induced cracking inside the reinforcement with tensile residual stress g = 175 MPa at its surface; w0 = 0; (e) branching of shear band along the reinforcement surface; w0 = 0.8; g = 0. (f) Shear banding at opposite sides of the reinforcement surface; w0 = 1; g = 0. KI = 27.5 MPa m1/2 in (c)–(f). u is the crack field of the tungsten fiber [41].
qij. Although the geometry of deformation defect is still not clear, the coupling between the defect density tensor with local displacement field u may be quantitatively determined by the first-principles calculation. Implementation of phase-field modeling by
considering the shape and size of deformation defects will facilitate the precise connection between the microscopic activities of defects with macroscopic deformation behaviors of metallic glasses under complicate loading conditions
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
5.3. Simulation of the length scale impact on the deformation of metallic glasses Recently, the effects of sample sizes on the mechanical strength and ductility of metallic glasses are of interests. In experiments the length scale impacts on the deformation of metallic glasses are found to occur in a range of tens of nanometers to hundreds of microns [35,36]. Phase-field modeling is the best methods to investigate such impacts and could provide a fundamental insight into such impacts. 5.4. Simulation of crystallization and deformation in in situ formed BMG composites The lack of tensile ductility is one of the major obstacles to the structural application of bulk metallic glasses (BMGs). Recently various experimental investigations have shown the feasibility of making BMG–matrix composites with excellent combination of mechanical properties from some glassy alloy systems [37,38]. Among those BMG–matrix composites some in situ-formed BMGs containing crystalline particles, which are precipitation of a crystalline phase during solidification with rest of the melt forming a glassy–alloy matrix, possess the most desirable mechanical properties, e.g., high fracture strength, impressive fracture toughness and ductility more than 10%. For example Zr-based BMG–matrix composites with an in situ dendritic b phase are found to be among the toughest bulk materials known to date [38]. The crystallization of the dendritic phase is determined by a lot of factors due to the multi-component nature of the BMGs. Phase-field modeling has been proved to be very robust in the simulation of solidification and the formation of dendritic phases [39,40]. When combined with that for dendritic phase modeling, the phase-field modeling for deformation of BMG–matrix composites will provide us with useful simulation approaches to predict and understand their mechanical behaviors, avoiding expensive trail-and-error materials preparation and mechanical tests. 6. Concluding remarks Although the length scale of phase-field modeling is typically mesoscopic, the phenomenological model for the modeling is based on theoretical and experimental understanding of the atomistic deformation defects of the metallic glasses. Hence it essentially bridges the scientific gap to correlate the atomistic deformation defects and the macroscopic mechanical properties of the metallic glasses. Application of the phase-field modeling to shear banding, local heating, shear band instability in metallic glasses and metallic-glass–matrix composites are discussed. Despite many challenges in the characterization of the deformation defects in metallic glasses, the energy landscape of metallic glasses consisting of the deformation defects could be quantitatively determined, facilitating the formulation of the free-energy functional of the system for phase-field modeling. The phase-field modeling could be very useful for the simulation of deformation behaviors of metallic glasses, with examples and research directions given or proposed in this paper. Acknowledgment The work described in this paper was supported by a grant from the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. PolyU 7196/06E). The author is grateful for the supports provided by the Research Funds of Hong Kong Polytechnic University (Project No. A-SA29).
123
References [1] Ashby MF, Greer AL. Metallic glasses as structural materials. Scripta Mater 2006;54:321–6. [2] Yavari AR, Lewandowski JJ, Eckert J. Mechanical properties of bulk metallic glasses. MRS Bull 2007;32:635–8. [3] Berry BS. Metallic glasses. Metals Park: ASM; 1978. [4] Zhang ZF, Wu FF, Gao W, Tan J, Wang ZG, Stoica M, et al. Wavy cleavage fracture of bulk metallic glass. Appl Phys Lett 2006;89:251917. [5] Lewandowski JJ, Greer AL. Temperature rise at shear bands in metallic glasses. Nat Mater 2006;5:15–8. [6] Zhang H, Subhash G, Maiti S. Local heating and viscosity drop during shear band evolution in bulk metallic glasses under quasistatic loading. J Appl Phys 2007;102:043519. [7] Zhu A, Shiflet GJ, Poon SJ. Unified approach to atomic transport phenomena in metallic glasses from the bond deficiency perspective. Phys Rev B 2010;81:224209. [8] Zachariason WH. The atomic arrangement in glass. J Am Chem Soc 1932;54:3841–51. [9] Spaepen F, Turnbull D. A mechanism for flow and fracture of metallic glasses. Scripta Metall 1974;8:563. [10] Lekka CE, Ibenskas A, Yavari AR, Evangelakis GA. Tensile deformation accommodation in microscopic metallic glasses via subnanocluster reconstructions. Appl Phys Lett 2007;91:214103. [11] Daub EG, Carlson JM. Stick-slip instabilities and shear strain localization in amorphous materials. Phys Rev E 2009;80:066113. [12] Langer JS. Shear-transformation-zone theory of deformation in metallic glasses. Scripta Mater 2006;54:375–9. [13] Manning ML, Langer JS, Carlson JM. Strain localization in a shear transformation zone model for amorphous solids. Phys Rev E 2007;76:056106. [14] Thamburaja P, Ekambaram R. Coupled thermo-mechanical modelling of bulkmetallic glasses: theory, finite-element simulations and experimental verification. J Mech Phys Solids 2007;55:1236–73. [15] Furukawa A, Tanaka H. Inhomogeneous flow and fracture of glassy materials. Nat Mater 2009;8:601–9. [16] Homer ER, Schuh CA. Mesoscale modeling of amorphous metals by shear transformation zone dynamics. Acta Mater 2009;57:2823–33. [17] Demetriou MD, Johnson WL, Samwer K. Coarse-grained description of localized inelastic deformation in amorphous metals. Appl Phys Lett 2009:94. [18] Onuki A. Nonlinear strain theory of plastic flow in solids. J Phys Condens Matter 2003;15:S891–901. [19] Ghoniem NM. Modeling the dynamics of dislocation ensembles. In: Yip S, editor. Handbook of materials modeling. New York: Springer; 2005. p. 2269–86. [20] Spaepen F. A macroscopic mechanism for steady state inhomogeneous flow in metallic glasses. Acta Metall 1977;25:407415. [21] Argon AS. Plastic deformation in metallic glasses. Acta Metall 1979;27:47–58. [22] Egami T, Levashov VA, Morris JR, Haruyama O. Statistical mechanics of metallic glasses and liquids. Metall Mater Trans 2010;41A:1628–33. [23] Egami T. Formation and deformation of metallic glasses: atomistic theory. Intermetallics 2006;14:882–7. [24] Falk ML, Langer JS, Pechenik L. Thermal effects in the shear-transformationzone theory of amorphous plasticity comparisons to metallic glass data. Phys Rev E 2004;70:011507. [25] Huang R, Suo Z, Prevost J, Nix W. Inhomogeneous deformation in metallic glasses. J Mech Phys Solids 2002;50:1011–27. [26] Dai LH, Yan M, Liu LF, Bai YL. Adiabatic shear banding instability in bulk metallic glasses. Appl Phys Lett 2005;87:141916. [27] Zheng GP. PhD thesis. The Johns Hopkins University; 2001. [28] Zheng GP, Li M. Mesoscopic theory of shear banding and crack propagation in metallic glasses. Phys Rev B 2009;80:104201. [29] Landau LD, Lifshitz EM. Theory of elasticity. 3rd ed. Oxford: Butterworth; 1986. p. 11–8. [30] Lu J, Ravichandran G, Johnson WL. Deformation behavior of the Zr41.2Ti13.8Cu12.5Ni10Be22.5 bulk metallic glass over a wide range of strain-rates and temperatures. Acta Mater 2003;51:3429–43. [31] Zheng GP, Shen Y. Multi-scale modeling of shear banding in iron-based metallic glasses. J Alloy Compd 2010;504S:56–9. [32] Shen Y, Zheng GP. Modeling of shear band multiplication and interaction in metallic glass matrix composites. Scripta Mater 2010;63:181–4. [33] Zheng GP, Shen Y. Simulation of crack propagation in fiber reinforced bulk metallic glasses. Int J Solids Struct 2009;47:320–9. [34] Ganesh P, Widom M. Ab initio simulations of geometrical frustration in supercooled liquid Fe and Fe-based metallic glass. Phys Rev B 2008;77:014205. [35] Conner RD, Johnson WL, Paton NE, Nix WD. Shear bands and cracking of metallic glass plates in bending. J Appl Phys 2003;94:904–11. [36] Delogu F. Molecular dynamics study of size effects in the compression of metallic glass nanowires. Phys Rev B 2009;79:184109. [37] Launey ME, Hofmann DC, Johnson WL, Ritchie RO. Solution to the problem of the poor cyclic fatigue resistance of bulk metallic glasses. PNAS 2009;106:4986–91. [38] Hofmann DC, Suh JY, Wiest A, Duan G, Lind ML, Demetriou MD, et al. Designing metallic glass matrix composites with high toughness and tensile ductility. Nature 2008;451:1085-U3.
124
G.P. Zheng / Current Opinion in Solid State and Materials Science 15 (2011) 116–124
[39] Gurevich S, Karma A, Plapp M, Trivedi R. Phase-field study of threedimensional steady-state growth shapes in directional solidification. Phys Rev E 2010;81:011603. [40] Galenko PK, Reutzel S, Herlach DM, Fries SG, Steinbach I, Apel M. Dendritic solidification in undercooled Ni–Zr–Al melts: experiments and modeling. Acta Mater 2009;57:6166–75.
[41] Zheng GP, Shen Y. Simulation of shear banding and crack propagation in bulk metallic glass matrix composites. Contributed paper in the 17th International Symposium on Metastable, Amorphous and Nanostructured Materials, Zurich, Switzerland; 2010.