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A controlled Poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis
Computational Materials Science 64 (2012) 84–89
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A controlled Poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis Pan Zhang a, Morad Karimpour a, Daniel Balint a,⇑, Jianguo Lin a, Didier Farrugia b a b
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK Swinden Technology Centre, Tata Steel UK Ltd., Rotherham S60 3AR, UK
a r t i c l e
i n f o
Article history: Received 31 October 2011 Received in revised form 12 February 2012 Accepted 14 February 2012 Available online 20 March 2012 Keywords: Crystal plasticity finite element Voronoi tessellation Cohesive zone model Cohesive junction Micromechanics
a b s t r a c t A novel computational framework is presented for the representation of virtual polycrystalline grain structures with cohesive boundaries for large-scale crystal plasticity finite element (CPFE) analyses. This framework consists of a grain structure generation model and cohesive zone (CZ) representation and junction partitioning scheme. The controlled Poisson Voronoi tessellation (CPVT) model is employed to generate virtual grain structures that are statistically equivalent to metallographic measurements in terms of grain size distribution. In the CPVT model, physical parameters including the mean grain size, a small grain size, a large grain size and the percentage of grains within this range are used to determine the grain size distribution. To study inter-granular crack initiation and evolution using the cohesive zone model, a novel grain boundary representation scheme is proposed for producing cohesive interfaces for Voronoi tessellations and automatically partitioning multiple junctions. The proposed virtual grain structure generation model and cohesive boundary generation method is demonstrated in a crystal plasticity simulation of polycrystal tension. The features of inter-granular crack initiation and propagation are presented and the mechanical response is discussed. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Computational studies of polycrystalline materials employing crystal plasticity theory combined with finite element methods, e.g., crystal plasticity finite element (CPFE) analyses, are widely used to study non-homogeneous plastic deformation, predict fracture and damage evolution, and simulate micro-forming processes. As rigorous studies have revealed, the mean grain size and the grain size distribution have a significant impact on both the macroscopic plastic flow stress and the local stain and stress development [1]. In addition to crystal plasticity, other mechanical properties such as creep rate [2], crack propagation rate [3], and creep damage evolution [4] also strongly relate to the grain size distribution characteristics of a polycrystalline microstructure. Therefore, grain structures must be explicitly modelled in CPFE analyses to understand the evolution of deformation and local damage. In addition, in order to perform micromechanics analyses of inter-granular crack initiation and evolution, there are essentially three geometrical features that need to be represented in a CPFE model: a realistic grain structure, a group of embedded cohesive zone elements and a feasible scheme for partitioning cohesive layer junctions. ⇑ Corresponding author. Tel.: +44 (0) 207 594 7084; fax: +44 (0) 207 594 7010. E-mail address: [email protected] (D. Balint). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2012.02.022
In simulating a micro-forming process, a CPFE model often employs a virtual grain structure containing a large number of grains. In this context, application of an experimentally-determined grain structure is generally difficult. For example, methods by means of sketching a large planar micrograph for a two-dimensional (2D) grain structure are extremely laborious, whilst approaches using a variety of serial-sectioning techniques to obtain a three-dimensional (3D) grain structure are very expensive and time-consuming [5]. The recently developed 3DXRD technique is capable of generating large grain structures efficiently and non-destructively [6], but it requires high-energy synchrotron sources, which are not generally accessible. Numerical models, e.g., the phase field models [7], the Monte–Carlo (Potts) models [8], the cellular automata (CA) [9], and the level-set methods [10], are capable of simulating the microstructural evolution at the grain level. A recent comprehensive review can be found in [11]. However, when using these models to generate a grain structure for a CPFE model, parameters or conditions, e.g., initial nucleation, are difficult to determine, which are necessary to obtain a desired grain structure. Finite element models have been developed to describe grain boundary mechanical behaviour, including aspects such as strengthening [12,13] and damage, for polycrystalline materials. Cohesive zone (CZ) models provide an effective and versatile means for simulating the fracture process [14,15], and have been increasingly employed in finite element analyses of delamination,
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P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
debonding and general crack initiation and propagation in polycrystals [16–18]. In CZ models, cohesive zone elements are embedded in or around materials elements, in which the cohesion is modelled by a traction–separation law. As the cohesive surfaces separate, the stress increases until a limiting value is reached locally, a damage process occurs and the traction decreases to zero. To simulate the grain boundary sliding and debonding, a grain boundary network in a Voronoi tessellation must be represented by non-zero thickness cohesive layers, where a cohesive interface between two adjacent grains can only be meshed by quadrilateral elements and a junction of multiple cohesive layers must be also partitioned by quadrilateral elements with at least one edge coincident with a boundary of an adjacent grain. Due to the geometrical complexity of multiple cohesive junctions, it is very important to provide a robust scheme capable of automatically meshing cohesive junctions, which can handle all possible special and degenerate cases. In this work, a virtual grain structure representation framework is presented for large-scale CPFE analyses of crystal plasticity with grain boundary sliding and debonding. The controlled Poisson Voronoi tessellation (CPVT) model is summarised in the following section, which is used to generate virtual grain structures with grain size distribution control. In Section 3, a novel cohesive zone representation scheme with automated junction partitioning is provided, and the entire virtual grain structure representation framework is presented. In Section 4, the proposed representation framework is demonstrated using a CPFE analysis of uniaxial tension, where two different virtual grain structures with identical grain size distribution properties were simulated. 2. Controlled Poisson Voronoi tessellation
Gamma distribution functions and lognormal functions have been widely used to fit grain size distributions of VTs. Note that, for the one-parameter gamma distribution, given by:
Px;xþdx ¼
cc c1 cx x e dx; CðcÞ
aðcÞ ¼ AðzðcÞ z0 ÞkþnzðcÞ ; c0 c
Under the assumption of a homogeneous crystallisation process described by a VT, the grain morphology and size distributions are completely determined by the initial seed distribution. Therefore, instead of directly evaluating the uniformity of a VT by examining its resultant grain structure, an alternative evaluation can be conducted by characterising properties of the seed lattice. In [23], a non-local parameter, a, was proposed to evaluate a VT’s regularity, defined by:
a ¼ d=dreg ;
ð1Þ
where dreg is the distance between two adjacent seeds in an equivalent regular tessellation, i.e., a regular hexagonal tessellation for a 2D VT or a regular truncated octahedral tessellation for a 3D structure [24,25]. Note that d = dreg, a = 1 represents a regular tessellation, and d < dreg, a < 1, corresponds to an irregular tessellation. As the regularity d decreases, VTs become more disordered. If d = 0, the VT is a completely random tessellation, i.e. the Poisson type.
ð3Þ
where z(c) = c/cm, z0 = c0/cm, and the identification of the constants are explained in [24,25] for 2D- and 3D-VTs, respectively. 2.2. CPVT model In the CPVT model, the desired grain size distribution is not specified directly by the distribution parameter, c, but by a set of physical parameters from metallographic measurements, including the mean grain size Dmean, a small grain size DL, a large grain size DR and the percentage of grains Pr within that range. According to the definition of a one-parameter gamma distribution, Pr is given by:
Z
Dmean =DR
Dmean =DL
2.1. Regularity and grain size distribution
ð2Þ
where the parameter c > 1 and C(c) is the gamma function, there is only a single parameter c. In [23,25], the one-parameter gamma distribution function has been employed to model grain size distribution features for 2D- and 3D-Voronoi tessellations with different regularities. The statistical results revealed a one-to-one relation between the single distribution parameter, c, and the regularity, a. As shown in Fig. 1, while a VT becomes more regular, the shape of the distribution function becomes narrower. This relation was established by a descriptive model given by:
Pr ¼
Poisson Voronoi tessellations have been traditionally used to represent polycrystalline materials in metallurgy [19,20], since such a tessellation is a natural geometrical description of a grain structure that originates from a homogeneous crystallisation process, where grains are randomly and simultaneously nucleated and grain growth is identical for all nuclei (seeds). In addition, other types of Voronoi tessellation (VT) models such as so-called hard-core models and regular hexagonal tessellations [21,22] have also been utilised to represent particular grain structures. Whilst these VTs exhibit variations of both morphology and grain size, they lack a consistent and quantitative approach to evaluate the grain distribution properties, by which methods can be developed to generate VTs with desired grain size distribution characteristics.
x>0
C c c1 cx x e dx: CðcÞ
ð4Þ
Given the values of Dmean, DL, DR and Pr, the distribution parameter, c, can be calculated by a Newton–Raphson search procedure as described in [24]. Once the distribution parameter is obtained, the regularity, a, of an expected virtual grain structure can be defined by the descriptive model in Eq. (3). Eventually, a control parameter, d, defined by:
d ¼ adreg ;
ð5Þ
is used to generate the seed lattice of a VT, whereby after the first seed is produced, each subsequent seed can only be accepted if the minimum distance between it and all the others is greater than the control parameter d. A VT generated using the CPVT model has the property that its grain size distribution is statistically equivalent to that defined by the physical parameter input. The CPVT model can be used to automatically produce large scale statistically equivalent virtual grain structures with high efficiency. A detailed description of the CPVT model can be found in earlier works [24,25]. 3. Cohesive boundary representation and junction partitioning scheme Utilisation of the cohesive zone model to simulate grain boundary sliding and debonding in polycrystals requires that the interface between a pair of grains be geometrically represented by a cohesive layer, to which a traction–separation law can be assigned. Generation of cohesive layers for a VT can be conducted by inward-offsetting of individual grains. In [26] a structured inward-offset algorithm was proposed to implement grain boundary offsetting with a specified non-zero thickness for VTs. This offset algorithm handles grains individually as follows: (1) Determination of the medial axis of an individual grain; (2) dividing the grain into triangular subregions by the medial axis, where each grain edge belongs to a sub-region; (3) for each triangular sub-region, inward offsetting of
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Fig. 1. Illustration of the relation between the VT’s regularity and the grain size distributions, modelled by a one-parameter gamma distribution function.
the corresponding grain boundary; and (4) combining all the offset edges into a closed boundary. A final grain structure with embedded cohesive zone representation can be achieved by collecting all the offset grain boundaries. In FE analyses, the quadrilateral element naturally identifies the normal and tangential directions, and commercial FE software, such as ABAQUS, typically use quadrilateral cohesive elements. Due to the complexity of the boundary network in a VT, a resultant cohesive layer network comprises a large variety of cohesive junctions, such as triple junctions, quadruple junctions, and five-fold junctions. Therefore, a feasible and efficient junction partitioning algorithm using quadrilaterals is critical for CPFE analyses with cohesive grain boundaries using FE/CAE platform. There are basically two reliable schemes to partition various cohesive zone junctions consistently. Fig. 2 shows both of the schemes implemented in the case of a triple junction. The first scheme simply divides the cohesive triple junction from its correlated cohesive layers by linking the ends of the junction, i.e. AB, BC, and CA. Since this isolated junction is a triangular region, it needs to be further subdivided into quadrilateral elements. As shown in Fig. 2a, three perpendicular line segments, O1, O2, and O3, are introduced cutting the triangular region into three quadrilateral elements. Although this scheme is capable of dealing with all types of junctions by adding a central point and related perpendicular line segments, this involves a group of free nodes, e.g., the nodes 1, 2 and 3 for the triple junction case. There-
Fig. 3. Illustration of a cohesive layer obtained by the second junction partitioning scheme.
fore, linear multi-point constraints (MPC) must be used to tie these free nodes in the FE model corresponding to the midpoints of the coincident element sides. In addition to the computational burden of the MPCs, this method lacks a natural rationale for identifying the normal and tangential directions for the junction elements. For the second scheme as shown in Fig. 2b, the cohesive junction is divided by line segments linking the centre and the vertices of the junction, e.g., OA, OB, and OC. A junction region, partitioned using the second scheme, is completely separated into elements that are part of the adjacent cohesive layers; hence the only further operation is the meshing of the layers themselves. A clearer illustration is depicted in Fig. 3, where a cohesive interface between
Fig. 2. Illustration of two junction partition methods.
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P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
Fig. 4. The framework of grain structure generation and cohesive zone representation.
the two grains is separated by edges AA0 , BB0 and the line segments OA, OB, OA0 , OB0 from the two correlated junctions. Typically, for each cohesive layer, an end consists of two line segments, e.g., OA, OB for end 1 in Fig. 3. A general rule for further meshing of this end involves a cut operation, e.g., the line segment B1, and trim operation, e.g., the line segment 2–3. The cut operation reduces the two-edge end into a single edge, and the trim operation makes the single edge more regular. After meshing the ends of a cohesive layer, the remainder has a rectangular shape and can be easily meshed by a single array of quadrilateral elements. Again, using this scheme, elements spanning a junction region are naturally correlated to their respective cohesive layers, allowing the normal and tangential directions to be identified. Fig. 4c depicts the automatic partitioning using the second scheme. It can be seen that all the junctions have been meshed uniformly in terms of both element size and shape, which correspondingly provide a smooth geometrical representation for reliable FE analysis. Fig. 4 summaries the computational framework for grain structure generation and cohesive zone representation. The procedure involves four steps to generate a CPFE model: (1) Assignment of the physical parameters and generation of a virtual grain structure using the CPVT model; (2) generation of cohesive layers for the grain interface network; (3) junction partitioning using the second scheme; and (4) specification of crystallographic orientations and mechanical properties for the virtual grain structure.
Table 1 Material parameters used for the crystal plasticity model, according to the nomenclature in [21]. E (GPa)
a_
n
g0 (MPa)
g1 (MPa)
h0
6.06
10
3
23
150
32
sions with an area fraction of 10% [27]. Inclusions were dispersed randomly in the austenitic matrix. A normal random distribution was used to assign grain and inclusion orientations. Austenitic matrix grains were modelled using the crystal plasticity material for an FCC crystalline structure whereas the MnS inclusions were assumed to have a simple cubic crystal structure. The crystal plasticity material model used is described in [21] and the material constants are presented in Table 1, which were determined based on data obtained from experiments conducted at 1100 °C. A factor of 1.7 was used for the properties of the MnS inclusions [28]. A damage initiation stress of 150 MPa and failure energy of 75 pJ were used for the traction–separation law. A CPFE analysis of plane strain uniaxial tension is performed; as shown in Fig. 5, for each specimen, a displacement, U = 280 lm, was applied to the right edge of the model, and the top and bottom faces are free of constraint. Although the model is subject to zero strain in the out-of-plane direction, the crystal plasticity model is three-dimensional, i.e. the slip systems are not constrained to be
4. Case study In this section, the proposed framework is used to build crystal plasticity finite element models by generating virtual grain structures, producing cohesive zones and automatically partitioning cohesive junctions. The specimen size, i.e., the grain structure domain, was 400 lm 800 lm. The physical parameter input for the CPVT model were: mean grain size, Dmean = 706 lm2, which corresponds to a diameter of 28.6 lm assuming a hexagonal grain shape, a small grain size, DL = 500 lm2, a large grain size, DR = 900 lm2, and a percentage value Pr = 77.7%. These physical parameters correspond to regularity a = 0.6. The generated virtual grain structure represents a sample taken from the centre of a free cutting steel bloom containing Manganese Sulphide (MnS) inclu-
Fig. 5. Schematic diagram of the plane strain uniaxial tension CPFE model.
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P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
Failure strain
0.25
80
strain to failure, εf
True stress, σ (MPa)
90 70 60 Grain structure A
50
Grain structure B
40
Grain structure C
30
Grain structure D
20
Grain structure E
10 0
0.12
0.3
100
0
0.02
0.04
0.06
0.2
0.08
0.15
0.06
0.1
0.04
0.05
0.02
0 0.08
0
Fig. 7. The stress–strain curves corresponding to the five statistically equivalent grain structures.
geometrically plane strain. Five statistically equivalent grain structures were generated based on the above physical parameter input, two of which are shown in Figs. 6a and c. Figs. 6b and d present the corresponding total cumulative plastic strain at the stage of failure, where the crack path is clearly shown in both cases; the near-zero contour areas were verified to be MnS inclusions in all cases. The different but statistically equivalent grain structure generations exhibited varied failure features, although in both cases a path or paths were established perpendicular to the tension axis, i.e. the exact locations and the number of these failures were different, depending on the local grain boundary and grain orientation distributions. The cracks were initiated in the vicinity of an inclusion; the simple cubic inclusions, unlike the FCC grains, have only six slip systems, which in certain orientations cannot accommodate the crystallographic slip required for plastic deformation, resulting in locally high stress levels which in turn cause damage initiation. Despite the differences in the grain structures and the local failure patterns, all simulations exhibit nearly identical macro response, which are presented in Fig. 7. The global response was obtained from the reaction forces measured on the face on which the boundary condition was applied. This indicates that the global response is independent of variations in the grain structures, provided they are statistically equivalent. This feature was not observed in models
100
200
300
400
0 500
Number of grains, n
0.1
True strain, ε
0.1
STD
standard deviation of ε f, σ (εf)
Fig. 6. Two statistically equivalent grain structures and their corresponding deformation: (a) and (c) are the original grain structures, where colour relates to grain orientation; (b) and (d) show the accumulated shear stains corresponding to the CPFE models (a) and (c).
Fig. 8. Variation of failure strain as a function of the number of grains in statistically equivalent grain structures.
with fewer grains and as shown in Fig. 8, the variance of failure strain is reduced with increasing number of simulated grains. Based on Figs. 7 and 8 it can be concluded that with 450 grains or more, the statistically equivalent grain structures have identical macro behaviour. This feature allows the cohesive zone properties to be independently calibrated using experimental data, i.e. any grain structure generation (for a given set of input parameters) can be used to calibrate the parameters of the cohesive zone and the macroscopic response will be repeatable upon re-generation of the grain structure. Further details of the calibration of parameters for cohesive zone traction–separation relations will be presented in a following work. 5. Conclusions In grain-level finite element analyses, e.g., CPFE, there is an increasing demand for generating high-fidelity grain structures uniquely based on physical parameter input. This work proposes a novel scheme for the generation of virtual grain structures with regularity control and the representation of grain boundaries automatically using cohesive zones; a virtual grain structure generated by this scheme has the desired grain size distribution properties and a high-fidelity grain morphology. Two modules are incorporated in this scheme: the CPVT model and the automatic cohesive
P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
layer partitioning model. The CPVT model is used to generate virtual grain structures using a set of physical parameter input, which is adopted from metallographic measurements. The cohesive layer partitioning model is then applied to a resultant VT for producing cohesive zone interfaces surrounding all grains including junctions for the study of grain boundary sliding and debonding. Two novel junction partitioning schemes have been presented for automatically meshing various complex junctions, with the superior choice adopted in the overall framework. This novel scheme is demonstrated by a CPFE analysis of plane strain uniaxial tension. Two virtual grain structures were generated based on the same set of physical parameter input, i.e., two statistically equivalent grain structures were used to study the fracture behaviour of specimens at the scale of grains and interfaces. The CPFE models were built by incorporating the generated grain structures with finite element meshes based on the automatic junction partitioning method. Simulation results showed the capability of the proposed junction partitioning technique in automatically dealing with various complex junction geometries. In addition, the mechanical behaviour suggested that different grain structures with approximately similar, or statistically equivalent grain size distribution properties exhibit nearly identical global stress–strain response to failure despite having local variations in the grain structure, and corresponding variations in the details of crack nucleation and propagation. References [1] S. Berbenni, V. Favier, M. Berveiller, International Journal of Plasticity 23 (1) (2007) 114–142. [2] B.N. Kim, K. Hiraga, K. Morita, I.W. Chen, Philosophical Magazine 85 (20) (2005) 2281–2292. [3] F. Yang, W. Yang, Journal of the Mechanics and Physics of Solids 57 (2) (2009) 305–324. [4] T. Yu, H. Shi, Journal of Physics D: Applied Physics 43 (2010) 165401. [5] A.C. Lewis, A.B. Geltmacher, Scripta Materialia 55 (1) (2006) 81–85.
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[6] H.F. Poulsen, 3DXRD – a new probe for materials science, Ph.D. thesis, Risø National Laboratory, Roskilde, Denmark (2004). [7] D. Fan, C. Geng, L.Q. Chen, Acta materialia 45 (3) (1997) 1115–1126. [8] K. Mehnert, P. Klimanek, Computational Materials Science 9 (1–2) (1997) 261– 266. [9] D. Raabe, Philosophical Magazine A 79 (10) (1999) 2339–2358. [10] M. Bernacki, H. Resk, T. Coupez, R.E. Logé, Modelling and Simulation in Materials Science and Engineering 17 (6) (2009) 064006. [11] L. Madej, J. Szyndler, K. Pasternak, M. Przenzak, L. Rauch, Tools for generation of digital material representations, in: Multi Scale Modeling of Microstructure Deformation in Material Processing, Materials Science & Technology 2011, 2011. [12] K. Kurzydlowski, Bulletin of the Polish Academy of Sciences: Technical Sciences 58 (2) (2010) 217–226. [13] R. Dobosz, M. Lewandowska, K. Kurzydlowski, Computational Materials Science 53 (1) (2012) 286–293. [14] A. Needleman, Journal of Applied Mechanics 54 (1987) 525. [15] J.R. Rice, J.S. Wang, Materials Science and Engineering: A 107 (1989) 23–40. [16] K. Helms, D.H. Allen, L.D. Hurtado, International Journal of Fracture 95 (1) (1999) 175–194. [17] P.D. Zavattieri, H.D. Espinosa, Acta Materialia 49 (20) (2001) 4291–4311. [18] S. Marfia, E. Sacco, International Journal for Numerical Methods in Engineering 57 (11) (2003) 1577–1602. [19] B.N. Boots, D.J. Murdoch, Computers and Geosciences 9 (3) (1983) 351–365. [20] D. Weaire, J.P. Kermode, J. Wejchert, Philosophical Magazine B 53 (5) (1986) 101–105. [21] F.J. Harewood, P.E. McHugh, Computational Materials Science 37 (4) (2006) 442–453. [22] V. Silberschmidt, E. Werner, C. Messner, Journal of Engineering Materials and Technology 125 (2003) 56–64. [23] H. Zhu, S. Thorpe, A. Windle, Philosophical Magazine A 81 (12) (2001) 2765– 2783. [24] P. Zhang, D. Balint, J. Lin, Computational Materials Science 50 (10) (2011) 2854–2864. [25] P. Zhang, M. Karimpour, D. Balint, J. Lin, Three-dimensional virtual grain structure generation with grain size control, Submitted to Mechanics of Materials. [26] P. Zhang, M. Karimpour, D. Balint, J. Lin, Cohesive zone representation and junction partitioning for crystal plasticity analyses, Submitted to International Journal for Numerical Methods in Engineering. [27] A. Foster, J. Lin, D. Farrugia, T. Dean, The Journal of Strain Analysis for Engineering Design 42 (4) (2007) 227–235. [28] J. Gaskell, F. Dunne, D. Farrugia, J. Lin, Journal of Multiscale Modelling 1 (1) (2009) 1–19.
Contents lists available at SciVerse ScienceDirect
Computational Materials Science journal homepage: www.elsevier.com/locate/commatsci
A controlled Poisson Voronoi tessellation for grain and cohesive boundary generation applied to crystal plasticity analysis Pan Zhang a, Morad Karimpour a, Daniel Balint a,⇑, Jianguo Lin a, Didier Farrugia b a b
Department of Mechanical Engineering, Imperial College, London SW7 2AZ, UK Swinden Technology Centre, Tata Steel UK Ltd., Rotherham S60 3AR, UK
a r t i c l e
i n f o
Article history: Received 31 October 2011 Received in revised form 12 February 2012 Accepted 14 February 2012 Available online 20 March 2012 Keywords: Crystal plasticity finite element Voronoi tessellation Cohesive zone model Cohesive junction Micromechanics
a b s t r a c t A novel computational framework is presented for the representation of virtual polycrystalline grain structures with cohesive boundaries for large-scale crystal plasticity finite element (CPFE) analyses. This framework consists of a grain structure generation model and cohesive zone (CZ) representation and junction partitioning scheme. The controlled Poisson Voronoi tessellation (CPVT) model is employed to generate virtual grain structures that are statistically equivalent to metallographic measurements in terms of grain size distribution. In the CPVT model, physical parameters including the mean grain size, a small grain size, a large grain size and the percentage of grains within this range are used to determine the grain size distribution. To study inter-granular crack initiation and evolution using the cohesive zone model, a novel grain boundary representation scheme is proposed for producing cohesive interfaces for Voronoi tessellations and automatically partitioning multiple junctions. The proposed virtual grain structure generation model and cohesive boundary generation method is demonstrated in a crystal plasticity simulation of polycrystal tension. The features of inter-granular crack initiation and propagation are presented and the mechanical response is discussed. Ó 2012 Elsevier B.V. All rights reserved.
1. Introduction Computational studies of polycrystalline materials employing crystal plasticity theory combined with finite element methods, e.g., crystal plasticity finite element (CPFE) analyses, are widely used to study non-homogeneous plastic deformation, predict fracture and damage evolution, and simulate micro-forming processes. As rigorous studies have revealed, the mean grain size and the grain size distribution have a significant impact on both the macroscopic plastic flow stress and the local stain and stress development [1]. In addition to crystal plasticity, other mechanical properties such as creep rate [2], crack propagation rate [3], and creep damage evolution [4] also strongly relate to the grain size distribution characteristics of a polycrystalline microstructure. Therefore, grain structures must be explicitly modelled in CPFE analyses to understand the evolution of deformation and local damage. In addition, in order to perform micromechanics analyses of inter-granular crack initiation and evolution, there are essentially three geometrical features that need to be represented in a CPFE model: a realistic grain structure, a group of embedded cohesive zone elements and a feasible scheme for partitioning cohesive layer junctions. ⇑ Corresponding author. Tel.: +44 (0) 207 594 7084; fax: +44 (0) 207 594 7010. E-mail address: [email protected] (D. Balint). 0927-0256/$ - see front matter Ó 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.commatsci.2012.02.022
In simulating a micro-forming process, a CPFE model often employs a virtual grain structure containing a large number of grains. In this context, application of an experimentally-determined grain structure is generally difficult. For example, methods by means of sketching a large planar micrograph for a two-dimensional (2D) grain structure are extremely laborious, whilst approaches using a variety of serial-sectioning techniques to obtain a three-dimensional (3D) grain structure are very expensive and time-consuming [5]. The recently developed 3DXRD technique is capable of generating large grain structures efficiently and non-destructively [6], but it requires high-energy synchrotron sources, which are not generally accessible. Numerical models, e.g., the phase field models [7], the Monte–Carlo (Potts) models [8], the cellular automata (CA) [9], and the level-set methods [10], are capable of simulating the microstructural evolution at the grain level. A recent comprehensive review can be found in [11]. However, when using these models to generate a grain structure for a CPFE model, parameters or conditions, e.g., initial nucleation, are difficult to determine, which are necessary to obtain a desired grain structure. Finite element models have been developed to describe grain boundary mechanical behaviour, including aspects such as strengthening [12,13] and damage, for polycrystalline materials. Cohesive zone (CZ) models provide an effective and versatile means for simulating the fracture process [14,15], and have been increasingly employed in finite element analyses of delamination,
85
P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
debonding and general crack initiation and propagation in polycrystals [16–18]. In CZ models, cohesive zone elements are embedded in or around materials elements, in which the cohesion is modelled by a traction–separation law. As the cohesive surfaces separate, the stress increases until a limiting value is reached locally, a damage process occurs and the traction decreases to zero. To simulate the grain boundary sliding and debonding, a grain boundary network in a Voronoi tessellation must be represented by non-zero thickness cohesive layers, where a cohesive interface between two adjacent grains can only be meshed by quadrilateral elements and a junction of multiple cohesive layers must be also partitioned by quadrilateral elements with at least one edge coincident with a boundary of an adjacent grain. Due to the geometrical complexity of multiple cohesive junctions, it is very important to provide a robust scheme capable of automatically meshing cohesive junctions, which can handle all possible special and degenerate cases. In this work, a virtual grain structure representation framework is presented for large-scale CPFE analyses of crystal plasticity with grain boundary sliding and debonding. The controlled Poisson Voronoi tessellation (CPVT) model is summarised in the following section, which is used to generate virtual grain structures with grain size distribution control. In Section 3, a novel cohesive zone representation scheme with automated junction partitioning is provided, and the entire virtual grain structure representation framework is presented. In Section 4, the proposed representation framework is demonstrated using a CPFE analysis of uniaxial tension, where two different virtual grain structures with identical grain size distribution properties were simulated. 2. Controlled Poisson Voronoi tessellation
Gamma distribution functions and lognormal functions have been widely used to fit grain size distributions of VTs. Note that, for the one-parameter gamma distribution, given by:
Px;xþdx ¼
cc c1 cx x e dx; CðcÞ
aðcÞ ¼ AðzðcÞ z0 ÞkþnzðcÞ ; c0 c
Under the assumption of a homogeneous crystallisation process described by a VT, the grain morphology and size distributions are completely determined by the initial seed distribution. Therefore, instead of directly evaluating the uniformity of a VT by examining its resultant grain structure, an alternative evaluation can be conducted by characterising properties of the seed lattice. In [23], a non-local parameter, a, was proposed to evaluate a VT’s regularity, defined by:
a ¼ d=dreg ;
ð1Þ
where dreg is the distance between two adjacent seeds in an equivalent regular tessellation, i.e., a regular hexagonal tessellation for a 2D VT or a regular truncated octahedral tessellation for a 3D structure [24,25]. Note that d = dreg, a = 1 represents a regular tessellation, and d < dreg, a < 1, corresponds to an irregular tessellation. As the regularity d decreases, VTs become more disordered. If d = 0, the VT is a completely random tessellation, i.e. the Poisson type.
ð3Þ
where z(c) = c/cm, z0 = c0/cm, and the identification of the constants are explained in [24,25] for 2D- and 3D-VTs, respectively. 2.2. CPVT model In the CPVT model, the desired grain size distribution is not specified directly by the distribution parameter, c, but by a set of physical parameters from metallographic measurements, including the mean grain size Dmean, a small grain size DL, a large grain size DR and the percentage of grains Pr within that range. According to the definition of a one-parameter gamma distribution, Pr is given by:
Z
Dmean =DR
Dmean =DL
2.1. Regularity and grain size distribution
ð2Þ
where the parameter c > 1 and C(c) is the gamma function, there is only a single parameter c. In [23,25], the one-parameter gamma distribution function has been employed to model grain size distribution features for 2D- and 3D-Voronoi tessellations with different regularities. The statistical results revealed a one-to-one relation between the single distribution parameter, c, and the regularity, a. As shown in Fig. 1, while a VT becomes more regular, the shape of the distribution function becomes narrower. This relation was established by a descriptive model given by:
Pr ¼
Poisson Voronoi tessellations have been traditionally used to represent polycrystalline materials in metallurgy [19,20], since such a tessellation is a natural geometrical description of a grain structure that originates from a homogeneous crystallisation process, where grains are randomly and simultaneously nucleated and grain growth is identical for all nuclei (seeds). In addition, other types of Voronoi tessellation (VT) models such as so-called hard-core models and regular hexagonal tessellations [21,22] have also been utilised to represent particular grain structures. Whilst these VTs exhibit variations of both morphology and grain size, they lack a consistent and quantitative approach to evaluate the grain distribution properties, by which methods can be developed to generate VTs with desired grain size distribution characteristics.
x>0
C c c1 cx x e dx: CðcÞ
ð4Þ
Given the values of Dmean, DL, DR and Pr, the distribution parameter, c, can be calculated by a Newton–Raphson search procedure as described in [24]. Once the distribution parameter is obtained, the regularity, a, of an expected virtual grain structure can be defined by the descriptive model in Eq. (3). Eventually, a control parameter, d, defined by:
d ¼ adreg ;
ð5Þ
is used to generate the seed lattice of a VT, whereby after the first seed is produced, each subsequent seed can only be accepted if the minimum distance between it and all the others is greater than the control parameter d. A VT generated using the CPVT model has the property that its grain size distribution is statistically equivalent to that defined by the physical parameter input. The CPVT model can be used to automatically produce large scale statistically equivalent virtual grain structures with high efficiency. A detailed description of the CPVT model can be found in earlier works [24,25]. 3. Cohesive boundary representation and junction partitioning scheme Utilisation of the cohesive zone model to simulate grain boundary sliding and debonding in polycrystals requires that the interface between a pair of grains be geometrically represented by a cohesive layer, to which a traction–separation law can be assigned. Generation of cohesive layers for a VT can be conducted by inward-offsetting of individual grains. In [26] a structured inward-offset algorithm was proposed to implement grain boundary offsetting with a specified non-zero thickness for VTs. This offset algorithm handles grains individually as follows: (1) Determination of the medial axis of an individual grain; (2) dividing the grain into triangular subregions by the medial axis, where each grain edge belongs to a sub-region; (3) for each triangular sub-region, inward offsetting of
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Fig. 1. Illustration of the relation between the VT’s regularity and the grain size distributions, modelled by a one-parameter gamma distribution function.
the corresponding grain boundary; and (4) combining all the offset edges into a closed boundary. A final grain structure with embedded cohesive zone representation can be achieved by collecting all the offset grain boundaries. In FE analyses, the quadrilateral element naturally identifies the normal and tangential directions, and commercial FE software, such as ABAQUS, typically use quadrilateral cohesive elements. Due to the complexity of the boundary network in a VT, a resultant cohesive layer network comprises a large variety of cohesive junctions, such as triple junctions, quadruple junctions, and five-fold junctions. Therefore, a feasible and efficient junction partitioning algorithm using quadrilaterals is critical for CPFE analyses with cohesive grain boundaries using FE/CAE platform. There are basically two reliable schemes to partition various cohesive zone junctions consistently. Fig. 2 shows both of the schemes implemented in the case of a triple junction. The first scheme simply divides the cohesive triple junction from its correlated cohesive layers by linking the ends of the junction, i.e. AB, BC, and CA. Since this isolated junction is a triangular region, it needs to be further subdivided into quadrilateral elements. As shown in Fig. 2a, three perpendicular line segments, O1, O2, and O3, are introduced cutting the triangular region into three quadrilateral elements. Although this scheme is capable of dealing with all types of junctions by adding a central point and related perpendicular line segments, this involves a group of free nodes, e.g., the nodes 1, 2 and 3 for the triple junction case. There-
Fig. 3. Illustration of a cohesive layer obtained by the second junction partitioning scheme.
fore, linear multi-point constraints (MPC) must be used to tie these free nodes in the FE model corresponding to the midpoints of the coincident element sides. In addition to the computational burden of the MPCs, this method lacks a natural rationale for identifying the normal and tangential directions for the junction elements. For the second scheme as shown in Fig. 2b, the cohesive junction is divided by line segments linking the centre and the vertices of the junction, e.g., OA, OB, and OC. A junction region, partitioned using the second scheme, is completely separated into elements that are part of the adjacent cohesive layers; hence the only further operation is the meshing of the layers themselves. A clearer illustration is depicted in Fig. 3, where a cohesive interface between
Fig. 2. Illustration of two junction partition methods.
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Fig. 4. The framework of grain structure generation and cohesive zone representation.
the two grains is separated by edges AA0 , BB0 and the line segments OA, OB, OA0 , OB0 from the two correlated junctions. Typically, for each cohesive layer, an end consists of two line segments, e.g., OA, OB for end 1 in Fig. 3. A general rule for further meshing of this end involves a cut operation, e.g., the line segment B1, and trim operation, e.g., the line segment 2–3. The cut operation reduces the two-edge end into a single edge, and the trim operation makes the single edge more regular. After meshing the ends of a cohesive layer, the remainder has a rectangular shape and can be easily meshed by a single array of quadrilateral elements. Again, using this scheme, elements spanning a junction region are naturally correlated to their respective cohesive layers, allowing the normal and tangential directions to be identified. Fig. 4c depicts the automatic partitioning using the second scheme. It can be seen that all the junctions have been meshed uniformly in terms of both element size and shape, which correspondingly provide a smooth geometrical representation for reliable FE analysis. Fig. 4 summaries the computational framework for grain structure generation and cohesive zone representation. The procedure involves four steps to generate a CPFE model: (1) Assignment of the physical parameters and generation of a virtual grain structure using the CPVT model; (2) generation of cohesive layers for the grain interface network; (3) junction partitioning using the second scheme; and (4) specification of crystallographic orientations and mechanical properties for the virtual grain structure.
Table 1 Material parameters used for the crystal plasticity model, according to the nomenclature in [21]. E (GPa)
a_
n
g0 (MPa)
g1 (MPa)
h0
6.06
10
3
23
150
32
sions with an area fraction of 10% [27]. Inclusions were dispersed randomly in the austenitic matrix. A normal random distribution was used to assign grain and inclusion orientations. Austenitic matrix grains were modelled using the crystal plasticity material for an FCC crystalline structure whereas the MnS inclusions were assumed to have a simple cubic crystal structure. The crystal plasticity material model used is described in [21] and the material constants are presented in Table 1, which were determined based on data obtained from experiments conducted at 1100 °C. A factor of 1.7 was used for the properties of the MnS inclusions [28]. A damage initiation stress of 150 MPa and failure energy of 75 pJ were used for the traction–separation law. A CPFE analysis of plane strain uniaxial tension is performed; as shown in Fig. 5, for each specimen, a displacement, U = 280 lm, was applied to the right edge of the model, and the top and bottom faces are free of constraint. Although the model is subject to zero strain in the out-of-plane direction, the crystal plasticity model is three-dimensional, i.e. the slip systems are not constrained to be
4. Case study In this section, the proposed framework is used to build crystal plasticity finite element models by generating virtual grain structures, producing cohesive zones and automatically partitioning cohesive junctions. The specimen size, i.e., the grain structure domain, was 400 lm 800 lm. The physical parameter input for the CPVT model were: mean grain size, Dmean = 706 lm2, which corresponds to a diameter of 28.6 lm assuming a hexagonal grain shape, a small grain size, DL = 500 lm2, a large grain size, DR = 900 lm2, and a percentage value Pr = 77.7%. These physical parameters correspond to regularity a = 0.6. The generated virtual grain structure represents a sample taken from the centre of a free cutting steel bloom containing Manganese Sulphide (MnS) inclu-
Fig. 5. Schematic diagram of the plane strain uniaxial tension CPFE model.
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Failure strain
0.25
80
strain to failure, εf
True stress, σ (MPa)
90 70 60 Grain structure A
50
Grain structure B
40
Grain structure C
30
Grain structure D
20
Grain structure E
10 0
0.12
0.3
100
0
0.02
0.04
0.06
0.2
0.08
0.15
0.06
0.1
0.04
0.05
0.02
0 0.08
0
Fig. 7. The stress–strain curves corresponding to the five statistically equivalent grain structures.
geometrically plane strain. Five statistically equivalent grain structures were generated based on the above physical parameter input, two of which are shown in Figs. 6a and c. Figs. 6b and d present the corresponding total cumulative plastic strain at the stage of failure, where the crack path is clearly shown in both cases; the near-zero contour areas were verified to be MnS inclusions in all cases. The different but statistically equivalent grain structure generations exhibited varied failure features, although in both cases a path or paths were established perpendicular to the tension axis, i.e. the exact locations and the number of these failures were different, depending on the local grain boundary and grain orientation distributions. The cracks were initiated in the vicinity of an inclusion; the simple cubic inclusions, unlike the FCC grains, have only six slip systems, which in certain orientations cannot accommodate the crystallographic slip required for plastic deformation, resulting in locally high stress levels which in turn cause damage initiation. Despite the differences in the grain structures and the local failure patterns, all simulations exhibit nearly identical macro response, which are presented in Fig. 7. The global response was obtained from the reaction forces measured on the face on which the boundary condition was applied. This indicates that the global response is independent of variations in the grain structures, provided they are statistically equivalent. This feature was not observed in models
100
200
300
400
0 500
Number of grains, n
0.1
True strain, ε
0.1
STD
standard deviation of ε f, σ (εf)
Fig. 6. Two statistically equivalent grain structures and their corresponding deformation: (a) and (c) are the original grain structures, where colour relates to grain orientation; (b) and (d) show the accumulated shear stains corresponding to the CPFE models (a) and (c).
Fig. 8. Variation of failure strain as a function of the number of grains in statistically equivalent grain structures.
with fewer grains and as shown in Fig. 8, the variance of failure strain is reduced with increasing number of simulated grains. Based on Figs. 7 and 8 it can be concluded that with 450 grains or more, the statistically equivalent grain structures have identical macro behaviour. This feature allows the cohesive zone properties to be independently calibrated using experimental data, i.e. any grain structure generation (for a given set of input parameters) can be used to calibrate the parameters of the cohesive zone and the macroscopic response will be repeatable upon re-generation of the grain structure. Further details of the calibration of parameters for cohesive zone traction–separation relations will be presented in a following work. 5. Conclusions In grain-level finite element analyses, e.g., CPFE, there is an increasing demand for generating high-fidelity grain structures uniquely based on physical parameter input. This work proposes a novel scheme for the generation of virtual grain structures with regularity control and the representation of grain boundaries automatically using cohesive zones; a virtual grain structure generated by this scheme has the desired grain size distribution properties and a high-fidelity grain morphology. Two modules are incorporated in this scheme: the CPVT model and the automatic cohesive
P. Zhang et al. / Computational Materials Science 64 (2012) 84–89
layer partitioning model. The CPVT model is used to generate virtual grain structures using a set of physical parameter input, which is adopted from metallographic measurements. The cohesive layer partitioning model is then applied to a resultant VT for producing cohesive zone interfaces surrounding all grains including junctions for the study of grain boundary sliding and debonding. Two novel junction partitioning schemes have been presented for automatically meshing various complex junctions, with the superior choice adopted in the overall framework. This novel scheme is demonstrated by a CPFE analysis of plane strain uniaxial tension. Two virtual grain structures were generated based on the same set of physical parameter input, i.e., two statistically equivalent grain structures were used to study the fracture behaviour of specimens at the scale of grains and interfaces. The CPFE models were built by incorporating the generated grain structures with finite element meshes based on the automatic junction partitioning method. Simulation results showed the capability of the proposed junction partitioning technique in automatically dealing with various complex junction geometries. In addition, the mechanical behaviour suggested that different grain structures with approximately similar, or statistically equivalent grain size distribution properties exhibit nearly identical global stress–strain response to failure despite having local variations in the grain structure, and corresponding variations in the details of crack nucleation and propagation. References [1] S. Berbenni, V. Favier, M. Berveiller, International Journal of Plasticity 23 (1) (2007) 114–142. [2] B.N. Kim, K. Hiraga, K. Morita, I.W. Chen, Philosophical Magazine 85 (20) (2005) 2281–2292. [3] F. Yang, W. Yang, Journal of the Mechanics and Physics of Solids 57 (2) (2009) 305–324. [4] T. Yu, H. Shi, Journal of Physics D: Applied Physics 43 (2010) 165401. [5] A.C. Lewis, A.B. Geltmacher, Scripta Materialia 55 (1) (2006) 81–85.
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