Microstructural simulation of adiabatic shear band formation in AISI 4340 steel using Voronoi Tessellation

Computational Materials Science 109 (2015) 157–171

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Microstructural simulation of adiabatic shear band formation in AISI 4340 steel using Voronoi Tessellation Ioannis Polyzois ⇑, Nabil Bassim Department of Mechanical Engineering, University of Manitoba, 15 Gillson St, R3T 5V6 Winnipeg, MB, Canada

a r t i c l e

i n f o

Article history: Received 23 October 2014 Received in revised form 23 June 2015 Accepted 28 June 2015 Available online 24 July 2015 Keywords: Adiabatic shear bands Grains Smooth Particle Hydrodynamics Grain elements Voronoi Tessellation Impact mechanics

a b s t r a c t Using the stress strain response data obtained from an experimental investigation carried out by the first author (Polyzois, 2014), a finite element model was developed in Matlab and explicit FEA software ANSYS and LSDYNA to simulate Adiabatic Shear Band (ASB) formation in AISI 4340 steel on the microstructural level under high strain rate impact. This FEA model uses the Voronoi Tessellation to generate geometry simulating the microstructure of heat treated AISI 4340 steel in a 2D axi-symmetric cross-section based on the prior austenite grain size and 2D geometric orientation of martensitic lath blocks. It also incorporates the microstructural material inhomogeneity between the grains and the grain boundaries. The model was discretized using the meshless Smooth Particle Hydrodynamics (SPH) method and utilizes the Johnson–Cook plasticity model parameters for various heat treatments of the steel as well as a rupture failure criterion. The model provides a good representation of the kinematics of ASB formation based on grain refinement showing a progressive reorientation and elongation of the grains in the direction of shear in 2D. Severe strain localization and ASB formation were shown to nucleate at the grain boundaries of the elongated grains, creating micro-voids, which grew and propagated as micro-cracks through the grains, separating them into smaller sizes. Under continued deformation, the grains continued to elongate and refine. Final refined grain size within the band is represented by the size of the smallest cluster of intact SPH particles. Ó 2015 Elsevier B.V. All rights reserved.

1. Introduction In this paper the development of a FEA model that can be used to simulate grain deformation behavior and ASB formation in AISI 4340 steel is presented. It accounts for the inhomogeneous physical and geometrical properties of steel at the microscopic level; i.e. grain size and grain composition, which are dependent on the heat treatment and pre-impact conditions of the metal. The geometry and properties of the material constituents were observed through an experimental investigation [1,39]. The simulation model was developed using MATLAB [2] and explicit dynamics software LSDYNA [3] and represents a 2D axi-symmetric area of grains through the cross section of a specimen. This area is tessellated by randomly oriented geometries representing the prior austenite grains whose initial size was based on the heat treatment temperature. The grain geometries were generated using a Voronoi tessellation [4–6]. Each grain was further subdivided into geometrical representations of parallel blocks of lath martensite, representative of the microstructure of quenched and ⇑ Corresponding author. Tel.: +1 204 510 2760. E-mail address: [email protected] (I. Polyzois). http://dx.doi.org/10.1016/j.commatsci.2015.06.041 0927-0256/Ó 2015 Elsevier B.V. All rights reserved.

tempered steel. The size and number of blocks varied according to heat treatment and tempering temperatures. The models were meshed using the mesh-free Smooth Particle Hydrodynamics (SPH) formulation [7–9], which allows for large localized deformation. The Johnson–Cook plasticity model parameters [10–12] for each heat treatment were calculated and implemented along with the rupture failure criterion that defines shear failure. The prior austenite grain boundaries were modeled by a thin layer separating the prior austenite grains and assigned physical properties having a higher hardness than the surrounding grains to account for the segregation of hard solute particles to the grain boundaries during heat treatment. The model was impacted using a rigid wall impactor of fixed mass at various initial velocities. The resulting grain deformation and ASB formation were compared to observations and predictions from experimentation. 2. Background 2.1. Modeling of ASB formation On a microscopic scale, polycrystalline metals are made up of grains or crystal structures with varying orientation separated by

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grain boundaries. These grains may contain impurities such as precipitates or secondary phases which may segregate to the grain boundary or to dislocations or can form carbides during heat treatment. The grain boundaries often have a higher concentration of segregant than the bulk grain since they are sites of lower energy [13]. Grain boundary segregation can lead to the formation of micro-cracks due to the increase in hardness and brittleness caused by the hard solute particles. Phenomena such as the formation of adiabatic shear bands (ASBs), which form in polycrystalline metals such as steel at particularly high strain rates and large strains are a discontinuous severely localized deformation which may initiate from these inhomogeneities. Thus the formation of ASBs and the mechanisms behind them are strongly driven by microstructure. The mechanical behavior of metals at the microscopic level is strongly influenced by chemical composition, phase distribution, and microstructural geometry such as grain size, shape, and orientation [6]. Modeling the microstructural mechanism of ASB formation must consider these factors. Several researchers have used macroscopic homogenous material modeling to simulate ASB formation in metals [14–22]. In these models ASB formation is assumed to be synonymous with material failure defined by a loss of load carrying capacity (stress collapse). Since metals contain a variety of characteristic defects and phase compositions induced though manufacturing and heat treatment processes it is necessary to consider these factors to properly simulate ASB formation. Models based on the importance of microstructure [23–25] and microscopic failure [26] that combine the advantages of continuum mechanics and classic fracture mechanics have been developed. The most prominent of such models is the GTN model first proposed by Gurson [27] and extended by Tvergaard and Needleman [28]. In this model the metallic materials are considered porous on the microstructural level. Tvergaard and Needleman showed that voids nucleate and grow until a local plastic fracture of the intervoid matrix occurs, which causes the coalescence of neighboring voids. Xue [29] expanded the GTN model to incorporate a void shearing mechanism of damage. This model also incorporates a damage model based on the void evolution in localized shear banding first proposed by McClintock et al. [30]. Several researchers have also shown that ASB formation may be simulated on the microscopic scale by modeling dynamic recrystallization and kinematics of grain deformation [15,31,32]. Hines et al. [15] developed a mechanical subgrain rotation model that accounts for the recrystallized grains which have been observed in ASBs based on testing of copper specimens under impact conditions. Various numerical models have been developed for grain-level microstructural material simulations. These include: (1) multi-phase field models [33]—used in modeling solid state phase transformations, grain growth, recrystallization where microstructure is described by a set of order parameters, each of which is related to a specific crystallographic orientation; (2) the Monte-Carlo (Cellular Potts) model which is a lattice based model used to simulate the collective behavior of cellular structures such as grains and is thought of as a generalized version of the Cellular Automata method; and, (3) Voronoi Tessellation [6,34,35] based models which are mathematical models that tessellate a space with seeds that are encircled with randomly shaped polygons whose interior consist of all points closer to that particular seed than any other seed.

numerical methods, however, are not suitable for situations where objects undergo localized discontinuous deformation or deformation outside of the continuum spectrum without modification— i.e. adaptive remeshing. This applies to materials which are better defined by a set of discrete physical particles rather than a continuum, e.g. movement of millions of atoms in equilibrium or non-equilibrium state, granular materials which flow like sand, etc. This lead to the development of mesh-free methods which provide an accurate and stable numerical solution to non-continuum problems. A popular mesh-free method is the Smoothed Particle Hydrodynamics (SPH) method first developed by Lucy [36] and Gingold and Monaghan [7] in 1977 to model astrophysical phenomena and was later extended to applications of solid and fluid mechanics by Libersky and Petchek [8,9] in 1990, and Randles and Libersky [37] in 1996. It was subsequently implemented into finite element software LS-DYNA [38]. Smooth Particle Hydrodynamics (SPH) is a meshless Lagrangian method that can be used to simulate problems with large irregular geometry. 2.3. Summary of experimental investigation and model parameters A series of high strain rate impact tests were carried out on AISI 4340 steel heat treated at various austenization and tempering temperatures, as well as impact strain rates to determine the effect of microstructure on the formation of ASBs. The results of the experimental investigation have been published by Polyzois and Bassim in [39] and presented extensively in the thesis of Polyzois [1]. Testing was performed using a modified Direct Impact Hopkinson Pressure Bar (DIHPB), shown in Fig. 1, on cylindrical specimens. The initial microstructure of the steel was controlled by varying the heat treatment temperatures (austenizing at 800 °C, 900 °C, and 1000 °C, quenching in oil and tempering at 400 °C and 550 °C, followed by air cooling) which controlled the initial prior austenite grain sizes, tempered martensite lath sizes, as well as the distribution of carbides, precipitates, and secondary phases. Grain sizes were measured as a function of heat treatment temperatures and grain size evolution was analyzed in and around the formation of ASBs as a function of the impact momentum of the striker bar, later translated to deformation strain rate. The dynamics stress–strain data gathered from experimentation allowed for the determination of the Johnson–Cook plasticity parameters for the various heat treatments of AISI 4340 steel. It is important to note that the Johnson–Cook plasticity model is a phenomenological (empirical) macroscopic model and assumes the material being modeled behaves like a homogenous continuous mass. It does not account for microstructural mechanisms of grain deformation based on dislocation theory such as dislocation glide. This model was used in the place of more accurate physical-based models (e.g. multi-phase, dislocation- and chemical composition-based

2.2. Meshing techniques Grid-based meshing techniques, i.e. Lagrangian and Eulerian, have been commonly used to discretize areas or volumes of material for numerical and computational FEA problems. Grid-based

Fig. 1. Schematic of the Direct Impact Hopkinson Pressure Bar.

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models) due to the limited availability of experimental techniques which could be used for obtaining the physical properties of the tested AISI 4340 steel and its microstructure. It provides a good macroscopic approximation to the microstructural behavior of AISI 4340 steel whose microstructural constituents, which were directly observable under optical microscopy (e.g. the prior austenite grain boundary and martensite blocks) were generated mathematically using the Voronoi Tessellation and modeled as continuous homogenous materials. The prior austenite grain boundaries and martensite blocks were each assigned different material properties, determined experimentally, simulating the material heterogeneity found in microstructure of the steel. In this paper a FEA microstructural model is presented which uses the Johnson–Cook plasticity parameters calculated for AISI 4340 steel austenized at 800 °C for 30 min, quenched in oil, followed by tempering at 400 °C for 90 min and air cooling. The physical properties of AISI 4340 steel with this heat treatment are presented in Table 1. The macroscopic and theoretically determined microscopic Johnson–Cook parameters for this heat treatment of steel are outlined in Table 2. The macroscopic material parameters are based on the behavior of all microstructural features and were determined from quasi-static and dynamic stress–strain responses and are based on the macroscopic behavior of all the microstructural features. The properties of various observable microstructural features (i.e. prior austenite grain boundary and martensite lath blocks) were estimated experimentally using a relationship between yield stress, hardness, and grain size (and subsequently grain boundary density) and modeled. This relationship is explained in more detail in Section 2.5. The Johnson–Cook plasticity model defines the flow stress as being dependent on strain hardening, strain rate, and temperature, shown in Eq. (1).

h

ro ¼ A þ Bðepl Þ

n

i

" 1 þ Cln

e_pl e_ o

!# ð1  Tb m Þ

ð1Þ

Table 1 Summary of the physical properties of heat treated AISI 4340 steel.

where A is the yield strength, B is the strain hardening coefficient, n is the strain hardening exponent, C is the strain rate sensitivity factor, e_pl is the equivalent plastic strain rate, e_ is the reference strain o

rate at which A, B, and n are evaluated, m is the thermal sensitivity b is a non-dimensional homologous temperature parameter and T defined in Eq. (2) as,

ðT  T ref Þ Tb ¼ ðT m  T ref Þ

where Tm is the melting temperature and Tref is the reference or ambient temperature. 2.4. Calibration of Johnson–Cook material parameters and failure criteria Calibration of the J–C material parameters were determined from direct impact tests of AISI 4340 steel specimens under various heat treatments performed at different strain rates and testing temperatures. A total of 48 stress strain curves were obtained from testing at various strain rates and temperatures, allowing the determination of the relationship between yield strength, hardening behavior, strain rate sensitivity and thermal sensitivity to be calculated for the purpose of obtaining the Johnson–Cook Parameters. A visual representation of the Johnson–Cook stress– strain curve approximation is shown in Fig. 2 for both quasi-static (e_ ¼ 1=s) and high strain rate dynamic responses. In Eq. (1), the yield strength of the steel, ‘A’, was determined from quasi-static tensile testing. The J–C parameters ‘B’ and ‘n’ describe the quasi-static plastic isotropic hardening behavior of a material as a power law relationship between effective plastic stress and effective plastic strain and are defined as the hardening coefficient and hardening sensitivity factor respectively, measured at a strain rate of 1 s1. They were determined empirically using a three term power-law regression fit of the plastic portion of the quasi-static stress–strain curve as follows:



rplQS ¼ A þ B epl

Physical properties of AISI 4340 steel Heat treatment (800 °C/30 min, OQ, 400 °C/90 min, AC) Property

Value

Source

Density (kg/m3) Young’s modulus (GPa) Poisson’s ratio Taylor Quinney coefficient Specific heat (J/kg/K) Thermal expansion coefficient (m/m/K) Reference strain rate (/s) e_ 0 Reference temperature (K) TR Melting temperature (K) Tm Initial temperature (K) T0

7850 205 0.29 0.9 465 0.000013 1 293 1793 293

[10]

ð2Þ

n

ð3Þ

The strain rate sensitivity parameter, ‘C’ defines the ‘‘average amplification’’ of the stress–strain response at higher strain rates and is the slope of the linear logarithmic strain rate dependence of flow stress. It was determined by plotting the ratio of dynamic plastic stress to quasi-static plastic stress at a specific value of strain, where the difference in stress values are minimal, against the natural log of the corresponding strain rate ratio, and then regression fitting the curve using a linear relationship of the form:

rpldyn rplQS

!

" ¼ 1 þ C ln

epl

e_pl dyn e_pl

!# ð4Þ

QS

Table 2 Summary of the macroscopic and microscopic Johnson–Cook parameters for AISI 4340 steel. Johnson Cook parameters of AISI 4340 steel [1] Heat treatment (800 °C/30 min, OQ, 400 °C/90 min, AC) Macroscopic parameters

Microscopic parameters

2

Vickers hardness (kg f/mm ) A (MPa) = ry B (MPa) n C m

ef

478 1366 867 0.1529 0.02581 1.03 0.78

Prior austenite grain boundary

Martensite blocks

502 (+5%) 1425 904 0.1296 0.02698 1.03 0.78

490 (+2.5%) 1394 884 0.1413 0.02623 1.03 0.78

466 (2.5%) 1324 840 0.1532 0.02543 1.03 0.78

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Fig. 2. Visual representation of the Johnson–Cook model approximation for quasistatic and dynamic stress–strain curves.

Normally, the deformation of a metal under high strain rates and large strains is not isotropic. In fact it undergoes thermal strain softening which is reflected on the stress strain curve as a gradual stress collapse. In other words, the temperature rise inside the steel due to the high strain rate deformation, induces a loss of stress carrying capacity. In Johnson–Cook’s definition of flow stress the decrease in flow stress by thermal softening is shown in Eqs. (1) and (2), and is observable immediately around the peak stress after hardening. The thermal sensitivity parameter ‘m’ was determined to be 1.03 for all heat treatments of AISI 4340 steel based on work performed by Lee and Yeh [50] who stated that thermal softening in metals during deformation is assumed to be constant for temperatures less than 298 K and the thermal softening parameter takes the value of m = 1. Above 298 K and lower than 1040 K, ‘m’ is assumed to be 1.03, and beyond 1040 K, ‘m’ is assumed to be equal to 0.5. Damage was defined using an instantaneous (rather than cumulative) damage rupture criterion [12,40], which is similar to the J–C damage parameter D used for tensile failure, but is often used to define the macroscopic dynamic localized shear rupture of metals under high strain rate deformation, such as blast damage. The rupture criterion is based on a critical value of the equivalent plastic strain at failure, ef, which is assumed to be independent of strain rate and stress triaxiality and is expressed in terms of a damage parameter x, as

x¼

Z

depl e

ef

ð5Þ

Failure is predicted when x reaches unity and the failure strain is calculated under quasi-static conditions. To simulate shear damage in impact deformation, a condition is added to elements whose damage parameter reaches unity such that they are only able support compressive stresses and cannot transmit shear or tensile stresses. This creates a mode II (shear) crack in elements in the zone of maximum shear. In simulating impact and penetration, Nashton et al. [40] showed that failure criteria which completely remove elements by setting their stress state to zero upon failure, removed excess amounts of material leading to unrealistic stiffness relaxation of the surrounding elements around the crack zone. The equivalent plastic strain at failure, ef, was determined based on experimentation, analysis of stress–strain curves at various impact strain rates and temperatures, microscopic examination of specimens which formed ASBs, and a review of the literature. To determine the failure strain, steel specimens of a specific heat treatment were impacted at increasing strain rates until fracture. Analysis of stress–strain curves showed a loss in stress carrying capacity in specimens which formed ASBs, which act as a precursor to shear failure, followed by a sudden drop in the stress during fracture. The first indication of stress collapse was determined at a strain of 0.75–0.80 for specimens austenized at 800 °C/30 min and tempered at 400 °C/90 min. This failure strain was consistent with the findings of Avyle [51] who found that the tensile fracture strain of specimens austenized at 840 °C and tempered at various temperatures, remained constant at ef = 0.78 for all tempering temperatures up to 427 °C and then increased to ef = 0.82 for specimens tempered at 510 °C. This jump in failure strain was attributed to the increase in toughness from tempering. A fracture strain of ef = 0.78 for the rupture failure criteria was used in the simulations presented in this paper. Similar to the Johnson–Cook Plasticity model, the rupture failure criterion is a phenomenological description of the underlying damage mechanisms, e.g. formation of voids and shear bands. It does not take into account microstructural damage mechanisms based on dislocation theory. Despite the limitations of the selected plasticity and damage models, simulation results were found to match closely with experimental findings. As presented later in the paper, simulation results show that damage initiation and localization form due to the heterogeneity in material properties in the model, which act as imperfections. The model simulates the material property heterogeneity found in the microstructures of AISI 4340 steel, which has been shown to be a key mechanism of microstructural damage initiation and localization in steel in the literature. Sun et al. [41] presents a quasi-static plastic localization theory for DP 980 based on the incompatible deformation between hard martensite and soft ferrite. The physical properties of the individual martensite and ferrite phases were determined using advanced synchrotron-based in-situ high energy X-ray diffraction (HEXRD) techniques. The authors model the 2D microstructure of DP 980 steel using photo-processing techniques to capture microscopic images and transform the shapes of the phases in the microstructure into meshable areas. No failure model or criteria was prescribed. During deformation under quasi-static conditions, the microstructural-level inhomogeneity in their model serves as the initial imperfection triggering the instability in the form of plastic strain localization during deformation. Failure occurs in the form of the coalescence of the highly strained regions in the modeled structure. Lian et al. [42] validates this model for various stress states by employing a microscopic hybrid damage plasticity model. The model is comprised of three parts: (1) a plasticity model to characterize the microscopic material behavior before damage initiation, (2) a phenomenological criterion to indicate the initiation of

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damage, and (3) a damage-induced softening part to characterize the post-damage material behavior to final fracture. The authors simulate damage initiation under quasi-static conditions for three stress states. The authors defined damage as the formation of microcracks, voids, and shear bands in ductile materials on the microstructural scale. The authors present in detail their damage initiation model which is based on a critical strain criterion proposed by Bai and Wierzbicki [43] which corresponds to the onset of material degradation from a microscopic level. Experimental characterization of damage initiation is carried out using a special technique, direct current potential drop (DCPD) method, and validated by interrupted tests. Microstructural plasticity behavior is modeled using a numerical estimation of the flow stress of each phase (martensite and ferrite) approximated on the basis of an empirical model whose parameters are determined from data obtained from HEXRD techniques. The authors conclude that the microstructural heterogeneity is considered the key factor that contributes to the damage initiation of dual phase steel. 2.5. Grain boundary modeling In the experimental investigation, ASB formation in steel was shown to be a microstructural phenomenon which depends on microstructural properties such as grain size, geometrical shape, orientation of the martensite lath blocks, and distribution of phases and hard particles—all controlled by the heat treatment process. Modeling grains requires the use of complex geometry and multiple material constitutive assignments to ensure accuracy in simulation. However, assumptions must be made to maintain a balance between accuracy and computational cost/time. For the current model a level of microstructural complexity was chosen based on optical microscopic observation in the experimental investigation. The geometrical microstructure of AISI 4340 steel was modeled mathematically using the Voronoi Tessellation technique in the place of photo-processing techniques, which produce complex geometries requiring a large number of finite elements to ensure accuracy in simulation. The FEA grain model, presented in this paper, is a 2D axi-symmetric geometrical representation of a rectangular area of prior-austenite grains in quenched and tempered AISI 4340 steel which are divided into randomly oriented blocks of tempered lath martensite. The initial size of grains was controlled by the heat treatment process. The material properties of the prior austenite grains were modeled using the Johnson–Cook plasticity parameters for each heat treatment. The prior austenite grain boundary acts as a low energy site for dislocations and the segregation of precipitates during heat treatment and deformation, giving it a higher strength and hardness compared with the grains themselves. The prior austenite grain boundary was modeled as a physical entity whose properties were determined from a relationship between yield strength, hardness, and grain size. The yield strengths of various heat treatments of steel were determined experimentally from stress–strain curves obtained by impact testing under various strain rates and temperatures [1,39]. Micro-hardness testing was performed on specimens before and after deformation, and both inside and around the formation of ASBs. Microscopic examination of the specimens was carried out to measure grain sizes before and after impact, with particular attention given to the grain sizes approaching the formation of ASBs [39]. Based on these experimentation a relationship, shown in Eq. (6), was established between macroscopic yield strength and mico-hardness for various heat treatments of steel whose prior austenite grain sizes ranged from 12 to 30 lm.

ry ¼ exp½A þ B  ðHVÞ þ C  ðHVÞ^ 2

161

ð6Þ

This three term exponential fit curve relates yield strength to Vickers hardness calculated for AISI 4340 steel, where A = 4.33287, B = 0.00993, and C = 8.15838E6, in Eq. (6). This relationship agrees closely with the data found for AISI 4340 steel in the ASM handbook [44] relating Rockwell-C hardness to yield strength. Optical microscopic examination of specimens after impact showed a significant reduction in observable grain size approaching the formation of an ASB which was minimized inside the ASB [39]. Consequently, hardness increased approaching the ASB to a maximum inside the ASB. Hardness was calculated to increase by a maximum of 5% inside the formation of ASBs in the heat treated steel presented in this paper (austenization at 800 °C/30 min; tempering at 400 °C/90 min). This maximum increase corresponds to a minimization of observable grain size caused by the refinement of grains inside the formation of ASBs. Therefore a theoretically determined hardness for the prior austenite grain boundary, assuming it to be a physical entity containing precipitates or impurities, was estimated by minimizing the grain size and maximizing the grain boundary density resulting in a maximum calculated increase in hardness of 5%. Using Eq. (6), the theoretically determined macroscopic yield strength of the grain boundary was determined from this 5% increased hardness. The theoretically determined hardness and yield strength of the prior austenite grain boundary was then compared with the stress–strain curves of specimens impacted at various strain rates and temperatures to estimate the hardening behavior and strain rate sensitivity and theoretically determine the microscopic Johnson–Cook material parameters. The thermal sensitivity parameter ‘m’ and failure strain ef could not be determined for the microstructural components and thus were assumed the same as the macroscopic parameters. For the heat treatment conditions presented in this paper (i.e. austenization at 800 °C/30 min and tempering at 400 °C/90 min), the microscopic Johnson–Cook material parameters for the prior austenite grain boundary estimated from a theoretically determined 5% increase in hardness are presented in Table 2. The properties of the martensite blocks were based on the macroscopic properties of the steel and assigned a ±2.5% variation in hardness in order to impart material property directionality to the simulated prior austenite grains, which would otherwise remain homogenous in simulation. This percentage was not based on any experimentally determined data. The Johnson–Cook parameters based on this variation in hardness for the martensite blocks is also presented in Table 2. The total area of the model measured (100 lm  100 lm), representative of the area viewed under maximum magnification of the optical microscope from experimentation. The model was deformed in compression using a rigid wall impactor with predefined initial velocities representative of the impact momenta of the striker bar of the DIHPB in the experimental investigation. A schematic of the model is shown in Fig 3. The final grain model was developed in three parts: (1) Generation of an evenly distributed prior austenite grain geometry in a 100 lm2 area using a Voronoi Tessellation algorithm [4–6] in MATLAB [2], given an initial grain size in lm. This area is equivalent to an optical micrograph at 1000 magnification. (2) Generation of 2D geometry representing the prior austenite grains, the prior austenite grain boundary, and martensite blocks using APDL (Ansys Parametric Design Language) code written in ANSYS [45].

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Fig. 5. Voronoi vertex and bounded cell.

Fig. 3. Schematic of model setup.

(3) Meshing and simulation in explicit dynamics FEA software LSDYNA [3] using the meshless Smooth Particle Hydrodynamics (SPH) formulation [7–9,36], and the Johnson Cook constitutive equations for various heat treatments of AISI 4340 steel. Simulated grain deformation was then compared with experimental observation. 2.6. Grain area generation using the Voronoi Tessellation A Voronoi Tessellation [4–6] resembles the natural geometry of a grain structure that originates from a homogeneous crystallization process. A given area is completely tessellated by grains. Grain shapes are usually non-uniform and exhibit large variability. The shape, geometric orientation, and size of the grains is governed largely by the distribution of ‘‘seeds’’ and the proximity of adjacent ‘‘seeds’’, or site points pi. For a set of points pi in a Euclidian space, the Voronoi tessellation of this set is the subdivision of the plane into n cells, one for each site. This is shown in Fig. 4. A Voronoi vertex is defined as the center of an empty circle touching three or more points, shown in Fig. 5. A vertex has a degree > 3. Four or more non-collinear sites are necessary to create a bounded cell. To generate the grain geometries, a function was written in MATLAB of the form [cells_total] = Grains2D(Gsize) that inputs a given grain size (Gsize) (average diameter) in lm and creates a random distribution of points representing the number of grains of that size which fit in a space measuring 100 lm2 termed ‘‘cells_total’’. This 100 lm2 area reflects an optical magnification of Fig. 6. Output of Matlab function Grains2D for various input grain sizes.

Fig. 4. Voronoi tessellation of a set of points pi.

approximately 1000. A Voronoi tessellation of those points is then generated whose average bounded areas corresponds to input grain size. The function also writes the Voronoi vertices and cells to a file which can be read by FEA software to generate geometry. Examples of output of grain distributions for various input grain sizes are shown in Fig. 6. The prior austenite grains observed under optical microscopy appear multi-sided, often having rounded grain-boundaries. The Voronoi Tessellation algorithm is limited in that it produces non-uniform areas, with high variability—from triangular areas to areas with very high aspect ratios. A special modification of the Voronoi Tessellation, called the Centroidal Voronoi Tessellation

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[46] produces more uniformly distributed grains where the generating point of each Voronoi cell is also its center of mass. The experimental results of Polyzois and Bassim [39] show that the prior austenite grain size increases with increasing austenization temperatures (50% increase in prior austenite grain size for every 100 °C above 800 °C). The authors also show that the prior austenite grain sizes vary by up to 25% at every austenization temperature. However the shapes of the prior austenite grains remained relatively uniform and equi-axed. The Voronoi algorithm could be modified to reduce area shape variability by modifying the location of the generated site points.

2.7. Martensitic lath block geometry generation Optical microscopy of the tested specimens reveals that the prior austenite grains contain several packets and blocks of lath martensite created by the heat treatment process. The prior austenite grain is divided into packets which are groups of martensite blocks with the same habit plate ({1 1 1}c-Fe) that consist of martensite laths with the same crystallographic orientation [47,48], as shown in Fig 7. The number and orientation of the packets and blocks are random. According to Totten and Howes [49], the morphology of martensite in steel is affected by both the carbon content and the size of the prior austenite grains. The strain energy produced by the transformation of BCT martensite during quenching of steel is minimized by a shape deformation along the habit plane categorized as either twinning or slip. Twinned martensite appears as plates and usually forms in high carbon steels while slip martensite appears as laths and forms in medium to low carbon steels. The size of the laths is on the scale of a few nanometers in width. The morphology of the martensite laths can also change with prior austenite grain size. According to Totten and Howes, for small prior austenite grains, produced from austenizing just above the minimum austenization temperature for the steel, the martensite laths are so small that they cannot be observed in a microsection and are called structureless. In large prior austenite grains produced by austenizing at higher temperatures and longer holding times, the martensite laths grow is size and appear plate-like. The change in the size of the martensite blocks with prior austenite grain size was confirmed in the results of the experimental investigation by Polyzois and Bassim [39], which showed that for every 100 °C increase in austenization temperature above 800 °C, the prior austenite grain size of quenched AISI 4340 steel increased by 50%. The size of the martensite blocks increased with increasing prior austenite grain size both in length (extending to the prior austenite grain boundary by a shear displacive transformation) and in width (from 1 lm at a prior austenite grain size of 12 lm to 2–4 lm for a prior austenite grain size of 29 lm). In the experimental investigation, AISI 4340 steel was quenched and tempered at various temperatures and deformed under high

Fig. 7. Schematic of a prior austenite grain in steel.

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strain rate impact at various strain rates. A specimen of steel austenized at 800 °C for 30 min, quenched in oil, then tempered at 400 °C for 90 min, followed by air cooling, was shown to be susceptible to the formation of ASBs at a minimum impact momentum of 30 kg m/s translating to a compressive strain rate of approximately 900/s. Under this heat treatment the steel had an average prior austenite grain size of 16 lm. Fig 8 shows a randomly generated Voronoi tessellation of grains of size 16 lm in a 100 lm2 area which was modeled into individual grain areas in ANSYS using ANSYS Parametric Design Language (APDL). Microstructural observation of heat treated steel showed that upon quenching, martensite block thickness ranged, approximately, between 1 and 2 lm for austenization at 800 °C, shown in Fig. 9. There was no measurable change in block size between the quenched state and the 400 °C tempered steel. The prior austenite grain boundary and martensite block structure were revealed using a picric acid etchant with small additions of hydrochloric acid and a wetting agent all heated to a temperature of 60 °C to avoid crystallization of the picric acid. The martensite blocks have sharp boundaries and appear as long, thin, nearly parallel grains, grouped together in packets within their larger parent prior austenite grains, whose grain boundaries are also clearly visible. The martensite blocks are the smallest microstructural grain visible under optical microscopy at 1000 magnification. The martensite packet grain boundaries, however, are not as clearly visible. Each packet is made up of groups of nearly parallel martensite blocks which share the same habit plane. The size of the martensite blocks is limited by the prior austenite grain boundary due to the diffusionless transformation. Martensite grows by a shear displacive transformation of austenite where atomic movement cannot be sustained across the austenite grain boundaries. There could be several packets in any given three-dimensional prior austenite grain. The optical micrograph, shown in Fig. 8, is essentially a two-dimensional cross-section through the three-dimensional grain structure of steel, which allows for the measurement of the number of packets which intersect that cross-section. The actual number of packets per three-dimensional prior austenite grain cannot be measured directly from the micrograph. For simplicity in simulation, only one packet was modeled per prior austenite grain. APDL code was written to generate a martensite block structure of a given size in each grain which varied in orientation between grains. Block sizes were calculated based on an average block size per prior austenite grain observed from experimentation. Additionally, the blocks, were assumed parallel and identical in size to reduce computation time. In experimentation the martensite block size varied and were near-parallel, varying minimally in orientation. For the heat treatment (800 °C/30 min and 400 °C/90 min) the martensite

Fig. 8. Model of grains in a 100 lm2 sample of AISI 4340 steel with heat treatment (800 °C/30 min, oil quenched, 400 °C/90 min, air cool).

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Fig. 9. Etched specimen of quenched steel austenized at 800 °C to reveal martensite blocks within the prior austenite grains. The Lath block packets are outlined in Section C.

Fig. 10. Simulated microstructure of a specimen with heat treatment 800 °C/ 30 min and 400 °C/90 min with martensite blocks of size 1.5 lm generated inside 16 lm average sized prior austenite grains in an area of 100 lm2.

blocks modeled to be 1.5 lm in width. A generation of this block structure in the grain model is shown in Fig. 10. In experimentation, martensite blocks in quenched and tempered steel varied in width and contain a varied amount of laths with a random distribution of carbides. This created a variation in the material properties of adjacent blocks. With this model, each block can be assigned to have a separate material model whose properties varies slightly from block to block. This imparts a level of heterogeneity in the prior austenite grain and also gives the grain a form of material property directionality so that the deformation of the model depends on the geometric shape, size, and orientation of the lath blocks. The effect of block geometry and physical properties on the formation of ASBs is examined in subsequent sections. 2.8. Grain boundary generation The prior austenite grain boundaries were modeled using the Johnson–Cook material model as thin layers separating the prior austenite grains and were assigned material properties which had a higher hardness than the surrounding prior austenite grains.

The prior austenite grain boundary was also bonded to the grains so that no sliding was allowed (f = 1) and no separation. The deformation and failure of the grain boundary depended solely on the material model parameters assigned to that boundary layer. In experimentation the grain boundary was not a physically separate entity but rather a zone of low energy separating the grains and to which precipitates or secondary phases segregate. In the simulation model, the modeled prior austenite grain boundary only acts to define the initial position of the hard particles, to track their movement during deformation, and to observe their effect on the formation of ASBs. In order to reduce computation time and instabilities from complex geometry, the prior austenite grain boundary layer was assigned to have a thickness of 5% of the average size of the surrounding grain. The grain boundary layers were generated in APDL. Fig. 10 shows the model geometry for AISI 4340 steel with heat treatment 800 °C/30 min and 400 °C/90 min. To reduce computation time, half of the model was simulated using axisymmetry, shown in Fig. 10. The geometry measures 50 lm  100 lm. The model shown in Fig. 11, is a geometrical representation of the microstructure of steel comprised of prior austenite grains filled with a single packet of parallel martensite blocks, separated by a prior austenite grain boundary, defined by a given thickness, which was based on both the size of the surrounding prior austenite grains and the smallest possible element size for meshing. The model is shown prior to meshing.

3. Impact simulations of the lath grain model The lath grain model is a two-dimensional axi-symmetric representation of the grain structure of AISI 4340 steel, with a specific heat treatment, observed under optical microscopy. The prior austenite grain boundaries were modeled as physical boundaries between prior austenite grains and were assigned to have a higher hardness and strength than the martensite blocks within the prior austenite grains. This material property heterogeneity created an incompatibility between the deformation behavior of the prior austenite grain boundary and the grains themselves simulating a microstructural defect within the model. This was found to be a good representation of behavior actual prior austenite grain boundaries, which is a curved two-dimensional defect in the prior austenite crystal structure, separating prior austenite grains with varying crystallographic orientations. To simulate a level of heterogeneity in the microstructure, the microstructural Johnson–Cook parameters of the prior austenite

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Fig. 12. Deformation of the bi-material lath grain model after 50 ls at 52 kg m/s using the Johnson–Cook parameters for steel with heat treatment (800 °C/30 min and 400 °C/90 min).

Fig. 11. Lath grain model with prior austenite grain boundary for a steel with heat treatment 800 °C/30 min and 400 °C/90 min and average initial grain size of 16 lm and lath width of 1.5 lm.

grain boundary and martensite lath blocks shown in Table 2, were implemented. Unlike the properties of the prior austenite grain boundary, which were estimated based on experimental results, the properties of the lath blocks were chosen to impart material property directionality in simulation while keeping the overall properties of the prior austenite grains close to that of the macroscopic properties of the steel. This variation was not based on any experimentally determined data. The variation in assigned material properties between martensite blocks simulated the variation in material properties due to the formation of carbides. Every lath block was assigned material properties which alternated between being 2.5% harder and 2.5% softer than the macroscopic Johnson– Cook parameters for the particular heat treatment of steel so that no two adjacent blocks shared the same properties. To test the stability of the model, two simulations were performed. The first simulation involved changing the properties of the prior austenite grain boundary to be 5% harder than the parent prior austenite grain while keeping the grains themselves homogenous and was called the bi-material lath model. The second simulation introduces a level of heterogeneity in the lath blocks by varying the hardness of the properties by +2.5%, called the heterogeneous lath model. Both models were subjected to an impact momentum of 52 kg m/s, which corresponds to the equivalent maximum air pressure capacity of the firing chamber of the Modified Direct Impact Hopkinson Pressure Bar to send the 2 kg striker bar into the specimen. Experimental results showed that an impact momentum of 52 kg m/s resulted in the formation of very well defined ASBs in the specimen cross sections of heat

treated AISI 4340 steel austenized at 800 °C/30 min and tempered at 400 °C/90 min. In the first simulation, the model was discretized using 30220 SPH nodes. The resulting deformation after 1 ls is shown in Fig. 12. Full deformation after 2 ls resulted in the complete crushing of the model after which the projectile wall separated from the impact face indicating the end of deformation. Experimentation carried out by these authors and published in [39] on cylindrical specimens under impact loading, showed that adiabatic shear bands formed in the vicinity of the maximum shear direction in an hour-glass pattern. In the two-dimensional model, the maximum shear directions are 45° from the axi-symmetric origin. The experimental investigation showed that the width of an ASB was several times larger than the width of the grains themselves. A complete crushing of the model under the given initial conditions indicates that the grain model is small enough to be contained within an ASB and the crushed grains can represent the final grain size inside that ASB. However a very fine SPH discretization is needed to calculate final grain size accurately, which is represented by the smallest cluster of SPH nodes after deformation. The simulated deformation gives a good indication of the source of ASB formation, by the location of severe localized strain, and its direction of propagation through the microstructure, which continues to grow under applied severe shear stresses, eventually consuming the grains. After 0.6 ls of impact, the deformed model shows an elongation of the grains in the direction of maximum shear through the cross section. Additionally, the harder grain boundary spalls and breaks up, pulling apart the surrounding softer grains and creating voids. Examining the progressive effective stress and strain responses, in Fig. 13, shows the propagation of a localization through the cross section which forms more favorably along the edges of the grain boundaries and is followed closely by the formation of microcracks (indicated by stress collapse) and void formation. The localization shears the bond between the hard grain boundary and soft surrounding grains. The overall direction of localization is along lines of maximum shear but is shown to also branch out along the grain boundaries as deformation progresses. Examining the stress and strain response during deformation, in Fig. 14, shows the extent of the branching effect of localization more clearly.

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Fig. 13. Effective plastic strain and effective (Von Mises) stress deformation in a grain model of steel with heat treatment (800 °C/30 min and 400 °C/90 min).

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Fig. 14. Effective Von Mises stress and Effective plastic strain response in a model of AISI 4340 steel with heat treatment (800 °C/30 min and 400 °C/90 min) impacted at 52 kg m/s after 1 ls.

The second simulation introduced a heterogeneity in material properties of the martensite lath blocks. This gave the grains material property directionality consistent with their deformation behavior, which not only depends on their size and shape but also on the orientation of the lath blocks. The small size and high aspect ratio of the lath blocks required a certain degree of SPH discretization across the width of each block. This increased geometrical and material complexity resulted in a large increase in computational costs. Consequently, the size of the model had to be reduced. The

final size of the model was reduced to 25% the original size and is shown in Fig. 15. The deformation time step was further reduced to accommodate the reduced model size. Due to the limitation of using macroscopic material models, the laths, and prior austenite grain boundary were modeled as homogenous and continuous masses. Accuracy in this type of modeling is limited by the smallest geometrically defined area and element size. As microstructural complexity is increased, so too is the computational costs for modeling. The grain model shown in

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Fig. 15. Setup of heterogeneous lath grain model discretized with 39100 SPH nodes.

Fig. 16. Deformation of the lath grain model after 0.25 ls of deformation at an impact momentum of 52 kg m/s showing reorientation of the lath blocks and spallation of the grain boundary.

Fig. 15 consists of three identifiable microstructural components (prior austenite grain, prior austenite grain boundary, and martensite blocks) and three different material models corresponding to the prior austenite grain boundary and variation in properties of alternating martensite laths. Each prior austenite grain is made up of a single packet of parallel blocks with alternating material properties, separated by a prior austenite grain boundary with a third material property assignment. The model was impacted at 52 kg m/s for a total deformation time of 0.5 ls. The resulting deformation after 0.25 ls, shown in Fig. 16, reveals a reorientation of the lath blocks toward the localization followed by a spallation of the grain boundary and void formation along lines of maximum shear. A progression in the strain and stress response, in Fig. 17, shows ASB localization form in the vicinity of the prior austenite grain boundary closest to the maximum shear direction, as a result of the material property incompatibility between the harder prior

austenite grain boundary and softer prior austenite grains. Localization was found to be more severe along the maximum shear direction which had a higher prior austenite grain boundary density (larger number of grain boundary intersections in the maximum shear direction). Severity was measured by a greater stress response and higher localized strain. This concept could be extended to describe the mechanism of ASB formation in a three-dimensional microstructure, where adiabatic shear bands are more likely to form along the maximum shear planes (hourglass pattern) which have the highest prior austenite grain boundary density. This concept was confirmed by the results of the experimental investigation [39], which showed that a high number of well-defined adiabatic shear bands formed in AISI 4340 steel with smaller initial prior austenite grain size (12 lm) compared specimens with large prior austenite grain size (29 lm). A close-up of the stress and strain distribution after 0.25 ls of deformation, shown in Fig. 18, shows the extent of the localization. Micro cracks represented by failed SPH nodes with zero stress capacity are more prominent along the sheared section between the prior austenite grain boundary and the surrounding grain. This may be an indication of the formation of a thin well-defined transformed ASB which are prone to micro-cracking while the localization which propagates through the grains, shown to be wider but less severe, by the lack of failed SPH nodes, may be indicative of a deformed type ASB. The stress–strain responses of the lath grain model using the microscopic Johnson–Cook parameters matched closely with experimental results, which were calculated from load-time data gathered from strain gauges attached to the transmitter bar of the apparatus. Fig. 19 shows a comparison between the stress– strain response taken from SPH nodes along the maximum shear directions, and the experimental response for a specimen of steel with heat treatment (800 °C/30 min and 400 °C/90 min) at an impact momentum of 52 kg m/s. The figure shows the stress– strain responses for the full deformation of the specimen. The Johnson–Cook approximation under-predicts the hardening portion of the response but the stress collapse resulting from the defined failure criteria is evident around a strain of 0.78. This matches closely to experimental stress collapse around this value of strain.

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Fig. 17. Progressive effective plastic strain and effective Von Mises stress response of heterogeneous lath grain model of AISI 4340 steel with heat treatment (800 °C/30 min and 400 °C/90 min) impacted at 52 kg m/s.

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4. Conclusions Simulation results using the lath grain model presented in this paper were compared with extensive high strain rate impact experimentation performed on 48 specimens under various heat treatments, strain rates, and testing temperatures, some of the results of which are shown in [39]. Since deformation under high strain rates is very fast, it is very difficult to measure the progressive deformation of grains and the dynamic formation of ASBs. But by understanding the prior conditions which lead to the formation of ASBs, and analyzing their formation microscopically, the mechanism of ASB formation was simulated. The main results from the theoretical investigation may be summarized as follows:

Fig. 18. Effective plastic strain and effective Von Mises stress distribution through lath grain model of AISI 4340 steel with heat treatment (800 °C/30 min and 400 °C/ 90 min) impacted at 52 kg m/s after 0.25 ls of deformation.

 The simulation models developed are a good representation of grain refinement and the kinematics of grain deformation in 2D, showing the mechanism of ASB formation on the microscopic scale.  The Voronoi Tessellation technique provides a fast and effective method for generating randomly oriented grain geometries, of a controlled size, in a given space.  Simulation results using the multi-material Lath Grain Model showed that ASBs formation is controlled by the size, geometric shape, and geometric orientation of the martensite lath blocks as well as the material property heterogeneity of the modeled microstructure.  Simulated grains were shown to rotate and elongate in the direction of maximum shear  In the simulation model, ASBs were characterized by either a thick localized zone of severe shear strain found to nucleate from the impact face and propagate through the grains or by a thin, more prominent zone of severe localization propagating along the prior austenite grain boundary. The thinner zones of severe localization contained failed SPH elements indicative of the formation of micro-cracks.  The final simulated grain size, upon complete compression of the lath grain model, can be determined by the smallest cluster of SPH particles, which have not failed after deformation.

Acknowledgements The author would like to acknowledge The Natural Sciences and Engineering Research Council of Canada (NSERC) grant to one of the authors, N. Bassim (RGPIN 1269-2010) and the University of Manitoba for their funding and support of this research. References

Fig. 19. Comparison between simulated stress–strain response and experimental response of AISI 4340 steel with heat treatment (800 °C/30 min and 400 °C/90 min) impacted at 52 kg m/s and tested at 20 °C.

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