metals Article
Grain Scale Representative Volume Element Simulation to Investigate the Effect of Crystal Orientation on Void Growth in Single and Multi-Crystals Woojin Jeong 1,4 , Chang-Hoon Lee 2 , Joonoh Moon 2 , Dongchan Jang 3 and Myoung-Gyu Lee 4, * 1 2 3 4
*
ID
Materials Science and Engineering, Korea University, Seoul 02841, Korea;
[email protected] Korea Institute of Materials Science, Changwon 51508, Korea;
[email protected] (C.-H.L.);
[email protected] (J.M.) Nuclear and Quantum Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Korea;
[email protected] Materials Science and Engineering & Research Institute of Advanced Materials, Seoul National University, Seoul 08829, Korea Correspondence:
[email protected]; Tel.: +82-28-801-711
Received: 27 April 2018; Accepted: 5 June 2018; Published: 8 June 2018
Abstract: Crystal plasticity finite element (CPFE) simulations were performed on the representative volume elements (RVE) modeling body centered cubic (bcc) single, bi- and tri-crystals. The RVE model was designed to include a void inside a grain, at a grain boundary and at a triple junction. The effect of single crystal orientation on the flow strength and growth rate of the void was discussed under prescribed boundary conditions for constant stress triaxialities. CPFE analyses could explain the effect of inter-grain orientations on the anisotropic growth of the void located at the grain boundaries. The results showed that the rate of void growth had preferred orientation in a single crystal, but the rate could be significantly different when other orientations of neighboring crystals were considered. Keywords: crystal plasticity; finite element method; void growth
1. Introduction Engineering structure materials used for the manufacturing processes generally require higher strength, prolonged plasticity with good ductility, weldability, creep resistance and enhanced fatigue properties among others. In general, the strength and ductility (or formability) of materials have an inverse relationship, which results from the higher possibility of the initiation of microscopic fracture along precipitate, grain and phase boundaries being responsible for the strengthening of materials. Therefore, modern structure material design aims to increase the ductility or to delay the fracture without deteriorating strength by controlling microstructures of materials. The challenge confronted during the shaping and forming of metallic materials is the fracture of materials. In ductile metals, the common observation is that cracks initiate at some critical locations such as at the interface between secondary particles and base matrix materials or existing voids introduced during the processing. These cracks grow in specific directions depending on the microstructure and external loading conditions. For example, Gurland [1] observed that the cracks grow in a perpendicular direction to the maximum tensile strain direction. In the review paper of Takuda et al. [2], the prediction capability of the frequently adopted ductile fracture criteria was evaluated based on the sheet formability. Four classical ductile fracture models were considered in their paper: Clift [3], Cockcroft and Latham [4], Brozzo [5] and Oyane model [6]. In these models, the plasticity and ductile fracture criteria are fully
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uncoupled, which can be used for criteria to estimate the initiation of the fracture after certain amount of plastic deformation. According to their works, all criteria except the Clift model could predict fracture initiations successfully for an axisymmetric deep drawing test when the models incorporated the dependence of fracture on plastic strain and stress state. More recently, a criterion named the Mohr–Coulomb model was proposed by Bao and Wierzbicki [7] to consider dependence of fracture initiation on the stress triaxiality and Lode angle parameter, which improved the prediction for the initiation of ductile fracture with different types of materials [8,9]. Gurson [10] proposed a constitutive law, which combines plastic strain and damage, based on plastic flow rule by considering voids in ductile steel for predicting fracture of material. Nucleation, growth and coalescence of voids in metals are the major sequence of the ductile failure. Voids can nucleate near secondary particles including precipitates during plastic deformation by decohesion caused by lattice mismatch between the secondary particles and base material. In the process of plastic deformation, a ligament between closely located voids causes coalescence, which repeatedly occurs until fracture. To simplify the constitutive modeling, Gurson assumed that the shape of a void is spherical. The Gurson model has no consideration of void nucleation and coalescence of voids. If initial void volume fraction is zero, the evolution of void volume fraction is not considered in the Gurson model. After Gurson’s pioneering work, Tvergaard and Needleman [11] further improved the original model by introducing new material parameters. For describing a void nucleation, they adopted the laws proposed by Chu and Needleman [12]. The so-called Gurson–Tvergaard–Needleman (GTN) model needs another parameter describing the coalescence of void when material reaches a critical void volume fraction. Usually, the critical void volume fraction is obtained by a numerical fit to the tensile test and has been assumed to be a constant independent of stress state. They analyzed ductile fracture model incorporating the concept of volume fraction with void growth and loss of load carrying capacity for axisymmetric tensile specimens. The GTN model doesn’t consider the microstructural features such as crystal orientation surrounding the void. Contrary to the phenomenological approaches for the determination of ductile fracture initiation and its propagation, analyses based on microscopic characteristics of materials are sometimes very useful for understanding the mechanism of the ductile fracture. For example, Yerra et al. [13] investigated the behavior of void in the material under different stress states (or different stress triaxiality) in a single crystal level. They simulated void coalescence behavior in the single crystal with different characteristic orientations using the crystal plasticity finite element method. Liu et al. [14] and Potirniche et al. [15] also studied the behavior of voids in single crystals with one or two idealized voids under prescribed boundary conditions. Their results showed that the rate of void growth and the onset of coalescence were dependent on the crystal orientation, which cannot be simulated using the conventional finite element approach. Srivastava and Needleman also adopted crystal plasticity and the RVE model with a single void to study the effect of crystal orientation on porosity and evolution of creep strain evolution. They found that the Lode parameter as a measure of stress state considerably affected the creep strain [16]. Similar work by Tekoglu studied the effect of stress triaxiality on the deformation behavior of the unit cell, but he additionally investigated the effect of shear ratio [17]. Lebensohn et al. proposed a new modeling approach with FFT (Fast Fourier Transformation)-based crystal plasticity to investigate the growth of multiple voids in the face centered cubic (fcc) polycrystals [18]. The hardening model used for crystal plasticity was usually a power-law type, but detailed study on the effect of stage III and IV strain hardening on the void growth and coalescence was provided by Lecarme et al., where they found that the stage IV hardening had significant influence in delaying fracture [19]. Along with numerous simulation based studies, there were experiment-based investigations on the void growth and its relationship with microstructure, especially with the grain orientations. Recently, Pushkareva et al. [20] and Nemcko et al. [21] studied the fracture process of pure titanium and magnesium, respectively, using experimentally introduced voids (or holes). They observed that
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the orientation effect was much stronger than other effects such as intervoid spacing. The numerical the orientation effect was much stronger than other effects such as intervoid spacing. The numerical method was also accompanied to validate their experimental observations in both works [20,21]. method was also accompanied to validate their experimental observations in both works [20,21]. The present study aims to investigate the effects of crystal orientation and existence of grain The present study aims to investigate the effects of crystal orientation and existence of grain boundary growth.The Theeffect effectofofgrain grainboundaries boundaries void boundaryorortriple triplejunction junctionon on the the rate rate of of void void growth. onon thethe void growth has not been well understood in the aspect of microstructural features such as grain orientations growth has not been well understood in the aspect of microstructural features such as grain and slip activities. fact, most ofIn thefact, existing the above-mentioned literature studied orientations and In slip activities. most works of the including existing works including the above-mentioned theliterature growth studied and coalescence of a single void inside grains under different stress states. Therefore, the growth and coalescence of a single void inside grains under different stress different of the crystal orientations arecrystal considered to studyare theconsidered preferred growth direction states. combinations Therefore, different combinations of the orientations to study the of preferred void (or anisotropy of the void growth) to mimicofthe of voidtolocated the grain boundary growth direction of void (or anisotropy thegrowth void growth) mimicat the growth of void in located polycrystalline metals. Three different characteristic grain Three orientations (100), (110) and (111) at the grain boundary in polycrystalline metals. different characteristic grainare orientations (100), (110) and (111) are plasticity introduced to the rate-dependent finite introduced to the rate-dependent crystal finite element simulationscrystal under plasticity different constant element simulations under different stress triaxialities. Analyses are made by theslip results for stress triaxialities. Analyses are madeconstant by the results for flow strength of single crystals, activities of single activities each constituent crystal and void growth rates of flow each strength constituent crystalcrystals, and voidslip growth ratesofbetween the bi- or tri-crystals. between the bi- or tri-crystals. 2. Motivation, Assumptions and Limitation of the Work 2. Motivation, Assumptions and Limitation of the Work Figure 1 shows the measured microstructures of a ferritic base steel. The EBSD (Electron backscatter Figure 1 shows the 1a measured microstructures of a ferritic base steel. Thegrains EBSDand (Electron diffraction) image in Figure shows the orientation distribution of the constituent Figure 1b backscatter diffraction) image in Figure 1a shows the orientation distribution of the constituent grainsand represents a detailed TEM image showing the second phase particles distributed both inside grains and Figure 1b represents a detailed TEM image showing the second phase particles distributed both at grain boundaries. These two figures became the motivation for the present study. The fundamental inside grains and at grain boundaries. These two figures became the motivation for the present study. assumption is that the fracture can be initiated from the second phase particles located either inside The fundamental assumption is that the fracture can be initiated from the second phase particles the grain or at the grain boundaries, and there is a strong relationship between void growth and located either inside the grain or at the grain boundaries, and there is a strong relationship between orientations of the neighboring grains. To simulate the effect of grain orientations, the crystal plasticity void growth and orientations of the neighboring grains. To simulate the effect of grain orientations, simulation was adopted. Though there are many similar crystal plasticity based approaches on the crystal plasticity simulation was adopted. Though there are many similar crystal plasticity based characterizing the fracture behavior of single or polycrystalline metals, the present study might be approaches on characterizing the fracture behavior of single or polycrystalline metals, the present able to contribute by including theby effect of multi-crystals which the idealized is locatedvoid at the study might be able to contribute including the effect ofin multi-crystals in whichvoid the idealized boundary of grains. is located at the boundary of grains.
(a)
(b)
Figure 1. (a) EBSD image showing random distribution of grain orientations, and (b) TEM image Figure 1. (a) EBSD image showing random distribution of grain orientations, and (b) TEM image showing the second phase particles located both inside grains and at grain boundaries. The material showing the second phase particles located both inside grains and at grain boundaries. The material is is ferritic based steel and both images were taken before deformation. ferritic based steel and both images were taken before deformation.
Because the mechanical deformation of real polycrystalline metals is complex, several Because mechanical deformation polycrystalline several simplifying simplifyingthe assumptions and limitations of in real the present study aremetals statedisascomplex, following: assumptions and limitations in the present study are stated as following: The microstructure of the single or multi-crystal is simplified as a representative volume element • The microstructure the single or multi-crystal is simplified as a representative volume element (RVE) in which anofidealized spherical void is located at the center of a single crystal or at the (RVE) in which an idealized spherical void is located at the center of a single crystal or at the grain grain boundary of multi-crystals. boundary of multi-crystals.
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•
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It is assumed that the initiation of void originates from the second phase particle (or precipitate), thus only void growth is considered from the initial size of the second phase particle. The crystallographic orientations of the microstructure are simplified as three characteristics primary orientations, (100), (110), and (111) along the normal direction of a sample. In real crystalline metals, there are complex interactions among grain boundaries and dislocations, which is highly dependent on the grain boundary characteristics. To simplify this, the perfect bonding between neighboring grains at the grain boundary is assumed and no special interactions such as transmission and absorption of dislocations at the boundary are considered. The material is assumed as a body centered cubic (bcc) crystal and a power-law type single crystal hardening model is adopted. In addition, the loading condition is quasi-static and isothermal at room temperature without strong strain rate and temperature effects.
The present work provides simulation based investigation, which should be further validated by experiments. However, even with this limitation, the current work is still meaningful considering recent advances in the computational approach based on crystal plasticity, which accurately predicts the anisotropic deformation response of cubic crystals under complex loading conditions [7,13–19]. 3. Crystal Plasticity Constitutive Equation Finite element simulations based on the crystal plasticity can represent the single and polycrystalline mechanical behaviors of metallic materials by considering their microstructure such as orientation and slip system of crystalline structures. In this study, the crystal plasticity based on rate-dependent elastic-viscoplasticity was adopted and implemented in the finite element code using a user material subroutine in ABAQUS. In the user material subroutine, a prescribed deformation gradient tensor at each time step is multiplicatively decomposed into its elastic Fe , and plastic part F p [22] (see Figure 2) F = Fe F p ,
(1)
where the elastic part of the deformation gradient accounts for elastic rotation and distortion, while the plastic part is activated by the dislocation slip on preferred slip systems. The second Piola–Kirchhoff stress Te can be calculated by the generalized Hooke’s law for anisotropic elastic materials as follows: Te = CEe , Te = det(Fe )Fe−1 TFe−T ,
(2)
where C is a fourth order anisotropic elasticity tensor, and T is the Cauchy stress tensor. The elastic Lagrangian strain tensor Ee is calculated from the elastic part of the deformation gradient tensor as Ee =
1 eT e F F −I . 2
(3)
The rate of plastic deformation gradient is expressed as follows: .p
F = LpFp,
(4)
where L p is the velocity gradient tensor that is calculated by the shear strain rates on preferred slip systems associated with slip direction d0α and slip plane normal n0α . If the dislocation glide on the .α activated slip systems can be represented by the shear strain rate γ for the slip system α, the velocity gradient tensor can be expressed as Lp =
.α αO α d0 n0 .
∑α γ
(5)
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Under the visco-plasticity scheme, the shear strain rate is expressed as an exponential function of the resolved shear stress and flow stress of the single crystal [23]: α 1 τ m γ = γ0 α sign(τ α ), s .α
.
(6)
.
8, 5 of 16 γ0 is 2018, where Metals a reference slip rate, sα is a slip resistance, m is the strain rate sensitive exponent and τ α is the resolved shear stress on the slip system α. is a reference slip rate, is a slip resistance, is the strain rate sensitive exponent and Towhere complete the constitutive equation for the single crystal plasticity, the rate of slip resistance is the resolved shear stress on the slip system . equivalent to the flow stress hardening in the macroscopic stress–strain curve is formulated To complete the constitutive equation for the single crystal plasticity, the rate of slip resistance as below [24]: . . α the macroscopic equivalent to the flow stress hardening in β stress–strain curve is formulated as below s = ∑ β hαβ γ , (7)
[24]:
where hαβ = qαβ h β , and hαβ is the hardening matrix that describes the latent hardening by qαβ . =∑ , (7) The function h β is a power-law function as below with self-hardening coefficient h0 , saturated slip where , and is the hardening . resistance ss and = strain hardening exponent a [25]:matrix that describes the latent hardening by The function is a power-law function as below with self-hardening coefficient resistance and strain hardening exponent a[25]: s β a
h β = h0 1 − =
ss
1−
, saturated slip
.
.
(8) (8)
The power-law type hardening for single crystals has been well adopted in the crystal plasticity The power-law type hardening for single crystals has been well adopted in the crystal plasticity simulation under quasi-static loading at room temperature. More detailed description on the single simulation under quasi-static loading at room temperature. More detailed description on the single crystal constitutive equations can be referred to [26]. crystal constitutive equations can be referred to [26].
Figure 2. Schematics of multiplicative decomposition of deformation gradient.
Figure 2. Schematics of multiplicative decomposition of deformation gradient.
4. Model Material and Identification of Crystal Plasticity Parameters
4. Model Material and Identification of Crystal Plasticity Parameters The purpose of the present study is to explore the relationship between grain scale deformation andpurpose the growth of voids embedded or multiple crystallinebetween solids. Therefore, for deformation simplistic The of the present study isintosingle explore the relationship grain scale numerical study, the existing material parameters for the crystal plasticity constitutive lawsimplistic were and the growth of voids embedded in single or multiple crystalline solids. Therefore, for adopted. The model material is selected from one of ferritic steel that has body centered cubic crystal numerical study, the existing material parameters for the crystal plasticity constitutive law were (bcc) structure. The anisotropic elastic properties for typical iron are referred to the material’s adopted. The model material is selected from one of ferritic steel that has body centered cubic crystal (bcc) handbook [27] and the three anisotropic constants are C11 = 230.4 GPa, C12 = 134.2 GPa and C44 = 115.9 structure. typical iron are referred to 12 the{110} material’s handbook GPa.The Theanisotropic slip systemselastic for theproperties bcc crystalfor have been often assumed to be and 12 {112}[27] and the threeslip anisotropic are C1.11The = 230.4 GPa, C12 systems = 134.2 for GPa and GPa.based The slip 44 = 115.9 systems asconstants listed in Table number of slip the bccCcrystal is chosen systems bcc crystal have be 12 {110} 12 {112}played aslip on for the the previous work by Lee been et al. often [28] inassumed which thetoadditional {123} and slip systems minor role in deformation texture.ofThe crystal properties in Equations (6)–(8) systems as listed in the Table 1. The number slipsingle systems for the bcc crystal is chosen basedwere on the identified as the best fitting parameters to the macroscopic stress–strain curve of polycrystalline steel assuming random orientations, which were adopted in the previous works [26,29,30]. The determined parameters are listed in Table 2.
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previous work by Lee et al. [28] in which the additional {123} slip systems played a minor role in the deformation texture. The single crystal properties in Equations (6)–(8) were identified as the best fitting parameters to the macroscopic stress–strain curve of polycrystalline steel assuming random orientations, which were adopted in the previous works [26,29,30]. The determined parameters are listed in Table 2. Table 1. 24 {110} and {112} slip systems for bcc crystal. No.
Plane Normal
1 2 3 4 5 6 7 8 9 10 11 12
(011) (101) 110 011 (101) (110) (011) 101 (110) 011 101 110
Slip Direction 111 111 111 111 111 111 111 111 111 [111] [111] [111]
No. 13 14 15 16 17 18 19 20 21 22 23 24
Plane Normal 211 121 (112) (211) 121 112 211 (121) 112 211 121 121
Slip Direction 111 111 111 111 111 111 111 111 111 [111] [111] [111]
Table 2. Single crystal parameters fitted for model material. Variables
Values
Anisotropic elastic property (C11 ) Anisotropic elastic property (C12 ) Anisotropic elastic property (C44 ) .α Reference shearing rate (γ0 ) Slip rate sensitivity parameter (m) Initial value of the slip resistance (s0 ) Value of the initial hardening rate (h0 ) Saturation value for the slip resistance (ss ) Exponent in hardening equation (a) Latent hardening parameter (qαβ )
230.4 GPa 134.1 GPa 115.9 GPa 0.001/s 0.05 50 MPa 200 MPa 150 MPa 2.25 1.4
5. Finite Element Modelling for RVE Analysis In this study, a representative volume element (RVE) analysis was conducted using the finite element simulations. A unit cell containing a void was modeled in the finite element simulations and different boundary conditions were applied to investigate the void behavior during plastic deformation. An implicit finite element software ABAQUS/Standard 6.12 was used to model the growth of the void in the RVE. The single crystal constitutive equations were coded in the user material subroutine, UMAT. The three-dimensional continuum element C3D8 was used to model the RVE and the total number of elements was 4024. In the RVE, a spherical void with an initial volume fraction of 0.01 is located in the single or multi-grain matrix as shown in Figure 3a. Each crystal has representative orientation among (100), (110) and (111) orientations. To represent the bi- or tri- crystals, the cell is divided into two or three regions with different crystal orientations. It is assumed that the boundary between two different orientations is perfectly bonded, which represents the grain boundary as the simplest approach without considering complex interactions among dislocations and grain boundaries. Therefore, the current RVE model can investigate the void behavior under plastic deformation when it is located at the center of single crystal, at the grain boundary of bi-crystal, and at the triple junction of the tri-crystal.
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(a)
(c)
(b)
Figure 3. (a) finite element model for an RVE analysis in X–Y plane; (b) cell division for a void crossing
Figure 3. (a) finite element model for an RVE analysis in X–Y plane; (b) cell division for a void crossing the grain boundary or triple junction; (c) 3D view of the RVE cell with a void inclusion. the grain boundary or triple junction; (c) 3D view of the RVE cell with a void inclusion.
In the RVE model, [0-10], [00-1], [100] in (100) orientation are set to coincide with global X, Y, Z coordinates, respectively. [100],in [01-1], in (110) orientation and [1-10], [11-2], [111]X, in Y, Z In the RVE model, [0-10],Likewise, [00-1], [100] (100)[011] orientation are set to coincide with global (111) orientation coincide with X, Y, Z direction, respectively. The RVE cell is divided into four coordinates, respectively. Likewise, [100], [01-1], [011] in (110) orientation and [1-10], [11-2], [111] in regions in the global XY-plane. Each region is assigned with single crystal orientation to allocate a (111) orientation coincide with X, Y, Z direction, respectively. The RVE cell is divided into four regions void either at the center of single crystal, grain boundary or triple junction of bi- and tri-crystals, in the global XY-plane. Each region is assigned with single crystal orientation to allocate a void either respectively. By the modeling approach, the effect of various combinations of primary orientations at the on center of single crystal, boundary or triple junction of bi- and tri-crystals, respectively. the void behavior can begrain investigated efficiently. By the modeling approach, thestress effectstates, of various combinations primary were orientations To obtain the different displacement boundaryofconditions applied. on Forthe this,void behavior be magnitude investigated the can same of efficiently. displacement along X and Y directions were prescribed, while the To obtain the along different stress was states, displacement boundary conditions werestate applied. displacement Z direction iteratively controlled to maintain constant stress duringFor the this, deformation. Though the matrix material incompressible, volume of the RVE increased due to the same magnitude of displacement along Xisand Y directionsthe were prescribed, while the displacement growth ofwas the void in the unit cell. along the Z direction iteratively controlled to maintain constant stress state during the deformation. Asmatrix a measure of stress state, the stress triaxiality of theof RVE calculated. The stress triaxiality Though the material is incompressible, the volume thewas RVE increased due to the growth of is defined as
the void in the unit cell. As a measure of stress state, the stress triaxiality of the RVE was calculated. The stress triaxiality = and Σ = (Σ Σ Σ ), (9) η is defined as 1 Σ where Σ is macroscopically averaged hydrostatic stress and Σ is an equivalent stress. Two (9) η = ¯h and Σh = (Σ11 + Σ22 + Σ33 ), representative stress triaxialities, 1 and 3, were selected in this study. 3 Σ 6. Result and Discussion
¯
where Σh is macroscopically averaged hydrostatic stress and Σ is an equivalent stress. Two representative stress triaxialities, 1 and 3, were selected inGrowth this study. 6.1. Effect of Grain Orientation on the Void in Single Crystals Figure 4a shows calculated stress–strain curves of single crystals with different orientations 6. Result and Discussion under a constant stress triaxiality of 1. For comparison purposes, the stress–strain curves of matrix (or those without a void) are presented in the figure. Crystals All three stress–strain curves corresponding 6.1. Effect of Grain Orientation onalso the Void Growth in Single to (100), (110) and (111) orientations exhibited softening by the inclusion of the void. However, this Figure 4a isshows calculated on stress–strain curves of and single withInterestingly, different orientations softening more pronounced the order of (110), (111) (100)crystals orientations. almost undernegligible a constant stress triaxiality of 1. For comparison purposes, the stress–strain curves of matrix softening is observed for (100) orientation by the void inclusion. The order of softening is (or those a void) areorder also presented in the figure. stress–strain curves alsowithout correlated with the of overall strength; i.e., if All the three strength of a grain (or singlecorresponding crystal) is higher, is predicted.exhibited In the figure, the ultimate strength (UTS) where the to (100), (110)more andsoftening (111) orientations softening by thetensile inclusion of the void. However, localization beginspronounced is also markedon and configuration voidand at the UTSorientations. point is shownInterestingly, for each this softening is more thethe order of (110), of (111) (100) of softening crystal. Theisconfiguration of void represented by thatThe surround almostorientation negligible observed for (100)is orientation byone thelayer voidelements inclusion. order of the void. The color in the figure represents the equivalent plastic strain. In Figure 4b, the rate of void softening is also correlated with the order of overall strength; i.e., if the strength of a grain (or single growth is calculated by the void volume fraction change as a function of RVE equivalent strain in the crystal) is higher, more softening is predicted. In the figure, the ultimate tensile strength (UTS) where loading direction. The RVE strain is defined as the nominal displacement divided by the initial length the localization begins is alsoFrom marked and the of void at the on UTS for each in the loading direction. the figure, theconfiguration rate of void growth is larger thepoint orderisofshown (110), (111) orientation of crystal. The configuration of void is represented by one layer elements that surround and (100) orientations. In addition, it is shown that the rate of void growth is accelerated after the
the void. The color in the figure represents the equivalent plastic strain. In Figure 4b, the rate of void growth is calculated by the void volume fraction change as a function of RVE equivalent strain in the loading direction. The RVE strain is defined as the nominal displacement divided by the initial length in the loading direction. From the figure, the rate of void growth is larger on the order of (110), (111)
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Metals 2018, 8, of 16 and (100) orientations. In addition, it is shown that the rate of void growth is accelerated after 8the UTS point and will result in failure by the coalescence with neighboring voids. The UTS points for (110), UTSand point andcrystals will result failure by the with neighboring voids. The UTS points forthe (111) (100) are in 0.3, 0.4, and 0.6,coalescence respectively, which has an inverse relationship with (110), (111) and (100) crystals are 0.3, 0.4, and 0.6, respectively, which has an inverse relationship with void growth rate. the void growth rate. In the simulation, the constant stress triaxiality was controlled by adjusting displacement In the simulation, the constant stress triaxiality was controlled by adjusting displacement boundary conditions along Z (loading) direction, which increased the RVE volume. Since the volume boundary conditions along Z (loading) direction, which increased the RVE volume. Since the volume of the matrix is conserved during the plastic deformation, most of the volume change is subject to the of the matrix is conserved during the plastic deformation, most of the volume change is subject to the void except small changes in the matrix by elastic deformation. For the three primary orientations, void except small changes in the matrix by elastic deformation. For the three primary orientations, the amount of displacement along the loading direction was on the order of (110), (111) and (100), the amount of displacement along the loading direction was on the order of (110), (111) and (100), which corresponds rate for for the theoriented orientedcrystals. crystals. which correspondstotothe theorder orderof ofvoid void growth growth rate
(a)
(b)
Figure 4. (a) stress–strain curves of RVEs with a void in a single crystal with different crystal Figure 4. (a) stress–strain curves of RVEs with a void in a single crystal with different crystal orientations. The figure indicates the deformed voids in which the color code represents the orientations. The figure indicates the deformed voids in which the color code represents the equivalent equivalent plastic strain at one-layer elements surrounding void under a constant stress triaxiality of plastic at one-layer void growth under a rate) constant triaxiality of 1; (b) evolution 1; (b)strain evolution of voidelements volume surrounding fraction (or void as astress function of macroscopic RVE ofequivalent void volume fraction (or void growth rate) as a function of macroscopic RVE equivalent strain with different crystal orientations under a constant stress triaxiality of 1. strain with different crystal orientations under a constant stress triaxiality of 1.
Similar observations were made for the void growths of the three considered orientations from theSimilar stress-strain curves with the stressfor triaxiality 3 as shown inthree Figure 5a. However, the growth observations were made the voidof growths of the considered orientations from rate is much faster than those thetriaxiality triaxialityofof31.asThe softening of crystal with (100) orientation the stress-strain curves with thewith stress shown in Figure 5a. However, the growth rate also faster criticalthan compared to that the triaxiality 1, and the strain at the(100) UTSorientation point is much is is much those with the with triaxiality of 1. The of softening of crystal with is also decreased. The order of growth rate for the three orientations is similar, but the difference critical compared to that with the triaxiality of 1, and the strain at the UTS point is muchbetween decreased. (110) andof(111) orientations is not very distinctive is compared to that the stress triaxiality of 1. The order growth rate for the three orientations similar, but the for difference between (110) and (111) Regarding the principal direction of the void growth, the current simulations resulted in the orientations is not very distinctive compared to that for the stress triaxiality of 1. same observation as the previous work of Gurland (1972) [1], in which the void growth rate is fastest Regarding the principal direction of the void growth, the current simulations resulted in the same in the direction perpendicular to loading direction; i.e., Z direction in this study. As shown in Figures observation as the previous work of Gurland (1972) [1], in which the void growth rate is fastest in the 4 and 5, there is no outstanding void growth along the Z direction. However, note that the relative direction perpendicular to loading direction; i.e., Z direction in this study. As shown in Figures 4 and 5, lengths of the cell along X, Y and Z directions differ according to the prescribed boundary condition there is no outstanding void growth along the Z direction. However, note that the relative lengths of and the length of the cell along Z direction is the shortest. the cell along X, Y and Z directions differ according to the prescribed boundary condition and the length of the cell along Z direction is the shortest.
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(b)
(a)
Figure 5. (a) stress–strain curves of RVEs with a void in a single crystal with different crystal
Figure 5. (a) stress–strain curves of RVEs with a void in a single crystal with different crystal orientations. The figure indicates the deformed void in which the color code represents the equivalent orientations. The figure indicates the deformed void in which the color code represents the equivalent plastic strain at one-layer surrounding void under a constant stress (b) triaxiality of 3; (b) (a) elements plastic strain at one-layer elements surrounding void under a constant stress triaxiality of 3; (b) evolution evolution of void volume fraction (or void growth rate) as a function of macroscopic RVE equivalent Figure 5. (a) stress–strain curves of RVEs with a void in single crystal with different crystal of void volume fraction (or void growth rate) as aa constant function ofa macroscopic strain with strain with different crystal orientations under stress triaxiality ofRVE 1. equivalent orientations. The figure indicates deformedstress void intriaxiality which the color different crystal orientations underthe a constant of 1. code represents the equivalent plastic strain at one-layer elements surrounding a constant triaxiality of 3; (b) 6.2. Effect of Inter-Grain Orientation on the Growth ofvoid Voidunder Located at Grainstress Boundaries
evolution of void volume fraction (or void growth rate) as a function of macroscopic RVE equivalent 6.2. Effect of Inter-Grain Orientation on the Growth of Void Located at Grain Boundaries
To investigate the crystal effect of inter-grain orientation on the strain with different orientations under a constant difference stress triaxiality of 1.growth of void, the void is
located at the grain boundaries among three primarydifference orientations explained previous To investigate the effect of inter-grain orientation onasthe growth in of the void, the void is 6.2. Effect of Inter-Grain on the Growth Void for Located at Grain Boundaries section. 6boundaries showsOrientation the deformed shapes of of voids different arrays of crystals in the XY plane, located at theFigure grain among three primary orientations as explained in the previous section. whichToisinvestigate the plane the normal to the loading direction Z in this study. Both loading conditions with effect of inter-grain difference growthin ofthe void, void is Figure 6 shows the deformed shapes of voidsorientation for different arraysonofthe crystals XYthe plane, which is triaxialities of 1 grain and 3 were simulated. Thethree orientation of each crystal region is indicated the figure. located at the boundaries among primary as explained in with theinprevious the plane normal to the loading direction Z in this study.orientations Both loading conditions triaxialities of The various grain configurations consist of three or two orientations, which is intentionally section. 6 shows the deformed shapes of voids for different different arrays of crystals in the XY plane, 1 anddesigned 3 were Figure simulated. The orientation of each crystal region isofindicated in the figure. The various voids to located either at the grain bi-crystal or at conditions the triple junction which is to therepresent plane normal the loading direction Z boundary in this study. Both loading with grainofconfigurations consist of three or two different orientations, which is intentionally designed to tri-crystal.ofIn1 the the arrowsThe with dots and of solid indicate directionsinfor triaxialities andfigures, 3 were simulated. orientation eachlines crystal regionthe is indicated theminimum figure. represent voids located either theconsist grain boundary of or at the triple junction of tri-crystal. and void growthatrates, respectively. For allbi-crystal cases, the void grows slowest toward (110) The maximum various grain configurations of three or two different orientations, which is intentionally In thegrain, figures, the with and solid indicate the directions minimum and maximum which isarrows the opposite observation compared to boundary the single crystal casefor with void located inside designed to represent voids dots located either at lines the grain of bi-crystal or at the triple junction the grain.rates, Note that the (110) grain showed the growth rate toward among the three orientations. void growth For all cases, themaximum voidsolid grows slowest (110) grain, which is the of tri-crystal. Inrespectively. the figures, the arrows with dots and lines indicate the directions for minimum Another interesting observation is respectively. that the maximum rate islocated always in the direction toward andobservation maximum void growth rates, For case all growth cases, void grows inside slowest toward (110) opposite compared to the single crystal withthe void the grain. Note that the (100) crystal, if included in the RVE. For the (111) crystals, it shows intermediate growth rate for grain, which is the opposite observation compared to the single crystal case with void located inside the (110) grain showed the maximum growth rate among the three orientations. Another interesting thetri-crystals. grain. Note The that color the (110) grain showed the maximum growth rate among the in Figure 6 represents the equivalent plastic strain. the three orientations. observation is that the maximum growth rate is always in the direction toward the (100) crystal, Another interesting observation is that the maximum growth rate is always in the direction toward if included in the RVE. For the (111) crystals, it shows intermediate growth rate for the tri-crystals. the (100) crystal, if included in the RVE. For the (111) crystals, it shows intermediate growth rate for The color in Figure The 6 represents the equivalent strain. plastic strain. the tri-crystals. color in Figure 6 representsplastic the equivalent
(a)
(a)
(b) Figure 6. Growth of voids embedded in bi- and tri- crystals. The dotted arrow indicates the slowest void growth direction, while the solid arrow(b) represents the fastest void growth direction. The orientations are denoted at the corner of each cell: (a) for a constant stress triaxiality of 1, (b) a constant Figure 6. Growth of voids embedded in bi- and tri- crystals. The dotted arrow indicates the slowest Figurevoid 6. Growth of voids embedded in bi- and tri- crystals. The dotted arrow indicates the slowest void growth direction, while the solid arrow represents the fastest void growth direction. The growth direction, while the at solid arrowofrepresents void growth direction. orientations orientations are denoted the corner each cell: (a)the for fastest a constant stress triaxiality of 1, (b) The a constant
are denoted at the corner of each cell: (a) for a constant stress triaxiality of 1, (b) a constant stress triaxiality of 3. The color code represents the equivalent plastic strain (red denotes high strain region).
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stress triaxiality of 3. The color code represents the equivalent plastic strain (red denotes high strain stress triaxiality of 3. The color code represents the equivalent plastic strain (red denotes high strain region). region). In Figure 7, the flow stresses simulated with the RVEs for different fractions of (100) orientation are
7, the flow stresses with the RVEs for different of (100) orientation shown In forFigure both triaxialities 1 and 3.simulated As the volume fraction of the (100) fractions crystal increases, the strength In Figure 7,both the flow stresses1simulated with the RVEsfraction for different fractions of (100) orientation are shown for triaxialities and 3. As the volume of the (100) crystal increases, thethe of the RVE decreases, while the void growth in (110) crystal is delayed. Therefore, the strain at are shown for both triaxialitieswhile 1 and 3.void As the volume fraction of is the (100) crystal increases, the strength of the RVE decreases, the growth in (110) crystal delayed. Therefore, the strain UTS point is elongated as the void growth is concentrated in the portion of (100) crystal that has strength of the RVE decreases, as while growth in (110) crystal delayed. Therefore, the strain the UTS point is elongated the the voidvoid growth is concentrated in is the of (100) crystal that theatminimum growth rate in the single crystal simulation. The result is portion consistent with a commonly at UTS point is elongated as the voidsingle growth is concentrated the portion (100) crystal thata hasthethe minimum growth rate in the crystal simulation.inThe result isofconsistent with observed inverse relationship between strength and ductility of metallic materials. has the minimum in the single crystal simulation. The result is consistent commonly observedgrowth inverserate relationship between strength and ductility of metallic materials.with a For detailed analysis, therelationship idealized bi-crystal, in which the (110) crystal is inmaterials. the upper region, commonly observed inverse between strength and ductility of metallic For detailed analysis, the idealized bi-crystal, in which the (110) crystal is in the upper region, while (100) crystal analysis, is in the lower regionbi-crystal, of the considered RVE, was chosen. The shape and void detailed idealized in which the (110) is in theshape upper region, whileFor (100) crystal is in thethe lower region of the considered RVE, wascrystal chosen. The and void growth rate were compared in the two orientations in Figure 8 where the solid line indicates the while (100) is in the lower of the considered RVE, was chosen. Theline shape and void growth ratecrystal were compared in theregion two orientations in Figure 8 where the solid indicates the boundary between the two regions. Figure 8a shows the amount of deformation in the Z direction growth rate were compared in the two orientations 8 where the solid in line indicates the boundary between the two regions. Figure 8a shows in theFigure amount of deformation the Z direction (loading direction) of the void in the RVE. Figure 8b represents elements surrounding the void in boundary between the two regions. Figure 8a shows the amount of deformation in the Z direction (loading direction) of the void in the RVE. Figure 8b represents elements surrounding the void in thethe RVE in in 3D3D view after The grain boundary regionwas wasexcluded excluded fora abetter better geometrical (loading direction) ofdeformation. the void in the RVE. Figure 8b represents elements surrounding thegeometrical void in the RVE view after deformation. The grain boundary region for comparison between upper (110) and lower (100) crystals. Figure 8c shows the void growth rate RVE in 3D view after deformation. Thelower grain (100) boundary region was 8c excluded for avoid better geometrical comparison between upper (110) and crystals. Figure shows the growth rate of of each crystal region RVE. The void growth rate in in8c (100) crystal almost two times comparison between upper (110) and lower (100) shows the void growth rate of each crystal regionininthe thebi-crystal bi-crystal RVE. The voidcrystals. growthFigure rate (100) crystal isisalmost two times each crystal region in the bi-crystal RVE. The void growth the rate in (100) crystal is strain almost two times higher than that of of (110) crystal. Again, thethe color represents equivalent plastic the higher than that (110) crystal. Again, color represents the equivalent plastic strainwhere where thered higher than thathigh ofthe (110) Again, the color represents the equivalent plastic strain where the color the strain region. reddenotes color denotes highcrystal. strain region. red color denotes the high strain region.
(b) (b)
(a) (a)
Figure 7. Stress-strain curves of (110)-(100) bi-crystals with different fraction of (100) portions for the Figure 7. Stress-strain curves of (110)-(100) bi-crystals with different fraction of (100) portions for the Figure 7. Stress-strain curves ofThe (110)-(100) bi-crystals with different fraction (100) portions for the stress triaxiality (a) 1, and (b) 3. orientations are denoted in the bracket andofthe number represents stress triaxiality (a) 1, and (b) 3. The orientations are denoted in the bracket and the number represents stress triaxiality (a) 1,For andexample, (b) 3. The0.5 orientations are50% denoted in the bracket and the number represents the volume fraction. (100) means of (100) orientation. the volume fraction. For example, 0.5 (100) means 50% of (100) orientation. the volume fraction. For example, 0.5 (100) means 50% of (100) orientation.
(a) (a) Figure 8. Cont.
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Figure 8. (a) a void located at the grain boundary between (110) and (100) crystal after deformation of Figure 8. (a) a void located at the grain boundary between (110) and (100) crystal after deformation of a macroscopic strain 0.15: (a) view on XY and YZ planes; (b) 3D representation of the void by drawing a macroscopic strain 0.15: (a) view on XY and YZ planes; (b) 3D representation of the void by drawing one layer elements surrounding the void after the deformation; (c) evolution of void volume fraction one layer elements surrounding the void after the deformation; (c) evolution of void volume fraction (or equivalently the void growth rate) in (110)-(100) bi-crystal. The plots (110) and (100) represent the (or equivalently the void growth rate) in (110)-(100) bi-crystal. The plots (110) and (100) represent the void volume fraction of each portion in the bi-crystal. The region with the red color represents highly void volume fraction of each portion in the bi-crystal. The region with the red color represents highly deformed regions. deformed regions.
Figure 9a shows the growth rate of void embedded in (110) single crystal, (100) single crystal and (110)-(100) bi-crystal as a function of RVE macroscopic strain. For voids (100) in single crystals, Figure 9a shows the growth rate of void embedded in (110) single crystal, single crystalthe and growth rate of void (110) crystal is much higher than that (100) in crystal, wasthe already (110)-(100) bi-crystal as in a function of RVE macroscopic strain. Forofvoids singlewhich crystals, growth the crystal previous result.higher Figurethan 9b shows the averaged effective single layer rateobserved of void inin(110) is much that of (100) crystal, which wasstrains alreadyinobserved in the elements surrounding each void with respect to the RVE macroscopic strain for the stress triaxiality previous result. Figure 9b shows the averaged effective strains in single layer elements surrounding of void 3. The average elements surrounding thethe voids aretriaxiality significantly than that of each with respectstrains to theof RVE macroscopic strain for stress of 3.larger The average strains corresponding macroscopic strain, which explains considerable local deformation due to the void of elements surrounding the voids are significantly larger than that of corresponding macroscopic growth effect. The average strains in surrounding elements of both (100) and (110) crystal regions in strain, which explains considerable local deformation due to the void growth effect. The average the (110)-(100) bi-crystal were also calculated as shown in Figure 9b. As clearly indicated in the figure, strains in surrounding elements of both (100) and (110) crystal regions in the (110)-(100) bi-crystal considerable increase of deformation in (100) crystal could be observed when the crystal co-exists were also calculated as shown in Figure 9b. As clearly indicated in the figure, considerable increase of with (110) crystal. On the contrary, the deformation of (110) decreased in the (110)-(100) bi-crystal deformation in (100) crystal could be observed when the crystal co-exists with (110) crystal. On the compared to that in (110) single crystal. As previously discussed, the flow stress (or strength) of (110) contrary, the deformation of (110) decreased in the (110)-(100) bi-crystal compared to that in (110) single crystal is larger than that of (100) crystal for a given triaxiality and boundary condition. single crystal. Asdeformation previously in discussed, the flow stressof(or strength) for of (110) single crystal is larger Therefore, the the bi-crystal consisting orientations two distinct strengths was than that of (100) crystal for a given triaxiality and boundary condition. Therefore, the deformation accelerated in the softer grain, which increased the relative growth rate in (100) crystal. Note that the in theoverall bi-crystal consisting of orientations distinct strengths accelerated in theThe softer grain, growth rate in the bi-crystal didfor nottwo exceed the growth ratewas in (110) single crystal. results which increased the relative growth rate in (100) crystal. Note that the overall growth rate in the from the current numerical analysis have great importance in understanding the void growth bi-crystal did not exceed the growth rate in (110) single crystal. The results from the current numerical mechanism in polycrystalline metals: the void grows due to the hydrostatic stress component of the analysis have great the void growth mechanism in polycrystalline metals: prescribed stressimportance state, but initunderstanding is highly cross-linked with the crystallographic orientation of In general, the rate of component void growthof has preferred orientation in a single thesurrounding void grows crystals. due to the hydrostatic stress the prescribed stress state, but it crystal, is highly for example, (110) in thisorientation study, but of the rate can be crystals. significantly different when cross-linked with the orientation crystallographic surrounding In general, the rate other of void orientations of neighboring crystals, especially (100) are considered. growth has preferred orientation in a single crystal, fororientation, example, (110) orientation in this study, but the rate can be significantly different when other orientations of neighboring crystals, especially (100) orientation, are considered.
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Figure 9. (a) evolutions of void volume fractions of (100), (110) single crystals and (100)-(110) bi-crystal Figure 9. (a) evolutions of void volume fractions of (100), (110) single crystals and (100)-(110) bi-crystal as a function of macroscopic RVE strain; (b) average strains of one layer elements surrounding voids as a function of macroscopic RVE strain; (b) average strains of one layer elements surrounding voids as as a function of macroscopic RVE strain. Solid lines represent the strains for single crystals, while a function of macroscopic RVE strain. Solid lines represent the strains for single crystals, while dotted dotted lines are those in the bi-crystal. lines are those in the bi-crystal.
6.3. Slip System Activity (Resolved Sheaer Strain on the Slip Systems) in the RVE 6.3. Slip System Activity (Resolved Sheaer Strain on the Slip Systems) in the RVE The flow stress of (110) single crystal in the simple tension is not higher than that of (100) single The flow stress of (110) single crystal in the simple tension is not higher than that of (100) single crystal. However, the boundary condition applied in this study is different from the simple tension. crystal. However, the boundary condition applied in this study is different from the simple tension. That is, a tensile load is prescribed along the Z direction, while the strain in the Y direction is That is, a tensile load is prescribed along the Z direction, while the strain in the Y direction is negligible negligible and the strain in the X direction is negative. Thus, the deformation mode is similar to the and the strain in the X direction is negative. Thus, the deformation mode is similar to the plane strain plane strain condition. This boundary condition was intentionally adopted to represent the favorable condition. This boundary condition was intentionally adopted to represent the favorable fracture at fracture at higher stress triaxiality. Unfavorable slip systems under simple tension for (110) crystal higher stress triaxiality. Unfavorable slip systems under simple tension for (110) crystal are activated are activated to satisfy the prescribed boundary condition, which increases the flow stress of (110) to satisfy the prescribed boundary condition, which increases the flow stress of (110) single crystal single crystal under the considered constant stress triaxiality. under the considered constant stress triaxiality. Further analysis is provided here by investigating the slip activities (or activation of shear stress Further analysis is provided here by investigating the slip activities (or activation of shear stress on on preferred slip systems) in different crystals neighboring the embedded void. For simple preferred slip systems) in different crystals neighboring the embedded void. For simple compression compression along the Z direction for (110) crystal without a void, six slip systems (5, 6, 11, 12, 16, 22 along the Z direction for (110) crystal without a void, six slip systems (5, 6, 11, 12, 16, 22 slip systems in slip systems in Table 1) are activated with the highest Schmid factor for the slip systems 16 and 22. Table 1) are activated with the highest Schmid factor for the slip systems 16 and 22. The number of The number of the activated slip systems increases as the stress triaxiality increases. For example, the activated slip systems increases as the stress triaxiality increases. For example, additional six slip additional six slip systems (2, 3, 8, 9, 13, 19 slip systems in Table 1) are activated when the stress systems (2, 3, 8, 9, 13, 19 slip systems in Table 1) are activated when the stress triaxiality is increased triaxiality is increased to 1. The activation of the slip system is further complicated when the void is to 1. The activation of the slip system is further complicated when the void is embedded in the embedded in the single crystal or at the boundary of bi-crystal. Almost all slip systems among 24 bcc single crystal or at the boundary of bi-crystal. Almost all slip systems among 24 bcc slip systems are slip systems are activated near the void region. For more detailed analysis on the deformation activated near the void region. For more detailed analysis on the deformation behavior in the RVE, behavior in the RVE, cumulative shear strains by the slip system activation in the elements cumulative shear strains by the slip system activation in the elements surrounding a void are analyzed. surrounding a void are analyzed. Three cases—(100) single crystal, (110) single crystal and (110)-(100) Three cases—(100) single crystal, (110) single crystal and (110)-(100) bi-crystal—are considered. bi-crystal—are considered. In Figure 10, accumulated shear strains of all 24 slip systems are presented In Figure 10, accumulated shear strains of all 24 slip systems are presented in the elements surrounding in the elements surrounding the void for the similar value of averaged RVE strain in the single and the void for the similar value of averaged RVE strain in the single and bi-crystals under the stress bi-crystals under the stress triaxiality of 3. Figure 10a,c show the accumulated shear strains of the triaxiality of 3. Figure 10a,c show the accumulated shear strains of the elements along a void in (100) elements along a void in (100) and (110) single crystals, respectively, while Figure 10b,d show those and (110) single crystals, respectively, while Figure 10b,d show those for the elements along a void for the elements along a void in (100) and (110) crystals in the (100)-(110) bi-crystal, respectively. The in (100) and (110) crystals in the (100)-(110) bi-crystal, respectively. The figures show that the slip figures show that the slip systems activated in the same crystal orientation, that is the (100) or (110) systems activated in the same crystal orientation, that is the (100) or (110) crystals in both single crystal crystals in both single crystal and bi-crystal, are the same, but the absolute magnitudes are quite and bi-crystal, are the same, but the absolute magnitudes are quite different whether it is included different whether it is included in the single or bi-crystal. An interesting observation is that the in the single or bi-crystal. An interesting observation is that the magnitude of the accumulated shear magnitude of the accumulated shear strains of (110) region in single crystal and (100) region in the strains of (110) region in single crystal and (100) region in the bi-crystal are similar in an average sense. bi-crystal are similar in an average sense. This elucidates that slip activities in the multi-crystals can This elucidates that slip activities in the multi-crystals can be different from those in single crystals be different from those in single crystals due to the relative difference in strength, and softer crystal due to the relative difference in strength, and softer crystal accommodates a larger part of plastic accommodates a larger part of plastic deformation corresponding to the prescribed macroscopic deformation. Figure 11 also supports the fact that the preferred slip systems in single crystals are
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corresponding to the prescribed macroscopic deformation. Figure 11 also supports 13 ofthe 16 fact that the preferred slip systems in single crystals are preserved in the region of the same orientation preserved in the region of the same orientation the bi-crystal. In this figure, as an example, of the bi-crystal. In this figure, as an example, theofaccumulated shear strains in three slip systems,the 14, accumulated strains three slipsurrounding systems, 14,a23 andfor 24(100), are shown in the elements surrounding 23 and 24 areshear shown in theinelements void (110) single crystals and (100)-(110) abi-crystal. void for (100), (110)that single (100)-(110) bi-crystal. It is clear thatactivated the intensity of the slip It is clear the crystals intensityand of the slip activity is different, but the slip systems are activity is different, but thesingle activated slip systems are almost the same between single and bi-crystal. almost the same between and bi-crystal.
(a)
(b)
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Figure 10. Average accumulated shear on the slip systems at (a) (100) single crystal, (b) (100) portion Figure 10. Average accumulated shear on the slip systems at (a) (100) single crystal, (b) (100) portion in in (110)-(100) bi-crystal, (c) (110) single crystal and (d) (110) portion in the bi-crystal. (110)-(100) bi-crystal, (c) (110) single crystal and (d) (110) portion in the bi-crystal.
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Figure 11. Activities of slip systems 14, 23, 24 for (100) single crystal, (110)-(100) bi-crystal, and (110) Figure 11. Activities of slip systems 14, 23, 24 for (100) single crystal, (110)-(100) bi-crystal, and (110) single crystal. The color represents the accumulated shear strain in each slip system. The red color single crystal. The color represents the accumulated shear strain in each slip system. The red color denotes highly deformed regions. denotes highly deformed regions.
7. Conclusions 7. Conclusions In this study, an RVE approach incorporating the rate-dependent crystal plasticity finite element In this study, an RVE approach incorporating the rate-dependent crystal plasticityinfinite element simulation was introduced to explore the characteristics of the void growth embedded the ductile simulation was introduced to explore the characteristics of the void growth embedded in the ductile metal matrix. Unlike the previous similar numerical studies, the current study investigated the effect metal matrix. Unlike the previous similar numerical studies, the current study investigated the effect of orientation relationship between two or three adjacent crystals on the growth of void during large of plastic orientation relationship between twotriaxialities or three adjacent crystalsas onboundary the growth of voidwhich during deformation. Two different stress were considered conditions, large plastic deformation. Two different stress triaxialities were considered as boundary conditions, were maintained as constants during the deformation of the RVE. The simulations showed that the which were maintained as preferred constantsorientation during thewhen deformation of the The simulations that rate of void growth had embedded in aRVE. single crystal, while itshowed could be thesignificantly rate of voiddeviated growth had preferred orientation embedded in awere single crystal,The while it could be when other orientations of when neighboring crystals included. followings are detaileddeviated conclusions made from the numerical simulations crystals for a generic model material. significantly when other orientations of neighboring werebcc included. The followings are• detailed conclusions made from the numerical simulations for a generic bcc model material. The RVE simulations with a rate-dependent crystal plasticity model could reproduce the orientation dependentwith singlea crystal flow stresscrystal behavior of the bcc metal could under reproduce two distinctthe The RVE simulations rate-dependent plasticity model constant stress triaxialities as reported in the previous work by Yerra et al. [13] and observed orientation dependent single crystal flow stress behavior of the bcc metal underastwo distinct in general bcc metals. In other words, the crystal with the highest flow stress resulted in the in constant stress triaxialities as reported in the previous work by Yerra et al. [13] and as observed earliestbcc localization least uniform to UTS while delayed general metals. In(or other words, the elongation crystal withcorresponding the highest flow stresspoint), resulted in the earliest localization for the lower flow stress when a void is included in the RVE. localization (or least uniform elongation corresponding to UTS point), while delayed localization • The growth rate of a void in the single crystal is highly dependent on the crystal orientation and for the lower flow stress when a void is included in the RVE. the rate is on the order of (110), (111) and (100) orientations for the present study. The growth • The growth rate of a void in the single crystal is highly dependent on the crystal orientation and rate is accelerated as the stress triaxiality increases, which has been commonly known in the the rate is on the order of (110), (111) and (100) orientations for the present study. The growth rate ductile fracture analysis. accelerated stress increases, whichorhas been commonly in the ductile • is When a voidas is the located at triaxiality the boundary of bi-crystal at the triple junctionknown of a tri-crystal, the fracture analysis. growth rate of a void shows considerable dependence on the orientation relationship among • When a voidcrystals. is located boundary of bi-crystal orand at the junctionthe of maximum a tri-crystal, constituent Forat allthe cases with mixed (100), (111), (110)triple orientations, thegrowth growthrate rateofofaavoid voidisshows considerable dependence on the orientation relationship among in the (100) crystal and the lowest is in the (110) direction, which constituent crystals. For all cases with mixed (100), (111), and (110) orientations, the maximum contradicts the single crystal cases. This difference of void growth rate in single crystal and growth rate of a void is in the (100) crystal and the lowest is in the (110) direction, which
•
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contradicts the single crystal cases. This difference of void growth rate in single crystal and multi-crystal is attributed to the relative strength differential between grains with specific orientations. For example, for the given boundary conditions, the flow stresses of (110) and (100) crystals represent highest and lowest values and the deformation is highly concentrated in the softer (100) crystal when the two crystals co-existed. For the prescribed macroscopic strain (in averaged sense), the strains of elements surrounding a void between (100) and (110) regions are significantly increased and decreased, respectively. The average strains of (100) and (110) portions in the bi-crystal are equivalent to those of (110) and (100) single crystals. This result could be explained by the analysis of slip system activation (or accumulated shear strain in the activated slip systems) in which the overall activation pattern of multiple slips is similar between the single and bi-crystal, but the magnitudes of the shear strains are quite different.
The present study focused on the numerical parametric study with a generic bcc metal, and limited experimental validation was provided based on results in the published references. However, considering the high fidelity of the modern crystal plasticity simulations, the results discussed in this study might be well applied to the fundamental material design stage to optimize the formability and strength. As a future work, this approach can be further extended to the polycrystalline case where the distribution of orientations or texture effect should be investigated. Author Contributions: M.G.L. and W.J. conceived and designed the computational modeling; C.-H.L., J.M., and D.J. analyzed the data; M.-G.L. and W.J. wrote the paper. Funding: This research was funded by the National Research Foundation of Korea grant number (NRF-2016M1A7A1A01005904). Acknowledgments: The EBSD and TEM works have been discussed with Prof. Hong at Changwon National University, and Prof. Shin at Myoungji University, which is highly acknowledged. Conflicts of Interest: The authors declare no conflict of interest.
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