Abstract
Deformation twinning is an essential plastic deformation mechanism that realizes the trade–off between strength and ductility. Twins nucleate and grow by the coordinated slip of partial dislocations on consecutive {111}–type slip planes. The route of twin growth is responsible for the evolution of twin morphology, which affects the non–coplanar dislocation slips by adjusting the twinning–associated mean free path. Incorporating such twinning mechanisms is critical for the accurate modelling and simulation of deformation behavior. In this study, a discrete dislocation plasticity (DDP) model was developed by integrating the source introduction methods and source activation criteria of Shockley partial dislocation. In the model, two twin nucleation mechanisms, i.e., the internal source and surface source, were considered concurrently, and the additional effect of stacking fault energy on the motion of partial dislocations was introduced. The evolution of partial dislocation slip–mediated deformation twins in micron–sized pillars of twinning–induced plasticity steel under uniaxial compression was investigated. The predicted twin morphologies and stress–strain curves from DDP simulation both agreed well with the experimental results, highlighting the inherent characteristics of partial–dislocation–based twinning behavior. The simulation results showed that the formation of nanometer–sized sharp twin tips was caused by the strong interaction between the front and rear dislocations on adjacent slip planes. In addition, a novel analytical model verified with the DDP simulation was proposed by considering the kinetics of the newly formed twin embryos. The competition between new twin activation and near–twin merging in determining the evolution of twin thickness was analyzed using the analytical model. The dependence of flow stress and twin morphology on the density and distribution of internal sources was demonstrated by considering the new twinning route. This research thus advances the understanding of partial dislocation slip–mediated twinning mechanisms.
| Original language | English |
|---|---|
| Article number | 103230 |
| Journal | International Journal of Plasticity |
| Volume | 152 |
| DOIs | |
| Publication status | Published - May 2022 |
Keywords
- Discrete dislocation plasticity
- Partial dislocation
- Twin nucleation
- Twinning
- Twinning–induced plasticity
ASJC Scopus subject areas
- General Materials Science
- Mechanics of Materials
- Mechanical Engineering
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In: International Journal of Plasticity, Vol. 152, 103230, 05.2022.
Research output: Journal article publication › Journal article › Academic research › peer-review
TY - JOUR
T1 - Evolution of partial dislocation slip–mediated deformation twins in single crystals: a discrete dislocation plasticity model and an analytical approach
AU - Wang, Chunhui
AU - Sun, Chaoyang
AU - Cai, Wang
AU - Qian, Lingyun
AU - Guo, Xiangru
AU - Fu, M. W.
N1 - Funding Information: Authors acknowledge the funding supported by National Natural Science Foundation of China (No. 52175285, 52161145407), International Exchanges Scheme of NSFC and Royal Society under Grant (No. 51911530209), and Fundamental Research Funds for the Central Universities (No. FRF-BD-20-08A, FRF-AT-20-09, FRF-TP-20-009A2). a. Surface sources, Deformation twins are believed to nucleate and thicken in submicron– and micron–sized pillars of TWIP steel under uniaxial deformation by successive partial dislocations emitted from the pillar surface and their subsequent propagation along the slip plane ( Choi et al. 2017; Liang and Huang, 2015; Wu et al. 2012). Fig. A1 provides insight into the physical meaning of twin surface nucleation to enable the introduction of the surface emission mechanism of partial dislocations into the 2D–DDP model. In Fig. A1(b), sectioned from a single cylindrical crystal pillar, the elliptical twinning plane (111) is enveloped by the crystal surface, and the slip direction [112¯] along the major axis of the ellipse is consistent with the Burgers vector a/6[112¯] of the arc partial dislocation. Activated when the resolved shear stress exceeds the nucleation strength, a leading partial dislocation is emitted from the acute corner of the ellipse, which subsequently slips to the opposite side leaving an ISF in mode 1. The conventional nucleation mechanism of the surface source for perfect dislocation is realized by the thermal activation of atoms in the pristine pillar ( Hu et al. 2017; Ryu et al. 2015). In contrast, the entangled dislocations are introduced as potential surface sources here, caused by defects on the crystal surface fabricated by a focused–ion beam ( Liu et al. 2020), with the criterion for surface emission of partial dislocations determined by the change in free energy before and after nucleation ( Jennings et al. 2013; Liang and Huang, 2015). The free energy change (ΔGs1) consists of the line energy of nucleation dislocations, the stacking fault energy, and the external work due to shear stress, expressed as ΔGs1(r)=Qline+QSF+Qwork Qline=∫−ΦΦ[Formula presented]ln(m[Formula presented])dφ QSF=γisf·A(r) Qwork=−τrssbp·A(r)where G and υ are the shear modulus and Poisson's ratio, respectively; bp is the Burgers vector of the partial dislocation; r is the initial radius of the arc partial dislocation centered on the end of the ellipse; α is the angle between the orientation of the dislocation line segment and the Burgers vector; Φ is half of the angle between the two intersections of the arc dislocation and the ellipse relative to the center of the circle; and r0 is the radius of the dislocation core region with a general value of bp ( Jennings et al. 2013). To incorporate the effect of the image force into the model, we introduce a constant m=0.55 that depends on Poisson's ratio and the dislocation configuration ( Beltz, 1993; Jennings et al. 2013). In Eqs. (A3) and ( A4), γisf is the ISF energy per unit area, τrss is the resolved shear stress, and A(r) is the area swept by the leading partial dislocation. In Eq. (A1), the free energy change first increases before decreasing with the radius of partial dislocation given a certain external stress, so there is a stress–dependent critical radius. This radius is exceeded by the slip distance, so the dislocation can be considered to propagate spontaneously, analogous to the activation–expansion mechanism of the Frank–Read source. Differentiating the free energy with respect to the radius of the arc partial dislocation and then letting the equation equal zero gives [Formula presented]|(τcrss)=0, The relationship between the surface source radius and the critical resolved shear stress can therefore be derived. As shown in Fig. A1(c), the critical resolved shear stress decreases with increasing source radius, i.e. a smaller surface source size requires higher activation stress. Thus, a radius of several tens of nanometers, consistent with the damage depth of ion–irradiated surfaces, is considered as an initial parameter. This is made to follow certain distribution conditions, such as the length distribution of the bulk source segment, so that its strength can be computed using the line tension approximation. The configuration of stacking faults or twins around the surface source also affects the emission of partial dislocations. As illustrated in Fig. A1(a), the surface–mediated nucleation mechanism of deformation twins consists of three modes for surface emission of partial dislocations, which form various morphologies of planar stacking faults. In mode 1, the surface source nucleates in an isolated region without stacking faults or twins on the adjacent slip planes, and the slip of a single Shockley partial leaves a single–layer ISF. Subsequently, the slip of the partial dislocations on the (111) slip plane with the boundary of the ISF gives rise to the formation of a two–layer ESF and then a TF, as shown in mode 2. Here, the additional partial dislocation slip has almost no effect on the stacking fault energy component of the free energy change, which can therefore be expressed as ΔGs2(r)=Qline+Qwork, Compared with mode 1, the resolved shear stress with the same source radius is significantly lower in mode 2, which enables further twin thickening in Fig. A1(c). When a partial dislocation is emitted between two adjacent stacking faults or twins, two independent stacking faults merge into a larger stacking fault, which is equivalent to eliminating a planar stacking fault in the twin domain in mode 3. Therefore, the expression of the free energy change is revised as ΔGs3(r)=Qline−QSF+Qwork, In this situation, removing the stacking fault energy acting against the expansion of the stacking fault region makes nucleation of the surface source more energetically favorable, requiring a lower stress level than for the first two modes in Fig. A1(c). In short, on the basis of the relationship between these generalized planar fault energies ( Liang et al. 2017), the same stacking fault energy is used in all the cases for simplicity, while the calculation of the free energy change under different modes needs to consider whether the stacking fault energy is introduced. Modes 2 and 3 also explain the preference for existing twin thickening instead of separate twin nucleation in TWIP steel micropillars. b. Internal sources, An isolated surface source nucleating through a partial dislocation emission with a radius of tens of nanometers requires a critical resolved shear stress of at least 200 MPa. However, the shear yield stress of an experimental micropillar was only 126 MPa, based on the Schmid factor (0.47) of the slip system corresponding to the leading partial dislocation for a [001]–oriented single crystal ( Choi et al. 2017; Liang and Huang, 2015). Therefore, the surface source for twin nucleation is unlikely to be activated at such a low stress level ( Liang et al. 2017). Many micropillars are non–pristine in micromechanics experiments ( Choi et al. 2018b), so perfect dislocations of considerable size remain in the crystal, which accommodate the initial plastic deformation as an internal source by dissociating into the leading partial and trailing dislocations connected by a stacking fault ribbon. The Schmid factor of the leading partial dislocation is almost twice that of the trailing dislocation (0.24) for a [001]–oriented single crystal under compression, so the resolved shear stress between the two kinds of dislocations is considerably different under an external force. Moreover, low stacking fault energy is beneficial to the increase of stacking fault width, and the leading dislocation can therefore be regarded as glissile while the trailing dislocation is sessile, as depicted in Fig. A3. As shown in Fig. A3(b), a stacking fault with a pair of partial dislocations as boundaries is created by the dissociation of a perfect dislocation as the internal source inside the pillar. The internal source for twin nucleation is simplified as a straight dislocation segment along the [1¯10] direction on the (111) elliptical twinning plane. Once dissociation occurs, the trailing dislocation with Burgers vector a/6[1¯21¯] is fixed at its initial position, while the straight leading dislocation with Burgers vector a/6[112¯] slips out of the free surface driven by the external shear stress, forming an ISF in mode 1. The free energy change (ΔGI1), comprising the dislocation line energy before and after nucleation, the interaction energy between leading and trailing dislocations, the stacking fault energy and the external work due to shear stress, is expressed as ΔGI1(L0,w)=Qline+Qinter+QSF+Qwork Qline=[Formula presented]ln[Formula presented]·L(L0,w)+[Formula presented]ln[Formula presented]L0−[Formula presented]ln[Formula presented]L0 Qinter=−∫0w[Formula presented]·L(L0,w)dw QSF=γisf·A(L0,w) Qwork=−τrssbp·A(L0,w)where L0 is the length of the internal source and sessile trailing partial, L is the length of the leading partial as a function of the stacking fault width w and the length of the internal source; R and R0 are the respective outer cut–off radii of the corresponding dislocation elastic fields, which are set to half the distance between the target dislocation and the nearer vertex of the major axis of the ellipse. Additionally, β is the angle between the Burgers vectors of the leading partial and the dragging partial and A(L0,w) is the area swept by the leading partial dislocation. Notably, the first two terms of Eq. (A9) are the line energy of the two partial dislocations after dissociation, and the last term is the line energy of the perfect dislocation before dissociation which is absent in prior research ( Liang and Huang, 2015). In the current coordinate system x-y in Fig. A3(b), a standard ellipse centered at the origin with width 2be and height 2ae. Thus, the coordinates (x, y) of any point on the contour of the elliptical twinning plane can be described in the following forms: [Formula presented]+[Formula presented]=1, For the length L0 of the internal source, one of the end point positions is (x0, y0) with the following relation: {y0=[Formula presented]x0=ae1−(y0/be)2, The position (x1, y1) of the end point of the leading partial dislocation dissociated from the internal source and sliding for a distance w is {x1=x0−wy1=be1−(x1/ae)2, As a result, the length L(L0,w) of the leading partial dislocation can be expressed by the combination of Eqs. (A14) and ( A15): L=2y1=2be1−(1−(L0/2be)2−w/ae)2, The area A(L0,w) swept by the leading partial dislocation is A=2be∫x1x01−(x/ae)2dx=2be∫ae1−(L0/2be)2−wae1−(L0/2be)21−(x/ae)2dx, Based on the exact expressions used to calculate L(L0,w) and A(L0,w), the free energy change for the dissociation of three perfect dislocations (L0=100,300,500nm) under a resolved shear stress of 150 MPa is plotted in Fig. A2(a). During the dissociation process, there is a peak of the free energy change. With the increase of the length, i.e. the pre-existing perfect dislocation located closer to the center of the pillar, the value of the peak will decrease until it is below zero, which means that the dissociation process will proceed spontaneously. Furthermore, as the free energy changes from the initial negative value to the positive value with increasing stacking fault energy, we can evaluate the critical slip spacing w, i.e. the stacking fault width of two partial dislocations after dissociation. Consequently, the evolution of L(L0,w) and A(L0,w) as the stacking fault energy grows from 18 mJ•m−2 to 35 mJ•m−2 is displayed in Fig. A2(b) for a certain internal source length (L0=500nm) and resolved shear stress (τrss=127MPa). The results show that both L and A decrease with increasing stacking fault energy, which is due to the increase of free energy change related to fault energy. As a function of the internal source length, the applied shear stress and the slip distance of the leading dislocation, the free energy change reaches a peak with the dissociation distance concerning the certain first two parameters in Eq. (A8). Hence, for a given internal source size, a critical resolved shear stress can be determined when the peak value of free energy change at the equilibrium distance is equal to 0, which implies that the nucleation of internal partial dislocations and subsequent propagation of stacking faults can occur spontaneously. Consequently, the critical resolved shear stress and equilibrium distance as a function of the internal source length are derived by: [Formula presented]|(wequil,τcrss)=0 ΔGI1|(wequil,τcrss)=0, In Fig. A3(c), the critical resolved shear stress decreases and the equilibrium distance increases with increasing length of the internal source when the perfect dislocation is located close to the center of the pillar before dissociation. In addition, the arrangements of stacking faults and twins around the internal sources in modes 1, 2 and 3 illustrated in Fig. A3(a) are equivalent to those of the surface sources, so a similar expression of free energy change revised considering the contribution of stacking fault energy is employed. When Eqs. (A18) and ( A19) are applied to modes 2 and 3, the critical resolved shear stress is lower than in mode 1, but the equilibrium distance is larger compared with mode 1. Especially in mode 3, the internal dislocation dissociation and the propagation of partial dislocations are spontaneous even without external shear stress if the event of partial slip occurs between two adjacent stacking faults or twins. Note that in the above case involving larger internal source length in mode 2, there is no peak value of free energy change and the equilibrium distance is set to a certain value, which indicates that the leading partial will slip out of the free surface of the pillar once the internal source nucleates. Funding Information: Authors acknowledge the funding supported by National Natural Science Foundation of China (No. 52175285, 52161145407), International Exchanges Scheme of NSFC and Royal Society under Grant (No. 51911530209), and Fundamental Research Funds for the Central Universities (No. FRF-BD-20-08A, FRF-AT-20-09, FRF-TP-20-009A2). Publisher Copyright: © 2022 Elsevier Ltd
PY - 2022/5
Y1 - 2022/5
N2 - Deformation twinning is an essential plastic deformation mechanism that realizes the trade–off between strength and ductility. Twins nucleate and grow by the coordinated slip of partial dislocations on consecutive {111}–type slip planes. The route of twin growth is responsible for the evolution of twin morphology, which affects the non–coplanar dislocation slips by adjusting the twinning–associated mean free path. Incorporating such twinning mechanisms is critical for the accurate modelling and simulation of deformation behavior. In this study, a discrete dislocation plasticity (DDP) model was developed by integrating the source introduction methods and source activation criteria of Shockley partial dislocation. In the model, two twin nucleation mechanisms, i.e., the internal source and surface source, were considered concurrently, and the additional effect of stacking fault energy on the motion of partial dislocations was introduced. The evolution of partial dislocation slip–mediated deformation twins in micron–sized pillars of twinning–induced plasticity steel under uniaxial compression was investigated. The predicted twin morphologies and stress–strain curves from DDP simulation both agreed well with the experimental results, highlighting the inherent characteristics of partial–dislocation–based twinning behavior. The simulation results showed that the formation of nanometer–sized sharp twin tips was caused by the strong interaction between the front and rear dislocations on adjacent slip planes. In addition, a novel analytical model verified with the DDP simulation was proposed by considering the kinetics of the newly formed twin embryos. The competition between new twin activation and near–twin merging in determining the evolution of twin thickness was analyzed using the analytical model. The dependence of flow stress and twin morphology on the density and distribution of internal sources was demonstrated by considering the new twinning route. This research thus advances the understanding of partial dislocation slip–mediated twinning mechanisms.
AB - Deformation twinning is an essential plastic deformation mechanism that realizes the trade–off between strength and ductility. Twins nucleate and grow by the coordinated slip of partial dislocations on consecutive {111}–type slip planes. The route of twin growth is responsible for the evolution of twin morphology, which affects the non–coplanar dislocation slips by adjusting the twinning–associated mean free path. Incorporating such twinning mechanisms is critical for the accurate modelling and simulation of deformation behavior. In this study, a discrete dislocation plasticity (DDP) model was developed by integrating the source introduction methods and source activation criteria of Shockley partial dislocation. In the model, two twin nucleation mechanisms, i.e., the internal source and surface source, were considered concurrently, and the additional effect of stacking fault energy on the motion of partial dislocations was introduced. The evolution of partial dislocation slip–mediated deformation twins in micron–sized pillars of twinning–induced plasticity steel under uniaxial compression was investigated. The predicted twin morphologies and stress–strain curves from DDP simulation both agreed well with the experimental results, highlighting the inherent characteristics of partial–dislocation–based twinning behavior. The simulation results showed that the formation of nanometer–sized sharp twin tips was caused by the strong interaction between the front and rear dislocations on adjacent slip planes. In addition, a novel analytical model verified with the DDP simulation was proposed by considering the kinetics of the newly formed twin embryos. The competition between new twin activation and near–twin merging in determining the evolution of twin thickness was analyzed using the analytical model. The dependence of flow stress and twin morphology on the density and distribution of internal sources was demonstrated by considering the new twinning route. This research thus advances the understanding of partial dislocation slip–mediated twinning mechanisms.
KW - Discrete dislocation plasticity
KW - Partial dislocation
KW - Twin nucleation
KW - Twinning
KW - Twinning–induced plasticity
UR - http://www.scopus.com/inward/record.url?scp=85123835834&partnerID=8YFLogxK
U2 - 10.1016/j.ijplas.2022.103230
DO - 10.1016/j.ijplas.2022.103230
M3 - Journal article
AN - SCOPUS:85123835834
SN - 0749-6419
VL - 152
JO - International Journal of Plasticity
JF - International Journal of Plasticity
M1 - 103230
ER -