Constitutive Modeling based on Evolutionary Multi-junctions of 

Dislocations 

Minh-Son Pham1,2, Anthony D. Rollett1

, Adam Creuziger2, 

Mark A. Iadicola2 and Timothy Foecke2 

1Dept. of Materials Sci. & Eng., Carnegie Mellon Univ., Pittsburgh, PA 15213, USA 

2

 Materials Performance Group, National Institute of Standards and Technology, 

US Department of Commerce, Gaithersburg, MD 20899-8553, USA 

*Email: minhson@andrew.cmu.edu 

Keywords: crystal plasticity, constitutive law, dislocation interactions, latent hardening, aluminum.

Abstract. A latent hardening model based on binary junction-induced hardening can effectively 

describe the anisotropy measured in multiaxial tests. However, this approach still has some 

descriptive and predictive limitations. Recent findings show that binary junctions generated by 

interactions of pairs of dislocations can only induce short-term hardening effect due to the 

unzipping process of binary junctions. By contrast, multi-junctions, which are formed via multiple 

interactions of dislocations, can exert a strong and enduring influence on the hardening of 

polycrystals. In this study, we extend the modeling of dislocation junctions from the binary to multijunctions, and implement this evolution into a self-consistent visco-plastic model. An application of 

this model for predicting the yield surface and texture evolution of AA5754 during uniaxial and 

plane strain loadings is given as a demonstration of the capabilities of the evolutionary binary-multi

junction approach. 

I. Introduction 

The movement and accumulation of dislocations are responsible for the plastic deformation of 

crystalline materials. The way dislocations interact during movement under specific boundary 

conditions influences the movement/accumulation of dislocations. It was first assumed that 

numerous dislocation binary junctions formed due to the 1st order interaction of dislocations (i.e., 

pairwise interactions) are responsible for deformation behavior of materials. The detailed 

incorporation of dislocation binary junctions into a crystal plasticity-based constitutive model can 

describe the evolution of texture and plastic anisotropic response reasonably well [1]. However, it 

still fails to accurately predict the intensity of main texture components and the evolution of the 

yield surface. Recent findings show that dislocation multi-junctions also exist, are stable at large 

deformation, and have a non-trivial influence on the plastic deformation response [2, 3]. The 

descriptive and predictive capabilities of constitutive models should be improved by incorporating 

dislocation higher-order interactions into constitutive models. In this study, we will implement the 

evolution of dislocation interaction from the 1st order to higher order interactions, and discuss the 

effectiveness of this approach on the modeling of texture and plastic anisotropy of AA5754. 

II. Dislocation interactions in Face-centered-cubic (FCC) materials 

There are twelve {111 110 } < > slip systems for FCC materials. Dislocation interactions between 

different slip systems form resultant junctions that hinder the movement of dislocations, leading to 

latent hardening. Beside the self-interactions of dislocations in the same slip system, there are four

categories of binary non-coplanar interactions between dislocations moving on different slip 

systems during plastic deformation. The most frequent interactions are Lomer-Cottrell interactions, 

which result in the formation of Lomer-Cottrell (LC) sessile junctions [4, 5]. The other dislocation 

interactions are Hirth (H), glissile (G) and colinear (COL) [6, 7], resulting in H, G, COL binary 

junctions, respectively. The binary interactions are often found to be prevalent in polycrystals. 

However, Bulatov et al. showed that higher order interactions (e.g. ternary interaction) also exist, 

resulting in the formation of multi-junctions which are very stable and greatly influence the strain 

hardening behavior [2]. Similarly, Madec and Kubin also showed that the ternary interactions can 

Key Engineering Materials Vols. 611-612 (2014) pp 1771-1776

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doi:10.4028/www.scientific.net/KEM.611-612.1771

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take place in FCC materials, but reported that the ternary junctions did not have as much influence 

on the plastic deformation behavior as they did in body-centered-cubic (BCC) materials [3]. It is 

interesting to note that a type of higher order dislocation interaction was studied in detail by Carter 

quite a long time ago [8]. Carter already discussed the interaction between co-planar multiple partial

dislocations and an inclined dissociated dislocation (Fig. 1 in [8]). This higher order interaction can 

change the character and configuration of both junction and junction nodes significantly. 

 

Figure 1: (a) Thompson tetrahedron and (b) its expansion to describe slip systems in FCC materials, 

(c) The assumed evolution of the strength of dislocation junctions. 

Moreover, a recent interesting treatment of dislocation interaction based on the Peierls model by 

Schoeck showed that there can be a higher-order interaction in a sequential fashion that transforms 

binary junctions into multi-junctions [9]. For example, a dislocation on slip system 1 interacts with 

another one on slip system 6 to form a LC lock γβ as follows, 

BD,a/2[0-1-1]+DC,a/2[110]=Bγ, a/6[1-2-1]+ γβ,a/6[10-1]+ βC,a/6[12-1] (1) 

Where A, B, C and D are vertices while α, β, γ, and δ are the midpoints of planes of Thompson 

tetrahedron opposite to A, B, C, and D, respectively (Fig. 1a); and a is the lattice parameter. 

The LC junction effectively locks other approaching edge dislocations. However, if a screw 

dislocation on the slip system 5 approaches the LC lock, the stress induced by the stress field of the

LC lock on the edge component of the screw dislocation increases. According to Shoeck‘s 

treatment, once the imposed stress reaches a critical value, the edge component of the approaching 

dislocation becomes unstable and collapses [9]. The stopping effect of the LC lock on the screw 

dislocation no longer holds [9]. A part of the approaching dislocation now can interact with the γβ 

lock to convert it into a Hirth junction: 

a/6[10-1]+a/6[-10-1]=a/3[00-1] (2) 

The Hirth junction is indeed a multi-junction with two 4-nodes at its ends where three interacting 

dislocations (on slip systems 1, 6 and 5) and the Hirth junction meet. Schoeck reported that this 

higher-order interaction would dramatically change the locking effect of the resultant junction since 

it introduces a screw character into the resultant junction. Along with what was reported in the 

Carter study [8], not only the character of dislocation junctions, but also the character and 

configuration of junction nodes are altered after interaction, e.g. expanded nodes can change into 

contracted nodes. All alternations of junction and nodes influence the locking effect of multijunctions. The situation becomes complex for super-jogs (i.e. super junctions whose Burgers vector 

is several times of unit binary junctions), where a junction can partially convert into another type, 

but its original part is still preserved after interaction. 

III. Constitutive equations 

At the grain level, the plastic strain rate ε

p

 is given by the sum of the shear strain rates ( )iγ from 

all the active slip systems 

( ) ( )

1

N

p i i

i

ε γ m

=

  = ∑

, 

(3) 

where N is the number of slip systems, ( )i m is the Schmid tensor of the slip system ( )i . The Schmid 

tensor is defined as 

1772 Material Forming ESAFORM 2014
( ) ( ) ( ) ( ) ( ) ( ) 1

2

i i i i i m b n n b = ⊗ + ⊗

 

(4) 

where b is the slip direction and n is the slip plane normal. The rate at which dislocations move 

under the influence of a shear stress on their glide plane depends on the magnitude of the shear 

stress. This rate sensitivity can be modeled as a power-law relationship between the resolved shear 

stress on slip systems τ

(i)

 and the shear rate γ

(i)

 [10], 

γ

(i)

= γ

0

(i) τ

(i)

τˆ

(i)

1/r

sgn(τ

(i)

) (5) 

with τ

(i)

, 

( )

ˆ

i

τ and ( )

0

i

γ are the Schmid resolved shear stress, the critical resolved shear stress and the 

reference shear rate for dislocation motion on slip system (i), respectively. Under imposed 

deviatoric Cauchy stress σ

’ 

, τ

(i)

 is given by 

τ

(i)

= m

(i)

:σ

'

, (6) 

The rate sensitivity r is assumed to be constant during isothermal deformation process and to be 

equal for all slip systems (r = 20 in this study). The critical stress of the slip system i (i.e., ( )

ˆ

i

τ ) 

evolves during plastic deformation due to the accumulation of dislocation density, which is a 

function of accumulated plastic strain. 

( )

( )

( )

, , 1

i

N i

j

ij k

i j k

d

h

d

τ

τ γ

γ

•

•

×

=

= ∑



(i j k N , , and 1, = ) (7) 

with ( )

1

N

i

i

γ γ

=

=∑ and ij k h ×

is a hardening matrix induced by dislocation interactions which evolves 

from binary into higher-order interactions. At the beginning of deformation, a single binary 

interaction type (e.g., COL hij ) is dominant and governs the material response. During deformation, 

higher-order interactions (involving the interaction of third dislocations with the existing binary 

dislocation junctions) occur to transform binary junctions into multi-junctions. This transformation 

can cause the strength of the initially strongest binary junctions to slightly decrease, while the 

strength of the new component can gradually increase after interaction. In this study, we first 

assume that (1) the COL is the strongest binary junction and (2) the strengths of other junctions 

equals to the strength of self-interaction at the beginning of plastic deformation. The second 

assumption is that once the multi-junctions form they are stable as reported by Bulatov et al.[2]. 

Stable multi-junctions can undergo a maximum number of three serial higher-order non-coplanar 

interaction types during deformation. The last assumption is that the strength of a multi-junction 

does not change if a junction experiences self-interactions or coplanar interactions with other 

dislocations. This sequential approach to dislocation interactions from first to higher order will 

cause the strength of dislocation junctions to asymptotically approach the saturation strength of 

stable multi-junctions. In this study, we assume the saturation strength is the arithmetic mean of the

strengths of all four non-coplanar binary junctions. Following the above-mentioned evolution of 

binary into multi-junctions, the evolution of hardening coefficients can be described by Eq. 8: 

( ) e

sat g sat

ij k ij ij k ij k h h h h − γ× × × = − + (8) 

ij h is the hardening rate associated with binary junctions of slip systems i and j, 1 ij h = for i j = and 

1 ij h > for non-coplanar binary intearctions with i j ≠ . ij k h ×is the hardening rate associated with 

ij k × multi-junctions which are resultants of interaction between the ij binary junction and another 

dislocation on system k. 

sat hij k ×

is the average of the strength of all binary junctions involving in the 

interaction chain. g is a constant to control the degree of deformation where the strength of all 

multi-junctions converges to sat hij k ×. We set g=100 to let all binary junctions develop into multijunctions after plastic deformation of about 5 % in this study (Fig. 1c). 

Key Engineering Materials Vols. 611-612 1773
IV.Results and Discussions 

The AA5754 is weakly textured in the as-received (AR) condition. The initial texture combines 

recrystallization components (mainly Cube whose intensity was about 4 times random) and some 

retained rolling components (Brass and Copper orientations) (Fig. 2a). The volume fractions of 

Brass and Copper components were almost identical, with intensities of about 2 times random [11]. 

The texture develops fastest during balanced biaxial loading (Fig. 2b) compared to that during other 

in-plane straining conditions (Fig. 2c, d). The Cube and Copper components are weakened and 

almost disappear, while the Brass is strengthened after balanced biaxial (BB) strain of 15 % (Fig. 

2b). 

Table1: Identified parameters for Voce-type law used in [12] 

0

τ 1τ θ0

θ1

37 30 100 33 

Figure 2: Experimental observation of texture condition (a) of the as-received AA5754 and (b) 15 % 

balanced biaxial (BB) straining, (c) after 15% unixial straining (U) in rolling direction (RD), (d) 15

% in plane straining (PS) in RD. 

Our previous studies showed that crystal plasticity models can predict the evolution of texture, 

but the models usually over-estimate the intensities even with a very detailed treatment of 

dislocation binary interaction [1, 13-15]. One possible reason is that the higher order dislocation 

interactions were neglected in the previous studies. In this study, we assume that at the onset of 

plastic deformation the COL is the strongest binary junction and equal to 5, while the strengths of 

other junctions relating to three non-coplanar interaction equal to that of self-hardening. The 

strengths of COL and other non-coplanar junctions evolve during plastic deformation due to the 

occurrence of higher order of dislocation interactions as explained in Section II. Finally, all 

dislocation strengths approach the strength of stable multi-junctions, which is the average of all 

non-coplanar binary junction types after 5 % (Fig. 1c). This evolution of the strength of dislocation 

junctions is implemented into visco-plastic self-consistent (VPSC) model [12]. Other parameters 

relating to the Voce hardening law are the same as used for the binary junction-based VPSC model 

(Tab. 1). This incorporation of evolutionary dislocation interaction gives a very good prediction of 

1774 Material Forming ESAFORM 2014
texture intensities after BB of 15 % (Fig. 3a) compared to that by the binary junction based VPSC 

(Fig. 3b). It also results in a reasonably good prediction of the texture after uniaxial straining and PS 

parallel to the RD of 15 % (Fig. 3c, d). 

Figure 3: Predictions of texture conditions of 5754 after 15 % of: (a) BB straining by VPSC multijunction based approach, (b) BB straining by VPSC binary-based approach, (c) U in RD by VPSC 

multi-junction based approach, (d) PS in the RD by VPSC multi-junction based approach. 

0 50 100 150 200 250 300 350

0

50

100

150

200

250

300

350

σRD (MPa)

σTD (MPa)

0 50 100 150 200 250 300 350

0

50

100

150

200

250

300

350

σRD (MPa)

σTD (MPa)

Hill 48

VPSC-multi

VPSC-binary

Exp. 15%

Hill 48

Exp. 1%

VPSC-binary

VPSC-multi

(a) (b)

Exp. 5% Exp. 10%

Figure 4: Simulations of (a) yield surface and (b) the stress paths of AA5754 by VPSC and Hill 48 

models. Note: The error bars associated with experimental data points in are measurement 

uncertainties described in Iadicola et al. [16]. 

AA5754 exhibits a complex plastic anisotropy during in-plane straining. The anisotropy is 

clearly manifested in the evolution of the yield surface as measured and reported in [16]. Compared 

to the binary junction-based approach, with the same parameters as used for the binary junctionbased model (Tab. 1), the multi-junction-based model gives a better prediction of the yield surface 

after in-plane straining of 15 % albeit it still slightly over-estimates the BB flow stress, and it can 

not predict well the BB stress path (Fig. 4). Interestingly, the Hill 48 model shows a very good fit 

regarding the shape of yield surface after 15 % compared to that by VPSC. However, the advantage 

of the crystal plasticity approach is that it gives a better description of stress paths during in-plane 

straining, in particular when the uncertainty of controlled strain is taken into account. If we can 

Key Engineering Materials Vols. 611-612 1775
assume the maximum uncertainty associated with controlled strain in Marciniak plane strain tests is 

of 10 %, flow stresses corresponding to the strain with an uncertainty of 7.5% being off the nominal 

value calculated by the multi-junction based VPSC show very good fits for observed stress paths in 

plane straining compared to those by binary junction based VPSC and by the Hill 48 model with the 

same uncertainty (Fig. 4b). Together with the improvement of the prediction of the texture 

development, it corroborates the idea that the multi-junctions not only govern the plastic response of

BCC materials but also play a significant role in the response of FCC materials. This finding differs 

from that of Madec et al [3] who suggested that the multi-junctions do not greatly influence the 

material response. In addition, our recent study [1] shows that by fitting the stress-strain curve using 

multi-junction based VPSC, the texture development in BB straining can be accurately modeled. An 

effort is being conducted to see if it also helps to improve the prediction of the yield surface. 

Conclusions 

The plastic anisotropy of a material can be better modeled by incorporating the higher-order 

interactions of dislocations into crystal plasticity model compared to a crystal plastic model based 

on the dislocation first order interactions. In particular, this study shows that the yield surface 

evolution and texture development of AA5754 during in-plane straining are reasonably well 

predicted by dislocation multi-junction based model. 

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1776 Material Forming ESAFORM 2014