Dislocation-Based Modeling and Numerical Analysis of Kink Deformations on theBasis of Linear Elasticity

Shunsuke Kobayashi+ and Ryuichi Tarumi

Graduate School of Engineering Science, Osaka University, Osaka 560-8531, Japan

This study conducts dislocation-based modeling and numerical analysis on the deformation fields of two-dimensional kink structures.Peierls-Nabarro model is used to express edge dislocations in the elastic medium which are implemented into the weak form stress equilibriumequation using the extended isogeometric analysis. Numerical analysis revealed that the macroscopic deformations of two types of ortho kinksand four types of ridge kinks, Cþ

2 , C�2 , D

þ1 and D�

1 , agree well with the experimental results. Although most of the stress components arelocalized near the dislocation cores, only the component ·22 in C�

2 type ridge kinks showed broad distribution within the kink boundaries. On theother hand, deformation fields in D�

1 type ridge kinks showed complementary rotation around the tip of kink structure. Such novel deformationswould contribute to kink strengthening mechanism. [doi:10.2320/matertrans.MT-MM2019006]

(Received September 6, 2019; Accepted December 17, 2019; Published January 31, 2020)

Keywords: kink deformation, long-period stacking ordered magnesium alloys, Peierls-Nabarro model, extended isogeometric analysis

1. Introduction

Long-period stacking ordered magnesium (LPSO-Mg) hasa dual phase microstructure consisting of intermetallic LPSOlayer embedded in ¡-Mg matrix.1) After a hot extrusion of as-cast ingot, this material shows notable mechanical propertiessuch as high yield strength and fair elongation.2,3) Theseproperties are considered to be related to kink deformationwhich is different from conventional plastic deformationmodes. In fact, recent experimental study suggests that thismaterial shows the kink strengthening effect in addition tothe conventional strengthening such as grain boundaryrefinement.4) Understanding of the mechanism is, however,still in open and further investigation is required.

The first study on the kink deformation in metallicmaterials was reported by Orowan in 1942.5) In this paper,it is demonstrated that Cd single crystal shows novel kinkdeformation under uniaxial compressive loading where theloading axis is parallel to the basal plane of hcp crystal.Following to the pioneering study, several researchersconfirmed the formation of kink structure in Zn singlecrystal.6­8) In this case, uniaxial compression is also subjectedparallel to the basal plane. Under the loading condition,Schmid factor of the primary slip systems of hcp crystals, i.e.,h11�20i directions on the basal plane (0001), are identicallyzero. Hence, the basal dislocations are inactive during theuniform compressive deformation. On the other hand, thesecondary slip systems or twinning are unable to movebecause of the strong plastic anisotropy of the hcp crystals.Consequently, the single crystal specimen experiences elasticbuckling before the activation of any plastic deformation andwhich induces non-uniform rotation of local crystallographicorientation. Such a rotation increases Schmid factor of theprimary slip system and eventually induces the one-dimensional motion of basal dislocations in the localizedregion so as to form the kink structure. This is thedislocation-based kink formation mechanism proposed by

Hess and Barrett.6) This model explains the morphology aswell as the structure of kink deformation from a phenome-nological viewpoint.

Several materials, including LPSO-Mg alloys, polymers,minerals, composite materials and bundled papers,9­16) showkink deformation in similar loading condition. A commonfeature of the materials is the strong plastic anisotropy dueto layered structures. That is, the strong resistance to inter-layer deformation restricts the effective plastic deformationto inner-layer deformation mode. Such a directional plasticdeformation is analogous to the one-dimensional motion ofbasal dislocations in hcp crystals. It is, therefore, reasonableto suppose that the kink formation as well as the resultingstrengthening in LPSO-Mg alloys are related to the plasticanisotropy due to the layered structure. More precisely, one-dimensional motion of edge dislocations along the compres-sive direction plays an important role to understand the kinkformation and resulting strengthening mechanism.

It is well known that interaction force between dislocationsthrough stress fields are explained by the Peach-Koehlerformula.17) The stress fields due to kink-forming dislocationsare, therefore, a clue for the understanding of kinkstrengthening mechanism. Up to date, atomistic-scalesimulations have been conducted to reveal the kink formationprocess.18,19) However, the stress fields around the kinkstructure has not been clarified yet.

This study aims to investigate the macroscopic deforma-tion and internal stress fields of kink structure under severalconfiguration of dislocations. Construction of the paper is asfollows. In the next section, we introduce a two-dimensionalelastic medium and consider the mechanics within theframework of the theory of linear elasticity and the calculusof variations. Edge dislocations are expressed by Peierls-Nabarro model17,20) and introduced into the medium usingextended isogeometric analysis (XIGA).21­24) Note thatXIGA is a generalization of IGA which solve the stressequilibrium equation in a weak form using the non-uniformrational B-spline (NURBS) basis function.25,26) Results ofnumerical analysis are presented in Sec. 3. After a briefexplanation on the simulation conditions, we present the

+Graduate Student, Osaka University. Corresponding author, E-mail:kobayashi@nlsm.me.es.osaka-u.ac.jp

Materials Transactions, Vol. 61, No. 5 (2020) pp. 862 to 869Special Issue on Materials Science on Mille-Feuille Structure©2020 The Japan Institute of Metals and Materials

results of numerical analysis including macroscopic defor-mation, elastic spin and two-dimensional stress componentsfor several configurations of dislocations. Some concludingremarks are given in Sec. 4.

2. Mathematical Modeling of Dislocations

2.1 Linear elasticity of dislocationsLet ³ be a simply connected open subset of real space R2

and let @³ be a piecewise smooth boundary of the domain³ which is consisted of two parts; ¥D and ¥N. Here, thedisplacement filed ui (i = 1, 2) is prescribed on the Dirichletboundary ¥D and surface traction ti is assigned on theNeumann boundary ¥N. To simplify the analysis, we employa two-dimensional rectangular domain ³ which is consistedof homogeneous and isotropic linear elastic material.Figure 1 shows a schematic illustration of the present model.As seen here, our model includes only one slip system whoseBurgers vector is parallel to the x1 direction. In order toexpress discontinuous displacement due to the dislocations,we introduce a set of interfaces ��D ¼ Sm

¡¼1��¡D representing

the slip plane of ¡-th dislocation and jump conditions½ui� ��¡

D¼ b¡i representing the Burgers vector. Such a

discontinuous displacement due to dislocations induceselastic fields.

According to the theory of linear elasticity, we introducethe strain tensor ¥ij defined by

¥ij ¼1

2ðui;j þ uj;iÞ; ð1Þ

where ui, j is partial derivative of the displacement function;ui, j = @ui/@xj. Here and after, we use a lower-case letter torepresent a spatial direction in R

2. We also employ thesummation convention for repeated lower-case indices. Theelastic constants tensor Cijkl is written in the following form

Cijkl ¼ ­¤ij¤kl þ ®ð¤ik¤jl þ ¤il¤jkÞ; ð2Þwhere ­ and ® are Lamé constants and ¤i j is Kronecker delta.The strain energy density W of the domain ³ is defined asthe inner product of ¥ij and stress tensor · ij ¼ Cijkl¥kl suchthat

W ¼ 1

2· ij¥ij: ð3Þ

Finally, we introduce the strain energy functional W as anintegration of the energy density W over the domain ³

excluding the interfaces ��D such that

W½ui� ¼Z�n ��D

WdV ¼Z�n ��D

1

2· ij¥ijdV: ð4Þ

According to the variational principle for the strain energyfunctional W [ui], we obtain the stress equilibrium equation ina weak form such thatZ

�n ��D

Cijklhi;j¥kldV ¼ 0; ð5Þ

where hi is the test function which satisfies hi = 0 on ¥D and��D. This is the theoretical framework of the present study.

2.2 Peierls-Nabarro dislocation model and XIGAIn the present analysis, elastic fields are induced by

discontinuous displacement due to the edge dislocations.Hence, to solve the stress equilibrium equation (5), we needa numerical scheme which is also suitable to represent boththe discontinuous displacement and resulting elastic fields. Inthe present study, we use the extended isogeometric analysis(XIGA). XIGA is a generalization of IGA which solve theweak form stress equilibrium equation (5) using a linearcombination of NURBS and enrichment functions.21­24) Acharacteristic feature of IGA is that it uses the NURBSfunctions both for the representation of a domain ³ and forthe basis function of weak form analysis.25,26) In other words,IGA is a Galerkin method which uses NURBS as the basisfunctions. On the other hand, XIGA introduces additionalenrichment functions into the basis function in order tosatisfy the discontinuous displacement conditions at the slipplane interfaces inside the medium.

Let us first introduce a two-dimensional parameter spacedefined by � ¼ fðt1; t2Þj0 < t1 < 1; 0 < t2 < 1g, where t1and t2 are independent variables which run on the parameterspace. The physical space ³ is obtained from the NURBSmap xi : � ! � such that

xiðt1; t2Þ ¼XmI¼1

NIðt1; t2ÞAIi; ð6Þ

where NI = NI (t1, t2) is the I-th NURBS function defined on� and AIi is the i-th component of the control point for theI-th NURBS function NI. After choosing a suitable set ofcontrol points, we can express geometrically exact shape ofthe domain ³ using NURBS map (6).

Let Sð ��¡DÞ be the sets of indices of NURBS functions

which are non-zero on the slip plane interface ��¡D. This set

for each dislocation is obtained from the quadrilateral four-point mesh.27) We approximate the displacement field ui andtest function hi by linear combinations of NURBS functionssuch that

ui ¼XnI¼1

NIðt1; t2ÞaIi þXm¡¼1

XJ2Sð ��¡

DÞNJ ðt1; t2Þb¡Ji¼¡ðx1; x2Þ;

ð7Þ

hi ¼XnI¼1

NIðt1; t2ÞcIi þXm¡¼1

XJ2Sð ��¡

DÞNJ ðt1; t2Þd¡Ji¼¡ðx1; x2Þ;

ð8Þwhere aIi, b¡Ji, cIi and d¡Ii are the real-valued coefficients ofNURBS functions and ¼¡ = ¼¡(x1, x2) is a normalizedfunction which represents the core shape of the ¡-th

Fig. 1 Schematic illustration on the elastic domain ³ and edgedislocations. ¥D denotes the Dirichlet boundary where displacement isfixed. ��¡

D represents an interface on a slip plane of ¡-th dislocation. Tosimplify the analysis, we set L1/L2 = 3.

Dislocation-Based Modeling and Numerical Analysis of Kink Deformations on the Basis of Linear Elasticity 863

dislocation. Roughly speaking, the first term in the right handside of eq. (7) represents the elastic displacement and that ofthe second term is the discontinuous displacement due to theedge dislocations. The same holds for the test function givenin eq. (8).

Some of the coefficients in eqs. (7) and (8) are determinedfrom the Dirichlet boundary condition and interfaceconditions, i.e., ui ¼ ui on ¥D and hi = 0 and ½ui� ��¡

D¼ b¡i

for all ¡ on ��D. Using the partition of unity of the NURBSfunctions,26) we obtain

b¡JiX

J2Sð ��¡DÞNJ¼

¡

24

35

��¡D

¼ b¡Ji½¼¡� ��¡D¼ b¡i ; ð9Þ

d¡JiX

J2Sð ��¡DÞNJ¼

¡

24

35

��¡D

¼ d¡Ji½¼¡� ��¡D¼ 0; ð10Þ

along the interface ��¡D. Clearly, d

¡Ji ¼ 0 satisfies eq. (10) for

any function ¼¡. Inserting d¡Ji ¼ 0 in eq. (8), we obtainhi ¼

PnI¼1 NIðt1; t2ÞcIi. In the present study, we restrict our

consideration on a single slip system for the x1 direction and,therefore, b¡i ¼ ðb¡1 ; 0Þ and b¡Ji ¼ ðb; 0Þ. Then, eq. (9) canbe rewritten as ½¼¡� ��¡

D¼ b¡1=b. In other words, ¼¡ can be

determined by the discontinuous displacement b¡1 on each��¡D.Let us express the position of the ¡-th dislocation by

ðx¡1 ; x¡2Þ. In this study, we employ Peierls-Nabarro dislocationmodel as it excludes the stress singularity at the dislocationcore.28) Then, b¡1=bðx1 � x¡1Þ represents the normalizeddistribution function of Burgers vector along ��¡

D such that

b¡1bðx1 � x¡1Þ ¼

1

2þ 1

³tan�1 x1 � x¡1

¦

� �: ð11Þ

Finally, the function ¼¡ of Peierls-Nabarro model is given ina form of separation of variable such that

¼¡ðx1; x2Þ ¼b¡1bðx1 � x¡1ÞHðx2 � x¡2Þ; ð12Þ

where Hðx2 � x¡2Þ is the Heaviside step function;

Hðx2 � x¡2Þ ¼1 x2 � x¡2 � 0

0 x2 � x¡2 < 0

�: ð13Þ

The parameter ¦ in eq. (11) plays an important role in thepresent modeling; it expresses the half width of thedislocation distribution function b¡1=bðx1Þ. For the case ofan edge dislocation, the parameter is expressed using thematerials constants such that ¦ = a/2(1 ¹ ¯), where a is theatomic spacing underlying the domain ³ and ¯ is the Poissonratio.29) Figure 2 shows the normalized dislocation distribu-tion function b¡1=bðx1Þ for ¦ = 0.02, 0.04, 0.06, 0.08 and0.10. As seen in a following section, we set ¦ = 1/150 µ0.067 in the numerical analysis.

The partial derivative of eqs. (7) and (8) reads

ui;j ¼XnI¼1

@NI

@tk

@xj@tk

� ��1

aIi

þXm¡¼1

XJ2Sð ��¡

DÞb¡Ji

@NJ

@tl

@xj@tl

� ��1

¼¡ þ NJ@¼¡

@xj

!; ð14Þ

hi;j ¼XnI¼1

@NI

@tk

@xj@tk

� ��1

cIi; ð15Þ

where (@xj/@tk)¹1 is obtained directly from eq. (6). Substitut-ing eqs. (14) and (15) into (5), we obtainXn

I¼1

XnJ¼1

cIjðKIjJkaJk þ FIjÞ ¼ 0; ð16Þ

where KIjJk and FIj are defined as follows;

KIiJk ¼Z�

Cijkl@NI

@tm

@xj@tm

� ��1@NJ

@tn

@xl@tn

� ��1

dV; ð17Þ

FIj ¼Z�

Cijkl@NI

@tp

@xj@tp

� ��1

�Xm¡¼1

XJ2Sð ��¡

DÞ

@NJ

@tq

@xl@tq

� ��1

¼¡ þ NJ@¼¡

@xl

!dV: ð18Þ

Equation (16) must be satisfied for any cIj. This conditionreads a system of linear algebraic equations for unknowncoefficients aJk such that

KIjJkaJk ¼ �FIj: ð19ÞThe total degrees of freedom of eq. (19) depends on thenumber of NURBS basis functions; » = 2n.

3. Results and Discussion

3.1 Simulation conditionsAs shown in Fig. 1, we consider the two-dimensional

rectangular domain ³ defined by ³ = {(x1, x2)«0 < x1 <L1, 0 < x2 < L2}. Peierls-Nabarro model includes atomicspacing a in the core width parameter ¦. Using the absolutelength scale, the size of the medium is normalized such thatL1 = 300a and L2 = 100a with a = 0.01. Similarly, thecoefficient b which defines the strength of the Burgers vectoris defined using the parameter a by b = a. Lamé constants arealso normalized to ­ = ® = 1 so as to satisfy the Cauchysolid condition with the Poisson ratio ¯ = ­/2(­ + ®) =0.25. Consequently, the core width parameter ¦ becomes¦ = 1/150 µ 0.067. The Dirichlet boundary condition is seton the left surface of the rectangular domain ³ such thatu1 ¼ u2 ¼ 0 on x1 = 0. The remaining boundary @� n �D is

Fig. 2 Distribution of Burgers vector around the core of an edgedislocation with respect to the x1 direction. The core width depends onthe parameter ¦.

S. Kobayashi and R. Tarumi864

classified as Neumann boundary, which is free from anysurface traction; ti = ·i jnj = 0.

The parameter domain � ¼ ð0; 1Þ � ð0; 1Þ is divided intosubdomains by the open and uniform knot vectors; thenumbers of knots are 1,497 in the x1 and 497 in the x2directions, respectively. The polynomial orders of B-splinefunctions are p1 = p2 = 3. It indicates that the NURBS basisfunction is C2 class in the entire domain ³. The weights ofcontrol points are taken to be unity. Consequently, thenumbers of B-spline basis functions in each direction aren1 = 1500 and n2 = 500, leading to the total number ofNURBS basis function to be n = n1n2 = 750,000. Hence, thedegree of freedom is » = 1,500,000.

3.2 Microstructure of ortho kink deformationAs mentioned in a previous section, Hess and Barrett

explained the kink structure on the basis of the distributionof edge dislocations.6) According to the modeling, there aretwo different kinds of kink structures called ortho kink andridge kink both of which are found in experiments on Zn6,8)

and LPSO-Mg alloys.30)

Let us first investigate the ortho kink structure. Figure 3(a)shows the configuration of dislocations for an ortho kinkstructure. We assume that five pairs of edge dislocations areemitted from five dislocation sources which are aligned atthe center of domain ³ with respect to the x1 direction.Essentially the similar dislocation configuration is consideredin Fig. 3(b). In this case, however, dislocations have theopposite signs against those in Fig. 3(a). Macroscopicdeformations due to the configurations in (a) and (b) arerespectively shown in Figs. 3(c) and (d). Note that thedeformations are magnified by a factor of 10. As expectedfrom the Hess-Barrett model, the macroscopic shapes showthe sharp inflections at the lines where dislocations align.This is a characteristic morphology of kink deformation. Thecolor in the figures show the magnitude of elastic spin ª

which is defined by the following form

ª ¼ ¥3ij½ji ¼1

2ðu2;1 � u1;2Þ: ð20Þ

Here ¥ijk is the alternating tensor. Roughly speaking, ª isthe measure of the lattice rotation due to the deformation. It is

evident from the numerical analysis that the sign ofdislocations determines the direction of elastic spin ª. Wecan also see from the figures that the spin is highly localizedwithin the dislocation arrays, namely within the kinkboundaries. It also shows that distribution of rotation isuniform. These features agree well with the experimentalresults.6,8)

Figures 4(a) and (b) show internal stress fields of ·11, ·22and ·12 obtained from the dislocation configuration given inFigs. 3(a) and (b). Note that the stress components arenormalized by shear modulus ®. Unlike the elastic spin ª

shown in Figs. 3(c) and (d), the stress fields are notdistributed within the kink boundaries but localized onlyaround the cores of dislocations. In addition, this feature isindependent of the stress components. This result impliesthat stress field around the ortho kink boundary is similar tolow angle boundary made by polygonization of edgedislocations. In other words, strengthening due to the orthokink structure would be comparable to conventional grainboundary refinement.

3.3 Microstructure of ridge kink deformationAnother kink structure we are interested in is the ridge

Fig. 3 (a) and (b) show dislocation configurations for an ortho kink structure. The red and blue circles in the figures show the dislocationsources assigned respectively to positive and negative. Resulting deformation fields are shown in (c) and (d), where the color distributionshows elastic spin ª. In order to avoid the large spin concentration at the dislocation cores, we restricted the range of color distribution tobe 60% of actual numerical result.

Fig. 4 Stress fields obtained from the dislocation configurations given inFigs. 3. In order to avoid the large stress concentration at the dislocationcores, we restricted the range of color distributions to be 20% of the actualnumerical result.

Dislocation-Based Modeling and Numerical Analysis of Kink Deformations on the Basis of Linear Elasticity 865

kink. Following the Hess-Barrett model,6) we consider theconfigurations of dislocations as illustrated in Fig. 5(a). Inthis case, ten pairs of dislocations are considered; five pairsare emitted from the positive sources (red circles) while theremaining five pairs come from the negative sources (bluecircles). Essentially the same dislocation configurations areshown in Fig. 5(b). Only the difference here is the sign ofBurgers vector. For the sake of convenience, we denote thetwo configurations by Cþ

2 and C�2 . Figures 5(c) and (d) show

the macroscopic deformation obtained from the dislocationconfigurations Cþ

2 and C�2 . Note that the deformation is

magnified by a factor of 10 and the color contour representsthe distribution of the elastic spin ª. As seen in (c), themacroscopic shape shows sharp extrusions on the top andbottom surfaces of the domain. This is a typical feature ofridge kink structure.6,8,30) We also see that each extrusion isconsisted of two sub-regions which have the opposite sign ofthe elastic spin ª. In addition, the spins are highly localizedwithin the ridge kink boundaries. Comparing the result toFigs. 3(a) and (b), this structure is explained by combiningthe two ortho kinks which have the opposite sign of spin. Ina previous study, Lei and Nakatani conducted numericalanalysis for two-dimensional atomistic system and found theformation of extrusion-type ridge kink structure undercompressive loading.19) They also revealed that the ridgekink is consisted of two sub-regions that have the oppositerotational displacement localized within the kink boundaries.These features agree qualitatively well with the presentnumerical analysis. We therefore conclude that the extrusiontype ridge kink structure is explained by a special config-uration of edge dislocations, i.e., Cþ

2 type configuration.The dislocation configuration of C�

2 is obtained by simplyreversing the signs of edge dislocations in Cþ

2 . Hence, theresulting macroscopic shape would be the opposite one; itbecomes intrusion-type both on the top and bottom surfacesof the domain. In fact, we can confirm the intrusion-typeridge kink structure in Fig. 5(d). Although the two kinkstructures C�

2 have essentially the same configuration ofdislocations, there is a remarkable difference between thestructures. The extrusion-type ridge kink is energeticallyfavorable under a compression while the intrusion-type kink

is favorable under an extension since they compensate thevolumetric change due to the external loads. In the actualexperiments, extrusion-type ridge kink appears frequentlybut the observation of intrusion-type is limited. Thisexperimental evidence implies that the formation of ridgekink structure is triggered by mechanical instability, suchas elastic buckling, as it appears only in the compressiveloading.

Figure 6(a) shows another configuration of dislocationsdenoted by Dþ

1 . In this case, the ten pairs of edge dislocationsare emitted from ten dislocation sources that have the samesign, say red circles. Consequently, at the middle part of aridge kink, edge dislocations of different signs are alignedand they counteract the displacement as well as the stressfields with each other. Hence, unlike the previous two casesof C�

2 , the resulting macroscopic deformation has uniformspin within the kink boundaries (see Fig. 6(c)). Figures 6(b)and (d) are similar configurations of dislocations except forthe signs of Burgers vectors. According to the macroscopicmorphology and elastic spin within the kink boundaries, thetwo configurations D�

1 are similar to ortho kink structuresshown in Figs. 3. In some way, the kink structures D�

1 areincomplete ortho kink which are terminated within thedomain. As shown in (b) and (d), such an incompletedeformation requires the complementary rotation outsidethe kink regions due to the requirement from the straincompatibility condition.

3.4 Strengthening by ridge kinkNumerical analysis given in previous sections revealed that

the morphologies of ridge kinks are explained from thearrays of edge dislocations as suggested by Hess-Barrett.6)

The remaining issue that we want to address here is theunderstanding of strengthening mechanism by the ridgekinks. Obviously, the most important information is includedin the stress field around the kink boundary since dislocationsinteract through stress fields in the elastic medium.17)

Figure 7 shows the internal stress fields obtained from thedislocation configurations Cþ

2 and C�2 . Similar to the cases

of ortho kinks, most of them are localized near the cores ofdislocations. However, the stress components ·22 shows

Fig. 5 (a) Cþ2 and (b) C�

2 show dislocation configurations for ridge kink structures. The red and blue circles in the figures show thedislocation sources assigned respectively to positive and negative. Resulting deformation fields are shown in (c) and (d), where the colordistribution shows the distribution of elastic spin ª. Although maximum values of spin reach ª = «0.190 around the cores of thedislocations, the color distribution is restricted to ª = «0.100.

S. Kobayashi and R. Tarumi866

broad distribution through the inner region of kinkboundaries. Especially, the stress concentration is notablearound the tip of ridge kink structures. Recently, Inamurainvestigated kink deformation in two-dimensional continuumsystem on the basis of rank-one connection.31) From thekinematical analysis, he concluded that there exists disclina-tion at the tip of ridge kink structures. Comparing the result topresent numerical analysis, the ·22 stress concentration foundin C�

2 might be related to the formation of disclination. Such along-range stress field would affect the motion of subsequentdislocations during the plastic deformation. It explains a partof kink strengthening mechanism.

Internal stress fields for dislocation configurations Dþ1 and

D�1 are summarized in Figs. 8. Unlike the previous case,

the stress concentration is insignificant and no additionalstrengthening is expected compared with the conventionallow-angle grain boundary refinement. On the other hand,these configurations show notable distortion around the

kink boundaries. As mentioned in the previous section, theridge kinks in D�

1 are terminated within the domain andtherefore induce the complementary rotation outside the kinkboundaries. Obviously, such a rotation influences on theapparent yield stress due to the change in Schmid factor andmight be effective for strengthening of kink-forming highlyanisotropic materials.

The last thing that we need to address here is theconstitutive equation. Within the framework of classicallinear elasticity, only the symmetric part of displacementgradient, namely the strain tensor, is considered in theconstitutive equation (3). In other words, rotational displace-ment produces no stress fields. This situation might bedifferent if we employ a non-classical constitutive equationsuch as strain gradient elasticity or micropolar elasticity.32,33)

In other words, the stress field analysis within the classicalelasticity might underestimate the strengthening mechanismdue to kink structure.

Fig. 6 (a) Dþ1 and (b) D�

1 show dislocation configurations for ridge kink structures. The red and blue circles in the figures show thedislocation sources assigned respectively to positive and negative. Resulting deformation fields are shown in (c) and (d), where the colordistribution shows the distribution of elastic spin ª. Although maximum values of spin reach ª = «0.228 around the cores of thedislocations, the color distribution is restricted to ª = «0.100.

Fig. 7 Stress fields obtained from (a) Cþ2 configuration and (b) C�

2 configuration. Although, most of the stress components are localizednear the cores of dislocations, only the stress component ·22 shows broad distribution within the kink boundaries. In order to avoid thelarge stress concentration at the dislocation cores, we restricted the range of color distributions to be 20% of the actual numerical result.

Dislocation-Based Modeling and Numerical Analysis of Kink Deformations on the Basis of Linear Elasticity 867

4. Conclusion

In the present study, we investigated the macroscopicdeformation as well as internal stress fields due to ortho andkink structures on the basis of Hess-Barrett edge dislocationmodel. Our formulation is based on the linear elasticcontinuum mechanics solved numerically by XIGA. Con-clusion of the present study is summarized as follows:(1) We consider two-dimensional continuum domain ³

which includes edge dislocations expressed by Peierls-Nabarro model. Weak form stress equilibrium equationis derived from the variational principle and solvednumerically using XIGA, i.e., Galerkin method whichuses NURBS as the basis functions.

(2) Numerical analysis for ortho kink structure revealedthat elastic spin is uniformly distributed within thekink boundaries. On the other hand, stress fields arehighly localized around the cores of the dislocations.Such a dislocation microstructure is similar to low anglegrain boundary and which implies that the resultingstrengthening is explained by conventional Hall-Petcheffect.

(3) Numerical analysis for Cþ2 type configurations of

dislocations revealed that the macroscopic deformationbecomes extrusion type ridge kink structure. On theother hand, C�

2 type configurations showed the oppositemanner, i.e., intrusion-type ridge kink. Although thetwo ridge kinks have dual configurations, onlyextrusion-type kink appears in the actual experiments.This result implies that the formation of ridge kink istriggered by elastic instability such as buckling.

(4) Numerical analysis for D�1 configurations revealed that

their spin distributions are similar to ortho kinkstructures. Only the difference is the termination ofkink structure and which induces the complementaryrotation outside the kink boundaries.

(5) Stress field analysis revealed that there exists notablestress concentration on ·22 components in Cþ

2

configurations. Comparing the previous results onkinematical analysis, it might be related to theformation of disclination at the tip of ridge structures.Crystallographic rotation due to the complementaryrotation in D�

1 is another effective strengtheningmechanism as it influences on Schmid factor in a localregion.

(6) Stress field analysis on the basis of classical elasticitymight underestimate of the actual stress fields as itexcludes the rotational contributions. It would beworthwhile to introduce a generalized constitutiveequation such as strain gradient elasticity or micropolarelasticity.

Acknowledgments

This study was supported by JSPS KAKENHI forScientific Research on Innovative Areas “MFS MaterialsScience (Grant Numbers JP18H05481)”. The authorsacknowledge Prof. T. Fujii (Tokyo Institute of Technology)for his valuable advice and comments.

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Fig. 8 Stress fields obtained from (a) Dþ1 configuration and (b)D�

1 configuration. All of the stress components are localized near the coresof dislocations. In order to avoid the large stress concentration at the dislocation cores, we restricted the range of color distributions to be20% of the actual numerical result.

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