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Acta Materialia 154 (2018) 147e160

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Acta Materialia

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Five-parameter intervariant boundary characterization of martensitein commercially pure titanium

Ehsan Farabi a, Peter D. Hodgson a, Gregory S. Rohrer b, Hossein Beladi a, *

a Institute for Frontier Materials, Deakin University, Geelong, Victoria, 3216, Australiab Department of Materials Science and Engineering, Carnegie Mellon University, Pittsburgh, PA 15213-3890, USA

a r t i c l e i n f o

Article history:Received 16 February 2018Received in revised form9 May 2018Accepted 10 May 2018Available online 15 May 2018

Keywords:Pure titaniumMartensite transformationCrystallographyVariant selectionIntervariant boundary plane characterdistribution

* Corresponding author.E-mail address: hossein.beladi@deakin.edu.au (H.

https://doi.org/10.1016/j.actamat.2018.05.0231359-6454/© 2018 Acta Materialia Inc. Published by E

a b s t r a c t

The intervariant boundary characteristics of a commercially pure Ti microstructure formed by the b/a

martensitic phase transformation were described according to the crystallography of the displacivetransformation and the boundary plane orientation. The martensitic transformation created a micro-structure whose grain boundary misorientation angle distribution had four distinct peaks that wereconsistent with the misorientations between the variants produced by the Burgers orientation rela-tionship. Interestingly, about 60% of population corresponded to 60�/½1120� intervariant boundaries.Three-variant clusters with a triangular morphology were observed frequently. This configuration isconsistent with the phenomenological theory of martensite, which predicts that these clusters, separatedby 60�/½1120� boundaries, have a lower transformation strain than other possible variant cluster ar-rangements. Other intervariant boundaries resulted from the impingement of different combinations ofdistinct three-variant clusters. The five-parameter boundary analysis revealed a strong anisotropy in theplane orientation distribution, showing that boundaries have a tendency to terminate on prismaticfhki0g and pyramidal f1011g planes, when misorientation was ignored. The dominant 60�/½1120�intervariant boundaries had symmetric tilt ð1101Þ boundary planes, which are a low energyconfiguration.

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1. Introduction

Grain boundaries have an undisputable effect on themechanicalresponse of polycrystalline materials. In titanium alloys, grainboundaries impede dislocation motion by imposing barriers thatdepend on grain boundary type, and this can result in localizedstresses [1]. Interestingly, grain boundary geometry and structureaffect the localized stress state and the dislocation/grain boundaryinteraction. Previous studies [2e4] assumed that high angle a/aboundaries in Ti alloys were resistant to dislocation movement andpotential sites for cavity nucleation and intergranular fracture.However, the extent of dislocation slip propagation depends on thea boundary characteristics and the type of dislocation. For example,f1012g twin boundaries with a low energy configuration [5]impede the prismatic slip dislocations [6], though it can transmitbasal dislocations, if the stress is large enough [7e9]. On the otherhand, f1011g twin boundaries can act as barriers to the basal slip

Beladi).

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dislocations [8]. Therefore, by engineering the microstructure andgrain boundary network of titanium alloys, it may be possible tooptimize the mechanical properties of interest.

Most Ti-alloys undergo the b/a phase transformation duringcooling, which is governed by a specific orientation relationship(OR), known as the Burgers OR [10]. The resultant orientationcorrespondence between the high temperature b and low tem-perature a phases provides specific crystallographic a variants,which in return affects the microstructure and texture. Interest-ingly, the impingement of possible a variants defines specific a/aintervariant boundary arrangements. The formation and arrange-ment of the variants are influenced by the crystallographic con-straints of the transformation, the related transformation strainenergy [11e13], and the externally applied strain energy that mightarise from deformation or precipitation [14e17].

The strain energy associated with the b/a martensitic phasetransformation stimulates specific a-variant cluster arrangementsto accommodate the transformation strain [11,12,18]. These clustersprovide a specific population of intervariant boundaries, as indi-cated byWang et al. [12] in commercially pure titanium. They have

E. Farabi et al. / Acta Materialia 154 (2018) 147e160148

shown that the formation of three and/or four a-variant clustersrelaxes the transformation strain energy and theoretically results ina high population of 60�=½1120� and 63:26�=½10 5 5 3� intervariantboundaries. However, despite the detailed description of themartensitic transformation and the resultant grain boundarynetwork, there is still a lack of information regarding the grainboundary plane character associated with the intervariant bound-aries formed in the martensitic microstructure of a commerciallypure titanium. A previous study [19] of a Ti-6Al-4V alloy found thatthe two most common intervariant boundaries have mis-orientations of 63:26�=½10 5 5 3� and 60�=½1120� and have grainboundary plane orientations of ð4130Þ and ð1011Þ, respectively.

In the current study, commercially pure Ti was subjected to amartensitic transformation and its intervariant boundary planecharacteristics were studied using a stereological interpretation ofconventional EBSD maps. This five-parameter crystallographicanalysis made it possible to measure the grain boundary planedistribution of all intervariant boundaries in the martensiticmicrostructure [20]. The microstructure characteristics wereconsistent with the phenomenological theory of martensite, whichwas used to calculate the transformation strain energy incommercially pure titanium to determine the nucleation possibil-ities for different a-variant cluster arrangements and the associatedintervariant boundaries.

2. Experimental procedure

2.1. Material and heat treatment procedure

The commercially pure titanium used for this study had a gradeII chemical composition [21], was received in an extruded condi-tion, and had an equiaxed microstructure. The sample was heattreated in an induction furnace embedded in a servo-testing ma-chine. To prevent the titanium from being oxidized at high tem-perature, it was coated with glass (a delta glaze coating). Thesample was reheated to 950 �C and held for 3min to transform theentire microstructure to the b-phase, before water-quenching it toroom temperature, to obtain a martensitic microstructure.

2.2. Microstructural and texture characterization

The microstructure and texture of the samples were studied byscanning electron microscopy (SEM) equipped with electron back-scattered diffraction (EBSD) technique along with transmissionelectron microscopy (TEM). Note that the microstructure was notuniform through the thickness of 10mm. At the surface, it wasmostly martensitic, but in the middle of the sample, the micro-structure was a mixture of martensitic and Widmanst€atten a-phase. Because of the low thermal conductivity of Ti alloys(K ¼ 19� 23 W :m�1:k�1 [22]), it is likely that the cooling rate wasslower in the centre of the sample than on the surface. To measurethe cooling rate, two holes with 1.1mmdiameterwere drilled alongthe diameter of 10 mm� 15mm cylindrical sample at depths of5mm (centre) and 9mm (sub-surface). K-type thermocouples wirewere brazed into the holes and temperature profile was recordedthroughout the heat treatment schedule for both positions. Theheat treatment was conducted 3 times to examine the reproduc-ibility of the result. The measured cooling rates were 135± 15 �C/sand 175 ± 9 �C/s at the centre and sub-surface of sample, respec-tively. Recognizing this gradient, all microstructure and texturemeasurements were conducted approximately 1000 mm beneaththe surface of the water-quenched sample, a region that was closeenough to the surface to obtain martensitic microstructure whilefar enough away not to be affected by oxidation.

EBSD analysis: A two-step procedure was used to prepare the

EBSD samples. First, the samples were ground to 4000 grit SiCpaper and then mechanically polished with a 0.4 mm OPS suspen-sion. In the second step, the surface was electro-polished by anappropriate electropolishing routine (i.e., a voltage of 30 kV and atthe temperature of 22 �C) and the A3 Struers™ electrolyte solution(i.e., 240ml Butoxyethanol, 400ml methanol, 40ml Perchloricacid). The microstructure was then examined by EBSD on a planeparallel to the extrusion direction. The EBSD measurements wereconducted using a FEG Quanta 3-D FEI SEM instrument equippedwith a fully automated EBSD device. The EBSD measurements weremade with an accelerating voltage of 20 kV, a working distance of10.5mm, and on a hexagonal grid with a step size of 1 mm (as-received sample) or 0.3 mm (martensite sample). Multiple scanscovering an area of ~875,000 mm2 have been measured by EBSD forthe martensitic microstructure. The average confidence index wasbetween 0.4 and 0.6. The EBSD post-processing was conductedusing the TexSEM Laboratories Inc., software (TSL).

Intervariant boundary analysis: A stereological analysis, whichhas been described previously [20,23], was used here to identify theintervariant boundary plane distribution in the martensitic struc-ture. This approach was employed extensively in earlier studies fordifferent materials [24e28]. The number of boundary segments/traces required for such a measurement depends on the crystalstructure (50,000 segments/traces for cubic and 200,000 for hex-agonal materials [20]). Before extracting the boundary segmentsfrom the EBSD maps, a multiple step cleaning procedure wasconducted. First, the ambiguous data was removed through thegrain dilation function. Then, a single average orientation wasassigned to the neighbouring groups of pixels with a disorientationangle of less than 5� (one single grain). To extract the boundary linetraces/segments from the EBSD data, the reconstructed grainboundary function with a boundary deviation limit of 2 pixels(0.6 mm) [29] was used. Here, a total of ~200,000 boundary linetraces were extracted from the EBSDmaps. After extracting the linesegments, the stereological procedure was applied to calculate thefive-parameter grain boundary character distribution (GBCD). Thefive-parameter GBCD was then used to determine the grainboundary plane distribution at different grain boundarymisorientations.

TEM analysis: To produce TEM foils, discs of 3mm in diameterwere taken from 1000 mm below the surface. They were themmechanically ground to a thickness of ~70 mm and then twin-jetelectro-polished using the solution employed for EBSD samplepreparation. The electro-polishing was conducted at a temperatureof about�40 �C and a voltage of 30 V. TEM examination of thin foilswas conducted using a JEM 2100 microscope operated at 200 kV.

Texture measurements: The overall texture of the as-received andtransformed samples was measured by EBSD using a Zeiss LEO 530FEG SEM instrument equipped with an Oxford Instruments EBSDattachment. The data were obtained using the Aztec software andprocessed using the HKL channel 5 software. As discussed later, theparent b phase undergoes significant grain growth during reheat-ing to 950 �C. To obtain unbiased and statistically reliable texturedata from many grains, the EBSD measurements were carried outusing a coarse step size of 30 mmon 30 different sections of both theas-received and martensitic samples, covering an area of4:17 mm � 3:12 mm.

3. Results

The as-received equiaxed microstructure had an average grainsize of 10.9± 0.1 mm (Fig. 1). The overall crystallographic texture ofthe as-received titanium revealed a strong fibre texture with anintensity of 6.2 multiples of a random distribution (MRD) (Fig. 2a),which is typical for extruded titanium alloys [30]. The intensity of

Fig. 1. The band contrast image of the as-extruded commercially pure titanium alloy.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 149

the texture arises from the alignment of the [0001] axes perpen-dicular to the extrusion direction. The (1 0 1 0Þ and ð1 1 2 0Þ polefigures revealedmaximamainly around extrusion directionwith anintensity of 3.5 MRD (Fig. 2a). The main texture component coversabout 43.2% of the orientation space.

3.1. The evolution of microstructure and texture in martensitetransformation

The data in Fig. 3 show that the heat treatment procedure re-sults in a microstructure that resembles martensite, as fullymartensitic microstructure is difficult to form in commercially pureTi due to the absence of alloying elements. A bright field TEM ex-amination also confirmed the presence of elongated martensiticlaths, typically contained only internal dislocations without anytransformation twinning (Fig. 4). This structure hereafter is referredto as a martensitic microstructure. The two-dimensional EBSDimages showed two distinct morphological features. The first wereelongated packets of martensitic laths stacked up in parallel arrays(Fig. 3bec). The parallel laths were separated either by lowmisorientation angle boundaries (�5�) (Fig. 3b) or by highmisorientation angle boundaries, mostly with a misorientation of60�/½1 1 2 0� (Fig. 3c). The second morphology consisted of themartensitic laths stacked up in a triangular shape (highlighted as ‘C’in Fig. 3a). The interior of the triangle was progressively filled withsmaller triangles (highlighted as ‘D’ in Fig. 3a and shown in Fig. 3d).This suggests that the first laths that nucleate fragment the parentb-grain and reduce the volume available for the nucleation of newvariants, leading to a smaller average lath size. The variants formedin the triangular morphology were mostly separated by boundarieswith a 60�/½1 1 2 0�misorientation (Fig. 3d). It should be noted thatthe observation of the triangular morphology depends on theorientation of the section plane through the grain.

Interestingly, the texture after the martensitic transformationwas nearly the same as the as-received condition (Fig. 2), thoughthe fibre consisted of multiple maxima. In general, the martensitictransformation led to a decrease in the overall texture intensityfrom 6.2 MRD to 5 MRD for the f0001g pole figure (Fig. 2b).Although the {1010g and f1120g pole figures showed similaroverall textures to the as-received condition, their texture strengthswere different. The overall texture in the {1010g pole figuredecreased from 3.5 MRD in the as-received condition to 2.6 MRD

after the martensitic transformation. By contrast, the intensity ofthe maximum in the f1120g pole figure increased from 3.5 MRD to6 MRD (Fig. 2).

3.2. The martensite intervariant boundary plane characterdistribution

The misorientation angle distribution of the martensiticmicrostructure revealed multiple peaks at the positions of ~10�,55e65� and ~90�, which was significantly different from what isexpected from a random distribution (Fig. 5a). The misorientationaxis distribution associated with each peak was clustered at aspecific axis. According to the Burgers orientation relationship, theb/a phase transformation in titanium can create 12 distinct a-phase variants from a given prior parent b-grain. These variants arelisted in Table 1. The impingement of these a-variants establishes 11intervariant boundaries (Table 1). However, because of crystalsymmetry, there are only 5 distinct misorientation angle/axis pairs(Table 1). These intervariant boundaries are consistent with theobserved misorientation angle/axis distribution (Fig. 5a), meaningthat the martensitic phase transformation closely followed theBurgers orientation relationship. However, the distribution ofobserved intervariant interfaces differed significantly from thoseexpected in a theoretical distribution calculated under theassumption that all variants occur in the product phase with equalprobability (Fig. 5b). Interestingly, the most common intervariantboundary has a misorientation of 60�/½1120�, and makes up 60% ofall boundary length. The 10.53�/½0001� intervariant was the leastcommon, making up only 2% of all boundary length. The lengthfractions of the 60.83�/½1:377 1 2:377 0:359�, 63.26�/½10 5 5 3� and90�/½1 2:38 1:38 0� intervariant boundaries were ~12%, 13% and 6%,respectively.

The relative areas of all boundary planes, independent ofmisorientation, are plotted in stereographic projection in thecrystal reference frame (Fig. 6). In the distribution, the f0001g basalplane is placed in the centre of the stereogram and the prismaticplanes (f1120g and f1010g) are located at the circumference of thestereogram. The distribution was anisotropic with a maximum in-tensity of 1.6 MRD, revealing two characteristic peaks at prismaticand pyramidal orientations. The first peak is positioned at thef4130g orientation and spreads towards f1010g. The second peakcorresponds to a pyramidal plane with the f1011g orientation.Interestingly, the distribution minimum was positioned at thef0001g orientation (Fig. 6).

The maxima in the misorientation angle/axis distribution(Fig. 5a) specify the misorientations where the distribution of grainboundary planes should be examined. Here, the distribution ofintervariant boundary planes for each misorientation was plottedin stereographic projection and compared with the schematicrepresentation of the characteristic grain boundaries drawn by theGlowinski's grain boundary toolbox software [31] (Fig. 7). Thedistribution of grain boundary planes at the 10.53�/[0001] misori-entation had no maxima greater than 0.5 MRD and was notconsidered significant. The distribution of intervariant boundaryplanes for grain boundaries with a misorientation of 60�/½1 1 2 0�had a single maximum near the pyramidal ð1101Þ plane with themaximum intensity of 500 MRD (Fig. 7a). The ideal symmetric tiltboundary, which also has 180�-twist and 180�-tilt character, has theð17 17 0 18Þ orientation. The orientation of the ideal plane differsfrom ð1101Þ by only 1.6�, which is less than the resolution of thedistribution, which is about 10�. The boundary plane distributionfor the 60.83�/½1:377 1 2:377 0:359� intervariant boundary has amaxima near ð9630Þ planes with an intensity of 14 MRD. Thisboundary was deviated by ~4� from the ð5320Þ tilt boundary, whichis also within the experimental resolution (Fig. 7b). For the 63.26�/

E. Farabi et al. / Acta Materialia 154 (2018) 147e160150

½10 5 5 3� intervariant boundary, there was a peak around theð19 13 6 3Þ planewith amaximum intensity of 200MRD. This planeis ~6� from the ð3 2 1 0Þ twist and 180�-tilt boundary plane (Fig. 7c).The intervariant plane distribution for the 90�/½1 2:38 1:38 0� has adiffuse maximum at the ð17 17 0 18Þ plane with an intensity of 30MRD. This boundary also has twist and 180�-tilt character, andspreads between the ð1 2 1 3Þ and ð4 3 1 0Þ planes (Fig. 7d).

4. Discussion

4.1. Transformation texture and variant selection

The martensite overall texture is qualitatively similar to theoverall texture of the extruded sample, revealing a fibre texturewith ½0001� perpendicular to the extrusion direction (Fig. 2). This ismostly due to the texture memory effect, which was also reportedby others when the material was subjected to a heat treatmentabove the transus temperature through both experiment andsimulation in steel [32] and Ti alloys [30,33,34]. The extruded a

grains in the as-received microstructure transform to the b parent

Fig. 2. The basal and prismatic pole figures of (a) the as-extruded and

during the reheating at 950 �C and this transforms back tomartensitic a during cooling. The presence of the Burgers OR forboth transformations to some extent leads the martensitic trans-formation texture to be similar to the starting extrusion texture.However, it is worth mentioning that the parent b grains growduring annealing at 950 �C, eliminating some pre-existing orien-tations and resulting in a relatively coarse grained structure beforethe martensitic transformation on rapid cooling.

Despite the texture similarity, the martensitic transformationproduces a weaker overall texture and the appearance of multiplepeaks in the ð0001Þ pole figure (Fig. 2). The texture weakening isexpected because rapid cooling reduces the difference in the phasetransformation driving force for distinct variants, promoting theformation of all possible variants associated with Burgers OR in agiven parent b grain (variant multiplication). This ultimately re-duces the overall texture strength. However, the presence of mul-tiple peaks suggests that some variant selection does occur in thetransformation to accommodate the transformation stress/strain.Variant selection should not promote specific orientation/s becauseof variant multiplication. On the other hand, the coarsening of the b

(b) the martensitic microstructures. ED is the extrusion direction.

Fig. 3. a) The image quality map of the martensitic microstructure revealing differentmorphologies; (b) parallel laths separated by low angle boundaries, highlighted by “A”in (a), c) parallel laths separated by the 60�/ ½1120� boundaries, shown by “B” in (a), andd) 3 differently oriented variants clustered in an indentation and\or triangular shape,separated by the 60�/ ½1120� boundaries, marked by “C” in (a). The triangle inset in (d)represents the colour codes referred to normal direction. The red lines indicates the60�/ ½1120� boundaries in (c) and (d). (For interpretation of the references to colour inthis figure legend, the reader is referred to the Web version of this article.)

Fig. 4. TEM bright-field micrograph of the martensitic a laths containing dislocationsubstructure.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 151

grains may lead to the strengthening of a specific b orientation and/or the promotion of low energy b grain boundaries [34,35], whichmay promote a specific a crystallographic variant/orientation oneither side of the b-boundary [33,36,37]. A similar observation ismade here through back-calculating the orientation of the hightemperature parent b phase from martensitic a-variants obeyingthe Burgers OR (Fig. 8), using the software developed in the Lor-raine university, as described in Ref. [38]. It appears that the pres-ence of special b boundaries provides preferred nucleation sites fora variants sharing same Burgers orientation relationship with bothb grains (i.e., having similar orientations) at either side of the parent

b grain boundary (Fig. 8). In other words, the a variants nucleated ateither side of the b/b boundary sharing a common (110)b pole havethe same (0001) pole and obey Burgers OR with both parent bgrains (i.e., double Burgers boundary) [36,39]. Similar observationshave been reported after the martensitic transformation of a rolledZr alloy [40]; limited variant selection on special b-boundaries ledto a texture memory effect and the promotion of a specific texturecomponent.

4.2. Rationalization of the martensitic microstructure in puretitanium

The martensitic transformation in materials usually results in arelatively large shape change, which is accommodated by specificvariant arrangement/s to minimise the associated strain energy. Inthe case of commercially pure Ti, where the martensite transitiontemperature is as high as 802 ± 10 �C, the shape change can beaccommodated by the plastic flow of the relatively soft matrix. Inthe current study, the martensitic laths have a wedge shapedmorphology, assembled into triangular clusters of three differentlyoriented variants (Fig. 3). This is a common characteristic of theself-accommodation morphology. Interestingly, there were nosigns of twins inside the assembled laths (Fig. 3), suggesting thatslip is the dominant operating mode for accommodation of

Fig. 5. a) The misorientation angle distribution of martensitic microstructure of CPtitanium alloy, and b) the length fraction of intervariant boundaries associated withBurgers orientation relationship. The theoretically calculated fractions are based on theassumption that all variants have equal statistical probability during the phasetransformation.

Fig. 6. The distribution of grain boundary planes for all misorientations in themartensitic microstructure. MRD is the multiples of a random distribution.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160152

inhomogeneous strain at high temperatures. This implies that thecritical resolved shear stress for slip is less than for twinning at thetransformation temperature [41]. The clustering of the martensitevariants to minimise the transformation induced shear strain isknown as the self-accommodation phenomenon [42], which in-fluences characteristics of the microstructure including the textureand the intervariant boundary characteristics. To test the idea thatthese clusters do minimise the transformation strain, the

Table 1Individual variants of Burgers OR corresponding to the b matrix and the a

Variants Orientation relationship

1 ð110Þb==ð0001Þa; ½111�b==½1122 ð101Þb==ð0001Þa; ½111�b==½1123 ð011Þb==ð0001Þa; ½111�b==½1124 ð110Þb==ð0001Þa; ½111�b==½1125 ð101Þb==ð0001Þa; ½111�b==½1126 ð011Þb==ð0001Þa; ½111�b==½1127 ð110Þb==ð0001Þa; ½111�b==½1128 ð101Þb==ð0001Þa; ½111�b==½1129 ð011Þb==ð0001Þa; ½111�b==½11210 ð110Þb==ð0001Þa; ½111�b==½11211 ð101Þb==ð0001Þa; ½111�b==½11212 ð011Þb==ð0001Þa; ½111�b==½112

phenomenological theory of the martensitic phase transformationpresented by Bowels and McKenzie [43] was employed to identifythe shape deformation relating to each variant and the character-istics of microstructure.

The crystallography of martensite transformation: The Burgerslattice correspondence between the b-phase (bcc crystal structure)and the a-phase (hcp crystal structure) results in the choice of aminimum distortion (the Bain deformation) and rigid body rotationof the parent lattice vectors. To calculate the Bain deformation, theBurgers lattice correspondence is defined based on the orthohex-agonal axis systems where a distorted hexagonal crystal can bedefined from the parent b-phase (Fig. 9). The Bain strain and therigid body rotation produce an invariant line strain, which is aresult of the implementation of two invariant plane strains (IPS).Therefore, the total transformation strain can be identified by theshape deformation and a shear strain (i.e., lattice invariantdeformation).

S ¼ BR ¼ PQ (1)

Where, S indicates the total transformation strain, B is the Bainstrain, R is the rigid body rotation, Q is the lattice-invariant shearand P is the shape deformation associated with the martensiticphase transformation. The complementary shear is to maintain theintegrity of the crystal structure (i.e., the deformation must belattice invariant), which results in reducing the shape changethrough different shear systems (i.e., twinning and slip). The

product phase [12,19].

Intervariant boundary (from V1)

0� e

0� ½1120�=60�0� ½1120�=60�0� ½1 2:38 1:38 0�=90�0� ½10 5 5 3�=63:26�0� ½1:377 1 2:377 0:359�=60:83�0� ½1 2:38 1:38 0�=90�0� ½1:377 1 2:377 0:359�=60:83�0� ½10 5 5 3�=63:26�0� ½0 0 0 1�=10:53�

0� ½1:377 1 2:377 0:359�=60:83�0� ½1:377 1 2:377 0:359�=60:83�

Fig. 7. Distribution of intervariant interface/boundary planes character for different intervariants and the corresponding calculated geometrically characteristic boundaries in themartensitic microstructure.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 153

Fig. 8. (a) The IPF map of martensitic microstructure of CP Ti alloy. b) the corre-sponding IPF map of reconstructed b-phase of (a), showing the position of prioraustenite grain boundaries. The white dash line in (a) represents a special b boundarywhere two b grains at either side of boundary have a common (110) pole. The triangleinsets in (a) and (b) represents the colour codes referred to normal direction. EDrepresents the extrusion direction. (For interpretation of the references to colour inthis figure legend, the reader is referred to the Web version of this article.)

Fig. 9. The orthogonal lattice correspondence between the parent b-phase anddaughter a-phase.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160154

current research assumed possible shear systems given by the Bilbyand Crocker inequality [44], provided in Ref. [45], to find outwhether the shear system is able to be a lattice invariant strain(LIS). Some of the possible shear systems in the parent and thecorresponding product phase, which have been employed in thecalculations, are listed in Table 2. These shear systems correspondsto possible slip deformation in titanium alloys [45,46], which makethe lattice invariant deformation invisible on themacroscopic scale.A four stage calculation method, presented by Bhadeshia [47], hasbeen employed for calculating the habit plane, and the shapedeformation associated with the phase transformation. Eachoperating shear system mentioned in Table 2 results in fourpossible habit planes in which the closest habit plane that lieswithin the experimental {334} habit plane [18] was considered forfurther calculations. Therefore, the shape deformation tensor, thedilatational (d) and shear components (s) of the transformation canbe calculated, as identified in Table 3. Furthermore, the shapedeformation tensors (P) of the possible 12 a-variants (mentioned inTable 1) were identified and the corresponding stress tensors (TÞ

were calculated in the following way:

T ¼ PtP � I2

(2)

Here, I is a unit matrix and the Pt is the transpose of shapedeformation tensor. Finally, the degree of the self-accommodationfor different variant combinations can be determined from theVon-Mises equivalent strain (εVM). The Von-Mises equivalent strainis computed from the overall transformation stress tensors fordifferent variant combinations (i.e., 2-, 3- and 4-variant clustercombinations) in the following way:

εVM ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23ðT211 þ T222 þ T333 þ

12

�4T212 þ 4T213 þ 4T223

�r(3)

The different possible combinations of stress tensors consideredin this study are shown in Fig. 10. There are 132, 1320, and 11880different possible 2-, 3-, and 4-variant combinations, respectively.For each type of variant cluster, the minimum εVM has been iden-tified and listed in Table 4. The ð110Þb½111�b shear system produces

Table 2The Bilby and Crocker criterion (1 and m) for 5 of the possible shear systems in theproduct hcp martensitic phase for variant 1 [44].

Shear system (S.S) No. b.c.c h.c.p l m

Plane Direction Plane Direction

1 ð011Þ ½111� ð1101Þ ½1213� <0 <02 ð101Þ ½111� ð1011Þ ½1210� <0 <03 ð110Þ ½111� ð1101Þ ½1213� <0 <0

4 ð011Þ ½111� ð1011Þ ½2113� <0 <05 ð101Þ ½111� ð1011Þ ½2113� <0 <0

Fig. 10. The calculation of εVM for different 2, 3 and 4 variant cluster combinationsusing Von Mises criteria.

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 155

the lowest Von-Mises equivalent strain for all types of variantcluster combinations. Interestingly, the three variant cluster has theminimum εVM (0.0011) resulting in 4 possible 3-variant clusters(i.e., V1V2V3, V4V5V6, V7V8V9 and V10V11V12). The nearest equiva-lent strain for the other shear systems mentioned in this studyprovided values of at least an order of magnitude higher than thecalculated values for the ð110Þb½111�b shear system. This meansthat the reduction of strain energy during the martensite trans-formation leads to a lattice invariant deformation, by theð110Þb½111�b slip system. Therefore, to minimise strain energy, thetransformation preferentially produces 3-variant cluster with atriangular morphology, as seen in Fig. 3. In different alloys that alsoundergo a bcc to hcp martensitic phase transformation, this isknown as a self-accommodation morphology [12,18,48].

According to Fig. 3, the triangular morphology is initially formedby the large martensitic laths and the space needed for the newlynucleated variants decreases as the transformation proceeds. Thisultimately leads to a smaller cluster size. In some parts of themicrostructure, 2-variant clusters are also observed as twin relatedparallel laths (Fig. 3c). However, according to Table 4, the lowestequivalent strain (εVM) for a 2-variant cluster is an order ofmagnitude higher than the three-variant cluster. Therefore, thetwinned lathmorphologymight be a sectioning artefact and appearwhen a pyramid shaped stack of 3-variant clusters is sectioned insuch a way that only two of the variants lie in the section plane.

Contrary to the current observations, Wang et al. [12] reportedtwo types of 3- and 4-variant clusters for the martensitic trans-formation in pure Ti. This discrepancy most likely arises from theway the transformation strain accommodation is calculated. In theprevious work, only the shear component of the strain reductionwas used to predict the possible variant combinations. In the cur-rent study, the reduction of both the dilatation and shear compo-nents of the shape strain (the Von-Mises criteria) are used to

Table 3The typical calculated crystallographic sets considering the activation of different shear

S.S No. Habit plane (p) Invariant line strain (S)

1 0@0:497

0:5190:696

1A

0@ 1:044 0:0371 0:0955

0:0452 1:0495 0:05090:0798 0:0808 0:8746

2 0@ 0:489

0:49480:7163

1A

0@1:0451 0:0431 0:0637

0:0426 1:0452 0:06250:0781 0:0766 0:8879

3 0@0:4258

0:56040:7115

1A

0@1:0454 0:0179 0:450

0:0680 1:0449 0:08090:0594 0:0965 0:8877

4 0@0:4903

0:3830:779

1A

0@1:0470 0:0625 0:0478

0:0275 1:0467 0:01870:0567 0:0255 0:8915

5 0@0:4707

0:49910:7280

1A

0@1:0369 0:0255 0:0262

0:0555 1:0468 0:05230:0468 0:0598 0:9054

determine the probable variant cluster combinations. The latterappears to be more consistent with the current martensiticmicrostructure, where it mostly consists of 3-variant clusters. Thissuggests that it is essential to consider the relaxation of all shapestrain components to better predict the variant selection mecha-nism in the martensitic transformation.

Theoretically, the combination of three consecutive variantsresults in the formation of 60�/½1120� intervariant boundaries(Table 1). Accordingly, the variants in the observed clusters areseparated by the 60�/½1120� boundaries (Fig. 11) leading to asignificant population (60%) in the martensitic microstructure(Fig. 5b). Interestingly, the most populated intervariant boundaryin a case of a Ti-6Al-4V alloy [19] was reported to be 63.85�/½10 5 5 3�. This may suggest that the chemical composition canalter the Bain deformation (by changing the lattice parameters)during the martensite transformation and consequently vary thevariant cluster arrangement/s. Moreover, as Fig. 11 suggests, theintersection of two clusters results in the formation of other

system numbers (S.S No.).

Shape deformation (P2) d and s1A

0@1:062 0:064 0:086

0:038 1:039 0:0530:088 0:092 0:877

1A d¼ 0.022

s¼ 0.228

1A

0@1:0418 0:0423 0:0611

0:0441 1:0446 0:06470:0749 0:0758 0:8903

1A d¼ 0.023

s¼ 0.1935

1A

0@1:02722 0:0036 0:0455

0:0486 1:0639 0:08110:0673 0:0885 0:8876

1A d¼ 0.021

s¼ 0.169

1A

0@ 0:9533 �0:036 �0:074

0:027 1:1021 0:044�0:056 �0:044 0:911

1A d¼ 0.0216

s¼ 0.1384

1A

0@ 0:945 0:0584 0:0851

0:0963 1:10207 0:14880:1253 0:1328 1:1937

1A d¼ 0.1467

s¼ 0.3237

Table

4Th

eVon

-Mises

equivalen

tstrain

calculatedfordifferentco

mbinationsof

varian

tcluster

arrange

men

tsin

themartensitictran

sformationassociated

withasp

ecificsh

earsystem

.

Shea

rsystem

TwoVariant

ThreeVariant

FourVariant

1Min

εave

0.04

61V1V3V4V6V7V9V10V12

Min

εave

0.01

18V1V2V3V4V5V6V7V8V9V10V11V12

Min

εave

0.01

71V3V5V2V8V5V8V11V12V8V2V9V11V2V3V5V8V2V5V11V6V2V8V9V11

2Min

εave

0.04

4V2V12V3V11V5V9V6V8

Min

εave

0.01

1V2V5V9V3V8V11V6V8V11V1V4V9V3V7V10V6V7V10

Min

εave

0.01

8V1V2V4V12V1V4V8V12V1V2V7V11

3Min

εave

0.02

88V5V6

V2V3V8V9V11V12

Min

εave

0.00

11V1V2V3V4V5V6V7V8V9V10V11V12

Min

εave

0.00

77V1V4V6V7V4V7V9V10V4V9V8V10

4Min

εave

0.03

21V2V11V5V11V8V10

Min

εave

0.01

79V1V5V11

V5V7V11

Min

εave

0.01

28V2V7V10V11

5Min

εave

0.06

83V1V7V4V10V1V10V4V7

Min

εave

0.06

06V4V7V8V1V2V10V1V10V11V4V5V7

Min

εave

0.05

30V1V5V7V11V2V4V8V10V1V7V8V11V1V2V5V7

E. Farabi et al. / Acta Materialia 154 (2018) 147e160156

intervariant boundaries, including 10.53�/[0001], 60.83�/½1:377;1; 2:377; 0:359�, 63.26�/½10553�; and 90�/½1; 2:38; 1:38; 0�, as illustrated in Fig. 11b. It should bementioned that some fraction of boundaries (~10%) do not followany misorientation angle/axis pairs associated with the Burgersorientation relationship, as they are the result of the intersectionof two crystallographic variants transformed from two distinctadjacent prior b grains (variants that impinge at the prior b grainboundaries).

4.3. Characteristics of martensitic intervariant boundaries

The intervariant boundary with the largest area fraction has a60�/ ½1120� misorientation and a ð1101Þ boundary plane orienta-tion (Fig. 7a). As noted before, this orientation is within experi-mental resolution of the more ideal ð17 17 0 18Þ boundary plane,which has symmetric and quasi-symmetric character. The inter-variant boundarywith the second largest area fraction has a 63.26�/½10 5 5 3� misorientation and a ð19 13 6 3Þ boundary plane orien-tation, which is only deviated by ~6� from ð3 2 1 0Þ twist and 180�-tilt boundary (Fig. 7c).

It is well demonstrated that the relative areas of grain bound-aries in microstructures correlate inversely with boundary energyin both simulations [49e51] and experiments [19,20,52e54]. Alimited number of grain boundary energies in Ti have been calcu-lated by molecular dynamics simulation, and they are mostly tiltboundaries [5]. These data, though, do not correspond with allintervariant boundaries associated with the present martensitictransformation. Therefore, the interplanar spacing is employedhere as another criteria to evaluate the grain boundary energy[55e57]. The flattest, smoothest surfaces have the fewest brokenbonds and the largest interplanar spacings. Because of this, theyhave relatively low surface energies and they form relatively lowenergy grain boundaries. Within this simple framework, grainboundary energy is inversely correlated to interplanar spacing. Forcommercially pure Ti, the basal (0001) plane has the greatestinterplanar spacing (i.e., 2.0283 A

), followed by the pyramidalð1011Þ plane (i.e., 1.8019 A

) and the prismatic planes (i.e., ð1010Þ,ð1120Þ and ð4130Þ planes with the interplanar spacing of 1.7038,1.4755, and 0.4725 A

, respectively, Table 5). Therefore, if one con-siders the interplanar spacing relationship with the relative energy,the basal plane should have the lowest energy and highest popu-lation among other boundary planes in Ti. However, in the distri-bution of the intervariant planes for themartensitic microstructure,the basal orientation is a minimum in the population and themaxima are at the prismatic and pyramidal orientations (Fig. 6).This suggests that the grain boundary population in the currentstudy does not have an inverse relationship with the boundaryenergy. A similar observation has been made for the martensiticmicrostructure in a Ti-6Al-4V alloy [19], showing the maximumpopulation around the prismatic f1010g planes. From such find-ings, it can be concluded that the crystallographic constraintsimposed by the Burgers orientation relationship can lead toboundaries with intervariant planes that do not necessarily have alow energy configuration (e.g., ð4130Þ boundary planes), as previ-ously demonstrated in lath martensitic steels [27,28] and titaniumalloys [19,58].

Through a similar approach [19,57], the interplanar spacings ofthe observed characteristic planes has also been computed andsummarised in Table 5. The intervariant boundary with the greatestrelative area (60%) has a misorientation of 60�=½1120� and is mostlyterminated on the pyramidal ð1101Þ plane (near the symmetric tiltð17 17 0 18Þ plane). This boundary is also both proper andimproper quasi symmetric (Fig. 7a). This is because the ½1120� axisis a two fold symmetry axis [59]. This plane has the second largest

Fig. 11. The IPF (a) and corresponding band contrast (b) images showing three variant clustering in the martensitic microstructure. Yellow, red, green, blue and white lines in (b) arethe 10.53�/½0001�, 60�/½1120� , 60.83�/½1:377 1 2:377 0:359�, 63:26�=½10 5 5 3� and 90�/[1�2.38 1.380] intervariant boundaries, respectively. A, B, C and D,E,F labels in (a) and (b)represent 3-variant clusters of 1 and 2, respectively, as plotted in (0001) pole figures (c). The triangle inset in (a) represents the colour codes referred to normal direction. (Forinterpretation of the references to colour in this figure legend, the reader is referred to the Web version of this article.)

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 157

interplanar spacing (i.e., 1.8019 A, Table 5), suggesting a relativelylow energy configuration. This is consistent with the moleculardynamics energy calculation [5], showing that the pyramidalð1101Þ plane has the minimum energy (168.7mJ/m2) among the½1120� tilt boundaries. Interestingly, the other sets of planes pre-dicted to have low energies by the calculation, such as ð1013Þ,

Table 5The interplanar spacing (dℎkl) for different measured intervariant planes and their corre

Intervariant boundary

Axis/Angle Plane

Fig. 6 All boundaries ð1 0 1 0Þð1 1 2 0Þð0 0 0 1Þð1 01 1Þð4 1 3 0Þ

Fig. 7 10.53�/½0 0 0 1� ð19 8 11 0Þ60�/½1 1 2 0� ð1 1 0 1Þ60.83�/½1:377 1 2:377 0:359� ð9 6 3 0Þ

ð5 3 2 0Þ63.26�/½10 5 5 3� ð3 2 1 0Þ

ð19 13 6 3Þ90�/½1 2:38 1:38 0� ð17 17 0 18

a Taking into account the structure factor as the plane passing through an additional

ð1012Þ and ð2021Þ at rotation angles of 31.39�, 42.47� and 74.7�,respectively, are absent from the misorientation angle distributionof the martensite sample. This suggests that the martensitictransformation in commercially pure Ti and the associated crys-tallographic constraints can simultaneously provide low energyconfigurations in the form of three-variant clusters terminating on

sponding population.

Intensity (MRD) Interplanar spacing (Å)

1.5 0.8519 or 1.7038a

1 1.47550.36 2.02831.62 0.3604 or 1.8019a

1.62 0.2363 or 0.4725a

0.49 0.1547

504.0 0.3604 or 1.8019a

14.01 0.321912.00 0.1954 or 0.3908a

160.73 0.3219 or 0.6439a

204.21 0.0251 or 0.1258a

Þ 30.68 0.0416 or 0.0833a

atom [57].

E. Farabi et al. / Acta Materialia 154 (2018) 147e160158

pyramidal planes. In other words, the 60�=½1120� with low energyð1101Þ plane accounts for ~60% of total boundary area in themicrostructure. The other intervariant boundaries reveal higherenergy configurations resulting from the intersection of thesetriangular clusters (e.g., 63.26�/½10553� intervariant boundariesterminated on f19 13 6 3g planes, Fig. 7c).

5. Conclusion

In the current study, the characteristics of intervariant bound-aries in commercially pure Ti subjected to the martensitic trans-formation were investigated by conventional EBSD mapping and afive parameter analysis of the boundary data. The following sum-marises themost important findings drawn from this investigation:

� The as-received and transformed microstructures had similartextures, with the (0001) pole figure revealing multiple peaksperpendicular to the extrusion direction. However, because ofvariant multiplication, the martensitic transformationdecreased the strength of the texture.

� The grain boundary misorientation angle distribution revealedfour distinct peaks closely associated with the theoreticalintervariant boundaries expected from the Burgers orientationrelationship. The 60�=½1120� intervariant boundary made up60% of the boundary area.

� Three-variant clusters separated by 60�/½1120� boundariesfrequently appeared in the martensitic microstructure. Accord-ing to the phenomenological theory of martensite, this specifictriangular morphology minimises the transformation straincompared to other possible cluster arrangements involving fouror fewer variants.

� The grain boundary plane distribution, ignoring misorientation,is strongly anisotropic with most boundaries terminated on theprismatic fhki0g and pyramidal ð1101Þ planes. The ð1101Þboundary orientations are associated with the symmetric tilt60�/½1120� intervariant boundaries. The other intervariantboundaries expected from the Burgers orientation relationshiplargely terminated on prismatic planes, but did not necessarilyhave low energy interface arrangements.

Acknowledgements

Deakin University's Advanced Characterization Facility isacknowledged for use of the EBSD instruments and assistance fromDr. Mark Nave. The authors thank Dr. Lionel Germain of the LorraineUniversity for his assistance on back-calculation of EBSD data.

Appendix

A.1 Phenomenological theory of martensite transformation

Based on the phenomenological theory of martensite intro-duced by Mackenzie and Bowels [60], it has been shown that thetotal atomic displacements and the transformation strain has thecharacteristic of an invariant plain strain, which leaves the habitplane invariant (i.e., undistorted and unrotated). The magnitude mof the shape deformation can be determined as its unit displace-ment vector d. The habit plane of the martensite (unit normal p) isthe invarianteplane of the shape deformation. The shape defor-mation can be represented bymeans of a shape deformationmatrixP:

P ¼ I þm½d�ðpÞ (4)

Where the basis is chosen to be orthonormal, although the

equation is valid for any basis.However, the invariant plane strain, IPS, cannot solely change

the structure of a bcc structure into a hcp structure. On the contrary,an invariant line strain (ILS) can accommodates the phase structurechange associated with the martensite transformation. The ILS canbe factorised into two plain strains. Therefore, another invariantplane Q, which is considered as a complementary simple shear,with the magnitude of n on a plane with unit normal of q and in aunit direction of e accompanies the shape deformation:

Q ¼ I þ n½e�ðqÞ (5)

Therefore, the invariant line strain is identified by two homo-geneous invariant plane strains in a way that S ¼ P:Q . However, theshape change due to the simple shear Q is rendered invisible on amacroscopic scale since there is also an inhomogeneous lattice-invariant deformation, which can be slip or twinning. This can-cels out the shape change due to Q , without altering the latticestructure. The macroscopic shape change is solely due to P andtherefore has the characteristics of an invarianteplane strain, asexperimentally observed. The transformation strain, S, can be fac-torised into a Bain strain, B, combined with an appropriate rigidbody rotation, R, such that S ¼ R:B. The invarianteline of trans-formation strain lies in the planes p and q, and the invariant-normalof S defines a plane containing d and e. Hence:

S ¼ R:B ¼ P:Q (6)

A.2. Determination of lattice transformation strain

The deformation matrix representing the Bain strain, whichcarries the bcc b-lattice to the hcp a-lattice in an orthonormal basis(Fig. 9) is given by:

B ¼8<:

h1 ¼ ch. ffiffiffi

2p

ab 0 0

0 h2 ¼ffiffiffi3

pah

. ffiffiffi2

pab 0

0 0 h3 ¼ ah=ab

9=; (7)

To find the rigid body rotation, first the invariant line andinvariant plane normal should be identified, based on the conditionthat the invariant line of Smust lie within the shear plane to cancelthe complementary shear Q. Therefore, the unit vector of½u� ¼ ½u1u2u3� must lie in the shear plane ðwÞ ¼ ðw1w2w3Þmeaning:

w1u1 þw2u2 þw3u3 ¼ 0 (8)

The magnitude of the ½F;u� prior to the deformation is:

���u2��� ¼ u21 þ u22 þ u33 ¼ 1 (9)

The Bain strain takes ½u� ¼ ½u1u2u3� in the orthonormal refer-ence frame into ½x� ¼ ½x1x2x3�. As a result the deformation u be-comes a new vector x but the magnitude of the vector remainsunchanged (i.e., ½u� ¼ B:½x�). Therefore;

u21 þ u22 þ u23 ¼ h21u21 þ h22u

22 þ h23u

23 (10)

Equations (8)e(10) can be solved and give two solutions for theundistorted line. Consequently, for the invariant normal of theinvariant lattice strain, the invariant plane ðhÞ ¼ ðh1h2h3Þ shouldcontain the shear direction ½r� ¼ ½r1r2r3�. Then;

E. Farabi et al. / Acta Materialia 154 (2018) 147e160 159

r1h1 þ r2h2 þ r3h3 ¼ 0 (11)

Furthermore;���h2��� ¼ h21 þ h22 þ h23 ¼ 1 (12)

The lattice deformation (i.e., Bain deformation) converts ðhÞ toðlÞ ¼ ðhÞB�1. Therefore;

h21 þ h22 þ h23 ¼ h21.h21 þ h22

.h22 þ h23

.h23 (13)

Consequently, equations (11)e(13) can be solved to give twosolutions for the undistorted normal. To convert the Bain strain intoan invariant line strain, S, the rigid body rotation, R, must beemployed, bringing an undistorted line (i.e., ½x�) and undistortednormal (i.e., ðhÞ) back into the original direction along with the ½u�and ðhÞ. For both ½x� and ðlÞ, there are two solutions. Therefore, therewould be four solutions for converting B into S.

x ¼ Bu l ¼ hB�1 (14)

Also, the cross product of ðhÞ and ½u� (i.e., u� h ¼ aÞ is converted(i.e., x� l ¼ bÞ into S after the Bain deformation [47]. Therefore, therigid body rotation can be obtained through the followingequations:

u ¼ R:x h ¼ R:l a ¼ R:b (15)

By converting these equations into a 3� 3 matrix, they can bedefined as:

0@u1 h1 a1

u2 h2 a2u3 h3 a3

1A ¼

0@R11 R12 R13

R21 R22 R23R31 R32 R33

1A0@ x1 l1 b1

x2 l2 b2x3 l3 b3

1A (16)

Here, R, B and S are explained in the orthogonal coordinate system(Fig. 9) and they can be defined in parent (i.e., b) coordinate, asbelow:

Bb ¼ T�1B0T Rb ¼ T�1R0T Sb ¼ RbBb (17)

The transformation matrix can be described as:

T ¼241. ffiffiffi

2p

�1. ffiffiffi

2p

0

1. ffiffiffi

2p

1. ffiffiffi

2p

00 0 1

35 (18)

A.3. Calculating the habit plane and the shape deformation

The habit plane (i.e., unit normal, p) can be solved throughequations (4)e(6), considering the fact that S is an invariant linestrain. In addition, the invariant line ½u� of S must lie in shear planeðwÞ and the invariant-normal ðhÞ, containing the shear direction ½r�.Therefore, the habit pane must be compatible with the chosenlattice shear system. Considering equations (4)e(6), the habit planecan be given by normalization of equation (19):

ðpÞ ¼ ðwÞ � ðqÞS�1 (19)

Now to completely define the shape deformation, the values of½d� and m in equation (6) should be identified, as below:

cm½d� ¼ S½e� � ½e� (20)

Here, c is dot product of the habit plane and ½e�, and m is the

magnitude of the involved displacement, which is factorised into ashear component s (s ¼ ðm2 � d2Þ parallel to the habit plane and adilatational component d ðd ¼ m½d�:ðpÞÞ. Note that e is the shear onthe directionwhere the Q (complementary shear) is being canceledout (½e� ¼ ½r�).

Hence, the homogenous shear (Q) can be identified, as below:

Q ¼ P�1:S (21)

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