Analysis of Crystal Rotation by Taylor Theory

ABSTRACT

Simple shear along specific slip plane in polycrystalline and rotation of grains was discussed. The Taylor theory was applied to bridge between macroscopic deformation behavior and crystal plasticity and to evaluate the orientation distribution. Its theoretical solution can hardly satisfy all of boundary condition and plastic dynamics so that the condition of dynamics was simplified and relaxed in the analysis. The path of crystal rotation due to slip deformation was quantitatively predicted by Taylor theory and gave an advantage on understanding of deformation texture. The analysis method can be applied to polycrystalline materials. Although good evaluation was available in fcc and bcc where the orientation distribution fitted well, no good fitting to experimental result in hcp materials was obtained.

Introduction

Texture

The arrangement of lattice is almost the same in a grain, and each grain in polycrystalline is usually distributed in random (Fig. 1(a)). On the other hand, the grains after working and/or heat-treatment reveal an arrangement with almost the same orientation (Fig. 1(b)). The arrangement of "deformation texture" is developed as the slip deformation progresses, because the grains are constrained by surroundings in polycrystalline. The deformation texture as well as "recrystallized texture" by heat treatment accompanies with an anisotropy in microstructure and affects on the properties of materials. The deformation texture mainly results from "crystal rotation" by slip deformation or twinning. Slip deformation occurs on the specific crystal planes and to the specific crystal directions. Each orientation of grain, therefore, changes to the specific one. The specific orientation is called as "preferred orientation". Since its formation mechanism makes an important role to control their microstructure, quantitative prediction of the deformation texture is needed. To understand the formation mechanism of the deformation texture, the active slip systems and crystal rotation in each of grain should be considered under large deformation. In order to bridge macroscopic deformation behavior and crystal plasticity, Taylor theory which was based on minimum internal work principle has been applied to analyze the slip deformation behavior under large deformation and to evaluate the orientation distribution .[1-3] The theory is applicable to finite element method.[4]


Illustration of polycrystalline with random orientation (a) and texture (b).

Fig. 1 Illustration of polycrystalline with random orientation (a) and texture (b).

Crystal Rotation Due to Slip Deformation

The plastic deformation in materials strain and the rotation is described as velocity gradient tensor Lij (8ui / 8xt). The velocity gradient tensor L* is described as

tmp11135_thumb

where ui is velocity of material point in current configuration xi (i =1,2,3). Strain velocity tensor DtJ and total spin WtJ are derived as

tmp11136_thumb

where WiJ shows rigid body rotation, and is determined by not only stress but also the constrained geometrical condition.

Figure 2 shows that single slip operation cause the crystal rotation under tension mode. When the tensile stress is applied to the body of single crystal, the deformation along a slip direction on a slip plane results in the change of its shape (Fig. 2 (a)). In the crystal coordinate, the slip deformation induces the crystal rotation Wj in the body. However, the body cannot rotate itself in the specimen coordinate, because the slip plane is invariant (Fig. 2(b)). The proportion of material axis is parallel to the tensile axis so that the body rotates about specimen coordinate as

tmp11137_thumb

wheretmp11138_thumbis plastic spin, and lattice spintmp11139_thumbmeans the crystal rotation. Lattice spin is dominant in the condition of constraint. When elastic component is ignored during deformation, operating slip system gives the strain ratetmp11140_thumband the rotation rate tmp11141_thumbat each point in the body.

When n-th slip system of the normal to slip planetmp11142_thumband slip directiontmp11143_thumbare described as

tmp11150_thumb

wheretmp11151_thumbis slip rate from n-th slip system. The equations in above are integrated as

tmp11152_thumbtmp11153_thumbtmp11154_thumb

The lattice spin Qmeans the infinitesimal rotation in At. Through the repeated calculation on the lattice spin, the path of crystal rotation can be analyzed.

Illustration of crystal rotation due to slip deformation under tensile deformation in single crystal.

Fig. 2 Illustration of crystal rotation due to slip deformation under tensile deformation in single crystal.

Taylor’s Full Constraints Model

A polycrystalline body deforms without defects at grain boundary during deformation, and all of grains are compatible each other in their strain. Taylor assumed that all of grains have the same strain. The Taylor model is called as full constraints model. In all of deformation modes, the compatibility in polycrystalline can be achieved by operating five independent slip systems.[5] When an uniaxial strain is parallel to the z-axis in the specimen’s coordinate system, XYZ,

grain deformation takes place under axial symmetry at a fixed volumetmp11156_thumb

A polycrystalline body deforms without defects at grain boundary during deformation, and all of grains are compatible each other in their strain. Taylor assumed that all of grains have the same strain. The Taylor model is called as full constraints model. In all of deformation modes, the compatibility in polycrystalline can be achieved by operating five independent slip systems.[5] When an uniaxial strain is parallel to the z-axis in the specimen’s coordinate system, XYZ,tmp11159_thumb

wheretmp11160_thumbare the plastic strain rate, andtmp11161_thumbare the plastic shear strain rate in a grain. The internal plastic work rate W is the increment of work per volume and it is the sum of the work of five independent slip systems in a grain:

tmp11164_thumb

where Tn is the CRSS (critical resolved shear stress) and j* is the slip rate in the n-th slip system. There are a number of combinations of operating slip systems that satisfy the external work constraint, but only one or few numbers combinations should be chosen. When the minimum rate of internal plastic work W can be obtained, the combination of operating slip systems is selected and the slip rate of their operating slip systems can be analyzed. W depends on the relationship between the tensile axis or compressive one (z-axis) and grain orientation.

Texture Formation in Magnesium-based Solid Solution

Taylor theory can hardly satisfy all of boundary condition and plastic dynamics so that the condition of dynamics was simplified and relaxed in the analysis. We have mentioned the problems in the description of slip deformation and the prediction of crystal rotation by Taylor theory. Simple shear along specific slip plane in polycrystalline and rotation of grains in magnesium alloy have been discussed.

Application of Taylor’s Full Constraints Model

Recently activities on research and development for magnesium alloys were very high. However their applications have been mostly for cast parts because of poor workability at low temperature. The poor workability in magnesium alloys intrinsically reflects on the behavior of plastic deformation in the hcp. To understand the deformation in magnesium alloys, deformation texture in magnesium-based solid solution was evaluated by Taylor’s full constraints model.

In the texture simulation, the crystal rotations in 300 grains were done. The grains were initially given as random orientation (Fig. 3(a)). In the case of fcc, the xn was taken into account, because only one slip system {111} < 110 > operates. In the case of hcp, three principal slip systems, {0001} < 1120 > , {10l0} < 1120 > and {1122} < 1123 >, were taken into account as deformation mode. However, the CRSSs were installed to the analysis, because they were different from each other (Eq. (13)) [6]:

tmp11165_thumb

Table 1 represents the conditions of CRSSs for the evaluation. The deformation becomes more homogeneous at higher temperature and all primary slip systems can operate sufficiently as Type 1.[7] In Type 2, the ratio of CRSSs at room temperature were installed. The primary slip was {0001} < 1120 > , because the CRSSs of {1010} < 1120 > and {1122} < 1123 > was higher than that of {0001} < 1120 > .[8-10] In addition, the CRSS of {1010} < 1120 > in magnesium alloys was lower than that of {1122} < 1123 > at room temperature.[8] In the present study, no deformation twinning was considered. The solution may be given at the ratios between Types 1 and 2.

Table 1 Ratios of critical resolved shear stress of principal slip systems in magnesium alloy.

Slip system

{0001}<1120>

{10! 0} < 1120 >

{1122} < 1123 >

Type 1

1

1

1

Type 2 (300 K)

1

n

40

80

Evaluation

The calculated deformation texture in magnesium alloy was insensitive to the condition of CRSSs so that the reasonable solution may be given. Figure 3 represents the rotation of the grains in the case of Type 2. The grains rotated to two orientations from e = 0.0 to 0.5 [-], although some grains still remained around (a, P) = (0, 0) (Fig. 3 (a) and (b)). At e = 1.0 [-], two preferred orientations at (a, P) = (27, 30) and (90, 30) were clear as shown in Fig. 3 (c). The grains around (a, P) = (0, 0) can hardly deformed but their number was lower than that in the preferred orientation. Therefore, (a, P) = (0, 0) direction gives quasi-stable orientation. The orientation distribution with e = 1.0 [-] at 573 K for AZ61 alloy is shown in Fig. 4.[11] In this experiment the most preferred orientation was detected at (a, P) = (33, 0). It was reported out that the preferred orientation resulted from slip deformation. In the present simulation, however, the preferred orientation due to slip deformation was not at (a, P) = (33, 0) but (a, P) = (27, 30) and (90, 30). The simulation suggested that the deformation texture at (a, P) = (33, 0) did not result from slip deformation. The path of crystal rotation was also shown in Fig. 3(d). According to the simulation, the rotation paths were divided into the regions of (a, P) = (25 ~ 35, 0 ~ 30) which were called as tradition bands [12]. It is pointed out that the recrystallization easily occur at the grains on the tradition band.[11] The recrystallization may cause the stronger texture at grains with (a, P) = (33, 0). The reason why the preferred orientation at (a, P) = (90, 30) was not observed in the experiment [11] may result in twinning.

Prediction of inverse pole figure for 300 grains in the Type 2: (a) e = 0 [-], (b) e = 0.5 [-], (c) e = 1.0 [-] and (d) illustration of rotation path. a is defined as the rotation angle from c-axis [0001] to a-axis [1120] and p is defined as the rotation angle about c-axis.

Fig. 3 Prediction of inverse pole figure for 300 grains in the Type 2: (a) e = 0 [-], (b) e = 0.5 [-], (c) e = 1.0 [-] and (d) illustration of rotation path. a is defined as the rotation angle from c-axis [0001] to a-axis [1120] and p is defined as the rotation angle about c-axis.

Contour map of orientation distribution at 573 K for AZ61 alloy (e = 1.0 [-], e = 1.0 x10 4 [/s]).

Fig. 4 Contour map of orientation distribution at 573 K for AZ61 alloy (e = 1.0 [-], e = 1.0 x10 4 [/s]).

Summary

The Taylor theory was applied to bridge between macroscopic deformation behavior and crystal plasticity and to evaluate the orientation distribution. Simple shear along specific slip plane in polycrystalline and rotation of grains was discussed. The path of crystal rotation due to slip deformation was quantitatively predicted by Taylor theory and gave an advantage on understanding of deformation texture. The analysis method can be applied to polycrystalline materials. Although good evaluation was available in fcc and bcc where the orientation distribution fitted well, no good fitting to experimental result in hcp materials was obtained. It is still difficult to understand the mechanism of texture formation in hcp metals.

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