Pencil glide formulation for polycrystal modelling

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PENCIL GLIDE FORMULATION FOR POLYCRYSTAL MODELLING R. Becker Alcoa Technical Center Alcoa Center, PA 15069 (Received December 13,1994) (Revised January 26,199s) Introduction Crystals deforming by pencil glide slip along distinct crystallographic directions, but slip is not restricted to any specific crystallographic plane. Slip will occur along the plane containing the slip direction on which the resolved shear stress is a maximum. Only four slip systems need to be considered for BCC crystals and six for possible pencil glide in FCC crystals (1) For rate independent materials, methods have been proposed for determining the proper slip planes and slip system activity which are compatible with the stress state and the applied strain rate. Although progress has been made in simplifying these algorithms (2,3), the methods are still considerably more complex than algorithms for deformation by restricted slip. Rollett and Rocks (4) proposed an algorithm for pencil glide in rate dependent crystals. They use results from restricted glide calculations as a trial solution, and distribute the resulting effective slip rates over the restricted glide systems. The computational requirements are only slightly greater than for restricted slip since the additional calculations are minimal. Here, an alternative method is described for pencil glide calculations in rate independent and rate dependent materials. When implemented as part of a forward gradient formulation, it is essentially the same as a restricted glide proceedure except that the slip planes are determined at the beginning of each time step as the plane which maximizes the resolved shear stress. The algorithm was easily implemented in a rate dependent model similar to that of Peirce et al. (5). Pencil Glide Formulation Determination of the slip plane involves finding the plane which maximizes the resolved shear stress. The resolved shear stress on a slip system, r”, can be expressed as a function of the applied stress, t, the slip direction, sa and the slip plane normal, ma. ra =mO.r.s”.

(1)

The slip vectors in the current configuration are orthogonal and can be expressed in terms of the corresponding orthonormal slip vectors in the initial configuration (se” and mg) by sa = F* . s;

and 2051

mQ = mt . F*-’ .

(2)

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PENCIL GLIDE FORMULATIONFOR POLYCRYSTALMODELLING

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Here, F* represents the elastic distortion and rotation of the crystal lattice. It is obtained from the multiplicative decomposition of the deformation gradient as F = F’ - FJ’ . This decomposition is standard in crystal plasticity formulations, and it has been described in detail in prior work ($6). The determination of the slip plane is facilitated by expressing the resolved shear stress in terms of the slip vectors in the initial configuration. TcL=m;

.I+

.z.F*

.s;

= rnt - t” .

(3)

The vector ta has been introduced for convenience. Working in the initial configuration, the goal is to find the direction of the slip plane, rnt, such that: r4 is a maximum; rnt is a unit vector; and rng is orthogonal to st. The solution can be constructed by taking rn$ coaxial with ta, which gives the maximum value, and then subtracting the portion which lies along the direction of $. After normalization, the slip plane normals are given by 4 - s; (s; * P) (4) mt=J&e

The evolution of the resolved shear stress must reflect the changes in stress, slip direction and slip plane normal. Here, the potential difference between restricted slip and pencil glide is the evolution of the slip plane. For restricted slip, ri$=O since the plane is fixed; but for pencil glide Eq. (4) must be employed. From Eq. (3), the time derivative of the resolved shear stress can be written as

am;

ta = p . _

ata

.ia++ .i*

where it is recognized that $ is constant. The product of t" with the derivative in the first term from Eq. (5) is identically zero. Thus, ri$ makes no contribution to the rate of the resolved shear stress, but this does not imply that tit = 0. The surviving term is the same as for the restricted glide formulation. Hence, no additional changes are needed to implement pencil glide in the framework of a preexisting restricted glide code. Results An existing rate dependent Taylor-like model, similar to that of Asaro and Needleman (7), was modified to include a call to a subroutine which determines the slip plane according to Eq. (4). This one subroutine call and reducing the number of slip systems were the only changes made to the code. Since calculations are done on each slip system, reducing the number of slip systems reduced the CPU time from that of the restricted glide calculations with twelve slip systems (( 111} < 110 > for FCC and {110) < 111 > for BCC). The speed was increased by approximately a factor of two with six slip systems {ijk} < 110 > for FCC pencil glide and a factor of three with four slip systems {ijk} < 111 > for BCC pencil glide. Results are given for 2000 random initial orientations subjected to uniaxial tension and plane strain compression. The strain rate sensitivity was modelled by a power law relation with a rate exponent of m=0.002, which is nearly rate independent. Uniaxial Tension For uniaxial tension there is little difference between the predicted textures for pencil glide and restricted glide to a strain of 1.0. The primary difference is a few more random orientations present in the restricted glide computations. The stress strain curves for the four analyses are given in Fig. 1. As expected, the responses of the FCC and BCC restricted glide analyses were similar initially, but the results diverged as

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PENCIL GLIDE FORMULATION FOR POLYCRYSTALMODELLING

FCC restncted

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glide

BCC pencil glide

FCC

pencil glide

0.5 -

i 0.0

I

0.0

0.2

”

’

,

”

’

0.4

I

0.6

”

‘I”’

0.8

1.0

STRAIN Figure 1. Stress-strain curves for uniaxial tension.

different textures evolved. There is more freedom in the pencil glide results, so the stress is lower. The two additional slip systems available to the FCC crystal in pencil glide result in a significantly lower stress. The Taylor factor for the random grain distribution was plotted against stress. Since the “yield” point is not well defined in rate dependent materials, the Taylor factor cannot be determined precisely. Approximate values are reported in Table 1 along with published results. Plane Strain Compression Predicted {111) pole figures after plane strain compression to a true strain of 1.0 are given in Fig. 2 for an FCC polycrystal deforming by both restricted glide and pencil glide. Unlike the results for uniaxial tension, here there are significant differences between restricted glide and pencil glide predictions. The results for BCC simulations show similar differences. The FCC pencil glide results more closely resemble experimental rolling textures than to the restricted glide predictions. However, because of other significant assumptions in the Taylor model, this does not imply that the pencil glide model is more appropriate. The stress strain curves for the four plane strain analyses are given in Fig. 3. As expected, the responses of the FCC and BCC restricted glide analyses were nearly identical throughout the deformation. As in

FCC restricted

glide

FCC pencil glide

Figure 2. {111) Pole figures for plane strain compression of art FCC polycrystal.

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PENCIL GLIDE FORMULATION FOR POLYCRYSTALMODELLING

2.0 FCC

+ BCC

restricted glide

0.5 -

0.0 0.0

I,

1,

0.2

I,,

I

I,

I

I

b,

0.6

0.4

I

I

0.8

I 1.0

STRAIN Figure 3. Stress-strain curves for plane strain compression.

the uniaxial simulations, there is more freedom in the pencil glide model, so the stress is lower. The two additional slip systems available to the FCC crystal in pencil glide result in a significantly lower stress. The approximate Taylor factors for these simulations are given in Table 1. The values from Gilormini, et al (3) are multiplied by 2/d to base them on applied stress rather than von Mises equivalent stress. Conclusion A simple expression, m (4), can be used to determine the slip planes for pencil glide. This result can be used directly in several polycrystal formulations with little modification to the computer codes. The approximate predicted Taylor factors using this pencil glide algorithm compare favorably with published results. References 1. 2. 3. 4. 5. 6. 7.

C. L. Maurice and J. H. Driver,Actu Metall. Mater., Al,1653 (1993) P. Penning, Metal. Trans., 7A, 1021(1976). P. Gilormini, B. Barcroix, and J. J. Jonas, Actu Metall., 36,231(1988). A. D. Rollett and U. E Kocks, Proceedings of ICOTOM 8, Ed J. S. Kallend and G. Gottstein, TMS, p. 375 (1988). D. Peirce, R. J. Asaro, and A. Needleman, Acfu Met&., 31,1951(1983). R. J. Asaro, Advances in Applied Mechanics, 2&l, Academic Press (1983). R. J. Asaro and A. Needleman, Acta Metall., 33,923, (1985).

TABLE Predicted Taylor Factors

Uniaxial Present Gilormini Plane strain Present Gilormini

FCC

BCC

FCC

BCC

Restricted

Restricted

Pencil

Pencil

3.07 3.067

3.07 3.067

2.26 -

2.75 2.740

3.32 3.324

3.32 3.324

2.53 -

3.08 3.067