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On stability of an interfacial edge dislocation wall in a symmetrical bicrystal
S c r i p t a METALLURGICA
Vol. i 0 , pp. 331-333, 1976 P r i n t e d in t h e U n i t e d S t a t e s
Pergamon P r e s s ,
Inc.
ON STABILITY OF AN INTERFACIAL EDGEDISLOCATIONWALL IN A SYMMETRICALBICRYSTAL Y. T. Chou Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 {Received February 6, 1976) {Revised March 1, 1976)
I.
Introduction
In the classical dislocation theory of grain boundaries [I-6], i t is demonstrated that an edge dislocation wallt has no long-range stress field, and geometrically f i t s a simple t i l t boundary. However, in the earlier analyses, the effect of the change in crystallographic orientations across the boundary was not taken into account. I t is also noted that i f a dislocation array in a grain boundary is energetically stable, Frank's formula [6] is a necessary consequence on the basis of geometry. Any dislocation array which does not obey Frank's formula is therefore not suitable for modelling a stable boundary. However, the stability of the dislocation array is implicitly assumed in Frank's formulation. In recent work on dislocations in two-phase media [7,8], i t was concluded that an edge dislocation wall has a long-range stress field, and hence i t is not suitable for modelling lowangle phase boundaries. Furthermore, since the special case considered in the anisotropic analysis [8] also applies to non-symmetrical bicrystals (single-phase), the above conclusion is then valid for such bicrystals as well. A simple question naturally arises as to whether such a long-range stress field would exist in a symmetrical bicrystal? The question cannot be resolved by the previous analysis [8] because i t is based on the assumption that the two half-crystals possess orthotropic symmetry which is not valid in a symmetrical bicrystal (see the analysis below). In this note, we shall provide a proof of the statement that in a symmetrical bicrystal, an interfacial edge dislocation wall has no long-range stress field. 2.
Analysis
To simplify the discussion, we construct a symmetrical bicrystal with simple crystallographic orientation. Let us consider an infinite crystal with orthotropic symmetry. Relative to a rectangular Cartesian coordinate system ox°y°z°, the elastic constants of the crystal are of the form:
On leave from Department of Metallurgy and Materials Science, Lehigh University, Bethlehem, Pennsylvania 18015. t i n t h i s note, the term "dislocation wall" is used to refer to a l i n e a r array of equally spaced, parallel dislocations with Burgers vectors equal and perpendicular to the plane of the array. 331
332
INTERFACIAL
EDGE DISLOCATION WALL
I
-Cyl c72 cY3 0
0
0
0
0
0
c)3 0
0
0
0
0
C~4
Vol. I0, No. 4
o 0 C55 CO 66 To form a symmetrical bicr ,stal, we divide the crystal into two halves along the plane x° = O, make a proper rotation about the z° axis to the right half-crystal (x°>O) and an improper rotation -e to the l e f t half-crystal (x°
--Cll
C12 C13 0
0
C16 -
C22 C23
0
0
C26
C33
0
0
C36
C44 C45 0 C55 0 C66 As a result of symmetry, the corresponding elastic constants in crystal I and crystal I I are
II
II
II
equal except that the signs of 6~6, C~6, C~6 and C~5 are opposite to those of C16, C26, C36 and C45II (the superscripts I and I I refer to crystal I and crystal I I , respectively). Even in such a simple case, the elastic f i e l d of an edge dislocation in the bicrystal is d i f f i c u l t to treat ana. lytically [9]. It is possible, however, to examine the elastic field of the dislocation on a qualitative basis, and subsequently to deduce whether there is a long-range stress field associated with a wall composed of these dislocations. To do this, it is helpful to consider the following correlation obtained from the analyses of edge dislocations in two-phase media and single-phase, nonsymmetrical bicrystals [7,8]. 2.1 Correlation Between Long-Range Stress and Stress Discontinuity Across the Boundary A close examination of the previous results [7,8] indicates that the magnitude of the longrange stress OLR produced by an edge dislocation wall is linearly proportional to the magnitude of the discontinuity in the in-plane nomal stress component across the boundary plane x=O, denoted by AOyy. (~Oxx = AOx~y = 0 at x=O as required by the boundary conditions). According to this correlation, no long-range stress would exist if AOyy = O. While the general validity of the above finding remains to be shown, its existence, however, does suggest the need for a careful examination of the term AOyy and its effect on the stress field of an Interfacial edge dislocation wall in a symmetrical bicrystal. 2.2 Validity of AOyy 0 and OLR 0 for Symmetrical Bicrystals A fomal mathematical proof for aOyy = 0 at x=O for an interfacial edge dislocation in a
Vol.
I0, No. 4
INTERFACIAL
EDGE DISLOCATION
WALL
333
symmetrical bicrystal is not available at present. Alternatively, we approach the problem from a physical viewpoint. When an edge dislocation is situated at the boundary in a symmetrical bicrystal (Burgers vector perpendicular to the boundary), its elastic fields on the two sides of the boundary are mirror images to one another. Thus, on approaching the boundary plane from the two half-crystals, all the corresponding stress components, including Oyy, must converge to single values. In other words, in a symmetrical bicrystal not only AOxx = Aaxy = 0 at x=O as required by the boundary condition, but also AOyy = O at x=O as required by the symmetry of the bicrystal. This conclusion in fact is quite general regardless of the actual symmetrical classes associated with the two half-crystals. Furthermore, because of the mirror symmetry, not only AOxx = A~xy = AOyy = 0 at x=O, but also the stress distributions in the entire region are symmetrical with respect to the boundary plane. This reflects that i f there is a sign change for a certain elastic constant from crystal I to crystal I I , the particular elastic constant must appear in even powers in the stress distributions. In other words, even though mathematically there are sign differences for certain elastic constants in the two half-crystals (a result due to the reference to the coordinate system), the bicrystal would respond elastically to a s~mmetrical disturbance as i f i t were a homogeneous single-crystal. In this sense, a slnmnetrical bicrystal may be considered as a "pseudo" single-crystal and the stress field of an edge dislocation situated at the boundary of the bicrystal (Burgers vector perpendicular to the boundary) would have the same characteristics as that in a homogeneous single-crystal. Consequently, following Stroh's analysis for single crystals [lO], we may conclude that an interfacial edge dislocation wall in a symmetrical bicrystal has no long-range stress field. I t is desirable that the above proof, based on physical deductions, be further strengthened by quantitative analysis. Although the mathematical proof for the general problem is d i f f i c u l t to achieve, analyses for certain simple cases are underway and the results will be reported shortly [9]. Acknowledgements The author wishes to thank Prof. J. P. Hirth and Prof. J. C. M. Li for helpful discussions. The work was supported by the National Science Foundation under Grant DMR72-03237 AOl. References I.
J.W. Burgers, Proc. Kon. Ned. Akad. Wet. 42, 293, 378 (1939).
2.
W.L. Bragg, Proc. Phys. Soc. 52, 54 (1940).
3.
W.T. Read and W. Shockley, Phys. Rev. 78, 275 (1950).
4.
W.T. Read, "Dislocations in Crystals", McGraw-Hill, New York 1953
5.
J.H. van der Merwe, Prec. Phys. Soc. A63, 616 (1950).
6.
F. C. Frank, in "Report of the Symposium on the Plastic Deformation of Crystalline Solids", Carnegie Institute of Technology, Pittsburgh, 1950, p. 15O.
7.
Y.T. Chou and L. S. Lin, Mater. Sci. Eng. 20, Ig (1975).
8.
Y.T. Chou, C. S. Pande and H. C. Yang, J. Appl. Phys. 46, 5 (1975).
9.
Y.T. Chou and H. C. Yang, research in progress.
10.
A. N. Stroh, Phil. Mag. 8, 625 (1958).
Vol. i 0 , pp. 331-333, 1976 P r i n t e d in t h e U n i t e d S t a t e s
Pergamon P r e s s ,
Inc.
ON STABILITY OF AN INTERFACIAL EDGEDISLOCATIONWALL IN A SYMMETRICALBICRYSTAL Y. T. Chou Department of Materials Science and Engineering Massachusetts Institute of Technology Cambridge, Massachusetts 02139 {Received February 6, 1976) {Revised March 1, 1976)
I.
Introduction
In the classical dislocation theory of grain boundaries [I-6], i t is demonstrated that an edge dislocation wallt has no long-range stress field, and geometrically f i t s a simple t i l t boundary. However, in the earlier analyses, the effect of the change in crystallographic orientations across the boundary was not taken into account. I t is also noted that i f a dislocation array in a grain boundary is energetically stable, Frank's formula [6] is a necessary consequence on the basis of geometry. Any dislocation array which does not obey Frank's formula is therefore not suitable for modelling a stable boundary. However, the stability of the dislocation array is implicitly assumed in Frank's formulation. In recent work on dislocations in two-phase media [7,8], i t was concluded that an edge dislocation wall has a long-range stress field, and hence i t is not suitable for modelling lowangle phase boundaries. Furthermore, since the special case considered in the anisotropic analysis [8] also applies to non-symmetrical bicrystals (single-phase), the above conclusion is then valid for such bicrystals as well. A simple question naturally arises as to whether such a long-range stress field would exist in a symmetrical bicrystal? The question cannot be resolved by the previous analysis [8] because i t is based on the assumption that the two half-crystals possess orthotropic symmetry which is not valid in a symmetrical bicrystal (see the analysis below). In this note, we shall provide a proof of the statement that in a symmetrical bicrystal, an interfacial edge dislocation wall has no long-range stress field. 2.
Analysis
To simplify the discussion, we construct a symmetrical bicrystal with simple crystallographic orientation. Let us consider an infinite crystal with orthotropic symmetry. Relative to a rectangular Cartesian coordinate system ox°y°z°, the elastic constants of the crystal are of the form:
On leave from Department of Metallurgy and Materials Science, Lehigh University, Bethlehem, Pennsylvania 18015. t i n t h i s note, the term "dislocation wall" is used to refer to a l i n e a r array of equally spaced, parallel dislocations with Burgers vectors equal and perpendicular to the plane of the array. 331
332
INTERFACIAL
EDGE DISLOCATION WALL
I
-Cyl c72 cY3 0
0
0
0
0
0
c)3 0
0
0
0
0
C~4
Vol. I0, No. 4
o 0 C55 CO 66 To form a symmetrical bicr ,stal, we divide the crystal into two halves along the plane x° = O, make a proper rotation about the z° axis to the right half-crystal (x°>O) and an improper rotation -e to the l e f t half-crystal (x°
--Cll
C12 C13 0
0
C16 -
C22 C23
0
0
C26
C33
0
0
C36
C44 C45 0 C55 0 C66 As a result of symmetry, the corresponding elastic constants in crystal I and crystal I I are
II
II
II
equal except that the signs of 6~6, C~6, C~6 and C~5 are opposite to those of C16, C26, C36 and C45II (the superscripts I and I I refer to crystal I and crystal I I , respectively). Even in such a simple case, the elastic f i e l d of an edge dislocation in the bicrystal is d i f f i c u l t to treat ana. lytically [9]. It is possible, however, to examine the elastic field of the dislocation on a qualitative basis, and subsequently to deduce whether there is a long-range stress field associated with a wall composed of these dislocations. To do this, it is helpful to consider the following correlation obtained from the analyses of edge dislocations in two-phase media and single-phase, nonsymmetrical bicrystals [7,8]. 2.1 Correlation Between Long-Range Stress and Stress Discontinuity Across the Boundary A close examination of the previous results [7,8] indicates that the magnitude of the longrange stress OLR produced by an edge dislocation wall is linearly proportional to the magnitude of the discontinuity in the in-plane nomal stress component across the boundary plane x=O, denoted by AOyy. (~Oxx = AOx~y = 0 at x=O as required by the boundary conditions). According to this correlation, no long-range stress would exist if AOyy = O. While the general validity of the above finding remains to be shown, its existence, however, does suggest the need for a careful examination of the term AOyy and its effect on the stress field of an Interfacial edge dislocation wall in a symmetrical bicrystal. 2.2 Validity of AOyy 0 and OLR 0 for Symmetrical Bicrystals A fomal mathematical proof for aOyy = 0 at x=O for an interfacial edge dislocation in a
Vol.
I0, No. 4
INTERFACIAL
EDGE DISLOCATION
WALL
333
symmetrical bicrystal is not available at present. Alternatively, we approach the problem from a physical viewpoint. When an edge dislocation is situated at the boundary in a symmetrical bicrystal (Burgers vector perpendicular to the boundary), its elastic fields on the two sides of the boundary are mirror images to one another. Thus, on approaching the boundary plane from the two half-crystals, all the corresponding stress components, including Oyy, must converge to single values. In other words, in a symmetrical bicrystal not only AOxx = Aaxy = 0 at x=O as required by the boundary condition, but also AOyy = O at x=O as required by the symmetry of the bicrystal. This conclusion in fact is quite general regardless of the actual symmetrical classes associated with the two half-crystals. Furthermore, because of the mirror symmetry, not only AOxx = A~xy = AOyy = 0 at x=O, but also the stress distributions in the entire region are symmetrical with respect to the boundary plane. This reflects that i f there is a sign change for a certain elastic constant from crystal I to crystal I I , the particular elastic constant must appear in even powers in the stress distributions. In other words, even though mathematically there are sign differences for certain elastic constants in the two half-crystals (a result due to the reference to the coordinate system), the bicrystal would respond elastically to a s~mmetrical disturbance as i f i t were a homogeneous single-crystal. In this sense, a slnmnetrical bicrystal may be considered as a "pseudo" single-crystal and the stress field of an edge dislocation situated at the boundary of the bicrystal (Burgers vector perpendicular to the boundary) would have the same characteristics as that in a homogeneous single-crystal. Consequently, following Stroh's analysis for single crystals [lO], we may conclude that an interfacial edge dislocation wall in a symmetrical bicrystal has no long-range stress field. I t is desirable that the above proof, based on physical deductions, be further strengthened by quantitative analysis. Although the mathematical proof for the general problem is d i f f i c u l t to achieve, analyses for certain simple cases are underway and the results will be reported shortly [9]. Acknowledgements The author wishes to thank Prof. J. P. Hirth and Prof. J. C. M. Li for helpful discussions. The work was supported by the National Science Foundation under Grant DMR72-03237 AOl. References I.
J.W. Burgers, Proc. Kon. Ned. Akad. Wet. 42, 293, 378 (1939).
2.
W.L. Bragg, Proc. Phys. Soc. 52, 54 (1940).
3.
W.T. Read and W. Shockley, Phys. Rev. 78, 275 (1950).
4.
W.T. Read, "Dislocations in Crystals", McGraw-Hill, New York 1953
5.
J.H. van der Merwe, Prec. Phys. Soc. A63, 616 (1950).
6.
F. C. Frank, in "Report of the Symposium on the Plastic Deformation of Crystalline Solids", Carnegie Institute of Technology, Pittsburgh, 1950, p. 15O.
7.
Y.T. Chou and L. S. Lin, Mater. Sci. Eng. 20, Ig (1975).
8.
Y.T. Chou, C. S. Pande and H. C. Yang, J. Appl. Phys. 46, 5 (1975).
9.
Y.T. Chou and H. C. Yang, research in progress.
10.
A. N. Stroh, Phil. Mag. 8, 625 (1958).