Interaction Between Dislocations in a Couple Stress Medium

Transcription

1 M. Ravi Shankar Srinivasan Chandrasekar School of Industrial Engineering, Purdue University, 315 N. Grant Street, West Lafayette, IN Thomas N. Farris School of Aeronautics and Astronautics, Purdue University, 315 N. Grant Street, West Lafayette, IN Interaction Between Dislocations in a Couple Stress Medium Taylor s theory of crystal plasticity is reformulated for dislocations in a couple stress medium. The divergence between Taylor s approach and an approach that includes the effects of couple stresses on dislocation interactions is demonstrated. It is shown that dislocations separated by a distance that is comparable to a characteristic material length scale, have mutual interaction somewhat weaker than that predicted by classical elasticity. DOI: / Introduction Classical elasticity has been traditionally used to model the stress field in the vicinity of a dislocation. This stress field has provided a basic understanding of the interaction between dislocations. Taylor 1, starting from the classical elasticity theory of dislocations, showed that the flow stress of a material is proportional to the square root of the dislocation density. A finite dislocation density may result either from a random trapping of dislocations statistically stored dislocations or due to an accumulation of dislocations for ensuring geometrical compatibility geometrically necessary dislocations,. The latter phenomenon is important when large strain-gradients occur in a material, 3. In problems involving a large strain gradient and/or large dislocation densities, the effect of couple stresses on elastic properties of dislocations may be significant. Mindlin 4,5 showed that the inclusion of couple stresses can introduce significant changes in the elastic stress state of a problem, especially at length scales on the same order as a characteristic material length scale. For instance, the elastic stress concentration factor of a circular hole in an infinite sheet under uniaxial tension is somewhat lower in a couple stress mediun than that predicted otherwise. This divergence becomes particularly conspicuous when the size of the hole is comparable to the characteristic length scale of the medium, 5. Dislocations, being an atomic scale phenomenon should, hence, be expected to be affected by the presence of couple stresses. Mindlin 4,5 suggested a number of possible sources of the characteristic material length scale depending on the type of the problem, with values of the length scale ranging any where from the lattice spacing in a perfect crystalline lattice to the grain size in a polycrystalline material. In this study, the length scale l is assumed to be equal to 500 nm Table 1, which is typical of the cell size in a highly work hardened material. While the characteristic length scale enters the calculations as a parameter that decides the range and strength of the couple stress terms, its specific value is not likely to affect the general conclusions that are arrived at in the ensuing analysis. Here, couple stresses are included in the formulation of the problem of an edge dislocation. It is shown that the presence of couple stresses introduces a so-called weak-interaction parameter in the elastic stress field that is operative only at very small Contributed by the Applied Mechanics Division of THE AMERICAN SOCIETY OF MECHANICAL ENGINEERS for publication in the ASME JOURNAL OF APPLIED ME- CHANICS. Manuscript received by the Applied Mechanics Division, May, 003; final revision; December 17, 003. Associate Editor: M.-J. Pindera. Discussion on the paper should be addressed to the Editor, Prof. Robert M. McMeeking, Journal of Applied Mechanics, Department of Mechanical and Environmental Engineering, University of California Santa Barbara, Santa Barbara, CA , and will be accepted until four months after final publication in the paper itself in the ASME JOURNAL OF APPLIED MECHANICS. distances and whose principal effect is to depress the stress field around the dislocation. On the basis of this observation, it is inferred that the stress required to operate a Frank-Read source in a couple stress medium is generally smaller than that required otherwise. This is not to mean that there is a softening of the material at large dislocation densities. What is suggested is that as the dislocation density increases, strain gradient effects appear to reduce the ability of dislocations to pin each other vis-a-vis the classical solution. Couple Stress Theory The stresses and couple stresses on an element in equilibrium are shown in Fig. 1. Couple stresses or couple per unit area are denoted by r and. The remaining stresses have the usual meaning. For this two-dimensional formulation, these stresses can be derived from stress functions and satisfying 5 : 4 0 (1) r l 1 l 1 r () 1 r l 1 l 1 r (3) l 4 0 (4) where, is the Poisson s ratio, and l is a length scale characteristic of the material defined as 5 : l M G. (5) G and M, respectively, are the shear modulus and bending modulus of the material. The stresses corresponding to and are r 1 r r 1 r r 1 r 1 r r 1 r r 1 r r 1 r (6) (7) r 1 r 1 r r 1 (8) r r 1 r r 1 r r (9) r r (10) 546 Õ Vol. 71, JULY 004 Copyright 004 by ASME Transactions of the ASME

2 Table 1 Material parameters assumed for copper Shear modulus G 44 GPa Poisson s ratio 0.34 Length of the dislocation 0.56 nm Material length scale l 500 nm 1 r. (11) Mindlin 4,5 has provided a detailed derivation of Eqs Stress Functions for a Dislocation in a Couple Stress Medium Consider the problem of a dislocation similar to the one considered by Taylor 1, i.e., an elastic isotropic cylinder with a dislocation as shown in Fig.. If the radius R of the cylinder is large, then the Airy stress function is B sin Dr log r sin. (1) r Couple stress effects can be included by introducing : Fig. Elastic cylinder with dislocation E r cos FK 1 r l cos (13) where, D, B, E, and F are constants that can be calculated subject to the boundary conditions. K n (x) is a modified Bessel function of the second kind and the nth order. The stress function in Eq. 13 is similar, if not the same as that postulated by Mindlin 5 for the problem of a circular hole in an infinite sheet subjected to uniaxial tension in a couple stress medium. At r a, the following boundary conditions hold: 1. r r a 0: It may be shown using Eqs. 13, 1, and 10 that in the limit a 0, F E/l.. r r a 0: It may be shown using Eqs. 13, 1, and 8 that this boundary condition and the following boundary condition lead to the same criterion. 3. r r a 0: It may be shown using Eqs. 13, 1, and 6 that in the limit a 0, B 0. Equations and 3 lead to a 4th relationship that is intrinsic to the ensemble: E 4 1 l D. (14) D may be evaluated from the fact that at, the relative displacement in the r-direction, u r u r 0 Fig.. Hence, D G 1. (15) The coefficients D, E, and F were calculated for material parameters typical of copper Table 1 and are given in Table. Note that the coefficients D and F are positive while E is negative. 4 Taylor s Theory in a Couple-Stress Medium Consider two edge dislocations of opposite signs at O and A in an infinite medium devoid of couple stress, whose glide-planes are spaced a distance h apart as shown in Fig. 3. Let us assume the dislocations are kept apart under the action of a remote shear stress S acting along the glide plane. It was suggested by Taylor that this shear stress would pull the two dislocations apart a distance x Fig. 3 such that S x h. (16) It was further argued that the maximum or critical value of S is D/(h) and if S were less than this critical value, then the two centers of dislocation cannot escape their mutual attraction, 1. A similar definition of the critical shear stress needed to pull apart two dislocations of opposite signs may be attempted for a couple stress medium as follows. Since r and r are not equal in a couple-stress medium, we define two stress parameters s and a as s r r / Table Coefficients for copper Fig. 1 Stress and couple stress in polar coordinates D E F.716 N/m Nm N Journal of Applied Mechanics JULY 004, Vol. 71 Õ 547

3 We can rewrite s Eq. 0 in Cartesian coordinates as s x y Ex x y. (1) The equivalent of Eq. 16 for Fig. 3 in a couple stress medium can be obtained by setting y h in Eq. 1 : S couple-stress x h Ex x h. () Fig. 3 Positive and negative dislocations in the presence of a remote shear stress and a r r /. Then, using the stress functions in Eqs. 1 and 13, weget s 1 E Dr r 3 FK 1 r/l FK r/l l lr cos (17) a FK 1 r/l cos. (18) l The Bessel function contribution to the shear stresses in Eq. 17 is negligible compared to the contribution from the other two terms involving D and E even at a distance of r 3l from the dislocation. Hence, we will not consider the contribution from terms involving the Bessel function K n (r/l). We take s to be the analog of S in Fig. 3. This analogy arises directly as a result of Mindlin s formulation, 4. In this formulation, the symmetric part of the shear stress ( s ) produces the usual shear strain ( r ), that in classical elasticity is produced by r of course, in classical elasticity r r ). That is r r s G. (19) Thus, to a first approximation, if a comparison is to be made between the classical result of Taylor and one involving couple stresses, it has to be between S and s. In this approximation we have, of course, neglected the terms arising from the interaction of the curvatures in the vicinity of the dislocation at A with the couple stresses produced by the dislocation at O in Fig. 3, since both of them are rapidly decaying quantities. By discarding the terms involving the Bessel functions in Eq. 17 we get D cos E cos s. (0) r r 3 It can be immediately seen using Eq. 0 and Table that, between two dislocations of opposite signs, the term involving E in Eq. 0 is a repulsive interaction. This term leads to a weak interaction that gains significance at large dislocation densities when the distances between dislocations are small. The term involving D in Eq. 0 is the Taylor interaction which causes two dislocations of opposite signs to always attract each other. 5 Weak Interaction Between Dislocations Equation can be broken into two parts, namely the Taylor interaction and the weak interaction. That is S couple-stress T x R x (3) where T(x) /(x h ), is the Taylor interaction same as Eq. 16, and R(x) Ex/(x h ), is the repulsive weak interaction. S and S couple-stress, the nondimensional interaction shear stresses, are defined as the nondimensionalized counterparts of S and S couple-stress such that S h G S (4) S couple-stress h G S couple-stress. (5) If the distance h, between the glide planes is very large compared to the material length scale parameter l, i.e., say h 10l, then R x is negligible and the solution for S couple-stress practically coincides with that obtained from Taylor s theory (S ). This is apparent from Fig. 4 where the variation of the nondimensional interaction shear stresses (S and S couple-stress ) with (x/h) is plotted for h 10l. However, when the distance h, between the glide planes becomes comparable to l, i.e., say h 3l, then the contribution of the repulsive weak interaction term, R(x), in Eq. 3 is seen to be significant Fig. 5. The effect of the R(x) term, hence, appears to be one of reducing the interaction stress between dislocations. For h 3l this effect is seen to be 5% based on the maximum value of the nondimensional interaction stress Fig. 5. The variation of the maximum value of S and S couple-stress with (h/l) is plotted in Fig. 6. Since Taylor s theory does not involve a characteristic material length scale, the maximum value of S has no dependence on the value of h/l, and hence, is a constant equal to 0.1. However, in the presence of couple stresses it can be seen Fig. 6 that the maximum value of S couple-stress decreases as h/l decreases. 6 Consequences of Weak Interaction for Strain- Gradient Plasticity Classically, the line tension of a dislocation line is proportional to the elastic self energy of the dislocation line. The stronger the elastic stress field around a dislocation, the stronger it interacts with other dislocations and the larger is the stress required to bend a dislocation line to the critical extent. Cottrell 6 has calculated the elastic self energy of an edge dislocation by treating it as the work done in displacing the faces of a cut made as in Fig. by against the resistance of the shear stress field of the dislocation. While such an approach yields the correct result in classical elasticity, similar simplistic calculations in the presence of couple stresses will lead to grossly misleading results. Since exact solutions for the self energy and line tension are extremely complicated, we will only attempt to qualitatively gauge the effect of couple stresses on dislocation interactions. 548 Õ Vol. 71, JULY 004 Transactions of the ASME

4 Fig. 4 Variation of nondimensional interaction shear stress with xõh for h Ä10l. The material parameters used are those for copper given in Tables 1 and. The maximum value of the interaction shear stress is representative of the stress required to disentangle a pair of dissimilar dislocations. As illustrated in Fig. 6, the maximum value of S couple-stress becomes smaller as h/l decreases. Hence, the stress required to disentangle a pair of dissimilar dislocations is generally smaller than that predicted by classical elasticity when the spacing between the dislocations becomes comparable to the characteristic material length scale l. This would imply that in the case of a large dislocation density, as is typical of highly strained materials or of many problems involving a significant strain gradient, when the distances between dislocations become comparable to the characteristic length scale of the material, the presence of couple stresses would tend to reduce the interaction between dislocations and reduce the ability of dislocations to pin each other. However, it is not likely that this softening produced by the reduced dislocation interactions will be comparable in magnitude to the hardening produced by the large dislocation density. While there have been attempts at understanding the effect of couple stresses close to the dislocation core by Eringen 7, Gutkin and Aifantis 8, and others, the importance of the corrections resulting from strain gradient effects vis-a-vis anharmonic effects Fig. 5 Variation of nondimensional interaction shear stress with xõh for h Ä3l. The material parameters used are those for copper in Tables 1 and. Journal of Applied Mechanics JULY 004, Vol. 71 Õ 549

5 Fig. 6 Variation of the maximum nondimensional interaction shear stress with hõl. The material properties used are those for copper given in Tables 1 and. close to the core remain questionable. Hence, we have confined ourselves here to trying to understand the effect of couple stresses on dislocation interactions occuring in regions far from the dislocation core where simplifying assumptions can be made to make the problem tractable. Due to the mathematical complexity involved, it is not attempted in this study to explicitly derive the expression for line tension in a couple stress medium. However, the results derived are sufficient to predict the nature of the change in interactions between dislocations in a couple stress medium at large dislocation densities. 7 Conclusions Taylor s theory of plasticity has been reformulated for a couplestress medium with a characteristic material length scale. The analysis has revealed the following: 1. The shear stress required to disentangle a pair of dissimilar dislocations is estimated to be smaller when couple stresses are considered. This reduction is attributed to the presence of a repulsive element R(x) Eq. 0 in the interaction shear stress.. R(x) becomes significant only at large dislocation densities (h 10l where l is a material length scale. 3. The proposed model degenerates to the Taylor model at low dislocation densities where the average distance between dislocations is much greater than the material length scale. 4. At large dislocation densities, the ability of dislocations to pin each other is estimated to be smaller than that predicted by classical elasticity. This by no means predicts any significant softening of materials at large dislocation densities. But, materials will behave somewhat softer than expected at large dislocation densities. Acknowledgment We would like to thank the reviewers for their very constructive comments and suggestions. The support of the National Science Foundation through grants CMS and DMI is gratefully acknowledged. References 1 Taylor, G. I., 1934, The Mechanism of Plastic Deformation of Crystals, Proc. R. Soc. London, Ser. A, 145, pp Ashby, M. F., 1970, The Deformation of Plastically Non-Homogeneous Alloys, Philos. Mag., 1, pp Fleck, N. A., Muller, G. M., Ashby, M. F., and Hutchinson, J. W., 1994, Strain Gradient Plasticity: Theory and Experiment, Acta Metall. Mater., 4, pp Mindlin, R. D., and Tiersten, H. F., 196, Effects of Couple Stresses in Linear Elasticity, Arch. Ration. Mech. Anal., 11, pp Mindlin, R. D., 1963, Influence of Couple-Stresses on Stress Concentrations, Exp. Mech., 3, pp Cottrell, A. H., 1956, Dislocations and Plastic Flow in Crystals, Oxford University Press, Oxford, UK. 7 Eringen, A. C., 1983, On Differential Equations of Nonlocal Elasticity and Solutions of Screw Dislocations and Surface Waves, J. Appl. Phys., 54 9, pp Gutkin, M. Y., and Aifantis, E. C., 1999, Dislocations in the Theory of Gradient Elasticity, Scr. Mater., 40, pp Õ Vol. 71, JULY 004 Transactions of the ASME