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Length scale effect on mechanical behavior due to strain gradient plasticity
Materials Science and Engineering A303 (2001) 241– 249 www.elsevier.com/locate/msea
Length scale effect on mechanical behavior due to strain gradient plasticity D.-M. Duan b, N.Q. Wu a, W.S. Slaughter a, Scott X. Mao a,* a
Department of Mechanical Engineering, The Uni6ersity of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261, USA b Department of Mechanical Engineering, The Uni6ersity of Calgary, Calgary, Alta, Canada Received 10 May 2000; received in revised form 15 September 2000
Abstract The characteristic material length scale l as introduced in the phenomenological theory of strain gradient plasticity by Fleck and Hutchinson [1] is crucial for the application of the theory. Three dislocation models are proposed in this paper for the derivation of the material length scales as there is a rational connection between dislocation theory and phenomenological theory. The length scale is determined as a function of material fundamental parameters and is independent of how the ‘overall effective strain’ is measured, or how the dislocations interact. The length scale is generally about 1.5 mm or less, agreeing with the observation that the size effect appears approximately at 1 mm and sub-micrometer levels for indentation deformations. The dislocation interaction is also discussed in the framework of strain gradient plasticity. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Strain gradient plasticity; Dislocation; Materials length scale
1. Introduction There is ample experimental evidence showing the existence of a strong size effect in the plastic flow of metals and ceramics. For example, the measured indentation hardness of metals and ceramics increases with decreasing indentation size by a factor of about two as the width decreased to 1 mm. This size effect is even more significant in the sub-micrometer region [2–4]. Another example of size effect was presented for the torsion response of copper wires [5] in which the scaled shear strength increases with diminishing wire diameter in the range 100 – 10 mm by almost a factor of three, while there is no evident size effect in the tensile data. Conventional theories of plasticity are unable to explain and predict these effects which in fact reflect some inherent properties of the materials. As discussed by Fleck and Hutchinson [1,5], these effects may not have the same explanation, but it is clear that all require a length scale for their interpretation. A natural way to include size effects in the constitutive law is to postulate * Corresponding author. Tel.: +1-412-6249602; fax: + 1-4126244846. E-mail address: [email protected] (S.X. Mao).
that the yield stress depends not only on the strain as in conventional plasticity theories but also on strain gradient. There have been attempts to include strain gradients within a plasticity formulation, notably those of Fleck and Hutchinson [1,6]. Fleck and Hutchinson introduced material length scales involving the concept of geometrically necessary dislocations. Consequently, as a material constitutive law concerning strain gradient effect, the theory must establish the characteristic material length scale that reflects a micro nature of the material. Dislocation theory provides a means for understanding the strain gradient effect in plastic hardening and the nature of characteristic material length scale. Dislocations become stored for two reasons: they accumulate by trapping each other in a random way or they are required for compatible deformation of various parts of the crystal. The randomly stored dislocations, which are created by homogeneous strain, are referred to as statistically stored dislocations [7,8]. The dislocations stored in the requirement for deformation compatibility, which are related to the curvature of the crystal lattice or to strain gradients, are called geometrically necessary dislocations [9,10]. The total hardening effect of a material should be a combined result of both the
0921-5093/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 9 2 1 - 5 0 9 3 ( 0 0 ) 0 1 9 0 7 - 9
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strain hardening and strain gradient hardening. Generally, the geometrically necessary dislocation density can be calculated for typical forms of deformation, and due to the close relation between the strain gradient and the geometrically necessary dislocation density [9,11], the strain gradient field can be obtained. This approach has been adopted by Nix and Gao [12] to characterize the size effect of indentation tests and to calculate the characteristic material length scale. This paper gives a further investigation of the characteristic material length scale of the strain gradient plasticity. It first reviews the strain gradient plasticity theory proposed by Fleck and Hutchinson [1,6] and then presents dislocation models for some basic forms of deformation to calculate the corresponding material length scales. The size effect results, predicted using the theory with use of the corresponding material length scale, are then presented and discussed. As a by product, some results of dislocation interaction associated with the strain gradient plasticity theory are presented and discussed.
2. Theoretical background In their phenomenological theory of strain gradient plasticity, Fleck and Hutchinson [1] constructed an ‘overall effective strain’: mcs = [m +l ] v e
v cs
v 1/v e
(1)
where me and e are the second von Mises invariant of strain and the analogous second invariant of the curvature: me =
' '
e =
2 mijmij 3
(2)
2 3 ij ij
(3a)
where the curvature is the spatial gradient of the material rotation q: ij =qi, j
(3b)
Here the material length scale lcs has been introduced as required for dimensional consistency, and the subscript ‘cs’ indicates that the equation is derived from ‘couple stress’ theory where only rotation gradient is included. A more general ‘combined strain quantity’ is also derived by Fleck and Hutchinson [6] which takes both rotation and stretch gradient into account as follows: m=
2 m% m% 3 ij ij
v/2
(1) (1) v/2 (2) (2) v/2 + (l 21p%(1) +(l 22p%(2) ijk p%ijk p%ijk ) ijk p%ijk p%ijk )
n
(3) (3) v/2 +(l 23p%(3) ijk p%ijk p%ijk )
1/v
(4)
where (%) denotes a quantity derived from an incompressible displacement field; p is the strain gradient tensor and has a unique orthogonal decomposition of: p% =p%(1) + p%(2) + p%(3)
(5)
Instead of introducing a single material length scale in Eq. (1), three length scales are introduced in Eq. (4). The couple stress theory expressed in Eq. (1) is a special case of the general theory in Eq. (4) when only curvature invariant is taken into account. Both theories assume that the strain energy w depends only upon the single scalar strain measure m while the dependence remains to be specified. The parameter v needs to be specified, which reflects the way in which the ‘overall effective strain’ is measured. Fleck and Hutchinson [6] pointed out, when Eq. (1) is used to deduce the value for the material length scale l with experimental data, the magnitude of l is sensitive to the value adopted for v. They suggested that v be chosen from 1 to 2 and observed numerical difficulties in the solution process for values close to 1. A parallel discussion for the definition of the ‘overall effective strain’ is the argument of the interaction between the statistically stored dislocation density and the geometrically necessary dislocation density. According to Eq. (1) the effective total dislocation density can be expressed as: z vT = z vS + z vG
(6)
Dividing the dislocation into two parts and then combining them in different ways implies changes in the strain history. Ashby [7] has pointed out that in general the presence of geometrically necessary dislocations will accelerate the rate of statistical storage and a linear sum of the two (v= 1 in Eq. (6)) only provides a lower limit of the effective total dislocation density. Thus, it is argued here that, to properly estimate the total dislocation density, v should be equal or less than 1 so that a higher total dislocation density can be obtained. It is assumed that the flow stress for a single-slip system of a single crystal depends upon the total dislocation density. With Eq. (6), the Taylor relation is expressed as: ~= hGb zT
(7)
where G is the shear modulus, b is the magnitude of Burger’s vector and h is a constant taken to be about 0.2 for fcc crystal metals and about 0.4 for bcc metals [13] and was taken to be 0.3 by Ashby [7]. By applying von Mises flow rule, |= 3~, and Eq. (6), we have: |= 3hGb(z vS + z vG)1/2v
(8)
The geometrically necessary dislocation density, zG, is closely related to strain gradient , and generally, if the higher order of strain gradient is neglected, can be expressed as:
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
zG =f ·
b
(9)
where the coefficient f is a function of higher order of strain gradient and for some special cases with constant strain gradient field, f is constant. Thus by considering the usual power hardening law in the absence of a strain gradient: |o =|refm N
(10a)
and noting |o = 3hGb zs
(10b)
Eq. (8) becomes | |ref
2
=[(m 2N)v +l v v]1/v
G |ref
(12)
Here one sees that the characteristic material length scale l is expressed in fundamental material parameters and is independent of v. A counterpart to Eq. (11) can be derived from Eq. (1). Assuming a power law relation between the overall effective strain and the effective stress:
S So
2
= m 2N =[m ve +l vcs ve ]2N/v
dV =2yr(c+ d)dr
(15)
where a is the radius of the contact area, b is the magnitude of Burger’s vector and c is the indentation deformation. The geometrically necessary dislocation density is therefore: zG =
du tan2 q = ; dV bh 1 − t 2
t= r/a
(16)
The vertical stretch strain at r is m=
−c =− c+ d
'
1− t · tan q 1+ t
(17)
(13)
the corresponding normalized effective stress can be expressed as:
dr h = 2yr dr; s ba
(11)
2
S=Som N
based on the dislocation theory. In their model, a constant strain gradient field is assumed in the defined hemispherical volume under the indenter. The model is modified here in such a way that the strain gradient is not a constant but varies with locations. The strain gradient field is derived directly from the strain field which is calculated from the geometry of the defined deformation volume. In the modified model shown in Fig. 1, considering a cylinder element with a thickness dr at position r, the total length of the injected loops in the element and the element volume are as follows: du = 2yr
where the material length scale l is l= 3h 2bf
243
(14)
and the strain gradient field is =
1 dm 1 dm tan2 q · = · = h dr a dt (1+t) 1 − t 2
(18)
Nix and Gao [12] tried to compare Eq. (11) with v = 1 in their case and Eq. (14), and concluded that the two equations match each other only under certain conditions. Note that the left sides of the two equations are different where |ref is a coefficient in the power law stress –strain relation without strain gradient while _o is the coefficient in an equi6alent power law relation with strain gradient taken into account. Exact match of the two theories is not possible due to some intrinsic incompatibility problems.
3. The models From Eqs. (9) and (12), the characteristic material length scale l can be obtained if the factor f is determined. In this section, dislocation models are proposed for three basic deformation forms to derive the corresponding geometrically necessary dislocation density zG and, thereafter, the factor f for each case.
3.1. A modified indentation model Nix and Gao [12] have proposed an indentation model
Fig. 1. The dislocation model for indentation deformation. The dislocation structure is idealized as circular dislocation loops and the deformation is assumed to be confined in a hemispherical volume.
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size-dependent hardness curves are plotted with various v values ranging from 0.2 to 1.0. The hardness is normalized by the reference stress and the indentation depth is normalized by the characteristic material length scale l. It is evident that while the v effect is significant especially at lower v values, all the curves show a strong size effect. The characteristic material length scale plays a key role in the size effect where, with the size fixed, a larger l corresponds to a higher value of hardness, and with l decreasing to zero all the curves approach a limit which is the result of conventional plasticity.
3.2. A bending model Fig. 2. Normalized strain gradient distribution for indentation deformation (Eq. (18)) compared with the result of Nix and Gao [12]. Strain gradient is normalized by 0 = tan q/a as = /0.
Eqs. (16) and (18) give the expression for f in Eq. (9) as: f = 1+t
(19)
It is clear that f is not a constant but is dependent on location. Nix and Gao [12] obtained a value of 1.5 for f from their model which is exactly the average value from Eq. (19). Since the final measured hardness is an average of integration of the stress over all the contact area, it is preferred here to use a weighed average value of f determined through the following equation: −
f=
1 y
&
1
f · 2yt dt =
0
5 3
(20)
By choosing Tabor’s factor of three, the final expression showing size effect for hardness can be derived from Eq. (11) as follows: H =3|= 6|ref
&
1
{ f vo(t) + [f1(t)/h]v}1/2vt · dt
A dislocation model for bending has been proposed in Ref. [9] and the geometrical necessary dislocation density has been calculated [8]. Here the dislocation density is derived in a more straightforward way. The model is shown in Fig. 4a where it is assumed that any cross section remains plane after deformation, slip planes are along with the beam length direction and there is no transverse shear effect on the deformation (pure bending). For the sake of easy calculation, Fig. 4a can be simplified as Fig. 4b for a half beam. From the relation: tanq =
b s
(23)
the total length of dislocation lines is: u=
2hB tan q = 2hB s b
(24)
where B is the thickness of the beam. Then the geometrically necessary dislocation density is
(21)
0
where fo(t)=
1−t 1+t
f1(t)=3h 2b
N
· tan2N q;
G |ref
2
·
tan2 q
(22)
1 −t 2 The major difference between the current model and the model proposed by Nix and Gao [12] is that a deformation zone with a non-uniform strain gradient field is introduced in the current model. This difference is shown in Fig. 2. The strain gradient is normalized by an overall measure of 0 =tan q/a. The current model reveals a strain gradient singularity at the boundary of the contact area. This is natural because immediately outside the contact area, the plastic strain is zero. The hardness H predicted from Eq. (21) is dependent on the value of v. In Fig. 3 a group of
Fig. 3. Normalized hardness versus normalized indentation depth with different values of v. Other parameter values used are: N= 0.2, h =0.3, tan q =0.358.
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245
Fig. 4. The dislocation model for pure bending deformation. Slip planes are assumed to be along the beam length direction, no transverse shear effect is considered; (a) is simplified as (b) for a half beam.
u u 2 tan q zG = = = V hLB bL
(25)
For small q, Eq. (25) can be written as zG =
2q s = (zb) − 1 = bL b
(26)
where s is the curvature of the beam. Since in pure bending, the stretch strain gradient is equal to the curvature, the value of the factor f in Eq. (9) is unity. Finally, the bending moment expression showing the size effect is obtained as follows: M0 =
1 M = 2 4Bh |ref 2
&
1
therefore leads to higher curves of bending moment. When h reaches unity, the value of the normalized moment response at every surface strain level is more than twice that without strain gradient effect.
3.3. A torsion model Unlike the indentation model and the bending model where the deformation is modeled using edge dislocations, the deformation of a round bar under torsion is,
[(m0 · t)2Nv +(l · m0/h)v]1/2v · t dt
0
(27) where m0 is the surface strain of the beam and l is the material length scale. Again, for a given material for which l is fixed, the normalized bending moment is a function of both the size of the beam, h, and v. These effects can be seen from Figs. 5 and 6. The load –strain curves with various v values shown in Fig. 5 indicate that an underestimate will be predicted for the deformation behavior if a higher value of v is chosen than the actual one of the material. More discussion of this will be given in later sections. The normalized load – strain curves in Fig. 6, where the beam thickness is normalized by the characteristic material length scale l, also show the role of l in the size effect. For a fixed beam thickness, increasing l will make h decrease and
Fig. 5. The effect of v on the load– strain curve for bending deformation. Parameter values used are: N =0.5, h= 0.3, h/l= 4.
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
246
Fig. 6. Size effect on the load–strain curve for bending deformation. The half beam thickness h is normalized by the characteristic material length scale l, h = h/l. Other parameter values used are N= 0.5, h = 0.3 and v =0.5.
by nature, due to screw dislocations. As shown in Fig. 7a, the following assumptions are made: the round bar deforms uniformly along its length, cross sections remain planes after deformation and the deformation is achieved through screw dislocations of which the slip planes are in accordance with cross sections. Consider the displacement of a point at r in a unit length of the bar in Fig. 7b. The total relative displacement d is an accumulation of Burger’s vector: d=
q ·r =q · r L
(28)
where q is the rotation angle per unit length which represents the shear strain gradient in torsion. The number of Burger’s vectors per unit length of the bar at r is
Fig. 8. Size effect on the load– strain curve for torsion deformation. The radius R is normalized by the characteristic material length scale l, R =R/l. Other parameter values used are: v= 0.7, N= 0.2 and h = 0.3.
d q ·r N= = b b
(29)
In another way, the dislocation number N is also an accumulation of screw dislocation from the center of the bar to the cylinder at r. The relation in Eq. (29) keeps the bar deformation in such a way that any diameter in the bar remains straight after deformation. Dislocation density is therefore given as the increment of dislocation number per unit thickness of the cylinder. Thus the density of geometrically necessary dislocation is: zG =
dN q = = dr b b
(30)
Again the value of the factor f in Eq. (9) is obtained as unity. Using Eqs. (10a) and (10b), the normalized torque is: T=
T = 2y 3 R ~ref
&
1
t 2{k v0 (t)+ [k1/R]v}1/2v dt
(31)
(32)
0
where k0(t)= (k0 · t)2N;
k1 = h 2b
G ~ref
2
· k0 = l · k0
and k0 is the surface strain of the bar. The reference shear stress ~ref in Eq. (32) has a relation with |ref in Eqs. (10a) and (10b) as follows: ~ref = |ref/(2× 0.75N)
Fig. 7. The dislocation model for torsion deformation. (a) A side view indicating the deformation assumptions, and (b) the displacement is assumed to be an accumulation of screw dislocation.
(see Appendix A). Eq. (31) is plotted in Fig. 8 with v= 0.7 and N= 0.2. In the same way as in Fig. 6 the radius of the bar is normalized by the characteristic material length scale, l. The figure reveals basically the same features as in Fig. 6 for bending deformation and indicates that when a torsion bar with a radius more than 30 times the characteristic material length scale l,
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Table 2 v values for three different nickel foils in bending
Thickness (mm)a Grain size (mm)a v a
Fig. 9. The normalized torque as a function of v plotted at different surface strain levels for torsion deformation. Other parameter values used are: N =0.2, h= 0.3, and normalized radius R = R/l= 1.
the size effect is negligible. This again confirms the role of the characteristic material length scale l in microscale plastic deformations. Instead of plotting the whole deformation curves for v effect investigation as in Figs. 3 and 5, the normalized torque is plotted here directly against v in Fig. 9 at different surface strain levels to reveal the enhanced effect in the region of lower v values. It is apparent that the torque response monotonically increases with decreasing v and the increasing rate is higher for a higher surface strain level (note the normalized torque is in a log scale). In summary, all the expressions of the material length scales derived above are tabulated in Table 1 together with the results from other work [12] for comparison. The numerical difference in the coefficient f reflects the difference in deformation form and also the constraint of the deformation of the corresponding model. 4. Determination of ¦ It has been shown above that, with the characteristic material length scale, l, determined, the predicted mateTable 1 Expressions for the characteristic material length scale l for different deformations Indentation
a
G |ref
Pure shear
Linear averagea
Weighted average 5h 2b
Pure bending
2
4.5h 2b
G |ref
2
3h 2b
G |ref
Same as the result of Nix and Gao [12].
2
h 2b
G ~ref
2
I
II
12.5 31 0.19
25 46 0.24
III 50 71 0.28
Data from Ref. [14].
rial behavior is strongly dependent on v. The parameter v was introduced before in two parallel ways from Eqs. (1) and (6). The difference in the implication between the two equations is that while Eq. (1) defines a way for pure geometrical measurement of the strain, Eq. (6) reflects the physical process of dislocation interaction. Thus, in Eq. (6), v should be a material parameter that is related to the material microstructure. In fact, since the measure of the overall strain in Eq. (1) is in the sense of strain energy, there is rationale for v also to be argued as a material parameter. With the models and relations of material behavior having been established in Eqs. (21), (27) and (31), it is now possible to determine the values of v from corresponding experimental data. Because of the complex relations between v and the normalized material response, it seems hard to use curve fitting to experimental data to determine the value of v. Nevertheless, curve matching is used in the following to make an estimation for our discussion. Experimental data have been presented [14] of both tensile deformation and bending deformation on polycrystalline nickel foils with thicknesses from 12.5 to 50 mm. It was reported in the paper that, to calculate the material length scale l, using v=1 in Eq. (1) provides a better fit to the micro-bend test data than using v=2. It has been pointed out in previous sections that v should generally be less than 1 in a sense of dislocation interaction; using v\ 1 simply under estimates the total dislocation density. By using the same experimental data and curve matching, values of v have been obtained in this paper for the materials of nickel with different thickness and grain size. The results are tabulated in Table 2. It is interesting to note that the values of v have a strong correlation with grain size or foil thickness. This correlation is not a possible result of numerical error since the load response is very sensitive to the value of v in this range as shown in Fig. 5. Fleck et al. [5] have presented experimental data for torsion of polycrystalline copper wires ranging in diameter from 12 to 170 mm. The grain sizes are between 5 and 25 mm, the larger diameter wires having the larger grain sizes. The corresponding tensile test results have also been presented [5] for each diameter of the wires. A very small size effect is observed on the tensile curves for wires of diameters from 12 to 30 mm but the strength for the wire of 170 mm diameter is generally
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about 10–15% lower than the others. Again, the values of v were obtained by matching the experimental data. The values of parameters used for the calculations are h =0.3, G =42 GPa, N = 0.2, ~ref = 120 MPa for the diameter of 170 mm and ~ref =130 MPa for other diameters. Details of the results are shown in Table 3. The results reveal again a correlation between v and grain size or wire thickness. This correlation holds if values of the above parameters change.
by statistical process. In like manner, ‘extra’ dislocations are created at the film/substrate interface in the thin film problem because of the constraint of the substrate. Thus there is rationale for us to argue that both the cases involve dislocation interaction and v is probably a function of grain size. Investigation on this argument deserve further attention.
6. Conclusions 5. Discussion As expected, dislocation interaction exists in deformation and v is less than 1, even as low as 0.19 for nickel bending. This is substantiated more clearly by checking the torsion responses of copper wires by Fleck et al. [5]. The torsion test curve for the diameter of 12 mm is more than twice as high as that for diameter 30 mm in value at every surface strain level. This observation cannot be explained as solely an effect of strain gradient since the strain gradient for the 12-mm wire is a little more than twice that of the 30-mm wire; otherwise the torsion curves would be almost purely due to the strain gradient and the strain and its hardening would contribute nothing to deformation resistance. This is obviously not the case and the only possible reason is the grain size effect of dislocation interaction. It is also interesting to note that v is strongly dependent on the grain size or thickness with a lower v for a smaller grain size. This behavior is hard to explain with the well known Hall –Petch effect because this effect is reflected in the tensile strength without strain gradient. It has also been reported that the grain-size effect on the strength is enhanced when the material exists in the form of a thin layer on an elastic substrate [15]. As it was argued [12] that though the thin film strength problem is not a strain gradient problem as in some cases there is no strain gradient, the two problems do have some characteristics in common. In both cases one has ‘extra’ dislocation storage as a consequence of the constraints on the plastic deformation. In the strain gradient problem the geometrically necessary dislocations are in excess of the ones that would form
Table 3 v values for five different copper wires in torsion
Diameter (mm)a Grain size (mm)a v a
I
II
III
IV
12 5 0.32
15
20 0.7
30
Data from Ref. [5].
0.4
0.8
From the above theoretical analysis and the discussion on experimental results the following conclusions can be drawn. The concept of geometrically necessary dislocation provides an excellent description of strain gradients and an effective means of deriving the characteristic material length scale, l. Models of three basic deformations, indentation, bending and torsion, have been proposed and the corresponding expression of the associated characteristic material length scale, l, has been derived. The expression of l has a unique form with different coefficient values for different deformations. l is a function of fundamental parameters of materials and is independent of strain levels and dislocation interaction. The values of l from Eq. (12) with h equal 0.3 as suggested by Ashby [7], are about 1.5 mm or less which is lower than the results of 5.84 –12 mm of Nix and Gao [12] for copper indentation, 3.7 mm of Fleck and Hutchinson [5] for copper torsion and 3–6 mm of Stolken [14] for nickel bending. The value of 1.5 mm or less is in accordance with the observation that size effect appears at about 1 mm or sub-micrometer levels as observed in indentation tests. There are some common features between the strain gradient plasticity theory derived from dislocation theory and the phenomenological strain gradient theory proposed by Fleck and Hutchinson [1]. v in Eq. (1) has a physical meaning through Eq. (6), which reflects the degree of interaction between the statistically stored dislocation and the geometrically necessary dislocation. v is found to be less than 1 and is dependent on the grain size for polycrystalline materials, with a smaller grain size having a lower v for a given material, indicating a higher degree of dislocation interaction.
Acknowledgements V 170 25 0.83
The authors would like to thank Professors W. Nix and H. Gao at Stanford University for their discussion and comments. This project is supported by The National Science and Engineering Research Council of Canada and US National Science Foundation.
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
Appendix A. Power law relations without strain gradient
References
Assuming a power law relation for uniaxial deformation |o =|ref · m N
(A1)
For one-dimensional deformation, von Mises yielding relation becomes |o =2~o
(A2)
and strains have a relation my =mz = − 0.5mx k45o =
(A3)
mx − my sin 2q 45o +mxy · con 2q 45o =0.75mx 2
(A4)
where the x direction is along the uniaxial deformation direction and the slip planes are in the 45° direction. From the above equations, we have the power law relation ~0 = ~ref · k N;
~ref =
|ref 2 ×0.75N
(A5)
.
249
[1] N.A. Fleck, J.W. Hutchinson, J. Mech. Phys. Solids 41 (1993) 1825. [2] N.A. Stelmashenko, M.G. Walls, L.M. Brown, Y.V. Milman, Mechanical properties and deformation behavior of materials having ultra-fine microstructures, in: M. Nastasi, D.M. Parkin, H. Gleiter (Eds.), NATO ASI Series E 233, p. 605. [3] Q. Ma, D.R. Clark, J. Mater. Res. 10 (1995) 853. [4] W.J. Poole, M.F. Ashby, N.A. Fleck, Scripta Metall. Mater. 34 (1996) 559. [5] N.A. Fleck, G.M. Muller, M.F. Ashby, J.W. Hutchinson, Acta Metall. Mater. 42 (1994) 475. [6] N.A. Fleck, J.W. Hutchinson, in: J.W. Hutchinson, T.Y. Wu (Eds.), Advances in Applied Mechanics, Vol. 33. Academic Press, New York, 1997, p. 295. [7] M.F. Ashby, Phil. Mag. 21 (1970) 399. [8] M.F. Ashby, in: A. Kelly, R.B. Nicholson (Eds.), Strengthening Methods in Crystals, Elsevier, Amsterdam, 1971, p. 137. [9] J.F. Nye, Acta Metall. 1 (1953) 153. [10] A.H. Cottrell, The Mechanical Properties of Matter, Wiley, New York, 1964, p. 227. [11] E. Kroner, Appl. Mech. Rev. 15 (1962) 599. [12] W.D. Nix, H. Gao, J. Mech. Phys. Solids 46 (1998) 411. [13] F.R.N. Nabarro, Z.S. Basinski, D.B. Holt, Adv. Phys. 13 (1964) 193. [14] J.S. Stolken, Ph.D. Dissertation, Materials Department, University of California, Santa Barbara, CA, 1997. [15] W.D. Nix, Metall. Trans. A 20A (1988) 2217.
Length scale effect on mechanical behavior due to strain gradient plasticity D.-M. Duan b, N.Q. Wu a, W.S. Slaughter a, Scott X. Mao a,* a
Department of Mechanical Engineering, The Uni6ersity of Pittsburgh, 3700 O’Hara Street, Pittsburgh, PA 15261, USA b Department of Mechanical Engineering, The Uni6ersity of Calgary, Calgary, Alta, Canada Received 10 May 2000; received in revised form 15 September 2000
Abstract The characteristic material length scale l as introduced in the phenomenological theory of strain gradient plasticity by Fleck and Hutchinson [1] is crucial for the application of the theory. Three dislocation models are proposed in this paper for the derivation of the material length scales as there is a rational connection between dislocation theory and phenomenological theory. The length scale is determined as a function of material fundamental parameters and is independent of how the ‘overall effective strain’ is measured, or how the dislocations interact. The length scale is generally about 1.5 mm or less, agreeing with the observation that the size effect appears approximately at 1 mm and sub-micrometer levels for indentation deformations. The dislocation interaction is also discussed in the framework of strain gradient plasticity. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Strain gradient plasticity; Dislocation; Materials length scale
1. Introduction There is ample experimental evidence showing the existence of a strong size effect in the plastic flow of metals and ceramics. For example, the measured indentation hardness of metals and ceramics increases with decreasing indentation size by a factor of about two as the width decreased to 1 mm. This size effect is even more significant in the sub-micrometer region [2–4]. Another example of size effect was presented for the torsion response of copper wires [5] in which the scaled shear strength increases with diminishing wire diameter in the range 100 – 10 mm by almost a factor of three, while there is no evident size effect in the tensile data. Conventional theories of plasticity are unable to explain and predict these effects which in fact reflect some inherent properties of the materials. As discussed by Fleck and Hutchinson [1,5], these effects may not have the same explanation, but it is clear that all require a length scale for their interpretation. A natural way to include size effects in the constitutive law is to postulate * Corresponding author. Tel.: +1-412-6249602; fax: + 1-4126244846. E-mail address: [email protected] (S.X. Mao).
that the yield stress depends not only on the strain as in conventional plasticity theories but also on strain gradient. There have been attempts to include strain gradients within a plasticity formulation, notably those of Fleck and Hutchinson [1,6]. Fleck and Hutchinson introduced material length scales involving the concept of geometrically necessary dislocations. Consequently, as a material constitutive law concerning strain gradient effect, the theory must establish the characteristic material length scale that reflects a micro nature of the material. Dislocation theory provides a means for understanding the strain gradient effect in plastic hardening and the nature of characteristic material length scale. Dislocations become stored for two reasons: they accumulate by trapping each other in a random way or they are required for compatible deformation of various parts of the crystal. The randomly stored dislocations, which are created by homogeneous strain, are referred to as statistically stored dislocations [7,8]. The dislocations stored in the requirement for deformation compatibility, which are related to the curvature of the crystal lattice or to strain gradients, are called geometrically necessary dislocations [9,10]. The total hardening effect of a material should be a combined result of both the
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strain hardening and strain gradient hardening. Generally, the geometrically necessary dislocation density can be calculated for typical forms of deformation, and due to the close relation between the strain gradient and the geometrically necessary dislocation density [9,11], the strain gradient field can be obtained. This approach has been adopted by Nix and Gao [12] to characterize the size effect of indentation tests and to calculate the characteristic material length scale. This paper gives a further investigation of the characteristic material length scale of the strain gradient plasticity. It first reviews the strain gradient plasticity theory proposed by Fleck and Hutchinson [1,6] and then presents dislocation models for some basic forms of deformation to calculate the corresponding material length scales. The size effect results, predicted using the theory with use of the corresponding material length scale, are then presented and discussed. As a by product, some results of dislocation interaction associated with the strain gradient plasticity theory are presented and discussed.
2. Theoretical background In their phenomenological theory of strain gradient plasticity, Fleck and Hutchinson [1] constructed an ‘overall effective strain’: mcs = [m +l ] v e
v cs
v 1/v e
(1)
where me and e are the second von Mises invariant of strain and the analogous second invariant of the curvature: me =
' '
e =
2 mijmij 3
(2)
2 3 ij ij
(3a)
where the curvature is the spatial gradient of the material rotation q: ij =qi, j
(3b)
Here the material length scale lcs has been introduced as required for dimensional consistency, and the subscript ‘cs’ indicates that the equation is derived from ‘couple stress’ theory where only rotation gradient is included. A more general ‘combined strain quantity’ is also derived by Fleck and Hutchinson [6] which takes both rotation and stretch gradient into account as follows: m=
2 m% m% 3 ij ij
v/2
(1) (1) v/2 (2) (2) v/2 + (l 21p%(1) +(l 22p%(2) ijk p%ijk p%ijk ) ijk p%ijk p%ijk )
n
(3) (3) v/2 +(l 23p%(3) ijk p%ijk p%ijk )
1/v
(4)
where (%) denotes a quantity derived from an incompressible displacement field; p is the strain gradient tensor and has a unique orthogonal decomposition of: p% =p%(1) + p%(2) + p%(3)
(5)
Instead of introducing a single material length scale in Eq. (1), three length scales are introduced in Eq. (4). The couple stress theory expressed in Eq. (1) is a special case of the general theory in Eq. (4) when only curvature invariant is taken into account. Both theories assume that the strain energy w depends only upon the single scalar strain measure m while the dependence remains to be specified. The parameter v needs to be specified, which reflects the way in which the ‘overall effective strain’ is measured. Fleck and Hutchinson [6] pointed out, when Eq. (1) is used to deduce the value for the material length scale l with experimental data, the magnitude of l is sensitive to the value adopted for v. They suggested that v be chosen from 1 to 2 and observed numerical difficulties in the solution process for values close to 1. A parallel discussion for the definition of the ‘overall effective strain’ is the argument of the interaction between the statistically stored dislocation density and the geometrically necessary dislocation density. According to Eq. (1) the effective total dislocation density can be expressed as: z vT = z vS + z vG
(6)
Dividing the dislocation into two parts and then combining them in different ways implies changes in the strain history. Ashby [7] has pointed out that in general the presence of geometrically necessary dislocations will accelerate the rate of statistical storage and a linear sum of the two (v= 1 in Eq. (6)) only provides a lower limit of the effective total dislocation density. Thus, it is argued here that, to properly estimate the total dislocation density, v should be equal or less than 1 so that a higher total dislocation density can be obtained. It is assumed that the flow stress for a single-slip system of a single crystal depends upon the total dislocation density. With Eq. (6), the Taylor relation is expressed as: ~= hGb zT
(7)
where G is the shear modulus, b is the magnitude of Burger’s vector and h is a constant taken to be about 0.2 for fcc crystal metals and about 0.4 for bcc metals [13] and was taken to be 0.3 by Ashby [7]. By applying von Mises flow rule, |= 3~, and Eq. (6), we have: |= 3hGb(z vS + z vG)1/2v
(8)
The geometrically necessary dislocation density, zG, is closely related to strain gradient , and generally, if the higher order of strain gradient is neglected, can be expressed as:
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
zG =f ·
b
(9)
where the coefficient f is a function of higher order of strain gradient and for some special cases with constant strain gradient field, f is constant. Thus by considering the usual power hardening law in the absence of a strain gradient: |o =|refm N
(10a)
and noting |o = 3hGb zs
(10b)
Eq. (8) becomes | |ref
2
=[(m 2N)v +l v v]1/v
G |ref
(12)
Here one sees that the characteristic material length scale l is expressed in fundamental material parameters and is independent of v. A counterpart to Eq. (11) can be derived from Eq. (1). Assuming a power law relation between the overall effective strain and the effective stress:
S So
2
= m 2N =[m ve +l vcs ve ]2N/v
dV =2yr(c+ d)dr
(15)
where a is the radius of the contact area, b is the magnitude of Burger’s vector and c is the indentation deformation. The geometrically necessary dislocation density is therefore: zG =
du tan2 q = ; dV bh 1 − t 2
t= r/a
(16)
The vertical stretch strain at r is m=
−c =− c+ d
'
1− t · tan q 1+ t
(17)
(13)
the corresponding normalized effective stress can be expressed as:
dr h = 2yr dr; s ba
(11)
2
S=Som N
based on the dislocation theory. In their model, a constant strain gradient field is assumed in the defined hemispherical volume under the indenter. The model is modified here in such a way that the strain gradient is not a constant but varies with locations. The strain gradient field is derived directly from the strain field which is calculated from the geometry of the defined deformation volume. In the modified model shown in Fig. 1, considering a cylinder element with a thickness dr at position r, the total length of the injected loops in the element and the element volume are as follows: du = 2yr
where the material length scale l is l= 3h 2bf
243
(14)
and the strain gradient field is =
1 dm 1 dm tan2 q · = · = h dr a dt (1+t) 1 − t 2
(18)
Nix and Gao [12] tried to compare Eq. (11) with v = 1 in their case and Eq. (14), and concluded that the two equations match each other only under certain conditions. Note that the left sides of the two equations are different where |ref is a coefficient in the power law stress –strain relation without strain gradient while _o is the coefficient in an equi6alent power law relation with strain gradient taken into account. Exact match of the two theories is not possible due to some intrinsic incompatibility problems.
3. The models From Eqs. (9) and (12), the characteristic material length scale l can be obtained if the factor f is determined. In this section, dislocation models are proposed for three basic deformation forms to derive the corresponding geometrically necessary dislocation density zG and, thereafter, the factor f for each case.
3.1. A modified indentation model Nix and Gao [12] have proposed an indentation model
Fig. 1. The dislocation model for indentation deformation. The dislocation structure is idealized as circular dislocation loops and the deformation is assumed to be confined in a hemispherical volume.
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size-dependent hardness curves are plotted with various v values ranging from 0.2 to 1.0. The hardness is normalized by the reference stress and the indentation depth is normalized by the characteristic material length scale l. It is evident that while the v effect is significant especially at lower v values, all the curves show a strong size effect. The characteristic material length scale plays a key role in the size effect where, with the size fixed, a larger l corresponds to a higher value of hardness, and with l decreasing to zero all the curves approach a limit which is the result of conventional plasticity.
3.2. A bending model Fig. 2. Normalized strain gradient distribution for indentation deformation (Eq. (18)) compared with the result of Nix and Gao [12]. Strain gradient is normalized by 0 = tan q/a as = /0.
Eqs. (16) and (18) give the expression for f in Eq. (9) as: f = 1+t
(19)
It is clear that f is not a constant but is dependent on location. Nix and Gao [12] obtained a value of 1.5 for f from their model which is exactly the average value from Eq. (19). Since the final measured hardness is an average of integration of the stress over all the contact area, it is preferred here to use a weighed average value of f determined through the following equation: −
f=
1 y
&
1
f · 2yt dt =
0
5 3
(20)
By choosing Tabor’s factor of three, the final expression showing size effect for hardness can be derived from Eq. (11) as follows: H =3|= 6|ref
&
1
{ f vo(t) + [f1(t)/h]v}1/2vt · dt
A dislocation model for bending has been proposed in Ref. [9] and the geometrical necessary dislocation density has been calculated [8]. Here the dislocation density is derived in a more straightforward way. The model is shown in Fig. 4a where it is assumed that any cross section remains plane after deformation, slip planes are along with the beam length direction and there is no transverse shear effect on the deformation (pure bending). For the sake of easy calculation, Fig. 4a can be simplified as Fig. 4b for a half beam. From the relation: tanq =
b s
(23)
the total length of dislocation lines is: u=
2hB tan q = 2hB s b
(24)
where B is the thickness of the beam. Then the geometrically necessary dislocation density is
(21)
0
where fo(t)=
1−t 1+t
f1(t)=3h 2b
N
· tan2N q;
G |ref
2
·
tan2 q
(22)
1 −t 2 The major difference between the current model and the model proposed by Nix and Gao [12] is that a deformation zone with a non-uniform strain gradient field is introduced in the current model. This difference is shown in Fig. 2. The strain gradient is normalized by an overall measure of 0 =tan q/a. The current model reveals a strain gradient singularity at the boundary of the contact area. This is natural because immediately outside the contact area, the plastic strain is zero. The hardness H predicted from Eq. (21) is dependent on the value of v. In Fig. 3 a group of
Fig. 3. Normalized hardness versus normalized indentation depth with different values of v. Other parameter values used are: N= 0.2, h =0.3, tan q =0.358.
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Fig. 4. The dislocation model for pure bending deformation. Slip planes are assumed to be along the beam length direction, no transverse shear effect is considered; (a) is simplified as (b) for a half beam.
u u 2 tan q zG = = = V hLB bL
(25)
For small q, Eq. (25) can be written as zG =
2q s = (zb) − 1 = bL b
(26)
where s is the curvature of the beam. Since in pure bending, the stretch strain gradient is equal to the curvature, the value of the factor f in Eq. (9) is unity. Finally, the bending moment expression showing the size effect is obtained as follows: M0 =
1 M = 2 4Bh |ref 2
&
1
therefore leads to higher curves of bending moment. When h reaches unity, the value of the normalized moment response at every surface strain level is more than twice that without strain gradient effect.
3.3. A torsion model Unlike the indentation model and the bending model where the deformation is modeled using edge dislocations, the deformation of a round bar under torsion is,
[(m0 · t)2Nv +(l · m0/h)v]1/2v · t dt
0
(27) where m0 is the surface strain of the beam and l is the material length scale. Again, for a given material for which l is fixed, the normalized bending moment is a function of both the size of the beam, h, and v. These effects can be seen from Figs. 5 and 6. The load –strain curves with various v values shown in Fig. 5 indicate that an underestimate will be predicted for the deformation behavior if a higher value of v is chosen than the actual one of the material. More discussion of this will be given in later sections. The normalized load – strain curves in Fig. 6, where the beam thickness is normalized by the characteristic material length scale l, also show the role of l in the size effect. For a fixed beam thickness, increasing l will make h decrease and
Fig. 5. The effect of v on the load– strain curve for bending deformation. Parameter values used are: N =0.5, h= 0.3, h/l= 4.
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
246
Fig. 6. Size effect on the load–strain curve for bending deformation. The half beam thickness h is normalized by the characteristic material length scale l, h = h/l. Other parameter values used are N= 0.5, h = 0.3 and v =0.5.
by nature, due to screw dislocations. As shown in Fig. 7a, the following assumptions are made: the round bar deforms uniformly along its length, cross sections remain planes after deformation and the deformation is achieved through screw dislocations of which the slip planes are in accordance with cross sections. Consider the displacement of a point at r in a unit length of the bar in Fig. 7b. The total relative displacement d is an accumulation of Burger’s vector: d=
q ·r =q · r L
(28)
where q is the rotation angle per unit length which represents the shear strain gradient in torsion. The number of Burger’s vectors per unit length of the bar at r is
Fig. 8. Size effect on the load– strain curve for torsion deformation. The radius R is normalized by the characteristic material length scale l, R =R/l. Other parameter values used are: v= 0.7, N= 0.2 and h = 0.3.
d q ·r N= = b b
(29)
In another way, the dislocation number N is also an accumulation of screw dislocation from the center of the bar to the cylinder at r. The relation in Eq. (29) keeps the bar deformation in such a way that any diameter in the bar remains straight after deformation. Dislocation density is therefore given as the increment of dislocation number per unit thickness of the cylinder. Thus the density of geometrically necessary dislocation is: zG =
dN q = = dr b b
(30)
Again the value of the factor f in Eq. (9) is obtained as unity. Using Eqs. (10a) and (10b), the normalized torque is: T=
T = 2y 3 R ~ref
&
1
t 2{k v0 (t)+ [k1/R]v}1/2v dt
(31)
(32)
0
where k0(t)= (k0 · t)2N;
k1 = h 2b
G ~ref
2
· k0 = l · k0
and k0 is the surface strain of the bar. The reference shear stress ~ref in Eq. (32) has a relation with |ref in Eqs. (10a) and (10b) as follows: ~ref = |ref/(2× 0.75N)
Fig. 7. The dislocation model for torsion deformation. (a) A side view indicating the deformation assumptions, and (b) the displacement is assumed to be an accumulation of screw dislocation.
(see Appendix A). Eq. (31) is plotted in Fig. 8 with v= 0.7 and N= 0.2. In the same way as in Fig. 6 the radius of the bar is normalized by the characteristic material length scale, l. The figure reveals basically the same features as in Fig. 6 for bending deformation and indicates that when a torsion bar with a radius more than 30 times the characteristic material length scale l,
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Table 2 v values for three different nickel foils in bending
Thickness (mm)a Grain size (mm)a v a
Fig. 9. The normalized torque as a function of v plotted at different surface strain levels for torsion deformation. Other parameter values used are: N =0.2, h= 0.3, and normalized radius R = R/l= 1.
the size effect is negligible. This again confirms the role of the characteristic material length scale l in microscale plastic deformations. Instead of plotting the whole deformation curves for v effect investigation as in Figs. 3 and 5, the normalized torque is plotted here directly against v in Fig. 9 at different surface strain levels to reveal the enhanced effect in the region of lower v values. It is apparent that the torque response monotonically increases with decreasing v and the increasing rate is higher for a higher surface strain level (note the normalized torque is in a log scale). In summary, all the expressions of the material length scales derived above are tabulated in Table 1 together with the results from other work [12] for comparison. The numerical difference in the coefficient f reflects the difference in deformation form and also the constraint of the deformation of the corresponding model. 4. Determination of ¦ It has been shown above that, with the characteristic material length scale, l, determined, the predicted mateTable 1 Expressions for the characteristic material length scale l for different deformations Indentation
a
G |ref
Pure shear
Linear averagea
Weighted average 5h 2b
Pure bending
2
4.5h 2b
G |ref
2
3h 2b
G |ref
Same as the result of Nix and Gao [12].
2
h 2b
G ~ref
2
I
II
12.5 31 0.19
25 46 0.24
III 50 71 0.28
Data from Ref. [14].
rial behavior is strongly dependent on v. The parameter v was introduced before in two parallel ways from Eqs. (1) and (6). The difference in the implication between the two equations is that while Eq. (1) defines a way for pure geometrical measurement of the strain, Eq. (6) reflects the physical process of dislocation interaction. Thus, in Eq. (6), v should be a material parameter that is related to the material microstructure. In fact, since the measure of the overall strain in Eq. (1) is in the sense of strain energy, there is rationale for v also to be argued as a material parameter. With the models and relations of material behavior having been established in Eqs. (21), (27) and (31), it is now possible to determine the values of v from corresponding experimental data. Because of the complex relations between v and the normalized material response, it seems hard to use curve fitting to experimental data to determine the value of v. Nevertheless, curve matching is used in the following to make an estimation for our discussion. Experimental data have been presented [14] of both tensile deformation and bending deformation on polycrystalline nickel foils with thicknesses from 12.5 to 50 mm. It was reported in the paper that, to calculate the material length scale l, using v=1 in Eq. (1) provides a better fit to the micro-bend test data than using v=2. It has been pointed out in previous sections that v should generally be less than 1 in a sense of dislocation interaction; using v\ 1 simply under estimates the total dislocation density. By using the same experimental data and curve matching, values of v have been obtained in this paper for the materials of nickel with different thickness and grain size. The results are tabulated in Table 2. It is interesting to note that the values of v have a strong correlation with grain size or foil thickness. This correlation is not a possible result of numerical error since the load response is very sensitive to the value of v in this range as shown in Fig. 5. Fleck et al. [5] have presented experimental data for torsion of polycrystalline copper wires ranging in diameter from 12 to 170 mm. The grain sizes are between 5 and 25 mm, the larger diameter wires having the larger grain sizes. The corresponding tensile test results have also been presented [5] for each diameter of the wires. A very small size effect is observed on the tensile curves for wires of diameters from 12 to 30 mm but the strength for the wire of 170 mm diameter is generally
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about 10–15% lower than the others. Again, the values of v were obtained by matching the experimental data. The values of parameters used for the calculations are h =0.3, G =42 GPa, N = 0.2, ~ref = 120 MPa for the diameter of 170 mm and ~ref =130 MPa for other diameters. Details of the results are shown in Table 3. The results reveal again a correlation between v and grain size or wire thickness. This correlation holds if values of the above parameters change.
by statistical process. In like manner, ‘extra’ dislocations are created at the film/substrate interface in the thin film problem because of the constraint of the substrate. Thus there is rationale for us to argue that both the cases involve dislocation interaction and v is probably a function of grain size. Investigation on this argument deserve further attention.
6. Conclusions 5. Discussion As expected, dislocation interaction exists in deformation and v is less than 1, even as low as 0.19 for nickel bending. This is substantiated more clearly by checking the torsion responses of copper wires by Fleck et al. [5]. The torsion test curve for the diameter of 12 mm is more than twice as high as that for diameter 30 mm in value at every surface strain level. This observation cannot be explained as solely an effect of strain gradient since the strain gradient for the 12-mm wire is a little more than twice that of the 30-mm wire; otherwise the torsion curves would be almost purely due to the strain gradient and the strain and its hardening would contribute nothing to deformation resistance. This is obviously not the case and the only possible reason is the grain size effect of dislocation interaction. It is also interesting to note that v is strongly dependent on the grain size or thickness with a lower v for a smaller grain size. This behavior is hard to explain with the well known Hall –Petch effect because this effect is reflected in the tensile strength without strain gradient. It has also been reported that the grain-size effect on the strength is enhanced when the material exists in the form of a thin layer on an elastic substrate [15]. As it was argued [12] that though the thin film strength problem is not a strain gradient problem as in some cases there is no strain gradient, the two problems do have some characteristics in common. In both cases one has ‘extra’ dislocation storage as a consequence of the constraints on the plastic deformation. In the strain gradient problem the geometrically necessary dislocations are in excess of the ones that would form
Table 3 v values for five different copper wires in torsion
Diameter (mm)a Grain size (mm)a v a
I
II
III
IV
12 5 0.32
15
20 0.7
30
Data from Ref. [5].
0.4
0.8
From the above theoretical analysis and the discussion on experimental results the following conclusions can be drawn. The concept of geometrically necessary dislocation provides an excellent description of strain gradients and an effective means of deriving the characteristic material length scale, l. Models of three basic deformations, indentation, bending and torsion, have been proposed and the corresponding expression of the associated characteristic material length scale, l, has been derived. The expression of l has a unique form with different coefficient values for different deformations. l is a function of fundamental parameters of materials and is independent of strain levels and dislocation interaction. The values of l from Eq. (12) with h equal 0.3 as suggested by Ashby [7], are about 1.5 mm or less which is lower than the results of 5.84 –12 mm of Nix and Gao [12] for copper indentation, 3.7 mm of Fleck and Hutchinson [5] for copper torsion and 3–6 mm of Stolken [14] for nickel bending. The value of 1.5 mm or less is in accordance with the observation that size effect appears at about 1 mm or sub-micrometer levels as observed in indentation tests. There are some common features between the strain gradient plasticity theory derived from dislocation theory and the phenomenological strain gradient theory proposed by Fleck and Hutchinson [1]. v in Eq. (1) has a physical meaning through Eq. (6), which reflects the degree of interaction between the statistically stored dislocation and the geometrically necessary dislocation. v is found to be less than 1 and is dependent on the grain size for polycrystalline materials, with a smaller grain size having a lower v for a given material, indicating a higher degree of dislocation interaction.
Acknowledgements V 170 25 0.83
The authors would like to thank Professors W. Nix and H. Gao at Stanford University for their discussion and comments. This project is supported by The National Science and Engineering Research Council of Canada and US National Science Foundation.
D.-M. Duan et al. / Materials Science and Engineering A303 (2001) 241–249
Appendix A. Power law relations without strain gradient
References
Assuming a power law relation for uniaxial deformation |o =|ref · m N
(A1)
For one-dimensional deformation, von Mises yielding relation becomes |o =2~o
(A2)
and strains have a relation my =mz = − 0.5mx k45o =
(A3)
mx − my sin 2q 45o +mxy · con 2q 45o =0.75mx 2
(A4)
where the x direction is along the uniaxial deformation direction and the slip planes are in the 45° direction. From the above equations, we have the power law relation ~0 = ~ref · k N;
~ref =
|ref 2 ×0.75N
(A5)
.
249
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