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A line-tension model of dislocation networks on several slip planes
Mechanics of Materials xxx (2015) xxx–xxx
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A line-tension model of dislocation networks on several slip planes Sergio Conti ⇑, Peter Gladbach Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
a r t i c l e
i n f o
Article history: Received 26 September 2014 Received in revised form 3 December 2014 Available online xxxx Keywords: Dislocation Peierls–Nabarro model Crystal plasticity Relaxation
a b s t r a c t Dislocations in crystals can be studied by a Peierls–Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing, and for dislocations restricted to planes, we show that it reduces to a line-tension model, with an energy depending on the local orientation and Burgers vector of the dislocation. This model predicts, for specific geometries, spontaneous formation of microstructure, in the sense that straight dislocations are unstable towards a zig-zag pattern. Coupling between dislocations in different planes can lead to microstructures over several length scales. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Dislocations are topological defects in crystals, which are responsible for plastic deformation. They can be interpreted as discontinuity lines of the plastic slip. Geometrically, a dislocation may be seen as an oriented line with a multiplicity, which corresponds to the Burgers vector. The Burgers vector is a lattice vector of the crystal and is a conserved quantity, in the sense that it is constant on the parts of the dislocation line which have no intersections, and at intersection points the sum of the Burgers vectors of the dislocations entering must equal the sum of the Burgers vectors of the dislocations exiting the point. For a more detailed description we refer for example to Hirth and Lothe (1968), Hull and Bacon (1984), Bacon et al. (1980) and Miura (1987). Dislocations are accompanied by long-range elastic stresses. To make this precise, we consider the variational problem of linear elasticity
Z X
1 Crs u rs u dx þ body forces þ surface tractions 2
⇑ Corresponding author.
ð1Þ
with suitable boundary conditions. Here X R3 is the reference configuration, u : X ! R3 is the (possibly discontinuous) elastic displacement, C the matrix of elastic coefficients, and rs u ¼ ðru þ ruT Þ=2 the elastic strain. Assuming for simplicity that plastic slip only takes place on the plane fx3 ¼ 0g, and denoting by v ðx1 ; x2 Þ the slip at the point ðx1 ; x2 ; 0Þ, we require u to be continuous away from the plane and the limits of u on both sides of the plane to differ by v,
uþ ðx1 ; x2 ; 0Þ u ðx1 ; x2 ; 0Þ ¼ v ðx1 ; x2 Þ: þ
ð2Þ
Here u and u denote the one-sided limits, taken on x3 > 0 and x3 < 0 respectively. A simple situation, corresponding to a straight dislocation with Burgers vector b along the x1 -axis, would have v ðx1 ; x2 Þ ¼ 0 for x2 < 0 and v ðx1 ; x2 Þ ¼ eb for x2 P 0. Here e represents the lattice spacing. The elastic strain rs u necessarily diverges at least as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ejbj= x22 þ x23 around the dislocation, and thus cannot be square integrable. In other words, all displacements u which obey the given jump condition have infinite elastic energy. The singularity in the strain field is normally regularized at the atomic scale e, at which the continuum elasticity approach loses validity. For example, one can
http://dx.doi.org/10.1016/j.mechmat.2015.01.013 0167-6636/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
truncate the energy in the core region removing a cylinder of radius e around the dislocation line from the domain. Alternatively, one can let the slip v pass continuously from 0 to eb over a length of order e. In both cases the elastic energy scales as e2 lnð1=eÞ, see for example (Hirth and Lothe, 1968). In this paper we make this computation quantitative and rigorous, deriving a model for the line-tension energy of dislocations, in the more general case that dislocations are contained in several parallel planes and without restricting to straight lines. One main finding will be that certain dislocations are unstable towards formation of oscillations. This will require replacing the line-tension model by its relaxation, as explained below. To illustrate our result we need to introduce a more precise notation. We denote the elastic energy by
E3D ½u; X ¼
Z X
1 Crs u rs u dx 2
ð3Þ
and characterize a distribution of dislocations by a set of
case was then studied in Cacace and Garroni (2009) and Conti et al. (2011), an extension to three dimensions was initiated in Conti et al. (in press a,b). A general discrete model for dislocations was introduced in Ariza and Ortiz (2005), its mathematical analysis is however not yet complete and in particular convergence to line-tension models has not yet been proven. In the simple case of screw dislocations in two dimensions both a discrete model (Ponsiglione, 2007) and a nonlinear regularization with subquadratic energy (Scardia and Zeppieri, 2012) have been shown to lead to the same e2 lnð1=eÞ scaling of the energy. This paper is organized as follows. In Section 2 we introduce our model for dislocations in a single plane, in Section 3 we study straight dislocations, in Section 4 we discuss the abstract theory of relaxation for dislocations, in Section 5 we extend the analysis to multiple planes, and finally in Section 6 we discuss some examples and predictions of our theory. 2. Peierls–Nabarro model
i
curves ci , with tangent vectors ti and Burgers vectors b . A line-tension model then takes the form i
ELT ½ðci ; t i ; b Þi ; X ¼
XZ i
uðbi ; ti Þdci
ð4Þ
X
where u is the line-tension energy, depending on the Burgi
.
ers vector b and the orientation t i of the dislocation, and dc denotes integration over the curve ci ; dci ¼ dH1 ci (in (4) we only integrate over the part inside X). We shall show that, in the limit of small lattice spacing e ! 0, and if the displacement field u corresponds to the dislocation i
i
distribution ðci ; t i ; b Þ, the elastic energy E3D ½u; Xe reduces, i
to leading order, to e2 lnð1=eÞELT ½ðci ; t i ; b Þi ; X for a suitable u. Here Xe denotes the set X minus a tube of radius e around the union of the dislocation lines ci . Let us now address the line-tension energy u entering (4). A straightforward computation, based on the assumption that dislocations are straight, infinite lines, gives an expression for the line-tension energy, which we call u0 , in terms of the elastic constants C of the crystal, as given for example in Hirth and Lothe (1968), see also Section 3 below. We shall show below that u – u0 . The difference can be understood on the basis of relaxation theory. Depending on the geometry, it may be energy-efficient for dislocations to spontaneously develop microstructures, which reduce the total energy. The integrand u entering (4) should then take into account that possibility, and indeed the appropriate function is the one that minimizes over all possible microstructures. This decay is related to the decay of dislocations into Shockley partials, but in our setting only ‘‘full’’ dislocations are present and there are no stacking faults involved. We shall use a Peierls–Nabarro-type model, as developed by Ortiz (1999), Koslowski et al. (2002) and Koslowski and Ortiz (2004) for the case of dislocations on a given slip plane, and generalize it to the case that more slip planes are active. Mathematically, the analysis of these models was initiated in the scalar case by Garroni and Müller (2005) and Garroni and Müller (2006), the vectorial
We start from the two-dimensional extension of the Peierls–Nabarro model that was introduced in Ortiz (1999), Koslowski et al. (2002) and Koslowski and Ortiz (2004). We denote by B the lattice of (normalized) Burgers vectors, which in the simple cases coincides with ZN ; N ¼ 2 or 3, and is assumed to be a subset of R3 (extending the vectors by 0 in the third component if B ¼ Z2 ). We assume for simplicity the reference configuration to have the form X ¼ x ðH; HÞ R3 , where H > 0 and x R2 is bounded, connected, with Lipschitz boundary. The key variable is the plastic slip v e : x ! RN . The local energy term penalizes the deviations of the plastic slip from the set of Burgers vectors, which correspond to the distance of v e from eB. This allows for smooth transitions from one Burgers vector to another. Adding the elastic energy E3D we obtain the functional
1
e
Z x
2
dist ðv e ; eBÞdx þ minfE3D ½u; X : uþ u ¼ v e g: u
ð5Þ
Here uþ u ¼ v e means that the jump of u over the plane x3 ¼ 0 equals v e , in the sense specified in (2). The elastic displacement u : X ! R3 is chosen as the one with the smallest elastic energy, among all those with the appropriate jump. The minimization in u in (5) is a linear problem, that can be solved explicitly if one knows the Green’s function of the relevant domain. If the domain is large, one can replace the Green’s function with the fundamental solution (corresponding to the case x ¼ R2 ; H ¼ 1), up to boundary terms which do not modify the leading-order behavior of the energy. One then replaces (5) by the energy
F e ½v ; x ¼
1
Z
2
dist ðv ; eBÞ
eZ x Z
þ
x
x
ðv i ðxÞ v i ðyÞÞCij ðx yÞðv j ðxÞ v j ðyÞÞdxdy ð6Þ
where we implicitly sum over i; j ¼ 1; . . . ; N; N being the dimension of the linear space spanned by B. The kernel
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
C : R2 ! RNN sym can be determined explicitly in Fourier
where ci are oriented Lipschitz curves in X with tangent ti
space from the matrix of elastic constants C (Koslowski and Ortiz, 2004) and has the form
and Burgers vector b 2 ZN . These measures are necessarily divergence-free, in the sense of distributions. We characterize the class of all dislocations in X as
CðzÞ ¼
1 ^ z C jzj3 jzj3
! ð7Þ
i
DislðXÞ ¼ fl ¼
X i b t i dci 2 MðX;RN3 Þjdiv l ¼ 0g:
ð13Þ
i
^ ðsÞ is uniformly bounded and positive definite, in where C the sense that there is c > 0 such that
1 2 ^ ðsÞn 6 cjnj2 jnj 6 n C c
for all n 2 RN ;
s 2 S1 :
ð8Þ
The extension to the case of several planes will be discussed in Section 5 below.
The divergence-free constraint translates to the fact that the Burgers vector is constant on each curve and obeys a balance condition at the endpoints of the curves ci : letting I ðzÞ be all indices of curves ending at z and Iþ ðzÞ all indices of curves starting at z; div l ¼ 0 if and only if
X i2I ðzÞ
3. Small-e limit: straight dislocations We consider a straight dislocation with Burgers vector
i
b ¼
X
b
j
ð14Þ
þ
j2I ðzÞ
at all endpoints z 2 X. The set DislðxÞ correspondingly is the set of dislocations localized in a two-dimensional set
b 2 B and orientation t 2 S1 , so that the dislocation line c is a straight line through the origin parallel to t; c ¼ tR R2 . We set, for x 2 R2 ,
x. We denote by E0LT the line-tension energy defined as in (4) with the energy density u0 from (10), which can be seen as a functional on the space DislðxÞ. If a dislocation
8 > <0 v e ðxÞ ¼ > bx t? : be
is completely contained within the ðe1 ; e2 Þ plane, the rotatP i ? ed measure l? ¼ i b ðti Þ dci is curl-free and consequently the gradient of a slip field v 2 SBVðx; BÞ. This
if
x t? < 0;
if if
0 6 x t? 6 e; x t? > e;
ð9Þ
where t? 2 R2 is the vector obtained rotating t counterclockwise by 90 degrees. To compute the energy per unit length of the dislocation we focus on the part contained in the ball of diameter one, B1=2 ¼ fx 2 R2 : jxj < 1=2g. The energy F e ½v e ; B1=2 of this segment of dislocation, which has unit length, is proportional to e2 lnð1=eÞ for e ! 0, as explained in the introduction. We consequently define
u0 ðb; tÞ ¼ lim
1
v
F e ½ e ; B1=2 : e!0 e2 ln 1 e
ð10Þ
The structure of C given in (7) permits a simpler representation for this limit (see, e.g., Conti et al. (2011, Lemma 5.1)),
u0 ðb; tÞ ¼ 2
Z
bi Cij ðt ? þ tsÞbj ds;
ð11Þ
R
where t ? denotes a unit vector orthogonal to t. Physically, the function u0 is the energy of a straight dislocation, which is not allowed to build oscillations. It can be shown that it is the same as obtained in classical theories by solving the equilibrium equations in R3 and computing the limit of E3D ½u; C e =ðe2 lnð1=eÞÞ where C e is a cylinder with axis along t and unit height, whose cross-section is a circular corona obtained taking away a core region of radius e from a circle of radius 1, see for example (Hirth and Lothe, 1968 and Conti et al., in press b). 4. Curved dislocations, relaxation A dislocation network in a reference configuration
X R3 is a RN3 -valued measure l 2 MðX; RN3 Þ of the form
l¼
X i b ti dci ; i
ð12Þ
identification and the L1 convergence of the slip fields gives the natural convergence concept for dislocation networks. Minimization of the functional E0LT leads, for some boundary conditions, to the formation of fine-scale oscillations. This is due to the fact that the energy of certain dislocations can be reduced by decomposing them into several components. For example, since E3D is quadratic, one obtains from (10) or directly from (11) that u0 ð2e1 ; tÞ ¼ 4u0 ðe1 ; tÞ. This implies that a dislocation with Burgers vector 2e1 will be unstable with respect to splitting into two parallel and very close dislocations with Burgers vector e1 each (see sketch in Fig. 1). For other Burgers vectors a more complex situation arises; for example in a cubic crystal the Burgers vector e1 þ e2 is stable for some orientations, and unstable towards splitting into e1 and e2 for other orientations, see Fig. 1 and Section 6 for details. Mathematically, one says that E0LT is not lower semicontinuous. The study of the relaxation of E0LT permits understanding the macroscopic material behavior without resolving the details of these microstructures. Precisely, one considers a similar functional with a different energy density urel . This is defined optimizing over all possible microstructures with given average,
n
o
urel ðb; tÞ ¼ inf E0LT ½l; B1=2 : l 2 Compðb; tÞ :
ð15Þ
Here the set of competitors Compðb; tÞ is the set of dislocation networks l 2 DislðR2 Þ which coincides with the straight dislocation with Burgers vector b and orientation t outside the ball of diameter 1, see Fig. 1 for some illustrations. The effective line-tension model will then be the functional Erel LT , defined as in (4) but using the relaxed energy urel . This means that Erel LT ½l; x gives the smallest possible asymptotic energy among all energies F e ½v e ; x of slip fields
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
Fig. 1. A straight dislocation can locally (inside a small ball) be replaced by a microstructure consisting of several dislocations. This is not always energetically convenient. Left panel: a straight dislocation with Burgers vector e1 is stable. Middle panel: A straight dislocation with Burgers vector 4e1 will spontaneously decay into four parallel dislocations with Burgers vector e1 , since u0 ð4e1 ; tÞ > 4u0 ðe1 ; tÞ. Right panel: A dislocation with Burgers vector e1 þ e2 may decay into two parallel dislocations with Burgers vector e1 and e2 ; this is only energetically favorable for certain orientations t.
v e which approximate the dislocation distribution l, in the sense that e1 Dv e converges weakly to l? . For example, if l is the straight dislocation discussed above, v e may take the form (9) but can also make use of any possible microstructure. Examples are discussed in Section 6 below. Mathematically the statement is phrased in terms of C-convergence (Dal Maso, 1993; Braides, 2002 and Conti et al., 2011),
C lim e!0
1
e2 ln 1e
F e ¼ Erel LT :
ð16Þ
entering (6). The multiplane kernel Ch is therefore characterized as follows:
1 80 Cðx yÞ 0 > > > C B > .. > C forjx yj h; B > > . A @ > > > < Cðx yÞ 0 1 Ch ðx yÞ 0 > Cðx yÞ . . . Cðx yÞ > > > C B > .. .. .. > C forjx yj h; >B > . . . A @ > > : Cðx yÞ . . . Cðx yÞ ð20Þ
5. Multiple planes In this section we extend the model to the case of multiple parallel planes. We denote by M the number of planes, by h the distance between two adjacent planes, and by v ¼ fv 1 ; . . . ; v M g 2 RMN the global slip, with v j 2 RN the slip in plane j, corresponding to the set xj ¼ x fjhg, and we assume X ¼ x ðH; HÞ with H > Mh. The corresponding multiplane Burgers vector b takes values in BM . The elastic displacement u : X ! R3 has jump v j on xj ,
uþ ðx; y; jhÞ u ðx; y; jhÞ ¼ v j ðx; yÞ:
ð17Þ
The generalization of the energy (5) is
1X
e
j
Z xj
see Gladbach for details. As in the single-plane case we start from the pointwise limit, in which e is made small keeping the dislocation straight and fixed. Precisely, we fix a multiplane Burgers vector b 2 BM , an orientation t 2 S1 , and define as in (10)
1
uM ðb; tÞ ¼ lim
e!0;h!0 e2 ln 1 e
F e;h ½v e ; B1=2
ð21Þ
where v e is defined as in (9). The corresponding line-tension R energy is denoted as before by E0LT ½l; x ¼ x uM ðb; tÞdc. The limit in (21) depends on the relation between the plane separation h and the lattice spacing e. Three regimes can be identified:
2
dist ðv j ; eBÞdxþminfE3D ½u; X : uþ u ¼ v j on xj g: u
ð18Þ The elastic problem is, as in the previous case, linear; the minimizer u thus depends linearly on v ¼ ðv 1 ; . . . ; v M Þ. Neglecting boundary terms the elastic energy can be characterized as
F e;h ½v ; x ¼
1X
e
þ
j
Z
Z Z x
x
uS ðb; tÞ ¼
M X
u0 ðbj ; tÞ:
ð22Þ
j¼1
In particular, this includes the cases where h is fixed or hðeÞ ¼ 1= lnð1=eÞ. In this case we only see the short-range kernel in (20), since the planes are widely separated.
2
xj
(i) If e is much smaller than hðeÞ, in the sense that for all b 2 ð0; 1Þ one has hðeÞ eb , then the planes do not interact and uM ¼ uS , where
dist ðv j ; eBÞdx
ðv ðxÞ v ðyÞÞ Ch ðx yÞðv ðxÞ v ðyÞÞdxdy:
(ii) If hðeÞ ¼ OðeÞ, then uM ¼ uL , where
ð19Þ A detailed analysis of the kernel Ch : R2 ! RNMNM shows that it has singular diagonal terms Cðx yÞ, corresponding to intra-plane interactions, and regular off-diagonal terms K h ðx yÞ, corresponding to inter-plane interactions. In fact, for jx yj h, the off-diagonal terms are negligible in comparison to Cðx yÞ, and for jx yj h we have K h ðx yÞ Cðx yÞ. Here C is the single-plane kernel
uL ðb; tÞ ¼ u0
! M X bj ; t :
ð23Þ
j¼1
In particular, this includes the case where hðeÞ ¼ e. In this case we only see the long-range kernel, and the limit only P depends on the total Burgers vector j bj . The planes are so close to each other that the M dislocations are not distinguished from a single one.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
(iii) If hðeÞ ¼ eb for some b 2 ð0; 1Þ, then we see a mixture of the short-range energy from (22) and the longrange one from (23),
uM ðb; tÞ ¼ ð1 bÞuS ðb; tÞ þ buL ðb; tÞ:
ð24Þ
ea ; a
Of the relevant length scales 2 ð0; 1Þ; 1 b many see the short-range kernel, the remaining b many see the long-range kernel. When only one dislocation (in one plane) is present, as for example in the case b ¼ ðb; 0; . . . ; 0Þ for some b 2 B, then uS ðb; tÞ ¼ uL ðb; tÞ ¼ u0 ðb; tÞ and all limit energies coincide. If dislocations in different planes have different orientation, then they only cross at isolated points (after projecting to one plane). Then to leading order there is no interaction in this case. The most interesting cases, which arise if dislocations are present in different planes which overlap over a finite length (and therefore have the same orientation over the overlapping segment) is discussed in detail below. See also Fig. 2. We now turn to the C-limit, which corresponds to the study of the asymptotic behavior along an ‘‘optimal’’ sequence. Optimal here means that it is the sequence which asymptotically has the smallest energy, among all those which converge to the same dislocation distribution. Our main result is
1
C lim e!0;h!0
F e;h ¼ EM; LT
ð25Þ
u M ðb; tÞdc;
ð26Þ
e2 ln 1e
where
EM; LT ½l; x ¼
Z x
see Gladbach for details. Using the C-limit result in Conti et al. (2011) we see, in case (i), that the planes are well separated and in the relaxation procedure different microstructures can be used in the different planes, leading to an independent relaxation of the energy in each plane,
u M ðb; tÞ ¼ urel S ðb; tÞ ¼
M X
urel 0 ðbj ; tÞ;
ð27Þ
j¼1
where urel 0 is the single-plane relaxed energy defined in (15). In case (ii) instead the planes are very close and interact strongly, so that one can only use a single microstructure, which affects all planes at the same time, leading to
u
M ðb; tÞ
¼u
rel L ðb; tÞ
rel 0
¼u
! M X bj ; t ;
where urel L is obtained from uL with the same procedure as in (15). In case (iii), using a separate microstructure in each plane, at a scale larger than hðeÞ, leads to the upper bound
u M 6 ½ð1 bÞuS þ buL rel :
ð29Þ
However, the actual C-limit is smaller and characterized by the integrand
u M ¼ ð1 bÞurel S þ buL
rel
:
ð30Þ
This iterated relaxation is achieved by dislocation networks realizing the exterior relaxation which are then perturbed at scale hðeÞ by microstructures realizing the interior relaxation. This corresponds to a combination of fine-scale microstructures, which are independent in each plane and have a length scale smaller than the separation between the planes, with a single large-scale microstructure, which is the same for all planes and has a length scale larger than the separation between the planes. See Fig. 3 for an example of such a two-scale microstructure. The lower bound rel ð1 bÞurel S þ buL 6 uM ;
ð31Þ
which corresponds to a separate relaxation of the shortrange and long-range terms, is generally unattainable because the short-range microstructure complies with the exterior one at intermediate scales. We show in the next section that the inequalities (31) and (29) can be strict, meaning the two-scale microstructure produces a lower energy than any single-scale structure in some cases. In order to explain the meaning of the iterated relaxation in (30) we explain how a possible microstructure for a straight dislocation with Burgers vector b 2 BM in direction t 2 S1 is constructed. One constructs an admissible exterior microstructure lext 2 Compðb; tÞ such that (with a slight abuse of notation)
ð1 bÞurel S þ buL ðlext Þ rel ¼ ð1 bÞurel ðb; tÞ: S þ buL
ð32Þ
One then shows that lext can be chosen to be polygonal, in the sense that all curves ci are segments. i
For every jump b 2 BM of lext in direction ti 2 S1 , one
ð28Þ
j¼1
Fig. 2. Dislocations in different planes interact only if they run in parallel for an extended length. The interaction increases the energy for equal Burgers vectors and decreases it for opposite ones, causing minimizing dislocations in different planes to repel or attract each other respectively.
i
finds an admissible microstructure liint 2 Compðb ; t i Þ with
Fig. 3. A dislocation network in two planes with global Burgers vector b ¼ ð1; 1; 1; 1Þ and orientation t ¼ e1 . Macroscopically, the lower dislocation line splits in two while the upper line follows one of the branches. The upper dislocation develops microstructure at scale hðeÞ. See Section 6 for details.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx i i uS ðliint Þ ¼ urel S ðb ; t Þ:
ð33Þ
Note that the interior structure can be chosen compoi
nentwise. We denote by b the Burgers vector along a segment of the dislocation network described by lext P i and by ti its orientation, lext ¼ i b t i dci . b Given e > 0 and hðeÞ e , one rescales lext to a fixed l < 1 and tiles along tR. Then for every line segment ci i
of lext with jump ðb ; ti Þ one rescales liint to hðeÞ and tiles along ci . The resulting measure is called lint . The corresponding slip v e;b;l : x ! RNM is then obtained by applying construction (9) to all curves making up lint . Then
1
lim
v
F e;hðeÞ ½ e;b;l ; B1=2 e!0 e2 ln 1 e
X ¼ ð1 bÞ jci juS ðliint Þ þ buL ðlext Þ i
¼ ð1 bÞurel S þ buL
rel
ðb; tÞ:
Letting lðeÞ ! 0, the normalized slip fields
ð34Þ
e1 v e;b;lðeÞ
1
converge in L to the slip associated with the straight dislocation with Burgers vector b in direction t. 6. Applications and examples We start from the case of a single plane. In the case of a simple cubic lattice the Burgers vectors in the ðx1 ; x2 Þ-plane are e1 and e2 , so that B ¼ Z2 . Assuming isotropic elasticity (and after rescaling) the limit u0 takes the form
u0 ðb; tÞ ¼ kðjbj2 þ gðb tÞ2 Þ;
ð35Þ
where k is the shear modulus divided by 2p and g ¼ m=ð1 mÞ 2 ½0; 1, with m the Poisson ratio, see Koslowski and Ortiz (2004). Since in the single-plane case this line-tension energy was already studied in Koslowski and Ortiz (2004), Cacace and Garroni (2009) and Conti et al. (in press) we only briefly sketch the key properties. The function u0 is convex in t, thus it generally prefers straight dislocations over curved ones. Further, since it is quadratic in b one can show that optimal microstructures contain only jumps b 2 Z2 with maxfjb1 j; jb2 jg 6 1 (see [Conti et al. (in press), Section 4]). From here one can establish that the only minimizing compatible microstructure for ðe1 ; tÞ is the single straight line with tangent t. In other words, straight dislocations with Burgers vector e1 are stable and
urel for i ¼ 1; 2 and t 2 S1 : 0 ðei ; tÞ ¼ u0 ðei ; tÞ
the straight dislocation with b ¼ ð1; 1Þ and t close to e1 is not stable, see Cacace and Garroni (2009) and Conti et al. (in press). Precisely, for any t h ¼ ðcos h; sin hÞ, two competitors in Compðb; th Þ are a pattern with a single straight dislocation and b ¼ ð1; 1Þ and a pattern with two very close parallel dislocations with Burgers vector ð1; 0Þ and ð0; 1Þ (as in the right panel of Fig. 1). The associated energies are
f ðhÞ ¼ u0 ðe1 þ e2 ; t h Þ ¼ kð2 þ gð1 þ sinð2hÞÞÞ
ð37Þ
and
^f ðhÞ ¼ u ðe1 ; t h Þ þ u ðe2 ; t h Þ ¼ kð2 þ gÞ: 0 0
ð38Þ
For t ¼ e1 , i.e. h ¼ 0, the two energies are equal. One can however construct a more complex competitor, which is sketched in Fig. 4, which achieves a smaller energy. We first replace the straight dislocation with orientation e1 with a zig-zag which alternates between the orientations th and th , for some angle h, and then separate the two dislocations in the second segment, see sketch in Fig. 4. The dislocation becomes longer by a factor of 1= cos h and has the energy
gðhÞ ¼
f ðhÞ þ ^f ðhÞ k g ¼ 2 þ g þ sinð2hÞ : 2 cos h cos h 2
ð39Þ
A simple computation shows that for h small and negative this is smaller than both f ð0Þ and ^f ð0Þ, see Fig. 4. In fact, a similar procedure gives the minimizer and the exact value of urel 0 , see Conti et al. (in press), Lemma 4.6. We now turn to the case of two planes, which is representative of the general situation of multiple planes. We show in an example that both of the inequalities (29) and (31) can be strict. For simplicity we focus here on the case b ¼ 1=2. Then for b ¼ ðb1 ; b2 Þ 2 Z4 we have from (24)
1 1 2 2 1 1 1 ¼ u0 ðb1 ; tÞ þ u0 ðb2 ; tÞ þ u0 ðb1 þ b2 ; tÞ: 2 2 2
uM ðb; tÞ ¼ uS ðb; tÞ þ uL ðb; tÞ
ð40Þ
ð36Þ
For multiples kei ; k 2 N, one can show that the only minimizing compatible microstructure consists of k parallel straight lines with tangent t and jumps ei , see Fig. 1 and Conti et al. (in press) for details. For b ¼ ð1; 1Þ however, the situation is different. Indeed, the numerical simulations in Koslowski and Ortiz (2004, Fig. 1) show that dislocations with orientation pffiffiffi pffiffiffi t ¼ ðe1 þ e2 Þ= 2 and t ¼ ðe1 e2 Þ= 2 give a different pattern; in one case the two dislocations are superimposed, in the other they are not. This fact can be used to show that
Fig. 4. One possible microstructure in a cubic crystal with Burgers vector b ¼ e1 þ e2 and orientation t ¼ e1 . The upper left panel shows the competitor in Comp; the lower left panel the corresponding microstructure, obtained by periodicity and scaling. The right panel shows the energy of this competitor as a function of the angle h of the inclined segments; the straight dislocation corresponds to the h ¼ 0 case, its energy is marked by the horizontal green line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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Our first example, which will show that (29) is strict, is the case b ¼ ð1; 1; 0; 1Þ; t ¼ e1 . Namely, we show that in this case
u M ðb; tÞ ¼
urel S þ uL
rel ðb; tÞ <
2
½uS þ uL rel ðb; tÞ: 2
ð41Þ
Then using the fact that the relaxation reduces the energy, we estimate
u M ðb; tÞ ¼
urel S þ uL
rel urel S þ uL
2
rel
2
one or two straight lines, but then E0LT ½l1 ; B1=2 cannot equal urel 0 ðe1 þ e2 ; tÞ, which is achieved only by a zig-zag pattern, a contradiction. We shall show in a similar fashion that (31) can be strict. We take b ¼ ð1; 0; 0; 1Þ and t ¼ e1 and show that for these values
ðb; tÞ 6
rel S
u þ uL 2
ðb; tÞ:
ð42Þ
rel urel 0 ðb1 ; tÞ þ u0 ðb2 ; tÞ þ u0 ðb1 þ b2 ; tÞ
2
urel 0 ðe1 þ e2 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 ; tÞ 2
:
ð44Þ
urel 0 ðe1 þ e2 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 ; tÞ 2 ð45Þ
(although E0LT does not, in general, have a minimizer in Comp, existence can be obtained in the finite-dimensional minimization problem discussed in Conti et al. (in press), Section 4, for simplicity of notation we do not distinguish the two problems here). If that were the case, then l1 2 Compðe1 þ e2 ; tÞ, l2 2 Compðe2 ; tÞ, and l1 þ l2 2 Compðe1 ; tÞ, therefore using (40) and the definition of urel 0 we obtain rel rel urel 0 ðe1 þ e2 ;tÞ þ u0 ðe2 ;tÞ þ u0 ðe1 ; tÞ
2
6 E0LT ½l; B1=2 :
ðb; tÞ ¼ u M ðb; tÞ:
ð47Þ
u þ urel L
ðb; tÞ 6
rel rel urel 0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 þ e2 ; tÞ
2
ð46Þ
Recalling (36) we see that equality holds throughout. But E0LT ½l2 ; B1=2 ¼ u0 ðe2 ; tÞ implies that l2 is a single straight line, and the same for l1 þ l2 . This implies that l1 is either
Fig. 5. One possible microstructure for the case of two planes. The total Burgers vector, ð1; 1; 0; 1Þ, corresponds to b ¼ e1 þ e2 in the upper plane, b ¼ e2 in the lower plane. At the microscale the lower plane has no microstructure, the upper plane instead develops a microstructure similar to the one illustrated in Fig. 4 (left panel). At the macroscopic scale only the average dislocation, with effective Burgers vector b ¼ e1 , is visible (right panel).
;
ð48Þ rel where we used (27) for urel S and (28) for uL and inserted b1 ¼ e1 ; b2 ¼ e2 ; b1 þ b2 ¼ e1 þ e2 . Recalling (36) one then obtains
;
2
rel
ð43Þ
see Fig. 5. If (29) were an equality, there would be some admissible microstructure l ¼ ðl1 ; l2 Þ 2 Compðb; tÞ with
E0LT ½l; B1=2 ¼ u M ðb;tÞ 6
rel S
2
Inserting b1 ¼ e1 þ e2 ; b2 ¼ e2 ; b1 þ b2 ¼ e1 , and using (36) one then obtains
u M ðb; tÞ 6
ðb; tÞ <
urel S þ uL
The left-hand side then obeys
Using (27) for urel S and (23) for uL gives
u M ðb; tÞ 6
rel urel S þ uL
2
ðb; tÞ 6
u0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ urel 0 ðe1 þ e2 ; tÞ 2
:
ð49Þ Assume that a microstructure l ¼ ðl1 ; l2 Þ 2 Compðb; tÞ exists with
E0LT ½l; B1=2 6
u0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ urel 0 ðe1 þ e2 ; tÞ 2
:
ð50Þ
Then, as before, the components l1 ; l2 ; l1 þ l2 are admissible competitors in the definition of urel 0 , therefore equality must hold throughout, E0LT ½l1 ; B1=2 ¼ u0 ðe1 ; tÞ, E0LT ½l2 ; B1=2 ¼ u0 ðe2 ; tÞ and E0LT ½l1 þ l2 ; B1=2 ¼ urel 0 ðe1 þ e2 ; tÞ. The first two equalities imply that both l1 and l2 must be single straight lines. Then, however, l1 þ l2 must be either one or two straight lines, which contradicts the third equality. 7. Conclusions We have presented a Peierls–Nabarro model for dislocations localized to several parallel planes and discussed its asymptotic behavior in the limit of small lattice spacing. To leading order we obtain a line-tension model, with a line-tension energy which depends on the dislocation’s Burgers vector and orientation, and the elastic constants of the crystal. The resulting line-tension model predicts that straight dislocations with certain orientations are unstable and are expected to spontaneously form zig-zag microstructures. In the case of several planes, complex microstructures are expected, which have a different structure at scale smaller and larger than the separation between the planes. We made specific predictions for the Burgers vectors and the orientations for which these microstructures can be observed in cubic crystals. Examples for fcc and bcc geometries remain to be investigated. Acknowledgements We gratefully acknowledge interesting discussions with Adriana Garroni and Michael Ortiz. This work was partially supported by the Deutsche Forschungsgemeinschaft
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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through the Forschergruppe 797 ‘‘Analysis and computation of microstructure in finite plasticity’’, project CO 304/4–2, and SFB 1060 ‘‘The Mathematics of emergent effects’’, project A5.
References Ariza, M.P., Ortiz, M., 2005. Discrete crystal elasticity and discrete dislocations in crystals. Arch. Ration. Mech. Anal. 178, 149–226. Bacon, D., Barnett, D., Scattergood, R., 1980. Anisotropic continuum theory of lattice defects. Progress Mater. Sci. 23, 51–262. Braides, A., 2002. C-Convergence for Beginners. Oxford Lecture Series in Mathematics and its Applications, vol. 22. Oxford University Press, Oxford. Cacace, S., Garroni, A., 2009. A multi-phase transition model for the dislocations with interfacial microstructure. Interfaces Free Bound. 11, 291–316. Conti, S., Garroni, A., Massaccesi, A., in press a. Modeling of dislocations and relaxation of functionals on 1-currents with discrete multiplicity. Calc. Var. PDE.. Conti, S., Garroni, A., Ortiz, M., in press b. The line-tension approximation as the dilute limit of linear-elastic dislocations. Conti, S., Garroni, A., Müller, S., 2011. Singular kernels, multiscale decomposition of microstructure, and dislocation models. Arch. Ration. Mech. Anal. 199, 779–819.
Dal Maso, G., 1993. An Introduction to C-Convergence. Progress in Nonlinear Differential Equations and their Applications, vol. 8. Birkhäuser Boston Inc., Boston, MA. Garroni, A., Müller, S., 2005. C-limit of a phase-field model of dislocations. SIAM J. Math. Anal. 36, 1943–1964. Garroni, A., Müller, S., 2006. A variational model for dislocations in the line tension limit. Arch. Ration. Mech. Anal. 181, 535–578. Gladbach, P., (in preparation). Line-tension models for dislocations in crystals (Ph.D. thesis), Universität Bonn. Hirth, J., Lothe, J., 1968. Theory of Dislocations. McGraw-Hill, New York. Hull, D., Bacon, D., 1984. Introduction to dislocations. Vol. 37 of International Series on Materials Science and Technology. Elsevier Science Inc.. Koslowski, M., Ortiz, M., 2004. A multi-phase field model of planar dislocation networks. Model. Simul. Mat. Sci. Eng. 12, 1087–1097. Koslowski, M., Cuitiño, A.M., Ortiz, M., 2002. A phase-field theory of dislocation dynamics, strain hardening and hysteresis in ductile single crystal. J. Mech. Phys. Solids 50, 2597–2635. Miura, T., 1987, . second ed. Micromechanics of Defects in Solids second ed. Kluwer. Ortiz, M., 1999. Plastic yielding as a phase transition. J. Appl. Mech.-Trans. ASME 66, 289–298. Ponsiglione, M., 2007. Elastic energy stored in a crystal induced by screw dislocations: from discrete to continuous. SIAM J. Math. Anal. 39, 449–469. Scardia, L., Zeppieri, C.I., 2012. Line-tension model for plasticity as the Climit of a nonlinear dislocation energy. SIAM J. Math. Anal. 44, 2372– 2400.
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A line-tension model of dislocation networks on several slip planes Sergio Conti ⇑, Peter Gladbach Institut für Angewandte Mathematik, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany
a r t i c l e
i n f o
Article history: Received 26 September 2014 Received in revised form 3 December 2014 Available online xxxx Keywords: Dislocation Peierls–Nabarro model Crystal plasticity Relaxation
a b s t r a c t Dislocations in crystals can be studied by a Peierls–Nabarro type model, which couples linear elasticity with a nonconvex term modeling plastic slip. In the limit of small lattice spacing, and for dislocations restricted to planes, we show that it reduces to a line-tension model, with an energy depending on the local orientation and Burgers vector of the dislocation. This model predicts, for specific geometries, spontaneous formation of microstructure, in the sense that straight dislocations are unstable towards a zig-zag pattern. Coupling between dislocations in different planes can lead to microstructures over several length scales. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Dislocations are topological defects in crystals, which are responsible for plastic deformation. They can be interpreted as discontinuity lines of the plastic slip. Geometrically, a dislocation may be seen as an oriented line with a multiplicity, which corresponds to the Burgers vector. The Burgers vector is a lattice vector of the crystal and is a conserved quantity, in the sense that it is constant on the parts of the dislocation line which have no intersections, and at intersection points the sum of the Burgers vectors of the dislocations entering must equal the sum of the Burgers vectors of the dislocations exiting the point. For a more detailed description we refer for example to Hirth and Lothe (1968), Hull and Bacon (1984), Bacon et al. (1980) and Miura (1987). Dislocations are accompanied by long-range elastic stresses. To make this precise, we consider the variational problem of linear elasticity
Z X
1 Crs u rs u dx þ body forces þ surface tractions 2
⇑ Corresponding author.
ð1Þ
with suitable boundary conditions. Here X R3 is the reference configuration, u : X ! R3 is the (possibly discontinuous) elastic displacement, C the matrix of elastic coefficients, and rs u ¼ ðru þ ruT Þ=2 the elastic strain. Assuming for simplicity that plastic slip only takes place on the plane fx3 ¼ 0g, and denoting by v ðx1 ; x2 Þ the slip at the point ðx1 ; x2 ; 0Þ, we require u to be continuous away from the plane and the limits of u on both sides of the plane to differ by v,
uþ ðx1 ; x2 ; 0Þ u ðx1 ; x2 ; 0Þ ¼ v ðx1 ; x2 Þ: þ
ð2Þ
Here u and u denote the one-sided limits, taken on x3 > 0 and x3 < 0 respectively. A simple situation, corresponding to a straight dislocation with Burgers vector b along the x1 -axis, would have v ðx1 ; x2 Þ ¼ 0 for x2 < 0 and v ðx1 ; x2 Þ ¼ eb for x2 P 0. Here e represents the lattice spacing. The elastic strain rs u necessarily diverges at least as qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ejbj= x22 þ x23 around the dislocation, and thus cannot be square integrable. In other words, all displacements u which obey the given jump condition have infinite elastic energy. The singularity in the strain field is normally regularized at the atomic scale e, at which the continuum elasticity approach loses validity. For example, one can
http://dx.doi.org/10.1016/j.mechmat.2015.01.013 0167-6636/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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truncate the energy in the core region removing a cylinder of radius e around the dislocation line from the domain. Alternatively, one can let the slip v pass continuously from 0 to eb over a length of order e. In both cases the elastic energy scales as e2 lnð1=eÞ, see for example (Hirth and Lothe, 1968). In this paper we make this computation quantitative and rigorous, deriving a model for the line-tension energy of dislocations, in the more general case that dislocations are contained in several parallel planes and without restricting to straight lines. One main finding will be that certain dislocations are unstable towards formation of oscillations. This will require replacing the line-tension model by its relaxation, as explained below. To illustrate our result we need to introduce a more precise notation. We denote the elastic energy by
E3D ½u; X ¼
Z X
1 Crs u rs u dx 2
ð3Þ
and characterize a distribution of dislocations by a set of
case was then studied in Cacace and Garroni (2009) and Conti et al. (2011), an extension to three dimensions was initiated in Conti et al. (in press a,b). A general discrete model for dislocations was introduced in Ariza and Ortiz (2005), its mathematical analysis is however not yet complete and in particular convergence to line-tension models has not yet been proven. In the simple case of screw dislocations in two dimensions both a discrete model (Ponsiglione, 2007) and a nonlinear regularization with subquadratic energy (Scardia and Zeppieri, 2012) have been shown to lead to the same e2 lnð1=eÞ scaling of the energy. This paper is organized as follows. In Section 2 we introduce our model for dislocations in a single plane, in Section 3 we study straight dislocations, in Section 4 we discuss the abstract theory of relaxation for dislocations, in Section 5 we extend the analysis to multiple planes, and finally in Section 6 we discuss some examples and predictions of our theory. 2. Peierls–Nabarro model
i
curves ci , with tangent vectors ti and Burgers vectors b . A line-tension model then takes the form i
ELT ½ðci ; t i ; b Þi ; X ¼
XZ i
uðbi ; ti Þdci
ð4Þ
X
where u is the line-tension energy, depending on the Burgi
.
ers vector b and the orientation t i of the dislocation, and dc denotes integration over the curve ci ; dci ¼ dH1 ci (in (4) we only integrate over the part inside X). We shall show that, in the limit of small lattice spacing e ! 0, and if the displacement field u corresponds to the dislocation i
i
distribution ðci ; t i ; b Þ, the elastic energy E3D ½u; Xe reduces, i
to leading order, to e2 lnð1=eÞELT ½ðci ; t i ; b Þi ; X for a suitable u. Here Xe denotes the set X minus a tube of radius e around the union of the dislocation lines ci . Let us now address the line-tension energy u entering (4). A straightforward computation, based on the assumption that dislocations are straight, infinite lines, gives an expression for the line-tension energy, which we call u0 , in terms of the elastic constants C of the crystal, as given for example in Hirth and Lothe (1968), see also Section 3 below. We shall show below that u – u0 . The difference can be understood on the basis of relaxation theory. Depending on the geometry, it may be energy-efficient for dislocations to spontaneously develop microstructures, which reduce the total energy. The integrand u entering (4) should then take into account that possibility, and indeed the appropriate function is the one that minimizes over all possible microstructures. This decay is related to the decay of dislocations into Shockley partials, but in our setting only ‘‘full’’ dislocations are present and there are no stacking faults involved. We shall use a Peierls–Nabarro-type model, as developed by Ortiz (1999), Koslowski et al. (2002) and Koslowski and Ortiz (2004) for the case of dislocations on a given slip plane, and generalize it to the case that more slip planes are active. Mathematically, the analysis of these models was initiated in the scalar case by Garroni and Müller (2005) and Garroni and Müller (2006), the vectorial
We start from the two-dimensional extension of the Peierls–Nabarro model that was introduced in Ortiz (1999), Koslowski et al. (2002) and Koslowski and Ortiz (2004). We denote by B the lattice of (normalized) Burgers vectors, which in the simple cases coincides with ZN ; N ¼ 2 or 3, and is assumed to be a subset of R3 (extending the vectors by 0 in the third component if B ¼ Z2 ). We assume for simplicity the reference configuration to have the form X ¼ x ðH; HÞ R3 , where H > 0 and x R2 is bounded, connected, with Lipschitz boundary. The key variable is the plastic slip v e : x ! RN . The local energy term penalizes the deviations of the plastic slip from the set of Burgers vectors, which correspond to the distance of v e from eB. This allows for smooth transitions from one Burgers vector to another. Adding the elastic energy E3D we obtain the functional
1
e
Z x
2
dist ðv e ; eBÞdx þ minfE3D ½u; X : uþ u ¼ v e g: u
ð5Þ
Here uþ u ¼ v e means that the jump of u over the plane x3 ¼ 0 equals v e , in the sense specified in (2). The elastic displacement u : X ! R3 is chosen as the one with the smallest elastic energy, among all those with the appropriate jump. The minimization in u in (5) is a linear problem, that can be solved explicitly if one knows the Green’s function of the relevant domain. If the domain is large, one can replace the Green’s function with the fundamental solution (corresponding to the case x ¼ R2 ; H ¼ 1), up to boundary terms which do not modify the leading-order behavior of the energy. One then replaces (5) by the energy
F e ½v ; x ¼
1
Z
2
dist ðv ; eBÞ
eZ x Z
þ
x
x
ðv i ðxÞ v i ðyÞÞCij ðx yÞðv j ðxÞ v j ðyÞÞdxdy ð6Þ
where we implicitly sum over i; j ¼ 1; . . . ; N; N being the dimension of the linear space spanned by B. The kernel
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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C : R2 ! RNN sym can be determined explicitly in Fourier
where ci are oriented Lipschitz curves in X with tangent ti
space from the matrix of elastic constants C (Koslowski and Ortiz, 2004) and has the form
and Burgers vector b 2 ZN . These measures are necessarily divergence-free, in the sense of distributions. We characterize the class of all dislocations in X as
CðzÞ ¼
1 ^ z C jzj3 jzj3
! ð7Þ
i
DislðXÞ ¼ fl ¼
X i b t i dci 2 MðX;RN3 Þjdiv l ¼ 0g:
ð13Þ
i
^ ðsÞ is uniformly bounded and positive definite, in where C the sense that there is c > 0 such that
1 2 ^ ðsÞn 6 cjnj2 jnj 6 n C c
for all n 2 RN ;
s 2 S1 :
ð8Þ
The extension to the case of several planes will be discussed in Section 5 below.
The divergence-free constraint translates to the fact that the Burgers vector is constant on each curve and obeys a balance condition at the endpoints of the curves ci : letting I ðzÞ be all indices of curves ending at z and Iþ ðzÞ all indices of curves starting at z; div l ¼ 0 if and only if
X i2I ðzÞ
3. Small-e limit: straight dislocations We consider a straight dislocation with Burgers vector
i
b ¼
X
b
j
ð14Þ
þ
j2I ðzÞ
at all endpoints z 2 X. The set DislðxÞ correspondingly is the set of dislocations localized in a two-dimensional set
b 2 B and orientation t 2 S1 , so that the dislocation line c is a straight line through the origin parallel to t; c ¼ tR R2 . We set, for x 2 R2 ,
x. We denote by E0LT the line-tension energy defined as in (4) with the energy density u0 from (10), which can be seen as a functional on the space DislðxÞ. If a dislocation
8 > <0 v e ðxÞ ¼ > bx t? : be
is completely contained within the ðe1 ; e2 Þ plane, the rotatP i ? ed measure l? ¼ i b ðti Þ dci is curl-free and consequently the gradient of a slip field v 2 SBVðx; BÞ. This
if
x t? < 0;
if if
0 6 x t? 6 e; x t? > e;
ð9Þ
where t? 2 R2 is the vector obtained rotating t counterclockwise by 90 degrees. To compute the energy per unit length of the dislocation we focus on the part contained in the ball of diameter one, B1=2 ¼ fx 2 R2 : jxj < 1=2g. The energy F e ½v e ; B1=2 of this segment of dislocation, which has unit length, is proportional to e2 lnð1=eÞ for e ! 0, as explained in the introduction. We consequently define
u0 ðb; tÞ ¼ lim
1
v
F e ½ e ; B1=2 : e!0 e2 ln 1 e
ð10Þ
The structure of C given in (7) permits a simpler representation for this limit (see, e.g., Conti et al. (2011, Lemma 5.1)),
u0 ðb; tÞ ¼ 2
Z
bi Cij ðt ? þ tsÞbj ds;
ð11Þ
R
where t ? denotes a unit vector orthogonal to t. Physically, the function u0 is the energy of a straight dislocation, which is not allowed to build oscillations. It can be shown that it is the same as obtained in classical theories by solving the equilibrium equations in R3 and computing the limit of E3D ½u; C e =ðe2 lnð1=eÞÞ where C e is a cylinder with axis along t and unit height, whose cross-section is a circular corona obtained taking away a core region of radius e from a circle of radius 1, see for example (Hirth and Lothe, 1968 and Conti et al., in press b). 4. Curved dislocations, relaxation A dislocation network in a reference configuration
X R3 is a RN3 -valued measure l 2 MðX; RN3 Þ of the form
l¼
X i b ti dci ; i
ð12Þ
identification and the L1 convergence of the slip fields gives the natural convergence concept for dislocation networks. Minimization of the functional E0LT leads, for some boundary conditions, to the formation of fine-scale oscillations. This is due to the fact that the energy of certain dislocations can be reduced by decomposing them into several components. For example, since E3D is quadratic, one obtains from (10) or directly from (11) that u0 ð2e1 ; tÞ ¼ 4u0 ðe1 ; tÞ. This implies that a dislocation with Burgers vector 2e1 will be unstable with respect to splitting into two parallel and very close dislocations with Burgers vector e1 each (see sketch in Fig. 1). For other Burgers vectors a more complex situation arises; for example in a cubic crystal the Burgers vector e1 þ e2 is stable for some orientations, and unstable towards splitting into e1 and e2 for other orientations, see Fig. 1 and Section 6 for details. Mathematically, one says that E0LT is not lower semicontinuous. The study of the relaxation of E0LT permits understanding the macroscopic material behavior without resolving the details of these microstructures. Precisely, one considers a similar functional with a different energy density urel . This is defined optimizing over all possible microstructures with given average,
n
o
urel ðb; tÞ ¼ inf E0LT ½l; B1=2 : l 2 Compðb; tÞ :
ð15Þ
Here the set of competitors Compðb; tÞ is the set of dislocation networks l 2 DislðR2 Þ which coincides with the straight dislocation with Burgers vector b and orientation t outside the ball of diameter 1, see Fig. 1 for some illustrations. The effective line-tension model will then be the functional Erel LT , defined as in (4) but using the relaxed energy urel . This means that Erel LT ½l; x gives the smallest possible asymptotic energy among all energies F e ½v e ; x of slip fields
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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Fig. 1. A straight dislocation can locally (inside a small ball) be replaced by a microstructure consisting of several dislocations. This is not always energetically convenient. Left panel: a straight dislocation with Burgers vector e1 is stable. Middle panel: A straight dislocation with Burgers vector 4e1 will spontaneously decay into four parallel dislocations with Burgers vector e1 , since u0 ð4e1 ; tÞ > 4u0 ðe1 ; tÞ. Right panel: A dislocation with Burgers vector e1 þ e2 may decay into two parallel dislocations with Burgers vector e1 and e2 ; this is only energetically favorable for certain orientations t.
v e which approximate the dislocation distribution l, in the sense that e1 Dv e converges weakly to l? . For example, if l is the straight dislocation discussed above, v e may take the form (9) but can also make use of any possible microstructure. Examples are discussed in Section 6 below. Mathematically the statement is phrased in terms of C-convergence (Dal Maso, 1993; Braides, 2002 and Conti et al., 2011),
C lim e!0
1
e2 ln 1e
F e ¼ Erel LT :
ð16Þ
entering (6). The multiplane kernel Ch is therefore characterized as follows:
1 80 Cðx yÞ 0 > > > C B > .. > C forjx yj h; B > > . A @ > > > < Cðx yÞ 0 1 Ch ðx yÞ 0 > Cðx yÞ . . . Cðx yÞ > > > C B > .. .. .. > C forjx yj h; >B > . . . A @ > > : Cðx yÞ . . . Cðx yÞ ð20Þ
5. Multiple planes In this section we extend the model to the case of multiple parallel planes. We denote by M the number of planes, by h the distance between two adjacent planes, and by v ¼ fv 1 ; . . . ; v M g 2 RMN the global slip, with v j 2 RN the slip in plane j, corresponding to the set xj ¼ x fjhg, and we assume X ¼ x ðH; HÞ with H > Mh. The corresponding multiplane Burgers vector b takes values in BM . The elastic displacement u : X ! R3 has jump v j on xj ,
uþ ðx; y; jhÞ u ðx; y; jhÞ ¼ v j ðx; yÞ:
ð17Þ
The generalization of the energy (5) is
1X
e
j
Z xj
see Gladbach for details. As in the single-plane case we start from the pointwise limit, in which e is made small keeping the dislocation straight and fixed. Precisely, we fix a multiplane Burgers vector b 2 BM , an orientation t 2 S1 , and define as in (10)
1
uM ðb; tÞ ¼ lim
e!0;h!0 e2 ln 1 e
F e;h ½v e ; B1=2
ð21Þ
where v e is defined as in (9). The corresponding line-tension R energy is denoted as before by E0LT ½l; x ¼ x uM ðb; tÞdc. The limit in (21) depends on the relation between the plane separation h and the lattice spacing e. Three regimes can be identified:
2
dist ðv j ; eBÞdxþminfE3D ½u; X : uþ u ¼ v j on xj g: u
ð18Þ The elastic problem is, as in the previous case, linear; the minimizer u thus depends linearly on v ¼ ðv 1 ; . . . ; v M Þ. Neglecting boundary terms the elastic energy can be characterized as
F e;h ½v ; x ¼
1X
e
þ
j
Z
Z Z x
x
uS ðb; tÞ ¼
M X
u0 ðbj ; tÞ:
ð22Þ
j¼1
In particular, this includes the cases where h is fixed or hðeÞ ¼ 1= lnð1=eÞ. In this case we only see the short-range kernel in (20), since the planes are widely separated.
2
xj
(i) If e is much smaller than hðeÞ, in the sense that for all b 2 ð0; 1Þ one has hðeÞ eb , then the planes do not interact and uM ¼ uS , where
dist ðv j ; eBÞdx
ðv ðxÞ v ðyÞÞ Ch ðx yÞðv ðxÞ v ðyÞÞdxdy:
(ii) If hðeÞ ¼ OðeÞ, then uM ¼ uL , where
ð19Þ A detailed analysis of the kernel Ch : R2 ! RNMNM shows that it has singular diagonal terms Cðx yÞ, corresponding to intra-plane interactions, and regular off-diagonal terms K h ðx yÞ, corresponding to inter-plane interactions. In fact, for jx yj h, the off-diagonal terms are negligible in comparison to Cðx yÞ, and for jx yj h we have K h ðx yÞ Cðx yÞ. Here C is the single-plane kernel
uL ðb; tÞ ¼ u0
! M X bj ; t :
ð23Þ
j¼1
In particular, this includes the case where hðeÞ ¼ e. In this case we only see the long-range kernel, and the limit only P depends on the total Burgers vector j bj . The planes are so close to each other that the M dislocations are not distinguished from a single one.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
(iii) If hðeÞ ¼ eb for some b 2 ð0; 1Þ, then we see a mixture of the short-range energy from (22) and the longrange one from (23),
uM ðb; tÞ ¼ ð1 bÞuS ðb; tÞ þ buL ðb; tÞ:
ð24Þ
ea ; a
Of the relevant length scales 2 ð0; 1Þ; 1 b many see the short-range kernel, the remaining b many see the long-range kernel. When only one dislocation (in one plane) is present, as for example in the case b ¼ ðb; 0; . . . ; 0Þ for some b 2 B, then uS ðb; tÞ ¼ uL ðb; tÞ ¼ u0 ðb; tÞ and all limit energies coincide. If dislocations in different planes have different orientation, then they only cross at isolated points (after projecting to one plane). Then to leading order there is no interaction in this case. The most interesting cases, which arise if dislocations are present in different planes which overlap over a finite length (and therefore have the same orientation over the overlapping segment) is discussed in detail below. See also Fig. 2. We now turn to the C-limit, which corresponds to the study of the asymptotic behavior along an ‘‘optimal’’ sequence. Optimal here means that it is the sequence which asymptotically has the smallest energy, among all those which converge to the same dislocation distribution. Our main result is
1
C lim e!0;h!0
F e;h ¼ EM; LT
ð25Þ
u M ðb; tÞdc;
ð26Þ
e2 ln 1e
where
EM; LT ½l; x ¼
Z x
see Gladbach for details. Using the C-limit result in Conti et al. (2011) we see, in case (i), that the planes are well separated and in the relaxation procedure different microstructures can be used in the different planes, leading to an independent relaxation of the energy in each plane,
u M ðb; tÞ ¼ urel S ðb; tÞ ¼
M X
urel 0 ðbj ; tÞ;
ð27Þ
j¼1
where urel 0 is the single-plane relaxed energy defined in (15). In case (ii) instead the planes are very close and interact strongly, so that one can only use a single microstructure, which affects all planes at the same time, leading to
u
M ðb; tÞ
¼u
rel L ðb; tÞ
rel 0
¼u
! M X bj ; t ;
where urel L is obtained from uL with the same procedure as in (15). In case (iii), using a separate microstructure in each plane, at a scale larger than hðeÞ, leads to the upper bound
u M 6 ½ð1 bÞuS þ buL rel :
ð29Þ
However, the actual C-limit is smaller and characterized by the integrand
u M ¼ ð1 bÞurel S þ buL
rel
:
ð30Þ
This iterated relaxation is achieved by dislocation networks realizing the exterior relaxation which are then perturbed at scale hðeÞ by microstructures realizing the interior relaxation. This corresponds to a combination of fine-scale microstructures, which are independent in each plane and have a length scale smaller than the separation between the planes, with a single large-scale microstructure, which is the same for all planes and has a length scale larger than the separation between the planes. See Fig. 3 for an example of such a two-scale microstructure. The lower bound rel ð1 bÞurel S þ buL 6 uM ;
ð31Þ
which corresponds to a separate relaxation of the shortrange and long-range terms, is generally unattainable because the short-range microstructure complies with the exterior one at intermediate scales. We show in the next section that the inequalities (31) and (29) can be strict, meaning the two-scale microstructure produces a lower energy than any single-scale structure in some cases. In order to explain the meaning of the iterated relaxation in (30) we explain how a possible microstructure for a straight dislocation with Burgers vector b 2 BM in direction t 2 S1 is constructed. One constructs an admissible exterior microstructure lext 2 Compðb; tÞ such that (with a slight abuse of notation)
ð1 bÞurel S þ buL ðlext Þ rel ¼ ð1 bÞurel ðb; tÞ: S þ buL
ð32Þ
One then shows that lext can be chosen to be polygonal, in the sense that all curves ci are segments. i
For every jump b 2 BM of lext in direction ti 2 S1 , one
ð28Þ
j¼1
Fig. 2. Dislocations in different planes interact only if they run in parallel for an extended length. The interaction increases the energy for equal Burgers vectors and decreases it for opposite ones, causing minimizing dislocations in different planes to repel or attract each other respectively.
i
finds an admissible microstructure liint 2 Compðb ; t i Þ with
Fig. 3. A dislocation network in two planes with global Burgers vector b ¼ ð1; 1; 1; 1Þ and orientation t ¼ e1 . Macroscopically, the lower dislocation line splits in two while the upper line follows one of the branches. The upper dislocation develops microstructure at scale hðeÞ. See Section 6 for details.
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
6
S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx i i uS ðliint Þ ¼ urel S ðb ; t Þ:
ð33Þ
Note that the interior structure can be chosen compoi
nentwise. We denote by b the Burgers vector along a segment of the dislocation network described by lext P i and by ti its orientation, lext ¼ i b t i dci . b Given e > 0 and hðeÞ e , one rescales lext to a fixed l < 1 and tiles along tR. Then for every line segment ci i
of lext with jump ðb ; ti Þ one rescales liint to hðeÞ and tiles along ci . The resulting measure is called lint . The corresponding slip v e;b;l : x ! RNM is then obtained by applying construction (9) to all curves making up lint . Then
1
lim
v
F e;hðeÞ ½ e;b;l ; B1=2 e!0 e2 ln 1 e
X ¼ ð1 bÞ jci juS ðliint Þ þ buL ðlext Þ i
¼ ð1 bÞurel S þ buL
rel
ðb; tÞ:
Letting lðeÞ ! 0, the normalized slip fields
ð34Þ
e1 v e;b;lðeÞ
1
converge in L to the slip associated with the straight dislocation with Burgers vector b in direction t. 6. Applications and examples We start from the case of a single plane. In the case of a simple cubic lattice the Burgers vectors in the ðx1 ; x2 Þ-plane are e1 and e2 , so that B ¼ Z2 . Assuming isotropic elasticity (and after rescaling) the limit u0 takes the form
u0 ðb; tÞ ¼ kðjbj2 þ gðb tÞ2 Þ;
ð35Þ
where k is the shear modulus divided by 2p and g ¼ m=ð1 mÞ 2 ½0; 1, with m the Poisson ratio, see Koslowski and Ortiz (2004). Since in the single-plane case this line-tension energy was already studied in Koslowski and Ortiz (2004), Cacace and Garroni (2009) and Conti et al. (in press) we only briefly sketch the key properties. The function u0 is convex in t, thus it generally prefers straight dislocations over curved ones. Further, since it is quadratic in b one can show that optimal microstructures contain only jumps b 2 Z2 with maxfjb1 j; jb2 jg 6 1 (see [Conti et al. (in press), Section 4]). From here one can establish that the only minimizing compatible microstructure for ðe1 ; tÞ is the single straight line with tangent t. In other words, straight dislocations with Burgers vector e1 are stable and
urel for i ¼ 1; 2 and t 2 S1 : 0 ðei ; tÞ ¼ u0 ðei ; tÞ
the straight dislocation with b ¼ ð1; 1Þ and t close to e1 is not stable, see Cacace and Garroni (2009) and Conti et al. (in press). Precisely, for any t h ¼ ðcos h; sin hÞ, two competitors in Compðb; th Þ are a pattern with a single straight dislocation and b ¼ ð1; 1Þ and a pattern with two very close parallel dislocations with Burgers vector ð1; 0Þ and ð0; 1Þ (as in the right panel of Fig. 1). The associated energies are
f ðhÞ ¼ u0 ðe1 þ e2 ; t h Þ ¼ kð2 þ gð1 þ sinð2hÞÞÞ
ð37Þ
and
^f ðhÞ ¼ u ðe1 ; t h Þ þ u ðe2 ; t h Þ ¼ kð2 þ gÞ: 0 0
ð38Þ
For t ¼ e1 , i.e. h ¼ 0, the two energies are equal. One can however construct a more complex competitor, which is sketched in Fig. 4, which achieves a smaller energy. We first replace the straight dislocation with orientation e1 with a zig-zag which alternates between the orientations th and th , for some angle h, and then separate the two dislocations in the second segment, see sketch in Fig. 4. The dislocation becomes longer by a factor of 1= cos h and has the energy
gðhÞ ¼
f ðhÞ þ ^f ðhÞ k g ¼ 2 þ g þ sinð2hÞ : 2 cos h cos h 2
ð39Þ
A simple computation shows that for h small and negative this is smaller than both f ð0Þ and ^f ð0Þ, see Fig. 4. In fact, a similar procedure gives the minimizer and the exact value of urel 0 , see Conti et al. (in press), Lemma 4.6. We now turn to the case of two planes, which is representative of the general situation of multiple planes. We show in an example that both of the inequalities (29) and (31) can be strict. For simplicity we focus here on the case b ¼ 1=2. Then for b ¼ ðb1 ; b2 Þ 2 Z4 we have from (24)
1 1 2 2 1 1 1 ¼ u0 ðb1 ; tÞ þ u0 ðb2 ; tÞ þ u0 ðb1 þ b2 ; tÞ: 2 2 2
uM ðb; tÞ ¼ uS ðb; tÞ þ uL ðb; tÞ
ð40Þ
ð36Þ
For multiples kei ; k 2 N, one can show that the only minimizing compatible microstructure consists of k parallel straight lines with tangent t and jumps ei , see Fig. 1 and Conti et al. (in press) for details. For b ¼ ð1; 1Þ however, the situation is different. Indeed, the numerical simulations in Koslowski and Ortiz (2004, Fig. 1) show that dislocations with orientation pffiffiffi pffiffiffi t ¼ ðe1 þ e2 Þ= 2 and t ¼ ðe1 e2 Þ= 2 give a different pattern; in one case the two dislocations are superimposed, in the other they are not. This fact can be used to show that
Fig. 4. One possible microstructure in a cubic crystal with Burgers vector b ¼ e1 þ e2 and orientation t ¼ e1 . The upper left panel shows the competitor in Comp; the lower left panel the corresponding microstructure, obtained by periodicity and scaling. The right panel shows the energy of this competitor as a function of the angle h of the inclined segments; the straight dislocation corresponds to the h ¼ 0 case, its energy is marked by the horizontal green line. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
Our first example, which will show that (29) is strict, is the case b ¼ ð1; 1; 0; 1Þ; t ¼ e1 . Namely, we show that in this case
u M ðb; tÞ ¼
urel S þ uL
rel ðb; tÞ <
2
½uS þ uL rel ðb; tÞ: 2
ð41Þ
Then using the fact that the relaxation reduces the energy, we estimate
u M ðb; tÞ ¼
urel S þ uL
rel urel S þ uL
2
rel
2
one or two straight lines, but then E0LT ½l1 ; B1=2 cannot equal urel 0 ðe1 þ e2 ; tÞ, which is achieved only by a zig-zag pattern, a contradiction. We shall show in a similar fashion that (31) can be strict. We take b ¼ ð1; 0; 0; 1Þ and t ¼ e1 and show that for these values
ðb; tÞ 6
rel S
u þ uL 2
ðb; tÞ:
ð42Þ
rel urel 0 ðb1 ; tÞ þ u0 ðb2 ; tÞ þ u0 ðb1 þ b2 ; tÞ
2
urel 0 ðe1 þ e2 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 ; tÞ 2
:
ð44Þ
urel 0 ðe1 þ e2 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 ; tÞ 2 ð45Þ
(although E0LT does not, in general, have a minimizer in Comp, existence can be obtained in the finite-dimensional minimization problem discussed in Conti et al. (in press), Section 4, for simplicity of notation we do not distinguish the two problems here). If that were the case, then l1 2 Compðe1 þ e2 ; tÞ, l2 2 Compðe2 ; tÞ, and l1 þ l2 2 Compðe1 ; tÞ, therefore using (40) and the definition of urel 0 we obtain rel rel urel 0 ðe1 þ e2 ;tÞ þ u0 ðe2 ;tÞ þ u0 ðe1 ; tÞ
2
6 E0LT ½l; B1=2 :
ðb; tÞ ¼ u M ðb; tÞ:
ð47Þ
u þ urel L
ðb; tÞ 6
rel rel urel 0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ u0 ðe1 þ e2 ; tÞ
2
ð46Þ
Recalling (36) we see that equality holds throughout. But E0LT ½l2 ; B1=2 ¼ u0 ðe2 ; tÞ implies that l2 is a single straight line, and the same for l1 þ l2 . This implies that l1 is either
Fig. 5. One possible microstructure for the case of two planes. The total Burgers vector, ð1; 1; 0; 1Þ, corresponds to b ¼ e1 þ e2 in the upper plane, b ¼ e2 in the lower plane. At the microscale the lower plane has no microstructure, the upper plane instead develops a microstructure similar to the one illustrated in Fig. 4 (left panel). At the macroscopic scale only the average dislocation, with effective Burgers vector b ¼ e1 , is visible (right panel).
;
ð48Þ rel where we used (27) for urel S and (28) for uL and inserted b1 ¼ e1 ; b2 ¼ e2 ; b1 þ b2 ¼ e1 þ e2 . Recalling (36) one then obtains
;
2
rel
ð43Þ
see Fig. 5. If (29) were an equality, there would be some admissible microstructure l ¼ ðl1 ; l2 Þ 2 Compðb; tÞ with
E0LT ½l; B1=2 ¼ u M ðb;tÞ 6
rel S
2
Inserting b1 ¼ e1 þ e2 ; b2 ¼ e2 ; b1 þ b2 ¼ e1 , and using (36) one then obtains
u M ðb; tÞ 6
ðb; tÞ <
urel S þ uL
The left-hand side then obeys
Using (27) for urel S and (23) for uL gives
u M ðb; tÞ 6
rel urel S þ uL
2
ðb; tÞ 6
u0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ urel 0 ðe1 þ e2 ; tÞ 2
:
ð49Þ Assume that a microstructure l ¼ ðl1 ; l2 Þ 2 Compðb; tÞ exists with
E0LT ½l; B1=2 6
u0 ðe1 ; tÞ þ u0 ðe2 ; tÞ þ urel 0 ðe1 þ e2 ; tÞ 2
:
ð50Þ
Then, as before, the components l1 ; l2 ; l1 þ l2 are admissible competitors in the definition of urel 0 , therefore equality must hold throughout, E0LT ½l1 ; B1=2 ¼ u0 ðe1 ; tÞ, E0LT ½l2 ; B1=2 ¼ u0 ðe2 ; tÞ and E0LT ½l1 þ l2 ; B1=2 ¼ urel 0 ðe1 þ e2 ; tÞ. The first two equalities imply that both l1 and l2 must be single straight lines. Then, however, l1 þ l2 must be either one or two straight lines, which contradicts the third equality. 7. Conclusions We have presented a Peierls–Nabarro model for dislocations localized to several parallel planes and discussed its asymptotic behavior in the limit of small lattice spacing. To leading order we obtain a line-tension model, with a line-tension energy which depends on the dislocation’s Burgers vector and orientation, and the elastic constants of the crystal. The resulting line-tension model predicts that straight dislocations with certain orientations are unstable and are expected to spontaneously form zig-zag microstructures. In the case of several planes, complex microstructures are expected, which have a different structure at scale smaller and larger than the separation between the planes. We made specific predictions for the Burgers vectors and the orientations for which these microstructures can be observed in cubic crystals. Examples for fcc and bcc geometries remain to be investigated. Acknowledgements We gratefully acknowledge interesting discussions with Adriana Garroni and Michael Ortiz. This work was partially supported by the Deutsche Forschungsgemeinschaft
Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013
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S. Conti, P. Gladbach / Mechanics of Materials xxx (2015) xxx–xxx
through the Forschergruppe 797 ‘‘Analysis and computation of microstructure in finite plasticity’’, project CO 304/4–2, and SFB 1060 ‘‘The Mathematics of emergent effects’’, project A5.
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Please cite this article in press as: Conti, S., Gladbach, P. A line-tension model of dislocation networks on several slip planes. Mech. Mater. (2015), http://dx.doi.org/10.1016/j.mechmat.2015.01.013