Derivation of a rod theory for biphase materials with dislocations at the interface
Starting from three-dimensional elasticity we derive a rod theory for biphase materials with a prescribed dislocation at the interface. The stored energy density is assumed to be non-negative and to vanish on a set consisting of two copies of . First, we rigorously justify the assumption of dislocations at the interface. Then, we consider the typical scaling of multiphase materials and we perform an asymptotic study of the rescaled energy, as the diameter of the rod goes to zero, in the framework of -convergence.
Key words: Nonlinear elasticity, Dimension reduction, Rod theory, Heterostructures, Crystals, Dislocations, Gamma-convergence.
2000 Mathematics Subject Classification: 74B20, 74K10, 74N05, 49J45, 46E40.
Hausdorff Center for Mathematics Institute for Applied Mathematics, Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germany.
SISSA,Via Beirut 2-4, 34014 Trieste, Italy. Email:email@example.com
We study the behavior of an elastic thin beam consisting of two parts made of different materials. The interface between the two parts of the beam is fixed. Our objective is, first, to rigorously prove that formation of dislocations on such interface is energetically more favorable than purely elastic deformation when the radius of the cross-section is sufficiently large. Second, to derive a one-dimensional theory of elastic thin beams with a prescribed dislocation on the interface.
The motivation to look at this problem relies on the connection with the study of nanowire heterostructures, which have important applications in semiconductor electronics. A heterostructure is a material obtained through an epitaxial growth process, where two materials featuring different lattice constants are brought together by deposition of one material (the overlayer) on top of the other (the underlayer). In general, lattice mismatch will prevent growth of defect-free epitaxial film over a substrate unless the thickness of the film is below certain critical thickness; in this last case lattice mismatch is compensated by the strain in the film. In contrast, as confirmed by experimental observations, one-dimensional systems, i.e., longitudinally heterostructured nanowires, can be grown defect-free more readily than their two-dimensional counterparts. A better understanding of nanowires is therefore crucial in the study and use of heterostructures.
A schematic of a heterostructured nanowire is showed in Figure 1. The radii of the unstrained underlayer and overlayer are denoted by and respectively. The lattice mismatch, , between the overlayer and the underlayer is defined as
For a given mismatch , if the radii and are sufficiently small, the system is elastically strained and no dislocation arises. Ultimately, as the radii increase, the mismatch strain is relieved by formation of misfit dislocations at the interface. In the dislocated system, a small portion of the total mismatch is accomodated by the dislocations, while the remainder (the residual mismatch) is accomodated by elastic strain both in the underlayer and overlayer (see Figure 1). Figure 2 represents a longitudinal section of a dislocated nanowire in the atomistic picture (where the crystalline lattice is assumed to be cubic): we observe an additional row of atoms in the overlayer.
A model for the critical radius for which the first dislocation appears has been developed, e.g., in  in the context of linearized elasticity. The critical radius is described in  as a function of the mismatch , and is shown to be roughly an order of magnitude larger than the critical thickness of the corresponding thin film/substrate system.
The purpose of this paper is, first, to rigorously justify formation of dislocations on the interface between the two parts of the beam; then, to derive a one dimensional model describing the deformations of the beam with a given dislocation. For the second part we consider the case of one misfit dislocation, though our analysis extends as well to the case of more dislocations without any additional difficulty.
More precisely, we consider a cylindrical region , which represents the reference configuration of the beam, where is the disk of radius in , i.e., , and is a small positive parameter, which, in the atomistic picture, is of the order of the atomic distance. Theorem 3.6 shows that when is sufficiently large, formation of dislocations is energetically more favorable than purely elastic deformation.
In the second part of the paper, we prescribe the dislocation. We assume that the dislocation line, , has the form
where is a Lipschitz, relatively closed curve in . The latter condition implies that is not simply connected.
We assume that the elastic energy (per unit cross-section) has the form
in the sense of distributions. In (1.3), the vector , with , denotes the Burgers vector, which, together with the dislocation line, uniquely characterizes the dislocation. We observe that any field satisfying (1.3) is locally the gradient of a Sobolev map. More precisely, if is simply connected, then there exists such that a.e. in . In particular, if is a closed loop in , one can take , where , and is the flat region enclosed by the curve ( is the shadowed set in Figure 3). Then, a.e. in , where and its distributional gradient satisfies
Therefore is the absolutely continuous part (with respect to Lebesgue measure) of the gradient . Following , we interpret as the elastic part of the deformation. More in general, may be regarded as the elastic part of a deformation which has a constant jump, equal to , across any surface having as its boundary.
The domain of the energy functional (1.2) is thus defined as
where . Indeed, because of (1.3), the fields of cannot expect to be in .
Furthermore, we assume that the density of energy has the form
where the functions and satisfy the following conditions
(i) , ;
(ii) is frame indifferent, i.e., for every and , ;
(iii) there exist , , with , such that for every
for some . (Remark that a typical is, for example, , where is the identity matrix and is defined by (1.1).) More generally in the following we assume that , with for .
In order to recast the functionals over varying domains into functionals with a fixed domain , we introduce in (1.2) the following change of variables:
and rescale the elements of accordingly
In (1.4) we used the notation
where stands for the th column of . We now rewrite (1.2) in terms of maps from to :
It will be convenient to define the set of admissible deformations in the fixed domain :
Our goal is to study the asymptotic behavior of the rescaled functionals in the framework of -convergence. This problem was already addressed in  in the dislocation-free case. The main difference here is that, due to the presence of a dislocation, one has to work with growth conditions slower than quadratic, as specified in (iii), which require suitable modifications of the methods introduced in .
In Theorem 4.1 we show that if a sequence is such that , then, up to subsequences, the sequence converges weakly in to some limit such that
where co denotes the convex hull of for any . Finally, in Theorem 5.3, we compute the -limit of the sequence .
2. Preliminary results
Throughout this paper the letter denotes various positive constants whose precise value may change from place to place. Its dependence on other variables will be emphasized only if necessary.
We will use the following two results from .
Let , and let . Suppose that is a bounded Lipschitz domain. There exists a constant such that for each there exists such that
Let , and let . Suppose that is a bounded Lipschitz domain. Then there exists a constant such that for each and each , there exists such that
The next proposition provides a generalization of the rigidity estimate (2.1), which cannot be applied as it is, due to the growth condition (iii) required for the function . It will be used to prove the compactness of sequences with equibounded energy.
Let , and let . Suppose that is a bounded Lipschitz domain. Then there exists a constant such that for each there exists such that
Since , we can find a constant , depending on , such that
Next we estimate the integral of in the set . In order to do this we use again the fact that, for sufficiently large , is equal to if , and is bounded by a constant if . Hence we write
The next proposition will be used in the proof of Proposition 4.2.
Let , and let . Suppose that is a bounded Lipschitz domain. Let satisfy and
with . Then there exists a constant such that
Let be solution of . We first provide an estimate for the norm of :
Finally, taking into account (2.16) and the fact that in , we obtain
Let . There exist such that for every
We first observe that there exist two positive constants , depending on and , such that
Indeed, let us fix such that for . Then, for , we have . The second inequality of (2.19) is trivial. In order to prove (2.18) it is enough to observe that if and , then, by (2.19), . If otherwise and , then (2.19) implies .
3. Competition between elastic deformation and formation of dislocations
We introduce the set
The cost associated with a transition of the elastic deformation from one well to the other is defined as
where, for each , and each , the set is defined as
It will be convenient to introduce the quantity , defined as the minimum cost of a transition in the case when no dislocation is present, i.e., is obtained by requiring, in (3.1), in the sense of distributions.
For each we have
For ease of notation we set and
Let us remark that such quantities also depend on the radius of the cross section
Notation. We will write , or to emphasize the dependance on when the radius of the cross section plays an essential role. In most of the cases however, the dependence on will be omitted not to overburden notation.
Suppose that for each and for each
Then and .
We will show that , the proof for being completely analogous. By contradiction suppose that . Then by definition of , there exists a sequence such that
As already remarked in the introduction, the fields are locally gradients of Sobolev functions. Therefore we can find a set and a sequence of functions such that is simply connected and in . We now apply the rigidity estimate (2.5) in combination with (3.3) and the growth condition (iii), to find sequences such that
The first formula of (3.4) implies that in measure. Moreover, since is a bounded sequence, we have that is bounded in , and therefore, up to subsequences (not relabeled), strongly in for each . Using the second formula of (3.4) and arguing in a similar way for the sequence , we deduce that strongly in for each . By the Poincaré inequality there exist such that
Finally, by the trace theorem we find that
which yields the contradiction for some , and .
It can be easily checked that . Indeed, if is a competitor for , then is a competitor for .
The next proposition provides an upper bound for the energy in the dislocation free case. We will denote by , with , any matrix with norm , i.e., .
Assume that . Then .
Let , where for , for and for . Then one easily checks that
Therefore , and
Under the assumptions of Proposition 3.4, we have that as .
In the next theorem we show that if the radius of the cross section is sufficiently large, than formation of dislocations is energetically more convenient than purely elastic deformation.
The following inequality holds
Let be the square of side centered at the origin, i.e.,
In analogy with , we define as the minimum cost associated with a transition of the elastic deformation from one well to the other when the cross section of the beam is , namely
We first prove (3.5) for , namely when the cross-section of the beam is .
From Remark 3.3 it follows that
We decompose into the union of four sub-squares of side . Set
and let , , , be the squares of side centered at , , and respectively (see Fig. 4). Next we decompose into the union of four sub-squares overlapping on stripes of thickness , with . Specifically, set
and let , , , be the squares of side centered at , , and respectively (see Fig. 4). Now fix . By definition of , there exist and such that in , in and