Discrete-to-continuum limits of planar disclinations

This paper is devoted to the mathematical analysis of a discrete model that describes frustrations in atomistic lattices induced by rotational mismatches. Such configurations are called wedge disclinations, which are angular defects. Disclinations are observed in solids in situations where the rotational symmetry is violated at the level of the crystal lattice. Figure 1 illustrates classical, simple examples of disclinations.

Historically, the existence of disclinations was predicted by Volterra alongside dislocations (translational defects) in a celebrated paper [Vol07]. However, it was not until the late 1960s that disclinations saw a systematical investigation both from an experimental and theoretical perspective. First examples of disclinations over planar lattices have been discovered in superconductors and reported in [TE68] and [ET67]. While a dislocation is a singularity of the deformation field which may be described by a lattice-valued vector, called Burgers vector, a disclination, as stated in [AEST68] (see also [Nab67]), is characterized by a closure failure of rotation … for a closed circuit round the disclination centre. A continuum theory for disclinations in linearized elasticity has been first derived by de Wit in [dW70] based on the idea of compatible elasticity and later elaborated by the same author in a series of articles [dW73a], [dW73b], [dW73c]. A comprehensive theory for disclinations (alongside dislocations) in non-linear elasticity has been developed by Zubov [Zub97]. We refer to [ZA18] for more recent developments on unified approaches to treat dislocations and disclinations as well as for a review of mechanical models of disclinations.

Our work is inspired by a number of experimental observations which have revealed formation of disclinations in metallic structures under a variety of mechanical loading and forces, geometrical regimes and kinematical constraints. Here, we elaborate on two such observations.

In austenite-to-martensite transformations, disclinations may emerge during the formation of rotated (and constant-strain) shear-bands [Bha03]. Such transformations are purely elastic and of the type solid-to-solid. They appear in a class of metals; in particular, in shape-memory alloys. Upon symmetry-break (typically triggered and driven by a negative temperature gradient), austenite, the high-symmetry and highly homogeneous crystal phase, turns into martensite, an anisotropic crystal phase with lower symmetry. The crystallographic lattice accommodates this phase change by forming a complicated microstructure composed of a mixture of thin plates and needle-shaped regions exhibiting differently rotated copied of martensitic phases. In a zero stress-microstructure, martensite can be described by a piecewise constant deformation gradient. Constant-strain regions, that is, regions of constant crystal orientation, are separated by sharp planar interfaces according to kinematical compatibility. This compatibility is called a rank-1 connection of the corresponding deformation tensors. However, such configurations are ideal and typically atomistic non-idealities such as dislocations and disclinations appear in large numbers. Outstanding examples of disclinations in martensite are represented by ”nested” star-shaped geometries observed in $\textrm{Pb}_{3}(\textrm{VO}_{4})_{2}$. Experiments are described in [KK91, MA80]; modeling work as well as numerical and exact constructions are described in [PL13] and [CPL14] respectively, and mathematical theories are developed in [CDPR⁺20]. Significantly more complicated microstructures rich in defects are described in [ILTH17] for Ti-Nb-Al-based alloys. Here, martensite nucleates and evolves in the form of thin plates embedded in an austenitic lattice. The evolution of these plates is complex due to plates colliding against surrounding structures, undergoing further branching into additional martensitic sub-plates or being reflected after hitting a grain boundary. Such evolution results in self-similar patterns resembling fractal structure which are rich in disclinations and dislocations [ILTH17]. The available models of such microstructures are essentially based on statistical analysis [IHM13] and probability [BCH15, CH], and thus the consistency with atomistic models remains elusive. This is where we aim to contribute.

The second experimental observation is the discovery of a superior (and, yet to date, largely unexplained) structural reinforcement mechanism which has recently spurred scientific interest on the experimental as well as on the theoretical investigation of kink formation in certain classes of metal alloys. Formation of kinks consisting of approximately constant-strain bands accompanied by high rotational stretches of the lattice are observed in classes of laminate ”mille-feuille” structures under uniaxial compression [HOI⁺16, HMH⁺16]. In one of their most typical morphologies, bands manifest themselves in the form of planar and sharp ridge-shaped regions which appear at various length-scales and are accompanied by localized plastic stresses and formation of disclinations [LN15]. Kinks of various morphologies have been described in [Ina19] where constructions of piecewise affine deformation maps based on the rank-1 connection rule and incorporating angular mismatches are presented. While [Ina19]’s analysis sheds light on the kinematics of the disclination-kinking mechanism for various morphologies of kinks, there is no available model based on atomistic descriptions which describes the energy of such disclination-kinking mechanisms.

A common aspect on both martensitic microstructures and kink formation is that planar regions of approximately constant strains and constant-orientation lattices need to rotate in order to preserve the continuity of the deformation field across their common border. As a result, these materials necessarily develop angular lattice misfits which are striking examples of wedge disclinations. This motivates the modeling assumption of this paper that wedge disclinations are caused by large (non-linear) rotational stretches in planar-confined geometries.

Our aim is to take the first step in the direction of a comprehensive variational theory that is suitable to simultaneously treat microscale and localized defects and to predict their effect on large (non-linear) elastic and plastic deformations including kinks and shear-bands. By designing a simple, nearest-neighbor-type interaction mechanism, we construct a model which we apply to describe a single disclination in a planar lattice and which is at the same time potentially adaptable to describe multi-disclination systems and to incorporate other lattice defects such as voids, dislocations and grain boundaries.

By pursuing this aim, we also fill a gap in the literature on atomistic modeling of planar lattice defects and the limit thereof as the lattice spacing tends to $0$. To identify this gap, we review related literature. First, in the framework of linearized elasticity, the formal asymptotic expansions in [SN88] provide a continuum model for lattice defects. Second, in [LPS13] two triangular lattices with different lattice spacings are attached together, which forces dislocations to form at the interface. The main result is a continuum limit of this model as the lattice spacing tends to 0. Third, based on the discrete calculus of lattices constructed in [AO05], the stress in a periodic crystal induced by parallel screw dislocations is computed in [Pon07, HO14, HO15, EOS16, BBO19]. These results provide quantitative estimates between the displacement in the lattice and the displacement computed from linear elasticity in the continuum counter part. In [BHO19] these techniques are extended to capture cracks. All these works deal with either localized or small deformation to the underlying two dimensional lattice, which is unfit for disclinations (see, e.g., the relatively large deformations in Figure 1).

For large deformations, the recent work in [KM18] presents and anlyzes an atomistic model of a stretchable hexagonal lattice defined over a smooth manifold, in which the main result is the continuum limit in the form of $\Gamma$-convergence. Since the approach in [KM18] is designed to describe the energetics of highly distorted membranes, we follow a similar approach. However, the choice in [KM18] that the number of bonds per atom is always $6$ makes the result not applicable to disclinations.

As in [KM18], we assign an energy $E_{\varepsilon}$ to a deformed lattice, where $\varepsilon$ is the lattice spacing. Two examples of such deformed lattices are illustrated by the dotted nodes in Figure 1. Given a deformed lattice, the energy $E_{\varepsilon}$ penalizes stretching and compression of the atomic bonds and changes in volume of the triangles formed by three neighboring atoms. To enforce the appearance of a $5$- or $7$-type disclination, we require the deformed lattice to satisfy a special boundary condition such that $5$ or $7$ rotated copies of it fit together such as in Figure 1.

Our main result, Theorem 3.1, is the exact derivation of $E$, the macroscale energy obtained in the limit as $\varepsilon\to 0$, that is,

in a suitable topology.

From the mathematical perspective, the interest of our analysis lies in the treatment of the special boundary condition. Conceptually, this boundary conditions requires us to incorporate a pointwise constraint in the space of traces on $W^{1,p}$-Sobolev vector maps which are characterized by means of a nonlocal norm. Our main contribution on this is Proposition 4.4.

The paper is organized as follows. In Section 2 we present the discrete mechanical model and the mathematical setting required for the related variational analysis. In Section 3 we state and prove our main result, Theorem 3.1, on the $\Gamma$-convergence of the lattice energy $E_{\varepsilon}$. In this proof, we postpone the proofs of technical lemmas to Section 4, which includes the proof of Proposition 4.4 on density in the space of admissible lattice displacements. In Section 5 we explore the physical implications of our $\Gamma$-convergence analysis in terms of energy and stress states of the minimizers of the continuum model. Finally in Section 6 we present several numerical realizations of both positive and negative disclinations.

Here we define the atomic lattice energy $E_{\varepsilon}$ as briefly introduced in the introduction. We start with the kinematics. Inspired by Figure 1, we consider a two-dimensional model. This corresponds conceptually to the mid-section of a 3-dimensional body. Furthermore, we impose rotational symmetry so that we may confine the domain to a single wedge in Figure 1 indicated by the black dots. The reference domain is given by the equilateral open triangle $\Omega\subset\mathbb{R}^{2}$ of size 1 with boundary $\Gamma=\cup_{i=1}^{3}\Gamma_{i}$ as depicted in Figure 2. We take the reference positions of the atoms as a triangular lattice $\mathcal{L}_{\varepsilon}$, where $\mathcal{L}_{\varepsilon}$ is such that it fits on top of $\Omega$ as in Figure 2, i.e., the lattice spacing $\varepsilon\in(0,1)$ is such that $\frac{1}{\varepsilon}\in\mathbb{N}$ and $\mathcal{L}_{\varepsilon}$ is positioned such that each closed line segment $\Gamma_{i}$ fits on top of the atoms in one of the lattice directions. We denote by

the set of the six outward-pointing bonds in $\mathcal{L}_{\varepsilon}$ from any lattice point, where $R_{\theta}\in SO(2)$ is the counter-clockwise rotation matrix by angle $\theta$. For later use, we sometimes interpret $\mathcal{L}_{\varepsilon}$ as a planar graph $\mathcal{G}_{\varepsilon}=(\mathcal{V}_{\varepsilon},\mathcal{E}_{\varepsilon},\mathcal{T}_{\varepsilon})$, where $\mathcal{V}_{\varepsilon}$ is the set of all vertices $x_{i}$, $\mathcal{E}_{\varepsilon}$ is the set of all edges $e_{ij}$ between neighboring vertices $x_{i},x_{j}\in\mathcal{V}_{\varepsilon}$, and $\mathcal{T}_{\varepsilon}$ is the set of all open triangles $T_{ijk}\in\Omega$ with sides given by the three edges $e_{ij},e_{ik},e_{jk}\in\mathcal{E}_{\varepsilon}$.

The set $\mathcal{A}_{\varepsilon}$ of admissible displacements is given by

where $\phi\in\{\frac{2\pi}{5},\frac{2\pi}{7}\}$ is the angle associated with a $5$- or $7$-type disclination (see Figure 1). Since $\mathcal{V}_{\varepsilon}$ is a finite set, the map $u_{\varepsilon}$ can be identified with a vector in $\mathbb{R}^{2|\mathcal{V}_{\varepsilon}|}$. The boundary condition in (2) is such that the rotated copy $R_{\phi}u_{\varepsilon}(\mathcal{L}_{\varepsilon})$ of $u_{\varepsilon}(\mathcal{L}_{\varepsilon})$ fits seamlessly to $u_{\varepsilon}(\mathcal{L}_{\varepsilon})$. Adding more rotated copies, we obtain a deformed lattice with rotational symmetry such as that in Figure 1, which has a $5$- or $7$-type disclination at the origin.

Next we define the lattice energy on $\mathcal{A}_{\varepsilon}$ as inspired by [KM18]. We start with a formal description. Given $u_{\varepsilon}\in\mathcal{A}_{\varepsilon}$, we set the lattice energy formally as

where the potential

penalizes atomic bonds which are not of length $\varepsilon$ (the parameter value $p=2$ corresponds to linear elasticity), the weight function

counts the outer edges as half¹¹1we model edges as part of the volume of the medium around them., and the potential $\Psi$ penalizes change in volume of the triangles $T\in\mathcal{T}_{\varepsilon}$ (here, $|T|$ is the volume of $T$, and $|u_{\varepsilon}(T)|$ is the volume of the triangle $T$ after applying the displacement $u_{\varepsilon}$), especially if the volume of $T$ gets inverted under $u_{\varepsilon}$. Note that while the identity map minimizes $\tilde{E}_{\varepsilon}$, it is not in $\mathcal{A}_{\varepsilon}$ because of the boundary condition. Hence, the boundary condition enforces mechanical frustration. For $p=2$ and $\Psi\equiv 0$, Figure 1 illustrates the deformed lattice of a local minimizer in $\mathcal{A}_{\varepsilon}$ of $\tilde{E}_{\varepsilon}$ which does not contain negative change in volume. In this paper we always assume $p\geq 2$ unless specified otherwise.

Since the term involving $\Psi$ is not derived from physical principles, we elaborate on this modelling choice. It is well-known that when $\Psi\equiv 0$, lattice energies with only nearest-neighbor interactions such as $\tilde{E}_{\varepsilon}$ do not penalize folding or any other negative change in volume, and as a result, the related continuum energy may not penalize compression. While our boundary condition in (2) is not standard, we show in Section 6 by means of numerical simulations that folded patterns appear as local minimizers of $\tilde{E}_{\varepsilon}$ when $\Psi\equiv 0$.

To penalize folding, we pose the following minimal requirements on $\Psi$:

• ($\Psi 1$)

  $\exists\,C>0\ \forall\,a,b\in\mathbb{R}:|\Psi(a)-\Psi(b)|\leq C\Big{(}|a|^{\tfrac{p}{2}-1}+|b|^{\tfrac{p}{2}-1}+1\Big{)}|a-b|$;

• ($\Psi 2$)

  $\Psi\geq 0$, and $\Psi(a)=0\Longleftrightarrow a=1$.

Condition ($\Psi 1$) is a continuity estimate, which implies both local Lipschitz continuity and $\frac{p}{2}$-growth. Condition ($\Psi 2$) ensures that any change in volume is penalized. A simple example of $\Psi$ is $\Psi(a)=|a-1|$. These conditions are more general than those in [KM18, Section 3.3].

The term in $\tilde{E}_{\varepsilon}$ related to $\Psi$ has an inconvenient form. We fix this by changing variables. The result is a rigorously defined energy functional $E_{\varepsilon}$, which we will use in the remainder of the paper.

We change variables by describing the state $u_{\varepsilon}$ as the deformation of a displacement $U_{\varepsilon}$ defined on $\Omega$. Given $u_{\varepsilon}\in\mathcal{A}_{\varepsilon}$, we set $U_{\varepsilon}:\Omega\to\mathbb{R}^{2}$ as the piecewise linear extension of $u_{\varepsilon}$ to $\Omega$, i.e.,

We note that

where $e\in B^{1}$ is any lattice bond of unit direction (see (1)), and the indices $\ell,m\in\{i,j,k\}$ depend on $e$. In particular, $\nabla U_{\varepsilon}$ is constant on each $T\in\mathcal{T}_{\varepsilon}$,

is the relative change in volume of $T$ under $U_{\varepsilon}$, and the sign of the determinant determines whether the volume of $T$ is inverted under $U_{\varepsilon}$. From these observations, we define the second term in (3) rigorously by

Next we rewrite $\mathcal{A}_{\varepsilon}$ and $\tilde{E}_{\varepsilon}$ in terms of $U_{\varepsilon}$. This yields

Using (5) and (6), we rewrite (3) as

where

In the computation above, the weight function $w(e_{ij})$ turns into the factor $\frac{1}{2}$ due to the fact that each edge in the interior of $\Omega$ borders two triangles in $\mathcal{T}_{\varepsilon}$. Eq. (7) motivates us to define

Having introduced the lattice energy $E_{\varepsilon}$ on the triangular lattice, we are now in a position to discuss the continuum limit as $\varepsilon\to 0$. To keep track of the asymptotic behavior of minima and minimizers of $E_{\varepsilon}$ we characterize the continuum model with $\Gamma$-convergence [DM93].

Let $p\geq 2$. The domain of the continuum energy is

Observe that $W_{\phi}^{1,p}(\Omega;\mathbb{R}^{2})$ is linear (and, in particular, convex), and non-empty since $\mathcal{B}_{\varepsilon}^{\phi}\cup W_{0}^{1,p}(\Omega;\mathbb{R}^{2})\subset W_{\phi}^{1,p}(\Omega;\mathbb{R}^{2})$. Moreover, if $U\in W_{\phi}^{1,p}(\Omega;\mathbb{R}^{2})$, there holds $RU\in W_{\phi}^{1,p}(\Omega;\mathbb{R}^{2})$ for any $R\in SO(2)$. Thanks to the properties of traces (see Lemma 4.6 below), we have that $W_{\phi}^{1,p}(\Omega;\mathbb{R}^{2})$ is strongly closed and, therefore, weakly closed as well thanks to convexity.

The continuum energy functional is given by

where $W$ is as in (8) and $QW$ is the quasiconvex envelop of $W$ defined by

where $f$ being quasiconvex means that $f:\mathbb{R}^{2\times 2}\to\mathbb{R}$ is Borel measurable, locally bounded and satisfies

for any bounded open set $\omega\subset\mathbb{R}^{2}$, any $A\in\mathbb{R}^{2\times 2}$ and any $\vartheta\in W^{1,\infty}_{0}(\omega,\mathbb{R}^{2})$.

We prove Theorem 3.1 in Section 3.2. Since the $\varepsilon$-dependence of $E_{\varepsilon}$ appears only in its domain $\mathcal{B}_{\phi}^{\varepsilon}$, proving a $\Gamma$-limit result reduces to proving a relaxation result, i.e., finding the lower semi-continuous envelope of

in the right functional framework. There is a large literature on such relaxation problems; in Section 3.1 we cite the relevant classical theory. While this theory gives a useful roadmap for proving Theorem 3.1, it does not capture Theorem 3.1 because of the periodic boundary condition in $\mathcal{B}_{\varepsilon}^{\phi}$. Therefore, in Section 3.2 we give the skeleton of the proof of Theorem 3.1 based on the classical theory, and identify the missing steps as technical lemmas which we prove in Section 4.

Here we state rigorously and prove the three statements mentioned at the start of Section 3.2. We start with the Poincaré Inequality.