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Semidiscrete Modeling of Dislocation-Disclination Systems: Finite Element Formulation

Pierluigi Cesana, Edoardo Fabbrini, Andrés A. León Baldelli, Marco Morandotti

arXiv:2609.29614v1math.NAcond-mat.mtrl-scics.NAmath.AP

TL;DR

Modeling interacting dislocations and disclinations is challenging because their mechanical fields span multiple length scales. This paper uses a finite-element Airy-potential formulation and reusable cell problems; its simulations examine how defect arrangements and geometry affect stress and energy.

  • Problem

    Modeling interacting dislocations and disclinations is challenging because their associated mechanical fields span a wide range of length scales and magnitudes.

    Source: §1 · p. 2

  • Method

    The model solves cell problems for defect fundamental solutions, then uses linear superposition in postprocessing to construct configurations in a finite-element framework.

    Source: Conclusion · p. 19

  • Results

    The examined configurations show that energy minima and maxima depend on defect arrangement and geometry.

    Source: Conclusion · p. 20

  • Takeaways & Limitations

    Once cell problems are solved for a geometry and topology, interactions among defects of any intensity, orientation, and type follow through postprocessing.

    Source: Conclusion · p. 19

    The present model is two-dimensional and planar-strain; nonlinear models and plate theories are identified as future extensions.

    Source: §2 · p. 4, Conclusion · p. 20

Abstract

from arXiv · show

We present a numerical formulation for the resolution of finite systems of interacting edge dislocations and wedge disclinations. The approach is based on a finite element discretization of a fourth-order elliptic boundary value problem arising from the mechanical equilibrium equations of plane-strain linear elasticity in the presence of kinematic incompatibilities. The numerical implementation follows a continuous interior penalty discontinuous Galerkin framework and relies on the solution of a finite set of local cell problems. Our method is validated against analytical benchmarks and used to explore several non-trivial dislocation-disclination configurations, illustrating how translational and rotational incompatibilities shape the stress state of the body.

1 Introduction

Interacting dislocations and disclinations shape material behavior, but modeling their multiscale stress fields is challenging. This paper develops a variational finite-element framework using Airy-potential cell problems to represent their interactions.

1 Introduction

  • Dislocations and disclinations interact in material phenomena including kinking, grain boundaries, and crack-tip plasticity, while their fields span challenging length scales and magnitudes.Dislocations are translational defects associated with infinite elastic energy; disclinations represent rotational lattice mismatches and have finite energy.Source: §1 · p. 2
  • The finite-element implementation solves cell problems for a fourth-order elliptic problem under linearized plane-strain isotropic elasticity, with singular stresses regularized by a core radius ε.The core radius represents the region around a defect where the continuum model breaks down.Source: §1 · p. 2
  • The method handles finite combinations of Volterra edge dislocations and wedge disclinations, treating them as idealized isolated geometric defects rather than statistically stored defects.The paper explores non-trivial constructive and destructive interactions.Source: §1 · pp. 2–3
  • A variational formulation uses the Airy stress potential as a scalar unknown to account for both traction-free boundaries and kinematic incompatibility.The formulation extends to a class of Lipschitz-type domains suited to finite elements and replaces the usual two-step Airy implementation described in engineering literature.Source: §1 · p. 3
  • After solving 3N cell problems for a given geometry, arbitrary configurations of K disclinations and J dislocations can be generated during postprocessing without rerunning the finite-element problem.Here N = K + J counts the defect cores.Source: §1 · p. 3
  • The paper presents the analytical well-posedness result and cell formulas before describing the numerical implementation and simulations of interacting configurations.The simulations examine how defect interactions shape stress states.Source: §1 · p. 3

2 Preliminaries and analytical model

The analytical model represents defect-induced incompatibilities in linearized plane-strain elasticity through an Airy stress potential. Core-radius regularization and a cell-problem characterization provide the basis for the numerical method.

2 Preliminaries and analytical model

  • Dislocations and disclinations enter linearized planar-strain elasticity through Burgers vectors and Frank angles, representing translational and rotational incompatibilities.The model considers edge dislocations and wedge disclinations at distinct points in a two-dimensional domain.Source: §2 · p. 4
  • The Airy potential converts equilibrium into a scalar biharmonic variational problem, with traction-free stress expressed as a boundary condition on its Hessian.The potential also encodes the incompatibility measure in the Euler–Lagrange equation.Source: §2 · pp. 4–5
  • Removing disjoint ε-radius cores regularizes defect stress singularities, and the resulting equilibrium configurations have smooth stress fields.The core boundaries approximate the defect measures in the regularized functional.Source: §2 · pp. 5–6
  • The variational theory extends to globally Lipschitz, piecewise-C2 domains, including polygonal domains, beyond the earlier smooth-boundary setting.The theorem's extension to weaker boundary regularity is established using an appendix proposition.Source: §2 · pp. 6, 9
  • The theorem characterizes a unique Airy-potential equilibrium using 3N cell problems, with stress/strain equivalence requiring the stated solution regularity.The cell problems provide the fundamental solutions used to reconstruct the potential.Source: §2 · pp. 7–8
  • For fixed geometry and core radius, solving 3N cell problems lets arbitrary Burgers vectors and Frank angles be reconstructed by finite-dimensional postprocessing.This decouples geometry from defect data and avoids solving the finite-element problem again for each configuration.Source: §2 · p. 9

3 Numerical formulation

The formulation reuses geometry-dependent cell problems to construct Airy stress potentials for defect configurations, using a C0 interior penalty discontinuous Galerkin method. Numerical solutions are compared with analytical benchmarks, including isolated defects and interacting disclinations.

3 Numerical formulation

  • For fixed geometry and core radius, solve 3N cell problems once, then reconstruct potentials for arbitrary Burgers vectors and Frank angles in postprocessing.The cell problems depend on defect locations and ε, not defect type or magnitude.Source: §3 · p. 9
  • C0–IPDG discretizes the fourth-order biharmonic problem with C0 elements and gradient-jump penalties, avoiding specialized H2-conforming elements.The penalty term accounts for jumps in shape-function gradients across mesh facets.Source: §3 · p. 9

3.1 Validation of the numerical implementation

  • Validation uses the unit disk, ν=0.15, E=1, third-order polynomial shape functions, and penalty parameter α=10.The penalty parameter was selected by trial and error.Source: §3.1 · p. 11
  • Affine boundary conditions for each Airy-function problem guarantee zero normal stress, which the simulations verify systematically.The zero-stress condition is checked in Figure 1.Source: §3.1 · p. 11
  • Numerical cell-formula solutions agree with exact solutions in energy norms and pointwise values across core radii ε∈(0,1).Figure 1 presents the analytical and numerical comparison.Source: §3.1 · p. 11, Figure 1 · p. 12
  • For an isolated disclination, ε of order 10^-2 or smaller accurately captures continuum behavior, while stress and Airy-potential magnitudes decrease toward the boundary.The benchmarks compare analytical and numerical solutions for different disclination positions.Source: §3.1 · pp. 12–13, Figure 2 · p. 13
  • For opposite-sign disclinations, the numerical model reproduces the stress field at ε=0.01; stress and mechanical energy tend to vanish near the boundary or upon coincidence at the origin.Coincident disclinations annihilate.Source: §3.1 · p. 13, Figure 3 · p. 14
  • As ε→0, the edge-dislocation Airy potential approaches the analytical expression, while squared stress and mechanical energy diverge proportionally to |log ε|.The right panel compares numerical profiles with the analytical solution and shows stress norms for different ε.Source: §3.1 · p. 14

4 Illustration of complex interactions

The simulations probe how defect arrangement, orientation, and cavity size shape stress profiles across four configurations. They show localized or long-range interactions and changing stress screening with geometry.

4.1 Stack of edge dislocations and isolated wedge disclination

  • For the isolated-disclination configuration, stress peaks near d ≈ 0.075 in case (i) and d ≈ 0.05 when the middle Burgers vector is reversed; negative Frank angle gives monotone stress behavior.The negative disclination tends to separate from the stack, and the interaction is localized when the dislocation spacing is comparable to the disclination–stack separation.Source: §4.1 · pp. 15–16, Figure 4 · p. 15

4.2 Stack of edge dislocations and dipole of wedge disclinations

  • A dislocation pile-up and disclination dipole interact over long ranges, unlike the strongly localized stress concentrations found with a single disclination.Cases (i) and (ii) peak at d ≈ 0.60 and d ≈ 0.55; inverted dipole orientations instead produce a broad plateau or monotone decrease.Source: §4.2 · pp. 16–17, Figure 5 · p. 16

4.3 Disclination between a stack of dislocations

  • Oppositely oriented Burgers vectors partially screen each other: the symmetric configuration has its stress maximum at the origin and is less stressed overall than the same-direction cases.In the symmetric case, the disclination core aligns with the two dislocation cores.Source: §4.3 · p. 17
  • Stress magnitude falls toward a minimum as the disclination approaches the boundary and moves farther from the dislocation pair; normal stress vanishes at the boundary.Tangential boundary stress components can remain significant when the disclination is central or intermediate.Source: §4.3 · p. 17

4.4 Interaction of disclinations with a cavity

  • With a cavity, each stress curve has a critical disclination separation: below it, smaller separations are favored; above it, larger separations are favored.For large separations near the domain boundaries, stress curves share the same trend across cavity radii.Source: §4.4 · pp. 18–19
  • As cavity radius R increases, the maximum normalized stress rises while screening between opposite-angle disclinations weakens; at R = 0.3, they behave as isolated defects in disconnected domains.The right panel shows stress fields at the maximizing separation for each R.Source: §4.4 · p. 19, Figure 7 · p. 19

Conclusion

The model resolves dislocation–disclination systems in Lipschitz-regular domains using core-regularized finite element cell problems and linear superposition. The examined configurations show that defect arrangements and geometry affect energy extrema, while translational and rotational incompatibilities influence stress states.

Conclusion

  • The framework represents defects through incompatibilities prescribed on ε-scale cores and combines cell-problem solutions by linear superposition.The model addresses finite systems in domains with Lipschitz-regular boundaries.Source: Conclusion · p. 19
  • After cell problems are solved for a given geometry and topology, defect interactions across intensities, orientations, and types follow through postprocessing.Source: Conclusion · p. 19
  • Energy minima and maxima in the examined configurations depend on defect arrangement and geometry.The simulations also highlight how translational and rotational incompatibilities influence the system’s stress states.Source: Conclusion · p. 20
  • The authors identify nonlinear large-deformation models and extensions to plate theories as future developments.Further directions include studying plasticity, damage, fracture, fragmentation, and coupled defect dynamics.Source: Conclusion · p. 20

A Technical results

The appendix collects technical results needed in the proof of Theorem 2.2.

A Technical results

A.1 Equivalence of boundary conditions for piecewise-C2 Lipschitz domains

For a piecewise-C2 domain and a C2 function, the boundary condition involving the tangential Hessian is equivalent to an affine-trace condition on each connected boundary component.

A.1 Equivalence of boundary conditions for piecewise-C2 Lipschitz domains

  • On each connected boundary component, the condition ∇²v t = 0 is equivalent to v being the trace of an affine function.The stated setting is a piecewise-C2 domain and v ∈ C2(A).Source: §A.1 · pp. 20–21
  • The proposition expresses the same boundary condition as v = a_q and ∂_n v = ∂_n a_q on each component, for affine functions a_q.Source: §A.1 · pp. 20–21
  • The equivalence applies separately to every connected component Γ_q of the boundary.Source: §A.1 · p. 20

A.2 A symmetry property of the Monge-Ampère operator

The appendix extends a Monge–Ampère-related lemma to Lipschitz boundaries and H2-regular functions, and establishes symmetry of the indicated expressions.

A.2 A symmetry property of the Monge-Ampère operator

  • The section extends a cited lemma to Lipschitz boundaries and functions with H2 regularity.Source: §A.2 · p. 21
  • The expressions in (A.8) are symmetric in η and χ.The passage states that both the first and last expressions are symmetric.Source: §A.2 · p. 22
  • The proof uses that Div(cof(∇²ψ_m)) = 0 pointwise in Ω_ε.Source: §A.2 · p. 22
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