[Paper Review] On structured surfaces with defects: geometry, strain incompatibility, internal stress, and natural shapes

Ayan Roychowdhury, Anurag Gupta|arXiv (Cornell University)|Feb 13, 2017

Advanced Materials and Mechanics50 references3 citations

TL;DR

This paper develops a differential geometric framework to model defects—translational, rotational, and metric—in 2D structured surfaces such as graphene, liquid crystals, and thin shells. By treating defect densities as sources of strain incompatibility and combining kinematic decomposition with elastic constitutive laws, the theory predicts internal stress fields and natural deformed shapes, yielding a unified formulation for Kirchhoff-Love shells with small in-plane strains and moderate rotations.

ABSTRACT

Given a distribution of defects on a structured surface, such as those represented by 2-dimensional crystalline materials, liquid crystalline surfaces, and thin sandwiched shells, what is the resulting stress field and the deformed shape? Motivated by this concern, we first classify, and quantify, the translational, rotational, and metrical defects allowable over a broad class of structured surfaces. With an appropriate notion of strain, the defect densities are then shown to appear as sources of strain incompatibility. The strain incompatibility relations, with appropriate kinematical assumptions on the decomposition of strain into elastic and plastic parts, and the stress equilibrium relations, with a suitable choice of material response, provide the necessary equations for determining both the internal stress field and the deformed shape. We demonstrate this by applying our theory to Kirchhoff-Love shells with a kinematics which allows for small in-surface strains but moderately large rotations.

Motivation & Objective

• To establish a unified geometric and mechanical framework for analyzing defects in 2D structured surfaces, including 2D crystals, liquid crystals, and thin shells.
• To classify and quantify translational (dislocations), rotational (disclinations), and metrical (vacancies, growth) defects on structured surfaces using differential geometry.
• To derive the relationship between defect densities and strain incompatibility, enabling prediction of internal stress and deformed shape.
• To extend classical plate theories to include continuous distributions of defects by incorporating plastic strain and curvature effects.
• To provide a systematic method for determining the natural (stress-free) shape and internal stress field of defective 2D surfaces under kinematic and constitutive assumptions.

Proposed method

• Uses non-Euclidean differential geometry to model defects as geometric incompatibilities in the surface metric, connection, and curvature.
• Introduces defect densities via the in-surface torsion (dislocations), curvature (disclinations), and non-metricity (metric anomalies) tensors.
• Applies strain decomposition into elastic and plastic parts, with strain incompatibility arising from defect sources in the Riemann curvature tensor.
• Employs the Föppl-von Kármán equations for thin shells by incorporating defect-induced plastic strain fields into the governing equations.
• Derives the Airy stress function formulation for 2D stress equilibrium, linking stress components to the biharmonic operator and defect sources.
• Solves the boundary-value problem using the compatibility conditions for total strain and the constitutive relations for bending and membrane stiffness.

Experimental results

Research questions

• RQ1 How can translational, rotational, and metrical defects be systematically classified and quantified on structured 2D surfaces using differential geometry?
• RQ2 What is the geometric origin of strain incompatibility in defective 2D surfaces, and how is it related to defect densities?
• RQ3 How do defect distributions influence the internal stress field and the natural (stress-free) deformed shape of thin elastic surfaces?
• RQ4 Can the Föppl-von Kármán equations be generalized to include continuous distributions of defects via plastic strain and curvature sources?
• RQ5 How do metric anomalies from growth or thermal effects contribute to shape instabilities and residual stresses in 2D materials?

Key findings

• Defects such as dislocations, disclinations, and metric anomalies are shown to act as sources of strain incompatibility through the Riemann curvature tensor.
• The strain incompatibility equations are derived from the Bianchi-Padova identities and are expressed in terms of defect density fields: torsion, curvature, and non-metricity.
• For Kirchhoff-Love shells, the governing equations reduce to the Föppl-von Kármán equations with defect-dependent source terms: $Dw_{,etaetaetaeta} + [w, heta] = D ilde{ heta}_p$ and $rac{1}{E} heta_{,etaetaetaeta} - rac{1}{2}[w,w] = - ilde{ heta}_p$.
• In membranes with zero bending stiffness, the natural shape is governed by the non-homogeneous Monge-Ampère equation: $[w,w] = 2 ilde{ heta}_p$, where $ ilde{ heta}_p$ is the plastic strain contribution.
• The theory successfully predicts the natural shape and internal stress for defective 2D materials such as graphene and lipid membranes, including cases with growth-induced metric anomalies.
• The framework is extendable to global topological defects (e.g., toroidal or Möbius surfaces) and non-convex material responses, enabling modeling of wrinkle and phase transformation patterns.