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Variational formulation of planar linearized elasticity with incompatible kinematics

Pierluigi Cesana, Edoardo Fabbrini, Marco Morandotti

Abstract

We present a variational characterization of mechanical equilibrium in the planar strain regime for systems with incompatible kinematics. For non-simply connected domains, we show that the equilibrium problem for a non-liftable strain-stress pair can be reformulated as a well-posed minimization problem for the Airy potential of the system. We characterize kinematic incompatibilities on internal boundaries as rotational or translational mismatches, in agreement with Volterra's modeling of disclinations and dislocations. Finally, we establish that the minimization problem for the Airy potential can be reduced to a finite-dimensional optimization involving cell formulas.

Variational formulation of planar linearized elasticity with incompatible kinematics

Abstract

We present a variational characterization of mechanical equilibrium in the planar strain regime for systems with incompatible kinematics. For non-simply connected domains, we show that the equilibrium problem for a non-liftable strain-stress pair can be reformulated as a well-posed minimization problem for the Airy potential of the system. We characterize kinematic incompatibilities on internal boundaries as rotational or translational mismatches, in agreement with Volterra's modeling of disclinations and dislocations. Finally, we establish that the minimization problem for the Airy potential can be reduced to a finite-dimensional optimization involving cell formulas.

Paper Structure

This paper contains 13 sections, 18 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Compatibility in non-simply connected domains: stress-strain formulation
  4. Compatibility in non-simply connected domains: Airy potential formulation
  5. Modeling disclinations and dislocations via kinematic incompatibility
  6. Core radius method
  7. Main results
  8. Orthogonal decomposition of $\mathop{\hbox{$\mathscr{C}\!\!$\calligra,,}}\nolimits(\Omega_\varepsilon(\mathsf{D}))$ and cell formulas
  9. Mechanical equilibrium in the Airy potential formulation
  10. Mechanical equilibrium in the stress-strain formulation
  11. Some useful results
  12. Proof of Proposition \ref{['prop_boundary_cond_incomp']}
  13. Useful properties of the Monge-Ampère operator

Key Result

Proposition 2.1

Let $\Omega\subset\mathbb{R}^2$ be as in eq_Omega and let $\epsilon\in L^2(\Omega;\mathbb{R}^{2\times 2}_{\mathrm{sym}})$. Then eq_SVincstrong holds in $H^{-2}(\Omega)$ if and only if there exists $u\in H^1(\Omega;\mathbb{R}^2)$ such that $\epsilon =\nabla^{\mathrm{sym}}u$. Moreover, $u$ is unique u

Theorems & Definitions (37)

  • Proposition 2.1
  • Remark 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4: properties of $v^p$
  • Proposition 2.5: properties of $\epsilon^p$
  • proof
  • Theorem 3.1
  • proof
  • Remark 3.2
  • ...and 27 more