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Comparison between solutions to the linear peridynamics model and solutions to the classical wave equation

Giuseppe Maria Coclite, Serena Dipierro, Francesco Maddalena, Gianluca Orlando, Enrico Valdinoci

Abstract

In this paper, we consider an equation inspired by linear peridynamics and we establish its connection with the classical wave equation. In particular, given a horizon $δ>0$ accounting for the region of influence around a material point, we prove existence and uniqueness of a solution $u_δ$ and demonstrate the convergence of $u_δ$ to solutions to the classical wave equation as $δ\to 0$. Moreover, we prove that the solutions to the peridynamics model with small frequency initial data are close to solutions to the classical wave equation.

Comparison between solutions to the linear peridynamics model and solutions to the classical wave equation

Abstract

In this paper, we consider an equation inspired by linear peridynamics and we establish its connection with the classical wave equation. In particular, given a horizon accounting for the region of influence around a material point, we prove existence and uniqueness of a solution and demonstrate the convergence of to solutions to the classical wave equation as . Moreover, we prove that the solutions to the peridynamics model with small frequency initial data are close to solutions to the classical wave equation.

Paper Structure

This paper contains 20 sections, 17 theorems, 172 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Basic notation and functional setting
  3. Test functions and distributions
  4. Fourier transform
  5. Fractional Sobolev spaces
  6. The peridynamics operator
  7. Peridynamics horizon
  8. Peridynamics operator
  9. Limit as $\delta \to 0$ of the operator
  10. Dual operator and symmetry
  11. Symbol of the peridynamics operator
  12. Energy space
  13. Well-posedness of the linear peridynamics model with low regularity
  14. The linear peridynamics model
  15. Total energy
  16. ...and 5 more sections

Key Result

Theorem 1.1

Let $s \in \mathbb{R}$. Let $u_0 \in H^s(\mathbb{R}^d;\mathbb{R}^d)$ and $v_0 \in H^{s-\alpha}(\mathbb{R}^d;\mathbb{R}^d)$. Then, there exists $u_\delta \in C([0,+\infty);H^s(\mathbb{R}^d;\mathbb{R}^d)) \cap C^1((0,+\infty);H^{s-\alpha}(\mathbb{R}^d;\mathbb{R}^d))$ such that, for every $t \in [0,+\i Moreover, $u_\delta$ is a distributional solution to the linear peridynamics model in eq:linear_per

Figures (1)

  • Figure 1: Aspect of the dispersion relation $\omega_\delta(\xi)$ and comparison with the dispersion relation $\gamma |\xi|$ of the wave equation with speed of propagation $\gamma$.

Theorems & Definitions (51)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 1.5
  • Remark 3.1
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Lemma 3.5
  • ...and 41 more