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Regularity of Solutions for Peridynamics Equilibrium and Evolution Equations on Periodic Distributions

Thinh Dang, Bacim Alali, Nathan Albin

Abstract

Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for a homogeneous material. Results for weakly singular (integrable) as well as singular integral kernels are developed. The asymptotic behavior of the eigenvalues of the peridynamic operator's Fourier multipliers and eigenvalues are characterized explicitly in terms of the nonlocality (peridynamic horizon), the integral kernel singularity, and the spatial dimension. We build on the asymptotic analysis to develop regularity of solutions results for the peridynamic equilibrium as well as the peridynamic evolution equations over periodic distribution. The regularity results are presented explicitly in terms of the data, the integral kernel singularity, and the spatial dimension. Nonlocal-to-local convergence results are presented for the eigenvalues of the peridynamic operator and for the solutions of the equilibrium and evolution equations. The local limiting behavior is shown for two types of limits as the peridynamic horizon (nonlocality) vanishes or as the integral kernel becomes hyper-singular.

Regularity of Solutions for Peridynamics Equilibrium and Evolution Equations on Periodic Distributions

Abstract

Results on the peridynamics equilibrium and evolution equations over the space of periodic vector-distributions in multi-spatial dimensions are presented. The associated operator considered is the linear state-based peridynamic operator for a homogeneous material. Results for weakly singular (integrable) as well as singular integral kernels are developed. The asymptotic behavior of the eigenvalues of the peridynamic operator's Fourier multipliers and eigenvalues are characterized explicitly in terms of the nonlocality (peridynamic horizon), the integral kernel singularity, and the spatial dimension. We build on the asymptotic analysis to develop regularity of solutions results for the peridynamic equilibrium as well as the peridynamic evolution equations over periodic distribution. The regularity results are presented explicitly in terms of the data, the integral kernel singularity, and the spatial dimension. Nonlocal-to-local convergence results are presented for the eigenvalues of the peridynamic operator and for the solutions of the equilibrium and evolution equations. The local limiting behavior is shown for two types of limits as the peridynamic horizon (nonlocality) vanishes or as the integral kernel becomes hyper-singular.

Paper Structure

This paper contains 19 sections, 38 theorems, 199 equations.

Table of Contents

  1. Introduction
  2. Linear peridynamic operators
  3. Eigenvalues' asymptotic analysis
  4. Local limits of the multipliers and the peridynamic operator
  5. The Sobolev space of periodic distributions
  6. The peridynamic equilibrium equation
  7. The peridynamic initial-value problem
  8. Spatial and temporal regularity over periodic distributions
  9. Convergence to the transient Navier-Cauchy equation
  10. Spatial and temporal regularity over
  11. Functions with absolutely summable Fourier coefficients
  12. Temporal regularity with respect to the Gateaux derivative
  13. Temporal regularity with respect to the classical derivative
  14. The peridynamic equation with a forcing term
  15. Spatial and temporal regularity over periodic distributions
  16. ...and 4 more sections

Key Result

Theorem 1

Expressing $\mathbf{u}$ through its Fourier transform as we have the representations where and In addition, the multipliers $M^{\delta,\beta}$ of $\mathcal{L}^{\delta,\beta}$ is given by $M^{\delta,\beta}=M_b+M_s$, which satisfies $\widehat{\mathcal{L}^{\delta,\beta}\mathbf{u}}=M^{\delta,\beta}\hat{\mathbf{u}}$.

Theorems & Definitions (65)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 7
  • proof
  • Theorem 8
  • ...and 55 more