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A midsurface elasticity model for a thin, nonlinear, gradient elastic plate

C. Rodriguez

TL;DR

This work derives a rigorous dynamic surface elasticity model for the midsurface of a thin gradient-elastic plate by projecting a three-dimensional gradient-elastic theory onto the two-dimensional surface. The key result is a leading cubic-in-$h$ surface energy density $U$ and a surface kinetic energy density $K$ that depend on five physical properties $\rho_R, E, \nu, h, d$ and on length scales $\ell_s = d/\sqrt{12}$ and $\ell_k = d/\sqrt{6}$, with $U$ satisfying strong ellipticity for $\ell_s>0$. In the limiting cases $\ell_s=0$ or $\ell_k=0$, the model recovers Koiter's plate energy and Hilgers-Pipkin kinetic energy, respectively, while plane-wave analysis reveals dispersive wave behavior and bounded longitudinal phase velocities. Importantly, the stored energy $U$ enables a fracture model on gradient-elastic boundaries that removes classical singularities, providing a robust framework for crack-front modeling in gradient-elastic materials.

Abstract

In this paper, we derive a dynamic surface elasticity model for the two-dimensional midsurface of a thin, three-dimensional, homogeneous, isotropic, nonlinear gradient elastic plate of thickness $h$. The resulting model is parameterized by five, conceivably measurable, physical properties of the plate, and the stored surface energy reduces to Koiter's plate energy in a singular limiting case. The model corrects a theoretical issue found in wave propagation in thin sheets and, when combined with the author's theory of Green elastic bodies possessing gradient elastic material boundary surfaces, removes the singularities present in fracture within traditional/classical models. Our approach diverges from previous research on thin shells and plates, which primarily concentrated on deriving elasticity theories for material surfaces from classical three-dimensional Green elasticity. This work is the first in rigorously developing a surface elasticity model based on a parent nonlinear gradient elasticity theory.

A midsurface elasticity model for a thin, nonlinear, gradient elastic plate

TL;DR

This work derives a rigorous dynamic surface elasticity model for the midsurface of a thin gradient-elastic plate by projecting a three-dimensional gradient-elastic theory onto the two-dimensional surface. The key result is a leading cubic-in- surface energy density and a surface kinetic energy density that depend on five physical properties and on length scales and , with satisfying strong ellipticity for . In the limiting cases or , the model recovers Koiter's plate energy and Hilgers-Pipkin kinetic energy, respectively, while plane-wave analysis reveals dispersive wave behavior and bounded longitudinal phase velocities. Importantly, the stored energy enables a fracture model on gradient-elastic boundaries that removes classical singularities, providing a robust framework for crack-front modeling in gradient-elastic materials.

Abstract

In this paper, we derive a dynamic surface elasticity model for the two-dimensional midsurface of a thin, three-dimensional, homogeneous, isotropic, nonlinear gradient elastic plate of thickness . The resulting model is parameterized by five, conceivably measurable, physical properties of the plate, and the stored surface energy reduces to Koiter's plate energy in a singular limiting case. The model corrects a theoretical issue found in wave propagation in thin sheets and, when combined with the author's theory of Green elastic bodies possessing gradient elastic material boundary surfaces, removes the singularities present in fracture within traditional/classical models. Our approach diverges from previous research on thin shells and plates, which primarily concentrated on deriving elasticity theories for material surfaces from classical three-dimensional Green elasticity. This work is the first in rigorously developing a surface elasticity model based on a parent nonlinear gradient elasticity theory.
Paper Structure (20 sections, 3 theorems, 119 equations, 2 figures)

This paper contains 20 sections, 3 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Classical thin plates and shells
  3. Gradient elasticity
  4. Main results and outline
  5. Preliminaries
  6. Kinematics
  7. Stored energy and kinetic energy assumptions
  8. Identification of the length parameters via averaging lattice dynamics
  9. Surface energies obtained as the leading order expressions from the plate energies
  10. Size assumption between the length parameters and plate thickness
  11. Parameterizing the motions
  12. Leading cubic order-in-$h$ expressions for the surface energies
  13. Strong ellipticity of the stored surface energy
  14. Surface dynamics associated to the energies $U$ and $K$
  15. Field equations
  16. ...and 5 more sections

Key Result

Theorem 3.1

Let $C_0, C_1$, and $C_2$ be fixed positive numbers, and assume that Let $\hat{\boldsymbol{\chi}} \in C^3(\mathcal{B} \times [t_0, t_1])$ be a motion such that for each $t \in [t_0, t_1]$, $\hat{\boldsymbol{\chi}}(\cdot,t)$ is an immersion satisfying eq:chiexp, eq:Erelations, eq:p3Erelations. Assume that $\hat{\boldsymbol{\chi}}$ satisfies the a priori bounds where $\boldsymbol{N} = \boldsymbol{

Figures (2)

  • Figure 1: A body primarily made of a Green elastic material with an additional thin, gradient elastic region of thickness $h$ extending from a section of its boundary and our modeling scheme.
  • Figure 2: The set-up for the mode-III problem with the crack $\mathcal{C}$ appearing in blue.

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Proposition 3.2
  • proof
  • Theorem 5.1: Theorem 4.4, Rodriguez2023StrainGradient