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New approach for elastic collisions with singular stress functions

Toyohiko Aiki, Chiharu Kosugi

TL;DR

The paper introduces a new one‑dimensional beam‑equation model for elastic collisions with singular stress functions to capture large deformations and obstacle contact. By combining a Signorini‑type boundary treatment with singular stresses $f$ and $\sigma_b$, and by formulating the system $P(\mu,u_0,v_0,f,\sigma_b)$, the authors establish existence and uniqueness results for strong and weak solutions, depending on the viscosity parameter $\mu$. The analysis uses linear auxiliary problems, truncation of nonlinearities, energy estimates, and fixed‑point arguments, with careful handling of the singular terms via (A1) and (AS)/(AW). The no‑viscosity limit ($\mu=0$) is obtained through compactness from the viscous case, yielding a robust weak solution framework for the Signorini‑type obstacle problem in elastic collisions. The work provides a rigorous foundation for modeling energy dissipation and restitution phenomena in flexible bodies encountering rigid boundaries, complemented by numerical observations of periodic vs. decaying behavior dependent on $\mu$.

Abstract

A collision of a rubber rod to a hard floor is regarded as a simple example of obstacle problems for elastic material. In this article we have proposed a new mathematical model for the collision phenomenon by applying beam equations with singular stress functions, which is investigated in our recent works. As in the works we have established a mathematical method to deal with the singular stress function. Here, we demonstrate the validity of our modeling through observation to the numerical results. Also, we present existence and uniqueness results of the model given as initial boundary value problems.

New approach for elastic collisions with singular stress functions

TL;DR

The paper introduces a new one‑dimensional beam‑equation model for elastic collisions with singular stress functions to capture large deformations and obstacle contact. By combining a Signorini‑type boundary treatment with singular stresses and , and by formulating the system , the authors establish existence and uniqueness results for strong and weak solutions, depending on the viscosity parameter . The analysis uses linear auxiliary problems, truncation of nonlinearities, energy estimates, and fixed‑point arguments, with careful handling of the singular terms via (A1) and (AS)/(AW). The no‑viscosity limit () is obtained through compactness from the viscous case, yielding a robust weak solution framework for the Signorini‑type obstacle problem in elastic collisions. The work provides a rigorous foundation for modeling energy dissipation and restitution phenomena in flexible bodies encountering rigid boundaries, complemented by numerical observations of periodic vs. decaying behavior dependent on .

Abstract

A collision of a rubber rod to a hard floor is regarded as a simple example of obstacle problems for elastic material. In this article we have proposed a new mathematical model for the collision phenomenon by applying beam equations with singular stress functions, which is investigated in our recent works. As in the works we have established a mathematical method to deal with the singular stress function. Here, we demonstrate the validity of our modeling through observation to the numerical results. Also, we present existence and uniqueness results of the model given as initial boundary value problems.
Paper Structure (4 sections, 7 theorems, 40 equations, 5 figures)

This paper contains 4 sections, 7 theorems, 40 equations, 5 figures.

Table of Contents

  1. Modeling
  2. Main result
  3. Problem with viscosity
  4. Problem without viscosity

Key Result

Theorem 2.1

Let $\gamma > 0$, $g \in {\mathbb R}$ and assume (A1). (1) Let $\mu > 0$ and assume (AS): Then the problem P$(\mu, u_0, v_0, f , \sigma_b)$ has a strong solution on $[0,T]$. (2) Let $\mu > 0$. If the following condition (AW) holds, then the problem P$(\mu, u_0, v_0, f, \sigma_b)$ has a unique weak solution on $[0,T]$. (3) If (AW) holds, then the problem P$(0, u_0, v_0, f, \sigma_b)$ has a u

Figures (5)

  • Figure 1: Collision of a rubber rod
  • Figure 2: Stress function $f$
  • Figure 3: $\sigma_b$
  • Figure 4: $\mu = 0$.
  • Figure 5: $\mu = 100$.

Theorems & Definitions (18)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.1
  • Definition 2.3
  • Theorem 2.1
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 3.1
  • Lemma 3.2
  • proof
  • ...and 8 more