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Piecewise linear constitutive relations for stretch-limited elastic strings

Roger Bustamante, K. R. Rajagopal, Casey Rodriguez

TL;DR

This work introduces a stretch-limiting, piecewise linear constitutive framework for perfectly flexible strings within an implicit elasticity setting. The core idea is a bounded stretch relation $ u= ilde{ u}(N)$ with saturation at $ u_0$ and $ u_1$, and different tangent moduli for extension versus compression, implemented via a piecewise linear form. The authors derive both static (catenary) and dynamic (piecewise-constant stretch with shocks) solutions, providing explicit expressions and conditions (including Rankine–Hugoniot and Lax criteria) for the new model. The results yield concrete, analyzable benchmarks that highlight qualitative differences from classical elastic strings and point to applications in civil engineering and biology, as well as future numerical developments.

Abstract

This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawing from modern advances in constitutive theory for elastic bodies. We investigate strings governed by constitutive relations where stretch is a bounded, piecewise linear function of tension, extending beyond the traditional Cauchy elasticity framework. Our analysis includes explicit solutions for both catenaries and longitudinal, piecewise constant stretched motions.

Piecewise linear constitutive relations for stretch-limited elastic strings

TL;DR

This work introduces a stretch-limiting, piecewise linear constitutive framework for perfectly flexible strings within an implicit elasticity setting. The core idea is a bounded stretch relation with saturation at and , and different tangent moduli for extension versus compression, implemented via a piecewise linear form. The authors derive both static (catenary) and dynamic (piecewise-constant stretch with shocks) solutions, providing explicit expressions and conditions (including Rankine–Hugoniot and Lax criteria) for the new model. The results yield concrete, analyzable benchmarks that highlight qualitative differences from classical elastic strings and point to applications in civil engineering and biology, as well as future numerical developments.

Abstract

This study proposes a simple and novel class of stretch-limiting constitutive relations for perfectly flexible elastic strings drawing from modern advances in constitutive theory for elastic bodies. We investigate strings governed by constitutive relations where stretch is a bounded, piecewise linear function of tension, extending beyond the traditional Cauchy elasticity framework. Our analysis includes explicit solutions for both catenaries and longitudinal, piecewise constant stretched motions.
Paper Structure (15 sections, 69 equations, 2 figures)

This paper contains 15 sections, 69 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Perfectly flexible strings
  3. Kinematics
  4. Balance laws and constitutive theory
  5. A new constitutive relation for elastic strings
  6. A piecewise linear constitutive relation
  7. Boundary value problems
  8. Alternative forms of the field equations
  9. Dimensionless equations
  10. Catenaries
  11. The case of tensile internal forces and inextensibility is not reached.
  12. The case of tensile external forces and inextensibility is reached for a part of the string.
  13. Motions with piecewise constant stretch
  14. Solution of the shock problem
  15. Final remarks

Figures (2)

  • Figure 1: Stretch limiting model for the string.
  • Figure 2: A string under the force of gravity.