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Uniqueness for Some Mixed Problems of Nonlinear Elastostatics

Phoebus Rosakis

TL;DR

The paper addresses the global uniqueness of classical equilibria for nonlinear elastostatic problems with mixed displacement/traction boundary data, focusing on homogeneous deformations. The authors extend Knops–Stuart’s pure displacement result by imposing a Boundary Partition Condition and leveraging Green's Identity, along with convexity assumptions on the stored energy $W$, to show that the homogeneous deformation $y(x)=F_0x$ is the unique equilibrium under suitable isotropy and convexity hypotheses. They develop distinct results for general and isotropic materials, including a mixed displacement/Cauchy-Pressure problem, by demonstrating that equilibria must satisfy strong structural constraints (e.g., constancy of principal stretches or conformality), which force uniqueness in many physically relevant cases. The work also clarifies the necessity of the Partition Condition by providing a nonuniqueness example when it is violated, and discusses implications for bifurcation analyses and boundary-geometric configurations relevant to elastostatic stability and material design.

Abstract

We show that certain mixed displacement/traction problems (including live pressure tractions) of nonlinear elastostatics that are solved by a homogeneous deformation, admit no other classical equilibrium solution under suitable constitutive inequalities and domain boundary restrictions. This extends a well known theorem of Knops and Stuart on the pure displacement problem.

Uniqueness for Some Mixed Problems of Nonlinear Elastostatics

TL;DR

The paper addresses the global uniqueness of classical equilibria for nonlinear elastostatic problems with mixed displacement/traction boundary data, focusing on homogeneous deformations. The authors extend Knops–Stuart’s pure displacement result by imposing a Boundary Partition Condition and leveraging Green's Identity, along with convexity assumptions on the stored energy , to show that the homogeneous deformation is the unique equilibrium under suitable isotropy and convexity hypotheses. They develop distinct results for general and isotropic materials, including a mixed displacement/Cauchy-Pressure problem, by demonstrating that equilibria must satisfy strong structural constraints (e.g., constancy of principal stretches or conformality), which force uniqueness in many physically relevant cases. The work also clarifies the necessity of the Partition Condition by providing a nonuniqueness example when it is violated, and discusses implications for bifurcation analyses and boundary-geometric configurations relevant to elastostatic stability and material design.

Abstract

We show that certain mixed displacement/traction problems (including live pressure tractions) of nonlinear elastostatics that are solved by a homogeneous deformation, admit no other classical equilibrium solution under suitable constitutive inequalities and domain boundary restrictions. This extends a well known theorem of Knops and Stuart on the pure displacement problem.

Paper Structure

This paper contains 10 sections, 9 theorems, 84 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The Mixed Displacement/Dead-Load Problem
  4. Isotropic Materials
  5. The Mixed Displacement/Cauchy-Pressure Problem
  6. Remarks and Examples
  7. Implications
  8. Partition Condition Examples
  9. Points of Convexity of the Stored Energy
  10. Partition Condition Omission and Nonuniqueness

Key Result

Lemma 2.2

(Green's Identity) If $y$ is a classical equilibrium, then

Figures (1)

  • Figure 1: Examples of Partition Condition satisfaction. Here ${\Omega}$ is shown grey. The heavy black portion of the boundary is ${\mathcal{D}}$, the remaining part of the boundary between grey and white is ${\mathcal{L}}$. The black dot is a possible choice of the origin so that the Partition Condition inequalities in Definition \ref{['coo']} hold. In all cases except (b,c,d,e), a revolution about a vertical axis through the origin produces a region in $\mathbb{R}^3$ with the Partition Condition in force. In (f,g,h) the cone boundary points can belong to either ${\mathcal{D}}$ or ${\mathcal{L}}$. In (c) a solid cube in $\mathbb{R}^3$ with ${\mathcal{D}}$ consisting of 3 faces that share a vertex is also admissible. The tapered column in (g) can be made arbitrarily slender for a fixed height and the taper can be made as small as desired.

Theorems & Definitions (22)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Definition 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof
  • Proposition 3.5
  • proof
  • ...and 12 more