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Irreversibility condition and stability of equilibria in the inverse-deformation approach to fracture

Arnav Gupta

TL;DR

This paper derives a thermodynamically enforced irreversibility condition for fracture within the inverse-deformation framework, showing crack motion in the material reference would produce negative entropy and is thus forbidden. It develops a rigorous second-order stability theory that accommodates the unilateral constraint $h'>0$ and the irreversibility constraint, and validates that all broken equilibria found in prior work are locally stable. The approach combines variational analysis with a gradient-regularized energy to capture surface energy and uses numerical FE/active-set methods to corroborate analytical predictions, including multiple-crack configurations. The results clarify how irreversibility shapes admissible variations and transitions under incremental loading, and point to potential extensions to higher dimensions and crack-tip dynamics.

Abstract

We derive the irreversibility condition in fracture for the inverse-deformation approach using the second law of thermodynamics. We consider the problem of brittle failure in an elastic bar previously solved in (Rosakis et al 2021). Despite the presence of a non-zero interfacial/surface energy, the third derivative of the inverse-deformation map is discontinuous at the crack faces. This is due to the presence of the inequality constraint ensuring the inverse strain is nonnegative and the orientation of matter is preserved. A change in the material location of a crack results in negative entropy production, violating the second law. Consequently, such changes are disallowed giving the irreversibility condition. The inequality constraint and the irreversibility condition limit the space of admissible variations. We prove necessary and sufficient conditions for local stability that incorporate these restrictions. Their numerical implementation shows that all broken equilibria found in (Rosakis et al 2021) are locally stable.

Irreversibility condition and stability of equilibria in the inverse-deformation approach to fracture

TL;DR

This paper derives a thermodynamically enforced irreversibility condition for fracture within the inverse-deformation framework, showing crack motion in the material reference would produce negative entropy and is thus forbidden. It develops a rigorous second-order stability theory that accommodates the unilateral constraint and the irreversibility constraint, and validates that all broken equilibria found in prior work are locally stable. The approach combines variational analysis with a gradient-regularized energy to capture surface energy and uses numerical FE/active-set methods to corroborate analytical predictions, including multiple-crack configurations. The results clarify how irreversibility shapes admissible variations and transitions under incremental loading, and point to potential extensions to higher dimensions and crack-tip dynamics.

Abstract

We derive the irreversibility condition in fracture for the inverse-deformation approach using the second law of thermodynamics. We consider the problem of brittle failure in an elastic bar previously solved in (Rosakis et al 2021). Despite the presence of a non-zero interfacial/surface energy, the third derivative of the inverse-deformation map is discontinuous at the crack faces. This is due to the presence of the inequality constraint ensuring the inverse strain is nonnegative and the orientation of matter is preserved. A change in the material location of a crack results in negative entropy production, violating the second law. Consequently, such changes are disallowed giving the irreversibility condition. The inequality constraint and the irreversibility condition limit the space of admissible variations. We prove necessary and sufficient conditions for local stability that incorporate these restrictions. Their numerical implementation shows that all broken equilibria found in (Rosakis et al 2021) are locally stable.

Paper Structure

This paper contains 13 sections, 8 theorems, 88 equations, 9 figures.

Table of Contents

  1. Introduction
  2. The inverse-deformation approach to fracture
  3. Smoothness of weak solutions
  4. Work done by external forces
  5. Entropy production during quasistatic motion
  6. End cracks
  7. Internal cracks
  8. Irreversability condition
  9. Stability of quasistatic equilibria
  10. Numerical results
  11. Stability analysis
  12. Results
  13. Concluding remarks

Key Result

Proposition 2.1

A point $h\in\mathcal{K}$ is a solution of ELineq if and only if there exist a nonegative Radon measure $\mu$ such that $\mathop{\mathrm{supp}}\nolimits \mu\subseteq \mathcal{B}_h$ and $h,\mu$ satisfy Furthermore, $h\in C^2(0,\lambda)$.

Figures (9)

  • Figure 1: The deformation $f$ maps the reference ${\Omega}$ onto the deformed configuration ${\Omega_\ast}$. The inverse deformation $h=f^{-1}$ maps the deformed configuration ${\Omega_\ast}$ onto the reference domain ${\Omega}$.
  • Figure 2: The domain $\mathcal{D}_*(t):=[\Delta_1(t),\Delta_2(t)]\subset {\Omega_\ast}$ with a simple crack face at $y=y_c(t)$.
  • Figure 3: The domain $[\Delta_1(t),\lambda(t)]\subset {\Omega_\ast}$ with a simple crack face at $y=y_c(t)$. The picture shows the deformation configuration of an end crack located at $x_c=1$.
  • Figure 4: The domain $[\Delta_1(t),\lambda(t)]\subset {\Omega_\ast}$ with a simple crack face at $y=y_c(t)$. The picture shows the deformation configuration of an internal crack located at $x=x_c$.
  • Figure 5: As defined in Theorem \ref{['thfam']}, the domain $G:=(r,s)\subseteq \mathcal{G}_h$ is unbroken region and region $R:=(q,t)$ such that $(q,r],\,[s,t)\in\mathcal{B}_h$.
  • ...and 4 more figures

Theorems & Definitions (18)

  • Proposition 2.1
  • proof
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.1
  • proof
  • Remark 2.1
  • Theorem 4.1: Necessary condition
  • proof
  • ...and 8 more