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Sharp-interface limits for brittle fracture via the inverse-deformation formulation

Timothy J. Healey, Roberto Paroni, Phoebus Rosakis

Abstract

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Gamma-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks; crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied load.

Sharp-interface limits for brittle fracture via the inverse-deformation formulation

Abstract

We derive sharp-interface models for one-dimensional brittle fracture via the inverse-deformation approach. Methods of Gamma-convergence are employed to obtain the singular limits of previously proposed models. The latter feature a local, non-convex stored energy of inverse strain, augmented by small interfacial energy, formulated in terms of the inverse-strain gradient. They predict spontaneous fracture with exact crack-opening discontinuities, without the use of damage (phase) fields or pre-existing cracks; crack faces are endowed with a thin layer of surface energy. The models obtained herewith inherit the same properties, except that surface energy is now concentrated at the crack faces. Accordingly, we construct energy-minimizing configurations. For a composite bar with a breakable layer, our results predict a pattern of equally spaced cracks whose number is given as an increasing function of applied load.
Paper Structure (5 sections, 4 theorems, 27 equations, 8 figures)

This paper contains 5 sections, 4 theorems, 27 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Formulation
  3. Sharp-Interface Models via $\Gamma$-Convergence
  4. Energy-minimizing configurations
  5. Concluding remarks

Key Result

Proposition 1

If $0<\lambda \le 1,$ then and are attained by the homogeneous inverse deformation $h(y)=y/\lambda$, with constant inverse stretch $H=1/\lambda$. In particular, the corresponding minima are $E_\varepsilon[1/\lambda]=U_\varepsilon[y/\lambda]=\lambda W^\ast (1/\lambda)=W(\lambda).$

Figures (8)

  • Figure 1: (a) Lennard Jones-type stored energy density $W$. (b) Corresponding inverse stored energy function $W^*$. The states corresponding to $H<0$ are inaccessible, rendering $W^*$ a two-well potential with minima at $H=0,1$.
  • Figure 2: Convex envelope of $W^{\ast }.$
  • Figure 3: (a) An inverse strain that minimizes $I$. (b) The associated broken configuration.
  • Figure 4: Several inverse strains having finite energy $I$.
  • Figure 5: (a) The graph of the inverse strain $h_{(1)}$. (b) The graph of $h_{(2)}$.
  • ...and 3 more figures

Theorems & Definitions (6)

  • Proposition 1
  • proof
  • Proposition 2
  • Proposition 3
  • proof
  • Theorem 4