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Energy Balance and Damage for Dynamic Fast Crack Growth from a Nonlocal Formulation

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Abstract

A nonlocal model for dynamic brittle damage is introduced consisting of two phases, one elastic and the other inelastic. Evolution from the elastic to the inelastic phase depends on material strength. Existence and uniqueness of the displacement-failure set pair follow from an initial value problem describing the evolution. The displacement-failure pair satisfies energy balance. The length of nonlocality \(\epsilon \) is taken to be small relative to the domain in \(\mathbb{R}^{d}\), \(d=2,3\). The strain is formulated as a difference quotient of the displacement in the nonlocal model. The two point force is expressed in terms of a weighted difference quotient and delivers an evolution on a subset of \(\mathbb{R}^{d}\times \mathbb{R}^{d}\). This evolution provides an energy balance between external energy, elastic energy, and damage energy including fracture energy. For any prescribed loading the deformation energy resulting in material failure over a region \(R\) is uniformly bounded as \(\epsilon \rightarrow 0\). For fixed \(\epsilon \), the failure energy is discovered to be is nonzero for \(d-1\) dimensional regions \(R\) associated with flat crack surfaces. Calculation shows, this failure energy is the Griffith fracture energy given by the energy release rate multiplied by area for \(d=3\) (or length for \(d=2\)). The nonlocal field theory is shown to recover a solution of Naiver’s equation outside a propagating flat traction free crack in the limit of vanishing spatial nonlocality. The theory and simulations presented here corroborate the recent experimental findings of (Rozen-Levy et al. in Phys. Rev. Lett. 125(17):175501, 2020) that cracks follow the location of maximum energy dissipation inside the intact material. Simulations show fracture evolution through the generation of a traction free internal boundary seen as a wake left behind a moving strain concentration.

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Acknowledgements

This material is based upon work supported by the U. S. Army Research Laboratory and the U. S. Army Research Office under Contract/Grant Number W911NF-19-1-0245 and W911NF-24-2-0184. Portions of this research were conducted with high performance computing resources provided by Louisiana State University (http://www.hpc.lsu.edu).

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Authors and Affiliations

  1. Department of Mathematics, Louisiana State University, Baton Rouge, LA, 70803, USA

    Robert P. Lipton

  2. Department of Mathematics, Grinnell College, Grinnell, IA, 50112, USA

    Debdeep Bhattacharya

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  1. Robert P. Lipton
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R.L and D.B. wrote and reviewed the manuscript.

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Correspondence to Robert P. Lipton.

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Lipton, R.P., Bhattacharya, D. Energy Balance and Damage for Dynamic Fast Crack Growth from a Nonlocal Formulation. J Elast 157, 5 (2025). https://doi.org/10.1007/s10659-024-10098-1

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