Defects and plasticity across scales

Rigorous bottom-up models of cracks and dislocations, from the discrete lattice to mesoscale plasticity.

An illustration: a stationary crack in a square crystal under rising anti-plane load. The contour lines are level sets of the displacement, modulo the lattice spacing; the crack tip emits screw dislocations (the squares), which glide ahead along the crack plane. click to add one

Overview

Cracks and dislocations are singular defects of a crystal. Far from the defect their elastic fields are described by continuum linearised elasticity; at the core the discreteness of the lattice dominates. We study defects directly as equilibria of atomistic energies, and derive what the continuum description captures, what it misses, and how it can be used as a boundary condition for atomistic simulation.

Cracks in a lattice. For an anti-plane crack on a square lattice with nearest-neighbour pair interactions, we establish existence, local uniqueness and stability of equilibria for small loads, together with sharp far-field decay estimates, which rest on decay estimates for the lattice Green’s function. Viewing crack propagation as a bifurcation problem, with the stress intensity factor as the parameter, gives a periodic snaking curve of equilibria, and we prove convergence rates for its finite-cell approximations.

Beyond linear elasticity. At leading order the displacement near a mode III crack is the continuum field

\hat u(r, \theta) = K \sqrt{r} \, \sin(\theta/2),

with $K$ the stress intensity factor. We derive the next term of the atomistic expansion, which requires an asymptotic expansion of the lattice Green’s function and a discrete geometry predictor for the lattice near the crack tip. It shows that Sinclair’s flexible boundary condition is incomplete, so in principle no better than boundary conditions from linear elasticity, and it gives boundary conditions for high-accuracy fracture simulations. For screw dislocations, exploiting the symmetries of the crystal improves the decay of the core correction beyond the generic $O(r^{-2})$ .

Near-crack-tip plasticity. A lattice manifold complex accounts for the crack surface while preserving duality. It yields locally stable equilibria that contain both a crack opening and dislocations, with no minimum separation between a dislocation core and the crack. This is the model in the animation above.

Towards mesoscale plasticity. In ongoing work with Patrick van Meurs, we upscale this model to describe the collective behaviour of many dislocations near a crack tip. With Lev Truskinovsky and Thomas Hudson, we study minimal models of plasticity: discrete models that keep only the essential lattice mechanisms, and so can be analysed and simulated at the scales where plastic flow emerges from many individual slip events (see the illustration below).

An illustration: a minimal model of plasticity, in which concentrated sources of dislocations relax under a single rounding rule, first into nested arcs and then into walls and clusters (red and grey: cores of opposite sign).

Publications

J. Braun and M. Buze. Incompleteness of Sinclair-type continuum flexible boundary conditions for atomistic fracture simulations. Multiscale Modeling & Simulation 23(2), 711–752 (2025). arXiv doi
bibtex
@article{braun2025incompleteness, title = {Incompleteness of Sinclair-type continuum flexible boundary conditions for atomistic fracture simulations}, author = {J. Braun and M. Buze}, journal = {Multiscale Modeling & Simulation}, volume = {23}, number = {2}, pages = {711--752}, year = {2025}, doi = {10.1137/24M1661078}, eprint = {2403.05462}, archivePrefix = {arXiv}, }
M. Buze, T. E. Woolley, and L. A. Mihai. A stochastic framework for atomistic fracture. SIAM Journal on Applied Mathematics 82(2), 526–548 (2022). doi pdf
bibtex
@article{buze2022stochastic, title = {A stochastic framework for atomistic fracture}, author = {M. Buze and T. E. Woolley and L. A. Mihai}, journal = {SIAM Journal on Applied Mathematics}, volume = {82}, number = {2}, pages = {526--548}, year = {2022}, doi = {10.1137/21M1416436}, }
M. Buze. Atomistic modelling of near-crack-tip plasticity. Nonlinearity 34(7), 4503–4542 (2021). arXiv doi
bibtex
@article{buze2021plasticity, title = {Atomistic modelling of near-crack-tip plasticity}, author = {M. Buze}, journal = {Nonlinearity}, volume = {34}, number = {7}, pages = {4503--4542}, year = {2021}, doi = {10.1088/1361-6544/abf33c}, eprint = {2007.02408}, archivePrefix = {arXiv}, }
M. Buze, T. Hudson, and C. Ortner. Analysis of cell size effects in atomistic crack propagation. ESAIM: Mathematical Modelling and Numerical Analysis 54(6), 1821–1847 (2020). arXiv doi
bibtex
@article{buze2020cellsize, title = {Analysis of cell size effects in atomistic crack propagation}, author = {M. Buze and T. Hudson and C. Ortner}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, volume = {54}, number = {6}, pages = {1821--1847}, year = {2020}, doi = {10.1051/m2an/2020005}, eprint = {1905.13328}, archivePrefix = {arXiv}, }
M. Buze, T. Hudson, and C. Ortner. Analysis of an atomistic model for anti-plane fracture. Mathematical Models and Methods in Applied Sciences 29(13), 2469–2521 (2019). arXiv doi
bibtex
@article{buze2019antiplane, title = {Analysis of an atomistic model for anti-plane fracture}, author = {M. Buze and T. Hudson and C. Ortner}, journal = {Mathematical Models and Methods in Applied Sciences}, volume = {29}, number = {13}, pages = {2469--2521}, year = {2019}, doi = {10.1142/S0218202519500520}, eprint = {1810.05501}, archivePrefix = {arXiv}, }
J. Braun, M. Buze, and C. Ortner. The effect of crystal symmetries on the locality of screw dislocation cores. SIAM Journal on Mathematical Analysis 51(2), 1108–1136 (2019). arXiv doi
bibtex
@article{braun2019screw, title = {The effect of crystal symmetries on the locality of screw dislocation cores}, author = {J. Braun and M. Buze and C. Ortner}, journal = {SIAM Journal on Mathematical Analysis}, volume = {51}, number = {2}, pages = {1108--1136}, year = {2019}, doi = {10.1137/17M1157520}, eprint = {1710.07708}, archivePrefix = {arXiv}, }

Collaborators

Julian Braun Heriot-Watt University
Thomas Hudson University of Warwick
Christoph Ortner University of British Columbia
Angela Mihai Cardiff University
Thomas Woolley Cardiff University
Lev Truskinovsky CNRS, ESPCI Paris
Patrick van Meurs Kanazawa University