arXiv subjects

Robert P. Lipton

Publications and source records attributed to Robert P. Lipton.

9 recordsLinked to original sources

arXiv2025

Crushing, Comminution and Fracture: Extreme Particle Deformation in Three-Dimensional Granular Aggregates

We present a high-fidelity three dimensional computational framework for simulating the bulk mechanical behavior of granular aggregates composed of deformable brittle grains. Departing from classical discrete element methods (DEM), our approach captures both inter-particle and intra-particle deformation using a nonlocal continuum formulation based on peridynamics. Each grain is individually meshed from level-set representations, enabling accurate modeling of elastic response and autonomous fracture evolution without requiring explicit crack tracking or fragment reconstruction. We validate the method through benchmark simulations, including the Kalthoff-Winkler fracture test, crushing of hollow spheres, and compound impact-crushing scenarios. The framework is further applied to large aggregates of up to 1000 sand grains of irregular shapes reconstructed from three dimensional X-ray computed tomography. Simulations reveal convergence of bulk stress response under compression, suggesting the feasibility of constructing representative volume elements (RVEs) for multiscale modeling. Finally, we investigate the role of grain geometry and topology on the macroscopic strength of the aggregate, providing insight into microstructure-driven failure mechanisms. The framework exhibits excellent strong and weak scaling behavior, with simulations executed on up to 1600 cores, demonstrating its suitability for high-performance computing environments and large-scale modeling.

cond-mat.soft↗

arXiv2024

Energy balance and damage for dynamic brittle fracture from a nonlocal formulation

A nonlocal model of peridynamic type 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 the initial value problem. The displacement-failure pair satisfies energy balance. The length of nonlocality $ε$ is taken to be small relative to the domain in $\mathbb{R}^d$, $d=2,3$. The new nonlocal model delivers a two point strain evolution on a subset of $\mathbb{R}^d\times\mathbb{R}^d$. This evolution provides an energy that interpolates between volume energy corresponding to elastic behavior and surface energy corresponding to failure. In general the deformation energy resulting in material failure over a region $R$ is given by a $d-1$ dimensional integral that is uniformly bounded as $ε\rightarrow 0$. For fixed $ε$, the failure energy is nonzero for $d-1$ dimensional regions $R$ associated with flat crack surfaces. 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. Simulations illustrate fracture evolution through generation of an internal traction free boundary as a wake left behind a moving strain concentration. Crack paths are seen to follow a maximal strain energy density criterion.

math.AP↗

arXiv2023

Quasistatic Evolution with Unstable Forces

We consider load controlled quasistatic evolution. Well posedness results for the nonlocal continuum model related to peridynamics are established. We show local existence and uniqueness of quasistatic evolution for load paths originating at stable critical points. These points can be associated with local energy minima among the convex set of deformations belonging to the strength domain of the material. The evolution of the displacements however is not constrained to lie inside the strength domain of the material. The load-controlled evolution is shown to exhibit energy balance.

math.AP↗

arXiv2022

Simulating grain shape effects and damage in granular media using PeriDEM

We provide a numerical platform for the analysis of particle shape and topology effect on the macroscopic behavior of granular media. We work within a Discrete Element Method (DEM) framework and apply a peridynamic model for deformable particles accounting for deformation and damage of individual particles. To accommodate arbitrary particle shapes including nonconvex ones as well as particle topology, an efficient method is developed to keep intra-particle peridynamic interaction within particle boundaries. Particle contact with the rigid boundary wall is computed analytically to improve accuracy. To speed up simulations with particles of different shapes and sizes the initial configuration is chosen using security disks containing different particle shapes that are placed in a jammed state using an optimization-based method. The effect of particle shape and topology on settling and compaction of the aggregate for deformable particles is analyzed.

cond-mat.soft↗

arXiv2021

Peridynamics for Quasistatic Fracture Modeling

Fracture involves interaction across large and small length scales. With the application of enough stress or strain to a brittle material, atomistic scale bonds will break, leading to fracture of the macroscopic specimen. From the perspective of mechanics fracture should appear as an emergent phenomena generated by a continuum field theory eliminating the need for a supplemental kinetic relation describing crack growth. We develop a new fast method for modeling quasi-static fracture using peridynamics. We apply fixed point theory and model stable crack evolution for hard and soft loading. For soft loading we recover unstable fracture. For hard loading we recover stable crack growth. We show existence of quasistatic fracture solutions in the neighborhood of stable critical points for appropriately defined energies. The numerical method uses an analytic stiffness matrix for fast numerical implementation. A rigorous mathematical analysis shows that the method converges for load paths associated with soft and hard loading. For soft loading the crack becomes unstable shortly after the stress at the tip of the pre-crack reaches the material strength.

cond-mat.soft↗

arXiv2020

Nonlocal elastodynamics and fracture

A nonlocal field theory of peridynamic type is applied to model the brittle fracture problem. The elastic fields obtained from the nonlocal model are shown to converge in the limit of vanishing non-locality to solutions of classic plane elastodynamics associated with a running crack.

math.AP↗

arXiv2013

Resonance and Double Negative Behavior in Metamaterials

A generic class of metamaterials is introduced and is shown to exhibit frequency dependent double negative effective properties. We develop a rigorous method for calculating the frequency intervals where either double negative or double positive effective properties appear and show how these intervals imply the existence of propagating Bloch waves inside sub-wavelength structures. The branches of the dispersion relation associated with Bloch modes are shown to be explicitly determined by the Dirichlet spectrum of the high dielectric phase and the generalized electrostatic spectra of the complement.

math.AP↗

arXiv2010

Convergent Power Series for Fields in Positive or Negative High-Contrast Periodic Media

We obtain convergent power series representations for Bloch waves in periodic high-contrast media. The material coefficient in the inclusions can be positive or negative. The small expansion parameter is the ratio of period cell width to wavelength, and the coefficient functions are solutions of the cell problems arising from formal asymptotic expansion. In the case of positive coefficient, the dispersion relation has an infinite sequence of branches, each represented by a convergent even power series whose leading term is a branch of the dispersion relation for the homogenized medium. In the negative case, there is a single branch.

math-ph↗

arXiv2010

Correctors and Field Fluctuations for the $p_ε(x)$-Laplacian with Rough Exponents

We provide a corrector theory for the strong approximation of fields inside composites made from two materials with different power law behavior. The correctors are used to develop bounds on the local singularity strength for gradient fields inside micro-structured media. The bounds are multi-scale in nature and can be used to measure the amplification of applied macroscopic fields by the microstructure.

math.AP↗