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The hybrid particle-based computational platform that couples peridynamics with the discrete element method (PeriDEM) is used to model vehicle mobility over roadbeds. We consider wheels rolling over gravel beds, where gravel is allowed to deform and fracture. The motion of particles are not constrained to translation and rotation as in DEM and grains can deform elastically or inelastically. This allows for more modes of inter-particle interaction. The effects of gravel shape and topology on the vehicle mobility are examined using the higher fidelity modeling. Here we study how these aspects affect vehicle range, average vehicle velocity, traction as measured by wheel slip, as well as the overall energy needed to travel a prescribed distance. When intraparticle fracture can occur, computations identify conditions on gravel particle topology that enhance vehicle mobility. In other computer simulations it is found that the driving torque is monotonically increasing with slip and capture trends seen in experiment Smith (Journal of Terramechanics, 2014).

The hybrid particle-based computational platform that couples peridynamics with the discrete element method (PeriDEM) is used to model vehicle mobility over roadbeds. We consider wheels rolling over gravel beds, where gravel is allowed to deform and fracture. The motion of particles are not constrained to translation and rotation as in DEM and grains can deform elastically or inelastically. This allows for more modes of inter-particle interaction. The effects of gravel shape and topology on the vehicle mobility are examined using the higher fidelity modeling. Here we study how these aspects affect vehicle range, average vehicle velocity, traction as measured by wheel slip, as well as the overall energy needed to travel a prescribed distance. When intraparticle fracture can occur, computations identify conditions on gravel particle topology that enhance vehicle mobility. In other computer simulations it is found that the driving torque is monotonically increasing with slip and capture trends seen in experiment Smith (Journal of Terramechanics, 2014). Graphical abstract

We introduce a nonlocal model of peridynamic type for fracture evolution in the quasistatic regime. Nonlocal quasistatic fracture evolution is developed and supporting numerical examples are presented. The approach is implicit and is based on local stationary and fixed point methods. Here a smooth cohesive force-strain model is used. Initially the force increases with strain then softens and decreases to zero. It is proved that the fracture evolution decreases stored elastic energy with each displacement step as the cracks advance; provided the displacement increments are chosen sufficiently small. These results apply to any system of multiple cracks. This is also seen in the numerical examples. The numerical examples include evolution of a straight crack, a crack propagating inside an L-shaped domain, and two offset inward propagating cracks.

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 ϵ is taken to be small relative to the domain in 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 R d × 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 ϵ → 0 . For fixed ϵ , 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.

We consider the modified Zakharov-Kuznetsov (mZK) equation in two and three space dimensions in both focusing and defocusing cases. Using the$I$ -method, for the 2D mZK equation, we prove the global well-posedness of the$H^s(\\R^2)$solutions for$s>\\frac{3}{4}$for any data in the defocusing case, and under the assumption that the mass of the initial data is less than the mass of the ground state solution of$\\Delta \\varphi - \\varphi + \\varphi^3 = 0$in the focusing case. This improves the global well-posedness result of Linares and Pastor. We also prove that low-regularity solutions to the focusing 2D mZK equation which blow up in finite time have the property that the mass of the solutions concentrates inside a moving ball of shrinking radius. For focusing 3D mZK equation, we prove a sufficient condition on the initial data that guarantees global well-posedness in$ H^1(\\R^{3}) $ . We also consider the multichannel deconvolution problem in the presence of additive white noise. We propose a hybrid algorithm employing a regularized Fourier based approach followed by wavelet thresholding. Minimizing a proposed cost function, we determine the optimum regularization parameter that balances the Fourier and wavelet shrinkage. Our method extends the ForWaRD algorithm introduced by Neelamani, Baranuik, and Choi to the multichannel setup.

We consider the focusing modified Zakharov-Kuznetsov (mZK) equation in two space dimensions. We prove that solutions which blow up in finite time in the \\(H^1(^2)\\) norm have the property that they concentrate a non-trivial portion of their mass (more precisely, at least the amount equal to the mass of the ground state) at blow-up time. For finite-time blow-up solutions in the \\(H^s(^2)\\) norm for \\(1718 < s < 1\\), we prove a slightly weaker result. Moreover, we prove that the stronger concentration result can be extended to the range \\( 1718 < s 1\\) under an additional assumption on the upper bound of the blow-up rate of the solution. The main tools used here are the \\(I\\)-method and a profile decomposition theorem for a bounded family of \\(H^1(^2)\\) functions.

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 \\(\\) is taken to be small relative to the domain in \\(R^d\\), \\(d=2,3\\). The new nonlocal model delivers a two point strain evolution on a subset of \\(R^d^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 \\( 0\\). For fixed \\(\\), 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.

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.

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.

This paper examines various ways of improving the impact resilience of protective structures. Such structures' purpose is to dissipate an impact's energy while avoiding cracking and failure. We have tested the reaction of plane elastic-brittle lattices to an impulse. Four topologies are compared: periodic triangular, square, and hexagonal topologies, and aperiodic Penrose topology. Then, structures with random variations of the links' stiffness, node positions, and random holes are compared. Combinations of these random factors are also considered, as well as the resilience of bistable elastic-brittle lattices with sacrificial links. Several parameters are introduced to measure the structural resilience of the compared designs: (i) the amount of dissipated impact energy, (ii) the size of broken clusters of links, and (iii) the spread of damage. The results suggest new routes for rationally designing protective structures using nonperiodic topology, bistability, and structural randomness. In particular, we find that some quantities of interest can be maximized by tuning the randomized design appropriately -- for example, randomly removing 8\\% of links maximizes energy dissipation. We also find that randomization of bistable lattices can offer superior energy dissipation while reducing the connectivity between broken clusters of links.