A Quasi-Brittle damage model in the framework of Bond-based Peridynamics with Adaptive Dynamic Relaxation method

Peridynamics (PD) is a new tool, based on the non-local theory for modelling fracture mechanics, where particles connected through physical interaction used to represent a domain. By using the PD theory, damage or crack in a material domain can be shown in much practical representation. This study compares between Prototype Microelastic Brittle (PMB) damage model and a new Quasi-Brittle (QBR) damage model in the framework of the Bond-based Peridynamics (BBPD) in terms of the damage plot. An in-house code using Matlab was developed for BBPD with inclusion of both damage models, and tested for a quasi-static problem with the implementation of Adaptive Dynamic Relaxation (ADR) method in the theory in order to get a faster steady state solutions. This paper is the first attempt to include ADR method in the framework of BBPD for QBR damage model. This paper analysed a numerical problem with the absence of failure and compared the displacement with literature result that used Finite Element Method (FEM). The obtained numerical results are in good agreement with the result from FEM. The same problem was used with the allowance of the failure to happen for both of the damage models; PMB and QBR, to observe the damage pattern between these two damage models. PMB damage model produced damage value of roughly twice compared to the damage value from QBR damage model. It is found that the QBR damage model with ADR under quasi-static loading significantly improves the prediction of the progressive failure process, and managed to model a more realistic damage model with respect to the PMB damage model.

Phase Field Modeling of Anisotropic Tension Failure of Rock-Like Materials

The tensile fracture is a widespread feature in rock excavation engineering, such as spalling around an opened tunnel. The phase field method (PFD) is a non-local theory to effectively simulate the quasi-brittle fracture of materials, especially for the propagation of a tensile crack. This work is dedicated to study the tensile failure characteristics of rock-like materials by the PFD simulation of the Brazilian test of the intact and fissure disk samples. The numerical results indicate that the tensile strength of the disk sample is anisotropic due to the influence of pre-existing cracks. The peak load decreases at first and then increases with the increase of the inclination angle, following the U-shaped trend. The simulation results also indicate that the wing crack growth is the main failure characteristic. Moreover, the crack propagation path initiates at the tip of the pre-existing crack when the inclination angle is less than 60°. Crack propagation initiates near the tip of the pre-existing crack when the angle is 75°, and it initiates at the middle of the pre-existing crack when the angle is 90°. Finally, all cracks extend to the loading position and approximately parallel to the loading direction. This process is in agreement with the Brazilian test of pre-existing cracks in the laboratory, which can validate the effectiveness of the PFD in simulating the tensile fracture of rock-like materials. This study can provide a reference for the fracture mechanism of the surrounding rock in the underground excavation.

Newton polygons for L-functions of generalized Kloosterman sums

In the present paper, we study the Newton polygons for the L-functions of n-variable generalized Kloosterman sums. Generally, the Newton polygon has a topological lower bound, called the Hodge polygon. In order to determine the Hodge polygon, we explicitly construct a basis of the top-dimensional Dwork cohomology. Using Wan’s decomposition theorem and diagonal local theory, we obtain when the Newton polygon coincides with the Hodge polygon. In particular, we concretely get the slope sequence for the L-function of F ¯ ⁢ ( λ ¯ , x ) := ∑ i = 1 n x i a i + λ ¯ ⁢ ∏ i = 1 n x i - 1 , \bar{F}(\bar{\lambda},x):=\sum_{i=1}^{n}x_{i}^{a_{i}}+\bar{\lambda}\prod_{i=1}% ^{n}x_{i}^{-1}, with a 1 , … , a n {a_{1},\ldots,a_{n}} being pairwise coprime for n ≥ 2 {n\geq 2} .

A Local-in-Time Theory for Singular SDEs with Applications to Fluid Models with Transport Noise

We establish a local theory, i.e., existence, uniqueness and blow-up criterion, for a general family of singular SDEs in Hilbert spaces. The key requirement relies on an approximation property that allows us to embed the singular drift and diffusion mappings into a hierarchy of regular mappings that are invariant with respect to the Hilbert space and enjoy a cancellation property. Various nonlinear models in fluid dynamics with transport noise belong to this type of singular SDEs. By establishing a cancellation estimate for certain differential operators of order one with suitable coefficients, we give the detailed constructions of such regular approximations for certain examples. In particular, we show novel local-in-time results for the stochastic two-component Camassa–Holm system and for the stochastic Córdoba–Córdoba–Fontelos model.

From locality and unitarity to cosmological correlators

In the standard approach to deriving inflationary predictions, we evolve a vacuum state in time according to the rules of a given model. Since the only observables are the future values of correlators and not their time evolution, this brings about a large degeneracy: a vast number of different models are mapped to the same minute number of observables. Furthermore, due to the lack of time-translation invariance, even tree-level calculations require an increasing number of nested integrals that quickly become intractable. Here we ask how much of the final observables can be “bootstrapped” directly from locality, unitarity and symmetries.To this end, we introduce two new “boostless” bootstrap tools to efficiently compute tree-level cosmological correlators/wavefunctions without any assumption about de Sitter boosts. The first is a Manifestly Local Test (MLT) that any n-point (wave)function of massless scalars or gravitons must satisfy if it is to arise from a manifestly local theory. When combined with a sub-set of the recently proposed Bootstrap Rules, this allows us to compute explicitly all bispectra to all orders in derivatives for a single scalar. Since we don’t invoke soft theorems, this can also be extended to multi-field inflation. The second is a partial energy recursion relation that allows us to compute exchange correlators. Combining a bespoke complex shift of the partial energies with Cauchy’s integral theorem and the Cosmological Optical Theorem, we fix exchange correlators up to a boundary term. The latter can be determined up to contact interactions using unitarity and manifest locality. As an illustration, we use these tools to bootstrap scalar inflationary trispectra due to graviton exchange and inflaton self-interactions.

Can classical electrodynamics predict nonlocal effects?

Classical electrodynamics is a local theory describing local interactions between charges and electromagnetic fields and therefore one would not expect that this theory could predict nonlocal effects. But this perception implicitly assumes that the electromagnetic configurations lie in simply connected regions. In this paper, we consider an electromagnetic configuration lying in a non-simply connected region, which consists of a charged particle encircling an infinitely long solenoid enclosing a uniform magnetic flux, and show that the electromagnetic angular momentum of this configuration describes a nonlocal interaction between the encircling charge outside the solenoid and the magnetic flux confined inside the solenoid. We argue that the nonlocality of this interaction is of topological nature by showing that the electromagnetic angular momentum of the configuration is proportional to a winding number. The magnitude of this electromagnetic angular momentum may be interpreted as the classical counterpart of the Aharonov–Bohm phase.

Nonlocal wrinkling instabilities in bilayered systems using peridynamics

Wrinkling instabilities occur when a stiff thin film bonded to an elastic substrate undergoes compression. Regardless of the nature of compression, this phenomenon has been extensively studied through local models based on classical continuum mechanics. However, the experimental behavior is not yet fully understood and the influence of nonlocal effects remains largely unexplored. The objective of this paper is to fill this gap from a computational perspective by investigating nonlocal wrinkling instabilities in a bilayered system. Peridynamics (PD), a nonlocal continuum formulation, serves as a tool to model nonlocal material behavior. This manuscript presents a methodology to precisely predict the critical conditions by employing an eigenvalue analysis. Our results approach the local solution when the nonlocality parameter, the horizon size, approaches zero. An experimentally observed influence of the boundaries on the wave pattern is reproduced with PD simulations which suggests nonlocal material behavior as a physical origin. The results suggest that the level of nonlocality of a material model has quantitative influence on the main wrinkling characteristics, while most trends qualitatively coincide with predictions from the local analytical solution. However, a relation between the film thickness and the critical compression is revealed that is not existent in the local theory. Moreover, an approach to determine the peridynamic material parameters across a material interface is established by introducing an interface weighting factor. This paper, for the first time, shows that adding a nonlocal perspective to the analysis of bilayer wrinkling by using PD can significantly advance our understanding of the phenomenon.

Diluted mass gap in strongly coupled non-local Yang-Mills

We investigate the non-perturbative regimes in the class of non-Abelian theories that have been proposed as an ultraviolet completion of 4-D Quantum Field Theory (QFT) generalizing the kinetic energy operators to an infinite series of higher-order derivatives inspired by string field theory. We prove that, at the non-perturbative level, the physical spectrum of the theory is actually corrected by the “infinite number of derivatives” present in the action. We derive a set of Dyson-Schwinger equations in differential form, for correlation functions till two-points, the solution for which are known in the local theory. We obtain that just like in the local theory, the non-local counterpart displays a mass gap, depending also on the mass scale of non-locality, and show that it is damped in the deep UV asymptotically. We point out some possible implications of our result in particle physics and cosmology and discuss aspects of non-local QCD-like scenarios.