Peridynamics review

Peridynamics (PD) is a novel continuum mechanics theory established by Stewart Silling in 2000 [1]. The roots of PD can be traced back to the early works of Gabrio Piola according to dell'Isola et al. [2]. PD has been attractive to researchers as it is a nonlocal formulation in an integral form; unlike the local differential form of classical continuum mechanics. Although the method is still in its infancy, the literature on PD is fairly rich and extensive. The prolific growth in PD applications has led to a tremendous number of contributions in various disciplines. This manuscript aims to provide a concise description of the peridynamic theory together with a review of its major applications and related studies in different fields to date. Moreover, we succinctly highlight some lines of research that are yet to be investigated

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors. The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics

Numerical Computation, Data Analysis and Software in Mathematics and Engineering

The present book contains 14 articles that were accepted for publication in the Special Issue “Numerical Computation, Data Analysis and Software in Mathematics and Engineering” of the MDPI journal Mathematics. The topics of these articles include the aspects of the meshless method, numerical simulation, mathematical models, deep learning and data analysis. Meshless methods, such as the improved element-free Galerkin method, the dimension-splitting, interpolating, moving, least-squares method, the dimension-splitting, generalized, interpolating, element-free Galerkin method and the improved interpolating, complex variable, element-free Galerkin method, are presented. Some complicated problems, such as tge cold roll-forming process, ceramsite compound insulation block, crack propagation and heavy-haul railway tunnel with defects, are numerically analyzed. Mathematical models, such as the lattice hydrodynamic model, extended car-following model and smart helmet-based PLS-BPNN error compensation model, are proposed. The use of the deep learning approach to predict the mechanical properties of single-network hydrogel is presented, and data analysis for land leasing is discussed. This book will be interesting and useful for those working in the meshless method, numerical simulation, mathematical model, deep learning and data analysis fields

Computational fracture modelling by an adaptive cracking particle method

Conventional element-based methods for crack modelling suffer from remeshing and mesh distortion, while the cracking particle method is meshless and requires only nodal data to discretise the problem domain so these issues are addressed. This method uses a set of crack segments to model crack paths, and crack discontinuities are obtained using the visibility criterion. It has simple implementation and is suitable for complex crack problems, but suffers from spurious cracking results and requires a large number of particles to maintain good accuracy. In this thesis, a modified cracking particle method has been developed for modelling fracture problems in 2D and 3D. To improve crack description quality, the orientations of crack segments are modified to record angular changes of crack paths, e.g. in 2D, bilinear segments replacing straight segments in the original method and in 3D, nonplanar triangular facets instead of planar circular segments, so continuous crack paths are obtained. An adaptivity approach is introduced to optimise the particle distribution, which is refined to capture high stress gradients around the crack tip and is coarsened when the crack propagates away to improve the efficiency. Based on the modified method, a multi-cracked particle method is proposed for problems with branched cracks or multiple cracks, where crack discontinuities at crack intersections are modelled by multi-split particles rather than complex enrichment functions. Different crack propagation criteria are discussed and a configurational-force-driven cracking particle method has been developed, where the crack propagating angle is directly given by the configuration force, and no decomposition of displacement and stress fields for mixed-mode fracture is required. The modified method has been applied to thermo-elastic crack problems, where the adaptivity approach is employed to capture the temperature gradients around the crack tip without using enrichment functions. Several numerical examples are used to validate the proposed methodology

On the coupling of peridynamics with the classical theory of continuum mechanics in a meshless framework

The classical theory of solid mechanics employs partial derivatives in the equation of motion and hence requires the differentiability of the displacement field. Such an assumption breaks down when simulation of problems containing discontinuities, such as cracks, comes into the picture. peridynamics is considered to be an alternative and promising nonlocal theory of solid mechanics that is formulated suitably for discontinuous problems. Peridynamics is well designed to cope with failure analysis as the theory deals with integral equations rather than partial differential equations. Indeed, peridynamics defines the equation of motion by substituting the divergence of the stress tensor, involved in the formulation of the classical theory, with an integral operator. One of the most common techniques to discretize and implement the theory is based on a meshless approach. However, the method is computationally more expensive than some meshless methods based on the classical theory. This originates from the fact that in peridynamics, similar to other nonlocal theories, each computational node interacts with many neighbors over a finite region. To this end, performing realistic numerical simulations with peridynamics entails a vast amount of computational resources. Moreover, the application of boundary conditions in peridynamics is nonlocal and hence it is more challenging than the application of boundary conditions adopted by methods based on the classical continuum theory. This issue is well-known to scientists working on peridynamics. Therefore, it is reasonable to couple computational methods based on classical continuum mechanics with others based on peridynamics to develop an approach that applies different computational techniques where they are most suited for. The main purpose of this dissertation is to develop an effective coupled nonlocal/local meshless technique for the solution of two-dimensional elastodynamic problems involving brittle crack propagation. This method is based on a coupling between the peridynamic meshless method, and other meshless methods based on the classical continuum theory. In this study, two different meshless methods, the Meshless Local Exponential Basis Functions and the Finite Point Method are chosen as both are classified within the category of strong form meshless methods, which are simple and computationally cheap. The coupling has been achieved in a completely meshless scheme. The domain is divided in three zones: one in which only peridynamics is applied, one in which only the meshless method is applied and a transition zone where a transition between the two approaches takes place. The coupling adopts a local/nonlocal framework that benefits from the full advantages of both methods while overcoming their limitations. The parts of the domain where cracks either exist or are likely to propagate are described by peridynamics; the remaining part of the domain is described by the meshless method that requires less computational effort. We shall show that the proposed approach is suited for adaptive coupling of the strategies in the solution of crack propagation problems. Several static and dynamic examples are performed to demonstrate the capabilities of the proposed approach

Discontinuous mechanical problems studied with a Peridynamics-based approach

The classical theory of solid mechanics is rooted in the assumption of a continuous distribution of mass within a body. It employs partial differential equations (PDEs) with significant smoothness to obtain displacements and internal forces of the body. Although classical theory has been applied to wide range of engineering problems, PDEs of the classical theory cannot be applied directly on a discontinuity such as cracks. Peridynamics is considered to be an alternative and promising nonlocal theory of solid mechanics that, by replacing PDEs of classical theory with integral or integro-differential equations, attempts to unite the mathematical modelling of continuous media, cracks and particles within a single framework. Indeed, the equations of peridynamic are based on the direct interaction of material points over finite distances. Another concept, derived from the peridynamic approach to cope with engineering problems with discontinuities, is that of the peridynamic differential operator (PDDO). The PDDO uses the non-local interaction of the material points in a way similar to that of peridynamics. PDDO is capable to recast partial derivatives of a function through a nonlocal integral operator whose kernel is free of using any correction function. In this dissertation, application of peridaynamics and PDDO, to three different important engineering problems including fatigue fracture, thermo-mechanics and sloshing phenomena, is examined comprehensively. To cope with fatigue fracture problems, an algorithm has been developed in such a way that the increment of damage due to fatigue is added to that due to the static increment of the opening displacement. A one degree of freedom cylinder model has been used to carry out an efficient comparison of the computational performance of three fatigue degradation strategies. The three laws have been implemented in a code using bond based peridynamics (BBPD) to simulate fatigue crack propagation. Both the cylinder model and the bond base peridynamics code provide the same assessment of the three fatigue degradation strategies. To deal with thermo-mechanical problems, an effective way is proposed to use a variable grid size in a weakly coupled thermal shock peridynamic model. The proposed numerical method is equipped with stretch control criterion to transform the grid discretization adaptively in time. Hence, finer grid spacing is only applied in limited zones where it is required. This method is capable of predicting complex crack patterns in the model. By introducing fine grid discretization over the boundaries of the model the surface (softening) effect can be reduced. The accuracy and performance of the model are examined through problems such as thermo-elastic and thermal-shock induced fracture in ceramics. Finally to investigate sloshing phenomena, the PDDO has been applied to the solution of problems of liquid sloshing in 2D and 3D tanks with potential flow theory and Lagrangian description. Moreover, liquid sloshing in rectangular tanks containing horizontal and vertical baffles are investigated to examine the robustness and accuracy of PDDO. With respect to other approaches such as meshless local Petrov-Galerkin (MLPG), volume of fluid (VOF) and and local polynomial collocation methods the examples are solved with a coarser grid of nodes. Using this new approach, one is able to obtain results with a high accuracy and low computational cost