A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized. © 2022, The Author(s)

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

Computational modeling of the initiation and propagation of complex fracture is central to the discipline of engineering fracture mechanics. This review focuses on two promising approaches: phase-field (PF) and peridynamic (PD) models applied to this class of problems. The basic concepts consisting of constitutive models, failure criteria, discretization schemes, and numerical analysis are briefly summarized for both models. Validation against experimental data is essential for all computational methods to demonstrate predictive accuracy. To that end, the Sandia Fracture Challenge and similar experimental data sets where both models could be benchmarked against are showcased. Emphasis is made to converge on common metrics for the evaluation of these two fracture modeling approaches. Both PD and PF models are assessed in terms of their computational effort and predictive capabilities, with their relative advantages and challenges are summarized

Modelling mixed-mode fracture in poly(methylmethacrylate) using peridynamics

AbstractPeridynamics (Silling (2000)) is a non-local continuum theory that is particularly suited to handle discontinuities in the displacement field, such as those arising during fracture. Peridynamics prescribes that each material point interacts with all its neighbors contained in a sphere of given radius; this assumption introduces a characteristic length scale in the continuum description. In a nutshell, the interactions between material points depend on their relative distance; in the peridynamics framework this distance is called the “bond length”. The equations of motion, holding at each material point, link the material point acceleration to the integral over the point neighborhood of a force density field, whose strength depend on bond-stretches, i.e. the ratio of the actual bond-length over the initial one. In these equations the displacement gradient does not appear, thus naturally allowing for discontinuities in the displacement field to occur. As to failure, the simplest possible damage description is provided by an interaction law prescribing the force to vanish when a critical bond-stretch threshold is crossed; this parameter can be related to the Mode I critical strain energy release rate. A single parameter is needed to describe failure, in principle under every possible loading condition.In this work the predictive abilities of peridynamics were checked against experimental results in the case of mixed-mode failure of brittle polymers. Pre-cracked poly(methylmethacrylate) (PMMA) samples were tested using different specimens, in order to obtain Mode I, Mixed-Mode and Mode II loading conditions. The material was assumed to behave according to a peridynamics brittle elastic material model; the parameters needed to calibrate the elastic behavior were determined from Mode I tests, as was the critical stretch.The peridynamics simulations of mixed-mode tests were able to catch the correct fracture initiation load and to provide a fair description of the crack path under different conditions. The peridynamics model was also able to qualitatively capture the typical “nail” shape assumed by the crack front during propagation

Continuum-kinematics-based peridynamics and phase-field approximation of non-local dynamic fracture

In this work, two non-local approaches to dynamic fracture are investigated: a novel peridynamic formulation and a variational phase-field approach. The chosen continuum-kinematics-based peridynamic model extends the current peridynamic models by introducing surface and volume-based interactions. The phase-field fracture approach optimizes the body’s potential energy and provides a reliable method for predicting fracture in finite element computations. Both methods are able to efficiently compute crack propagation even when the cracks have arbitrary or complex patterns. We discuss the relations of critical fracture parameters in the two methods and show that our novel damage model for the continuum-kinematics-based peridynamics effectively manages fracture under dynamic loading conditions. Numerical examples demonstrate a good agreement between both methods in terms of crack propagation, fracture pattern, and in part, critical loading. We also show the limitations of the methods and discuss possible reasons for deviations

An implicit non-ordinary state-based peridynamics for large deformation solid mechanics problems

The numerical simulation of the cracking process remains one of the most significant challenges in solid mechanics. Compared classical approaches, peridynamics(PD) has some attractive features because the basic equations remain applicable even when singularities appear in the deformation. Numerical time-integration plays a big role in any computational framework and unlike explicit time-integration, implicit time-integration methods can be much more efficient because of the ability to adopt fairly large time increments, making it a suitable option for PD analyses of large deformation problems. The objective of this thesis is to propose an implicit non-ordinary state-based peridynamics (NOSB PD) approach focusing on quasistatic analyses with large deformation mechanics. Firstly, the use of the adaptive dynamic relaxation (ADR) method as a solution strategy for quasi-static analyses with large deformation mechanics is discussed. Next, an analytical expression of the Jacobian matrix based on the equation of motion of NOSB PD is formulated to ensure optimum convergence of the global residual force. To address some instability issues in the existing “corresponding material” model, caused by zero-energy modes instability, recent approaches proposed by Silling (2017) are used to control the spurious deformation modes. An additional stabilisation term with respect to displacement is included in the derivatives for Jacobian formulation. This allows a more accurate NOSB PD approach to model material behaviour where correspondence materials have previously failed due to instability. Finally, to validate the proposed methodology, several numerical examples of 2D damage problems model using a stabilised correspondence model are verified, and suggestions are made for future implementation. The novelty of this thesis lies in providing theoretical development and numerical implementation of an implicit non-linear NOSB PD focusing on quasi-static analyses with large deformation mechanics. Findings from this thesis will interest researchers working in numerical methods, along with those solving discontinuous solid mechanics problems

Three-dimensional peridynamic modelling of quasi-brittle structural elements

The peridynamic theory provides a promising theoretical framework for developing robust numerical models capable of simulating the complex fracture processes in quasi-brittle materials. However, there is a lack of detailed validation studies in the literature, and significant work remains to quantify the predictive accuracy and generality of a peridynamic model. This thesis presents the development and validation of a three-dimensional bond-based peridynamic framework for modelling quasi-brittle structural elements. By following a rigorous validation process and carefully selecting validation problems that test a wide range of fundamental behaviours, a robust examination of the model is provided, and new insights into the capabilities of the bond-based model are gained. This thesis begins with an examination of existing constitutive laws and a new non-linear softening model is introduced. Predictions with the newly proposed non-linear model improve upon existing laws. In an attempt to explain the cause of discrepancies between experimental and numerical results, it was determined that the application of surface correction factors increases the energy required to produce a fracture surface. This is the first time that this effect has been described, and a correction scheme is proposed that is simple to implement and yields improved results. It is demonstrated that a bond-based peridynamic model can accurately capture the size effect in quasi-brittle materials. This is the first time that a peridynamic model has been used to examine the size effect and provides an important check on the validity of the numerical model. The thesis ends with an examination of the predictive accuracy and generality of the model against nine reinforced concrete beams that exhibit a wide range of failure modes. The shear-span-to-depth ratio is systematically varied from 1 to 8 to facilitate a study of different load-transfer mechanisms and failure modes. This is the first study to rigorously validate the predictive capability of a peridynamic model against a series of problems. The model is validated using published experimental data, and the predictive accuracy is equivalent to well-established numerical methods whilst offering several benefits that justify further research and development.UK Engineering and Physical Sciences Research Council (EPSRC), grant no. EP/L016095/1 - University of Cambridge Centre for Doctoral Training in Future Infrastructure and Built Environmen

Master of Science

thesisTight shale reservoirs have recently emerged as potential game changers in oil and gas and energy sectors worldwide. Consequently, exploration and exploitation of unconventional reservoirs has significantly increased over the last decade. Currently used stimulation designs are based on conventional planar fracture models that cannot realistically simulate the geometry and the extent of hydraulically induced fractures. For that reason, developing models that can thoroughly and accurately describe fracture network initiation and propagation plays a significant role in evaluating well production. The main goal of this work is to evaluate the utility of the peridynamic theory (PD) in modeling the process of hydraulic fracturing. Peridynamics is a nonlocal theory of continuum media that can facilitate a direct coupling between classical continuum mechanics and molecular dynamics. A linear-viscoelastic PD model was applied to a three-dimensional domain that was discretized with cubic lattices of particles. Damage in the model is represented by the bond breakage; as the stretch between two lattices reaches its critical limit, s_0, the bond breaks. The validity of the peridynamic simulation was tested by comparing results obtained in this project against the results obtained in a study performed by Zhou et al. Therefore, six sets of experimental tests were conducted to simulate hydraulic fracturing based on the peridynamic method. Five sets of the simulation results produced in this work were in good agreement with the experimental results. The investigation examined the influences of the differential horizontal stress and preexisting fracture, along with different approach angles, on the geometry of the hydraulic fracture. Different injection rates were applied to the model in order to compare the fractured area that resulted from different injection rates. The simulation showed that the maximum dilatation and fractured zone occurred at the injection rate of 0.61 m3?min. The 0.61 m3/min injection rate caused the highest complete damage (0.9-1) with 5.24 % of the total number of atoms. As a result, the peridynamic approach presents promising results in predicting fracture propagation and damage area

Parallel simulation of particle dynamics with application to micropolar peridynamic lattice modeling of reinforced concrete Structures

As the first goal of this thesis, we will explain a general purpose parallel particle dynamics code (pdQ2). We describe the re-architecting of pdQ (the MD/PD code that was developed in [Sakhavand 2011]) as pdQ2. pdQ2 is completely non-domain-specific in that user files are clearly separated from non-user files and no #ifdefs exist in the code. Thus, it operates as a particle simulation engine that is capable of executing any parallel particle dynamics model. As in the original pdQ, users can customize their own physical models without having to deal with complexities such as parallelization, but the ease of extensibility has been significantly improved. It is shown that pdQ2 is about four times as fast as pdQ using parallel supercomputers. In the second part of the thesis, we will model reinforced concrete structures based on peridynamic theory [Silling 1998]. We discard the continuum mechanics paradigm completely, and model reinforced concrete by introducing the micropolar peridynamic lattice model (MPLM)\u27. The MPLM models a structure as a close-packed particle lattice. In the MPLM, rather than viewing the structure as collection of truss or beam elements (as with traditional lattice models), the model is viewed as collection of particle masses (as with peridynamic models). The MPLM uses a finite number of equally-spaced interacting particles of finite mass. Thus, it does not need any ad hoc discretization and it is more straightforward to implement computationally. Also, the MPLM is conceptually simpler than both the lattice and peridynamic models [Gerstle et al. 2012]. After defining the MPLM, its application to reinforced concrete structures is investigated through several examples using pdQ2.\u2