1河海大学力学与材料学院工程力学系, 南京 211100
2 Department of Aerospace and Mechanical Engineering,University of Arizona,Tucson 85721, USA
Review of peridynamics for multi-physics coupling modeling
GU Xin1,ZHANG Qing1,†,,MADENCI Erdogan2
1 Department of Engineering Mechanics,College of Mechanics and Material,Hohai University, Nanjing 211100, China
2 Department of Aerospace and Mechanical Engineering,University of Arizona, Tucson 85721, USA
版权声明:2019 开云棋牌官方 This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Generally, peridynamics is a theory focusing on the evolution of a physical system, which is based on the assumption that each material point interacts with the other material points within a certain domain through non-contact or nonlocal interactions. It provides a unified mathematical framework for analyzing problems involving the evolving discontinuities and nonlocality. After a brief introduction of the peridynamic solid models and the urgent requirements on multi-physics models and corresponding commercial software, which have the capability of dealing with the evolving discontinuities, we made a systematic review on peridynamic nonlocal diffusion and peridynamic multi-physics coupled modeling. It can be found that the existing multi-physics coupled modeling studies mostly concentrated on the problems in the electronic components, electronic packaging and geotechnical engineering fields, including the un-coupled, partial coupled and fully coupled models about thermo-mechanics, hygro-thermo-mechanics, thermo-oxidative, thermo-mechanics-oxidative, mechanics-electronics, thermo-electronics, thermo-mechanics-electronics, fluid-solid interaction model for porous medium. Finally, several potential problems in the theoretical model, numerical algorithm and engineering application of peridynamic diffusion modeling and multi-physics coupled modeling are suggested.
Keywords：
peridynamics
;
nonlocal theory
;
multi-physics field
;
coupled modeling
Han 等(2016)与Diyaroglu 等 (2017a,2017b)给出多种物理场的键型近场动力学扩散方程形式,通过LINK33三维热传导杆单元和MASS71热质量单元,在ANSYS软件中分别实现了PD热扩散、水分浓度扩散、电传导、归一化湿度场扩散问题的数值模拟,以充分利用ANSYS软件的高效隐式求解器,得到的一维杆和三维交叉堆叠封装结构浓度扩散问题的PD预测结果具有很高的精度.
Numerical modeling of 1D corrosion pit propagation under different overpotentials using peridynamic method. Journal of Nanjing Tech University (Natural Science Edition)
Peridynamics (PD) is arecently developed theory in solid mechanics that employs anonlocal model of force interaction, and replaces the partialdifferential equations, such as the stress-strain relationship ofthe classical continuum theory, by an integral operator that sumsup internal forces separated by finite distances. It allowsdiscontinuities of various types to be modeled without applicationof special techniques to the discontinuous field and shows apromising prospect for solving practical problems in whichdiscontinuities form and grow spontaneously. Peridynamics has beensuccessfully applied to damage and failure problems at both macro-and micro-scales with satisfactory solution precision andnumerical efficiency. Without pathological defects of traditionalmethods when facing discontinuous problems, and excluding thecomputational limitations in length and time scales, peridynamicsshows great potential analogous but advantageous to both classicalmeshfree and molecular dynamic methods. The present paper firstreviews its theoretical basis, numerical scheme and modelingmethod, and then elucidates its application to discontinuousproblems at different scale, including damage, fracture, impact,penetration and stability analysis for macro-scale homogeneous andheterogeneous materials and structures, kinetics analysis forphase transformations and atomistic analysis for nanoscalematerials. Finally, some unsolved problems and future researchtrends in PD are discussed.
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Peridynamics (PD) is arecently developed theory in solid mechanics that employs anonlocal model of force interaction, and replaces the partialdifferential equations, such as the stress-strain relationship ofthe classical continuum theory, by an integral operator that sumsup internal forces separated by finite distances. It allowsdiscontinuities of various types to be modeled without applicationof special techniques to the discontinuous field and shows apromising prospect for solving practical problems in whichdiscontinuities form and grow spontaneously. Peridynamics has beensuccessfully applied to damage and failure problems at both macro-and micro-scales with satisfactory solution precision andnumerical efficiency. Without pathological defects of traditionalmethods when facing discontinuous problems, and excluding thecomputational limitations in length and time scales, peridynamicsshows great potential analogous but advantageous to both classicalmeshfree and molecular dynamic methods. The present paper firstreviews its theoretical basis, numerical scheme and modelingmethod, and then elucidates its application to discontinuousproblems at different scale, including damage, fracture, impact,penetration and stability analysis for macro-scale homogeneous andheterogeneous materials and structures, kinetics analysis forphase transformations and atomistic analysis for nanoscalematerials. Finally, some unsolved problems and future researchtrends in PD are discussed.
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The dynamic mechanical behavior of granular materials under impact load is a complex issue. Peridynamics as a new theory based on discontinuous and nonlocal hypothesis regards materials as compositions of massive material points with finite volume and finite mass, and builds an integral governing equation to reflect the motion law of material points. For all the features mentioned above, peridynamics is certainly suitable for describing and analyzing the dynamic behavior of particles. An improved PMB model considering the feature of nonlocal long range force and eliminating the oundary e ect and a repulsive force model at material point level to describe the inter-particle contact interaction are proposed. Then the method is applied to analyze the dynamics responses of tungsten carbide (WC) ceramic granular system su ering from impact loading. Wave velocities of the system were calculated accurately under di erent impact velocities compared with the experiment results. Phenomena of the motion, including translation and rotation, deformation and crushing of particles are reappeared. There are both total damaged particle and slight damaged particle near the impactor, and there are also particles far out from the impactor which are damaged. The extrusion, collision and shear slide between particles result in the particle crushing. The results indicate that the calculation model and analysis method developed here can well reflect the dynamic behavior of granular materials and have large application value.
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The dynamic mechanical behavior of granular materials under impact load is a complex issue. Peridynamics as a new theory based on discontinuous and nonlocal hypothesis regards materials as compositions of massive material points with finite volume and finite mass, and builds an integral governing equation to reflect the motion law of material points. For all the features mentioned above, peridynamics is certainly suitable for describing and analyzing the dynamic behavior of particles. An improved PMB model considering the feature of nonlocal long range force and eliminating the oundary e ect and a repulsive force model at material point level to describe the inter-particle contact interaction are proposed. Then the method is applied to analyze the dynamics responses of tungsten carbide (WC) ceramic granular system su ering from impact loading. Wave velocities of the system were calculated accurately under di erent impact velocities compared with the experiment results. Phenomena of the motion, including translation and rotation, deformation and crushing of particles are reappeared. There are both total damaged particle and slight damaged particle near the impactor, and there are also particles far out from the impactor which are damaged. The extrusion, collision and shear slide between particles result in the particle crushing. The results indicate that the calculation model and analysis method developed here can well reflect the dynamic behavior of granular materials and have large application value.
Modeling of heat and electrical current flow simultaneously in thermoelectric convertor using classical theories do not consider the influence of defects in the material. This is because traditional methods are developed based on partial differential equations (PDEs) and lead to infinite fluxes and stress fields at the crack tips. The usual way of solving such PDEs is by using numerical technique, like Finite Element Method (FEM). Although FEM is robust and versatile, it is not suitable to model evolving discontinuities since discontinuous fields are mathematically singular at the crack tip and required an external criterion for the prediction of crack growth. In this paper, we follow the concept of peridynamic (PD) theory to overcome the shortcomings above. Therefore, the main aim of this paper is to develop the peridynamic equations for the generalized Fourier090005s and Ohm090005s laws. Furthermore, we derived the peridynamic equations for the conservation of energy and charge for the coupled thermoelectric phenomena.
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In bodies where discontinuities, like cracks, emerge and interact, the classical equations for heat and mass transfer are not well suited. We propose a peridynamic model for transient heat (or mass) transfer which is valid when the body undergoes damage or evolving cracks. We use a constructive approach to find the peridynamic formulation for heat transfer and test the numerical convergence to the classical solutions in the limit of the horizon (the nonlocal parameter) going to zero for several one-dimensional problems with different types of boundary conditions. We observe an interesting property of the peridynamic solution: when two m-convergence curves, corresponding to two different horizons, for the solution at a point and an instant intersect, the intersection point is also the exact classical (local) solution. The present formulation can be easily extended to higher dimensions and be coupled with the mechanical peridynamic description for thermomechanical analyses of fracturing bodies, or for heat and mass transfer in bodies with evolving material discontinuities.
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The kernel in a peridynamic diffusion model represents the detailed interaction between points inside the nonlocal region around each material point. Several versions of the kernel function have been proposed. Although solutions associated with different kernels may all converge, under the appropriate discretization scheme, to the classical model when the horizon goes to zero, their convergence behavior varies. In this paper, we focus on the particular one-point Gauss quadrature method of spatial discretization of the peridynamic diffusion model and study the convergence properties of different kernels with respect to convergence to the classical, local, model for transient heat transfer equation in 1D, where exact representation of geometry is available. The one-point Gauss quadrature is the preferred method for discretizing peridynamic models because it leads to a meshfree model, well suited for problems with damage and fracture. We show the equivalency of two definitions for the peridynamic heat flux. We explain an apparent paradox and discuss a common pitfall in numerical approximations of nonlocal models and their convergence to local models. We also analyze the influence of two ways of imposing boundary conditions and that of the “skin effect” on the solution. We explain an interesting behavior of the peridynamic solutions for different horizon sizes, the crossing ofm-convergence curves at the classical solution value that happens for one of the ways of implementing the classical boundary conditions. The results presented here provide practical guidance in selecting the appropriate peridynamic kernel that makes the one-point Gauss quadrature an “asymptotically compatible” scheme. These results are directly applicable to any diffusion-type model, including mass diffusion problems.
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61First model to simulate sub-surface corrosion damage in corrosion.61Corrosion reaction as nonlocal-diffusion plus phase-change in metal/electrolyte.61Activation, IR and diffusion-controlled corrosion regimes predicted with single model.61Damage evolution coupled with metal corrosion is modeled in 1D, 2D and 3D.61Mircostructural heterogeneities and overpotential shown to affect corrosion damage.
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61A novel multiphysics peridynamic framework is used to predict crack initiation, propagation and branching due to stress corrosion cracking.61The model consists of a 2D polycrystalline pre-cracked thin steel plate subjected to mechanical load and exposed to a corrosive aqueous solution.61Novel micro-mechanical and hydrogen grain boundary diffusion peridynamic formulations are introduced.61The material is modelled at the microscale and first principle calculations are used to predict the toughness reduction of the material due to grain boundary hydrogen diffusion.61A good agreement between numerical and experimental results is found.
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The multiscale nature of geological formations is reflected in the flow and transport behaviors of the pore fluids. For example, multiple pathways between different locations in the porous medium are usually present. The topology, length, and strength of these flow paths can vary significantly, and the total flow at a given location can be the result of contributions from a wide range of pathways between the points of interest. We use a high-resolution pore network of a natural porous formation as an example of the multiscale connectivity of the pore space. A single continuum model can capture the contributions from all the flow paths properly only if the control volume (computational cell) is much larger than the longest pathway. However, depending on the densities and lengths of these long pathways, choosing the appropriate size of the control volume that allows for a single continuum description of the properties, such as conductivity and transmissibility, may conflict with the desire to resolve the flow field properly. To capture the effects of the multiscale pathways on the flow, a non-local continuum model is described. The model can represent non-local effects, for which Darcy law is not valid. In the limit where the longest connections are much smaller than the size of the control volume, the model is consistent with Darcy law. The non-local model is used to describe the flow in complex pore networks. The pressure distributions obtained from the non-local model are compared with pore-network flow simulations, and the results are in excellent agreement. Importantly, such multiscale flow behaviors cannot be represented using the local Darcy law.
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Diffusion modeling is essential in understanding many physical phenomena such as heat transfer, moisture concentration, and electrical conductivity. In the presence of material and geometric discontinuities and nonlocal effects, a nonlocal continuum approach, named peridynamics (PD), can be advantageous over the traditional local approaches. PD is based on integro-differential equations without including any spatial derivatives. In general, these equations are solved numerically by employing meshless discretization techniques. Although fundamentally different, commercial finite-element software can be a suitable platform for PD simulations that may result in several computational benefits. Hence, this paper presents the PD diffusion modeling and implementation procedure in a widely used commercial finite-element analysis software, ANSYS. The accuracy and capability of this approach is demonstrated by considering several benchmark problems.
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Peridynamic wetness approach for moisture concentration analysis in electronic packages
Within the finite element framework, a commonly accepted indirect approach employs the concept of normalized concentration to compute moisture concentration. It is referred to as etness approach. If the saturated concentration value is not dependent on temperature or time, the wetness equation is analogous to the standard diffusion equation whose solution can be constructed by using any commercial finite element analysis software such as ANSYS. However, the time dependency of saturated concentration requires special treatment under temperature dependent environmental conditions such as reflow process. As a result, the wetness equation is not directly analogous to the standard diffusion equation. This study presents the peridynamic wetness modeling for time dependent saturated concentration for computation of moisture concentration in electronic packages. It is computationally efficient as well as easy to implement without any iterations in each time step. Numerical results concerning the one-dimensional analysis illustrate the accuracy of this approach. Moisture concentration calculation in a three-dimensional electronic package configuration with many different material layers demonstrates its robustness.
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An unconditionally stable second order accurate implicit-implicit staggered procedure for the finite element solution of fully coupled thermoelasticity transient problems is proposed. The procedure is stabilized with a semi-algebraic augmentation technique. A comparative cost analysis reveals the superiority of the proposed computational strategy to other conventional staggered procedures. Numerical examples of one- and two-dimensional thermomechanical coupled problems demonstrate the accuracy of the proposed numerical solution algorithm.
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The Split Hopkinson Pressure Bar (SHPB) technique is a widely used method for measuring mechanical properties of materials subjected to high-strain-rate loads, while it is difficult to simulate the whole testing process including high-rate deformation, local damage and failure of materials by the common numerical methods. In this paper, an improved numerical approach based on the non-local peridynamic (PD) theory is employed to study the elastic wave dispersion and propagation and the impact failure of concrete Brazilian discs in SHPB test. Meanwhile, an improved PMB (Prototype Microelastic Brittle) model and an implementation method of the contact-impact process are introduced. In PD, the nonlocal long-range force controls the numerical dispersion of wave through different material point sizes and horizon sizes. The numerical dispersion can result in a slight distortion of the wave speed and crack propagation speed, which is not conductive to failure analysis of solids. The improved PMB model can effectively lessen the numerical dispersion compared with the original PMB model. Furthermore, the PD simulation of concrete Brazilian disc SHPB test can reproduce damage accumulation and progressive failure of a specimen, and produce typical final failure pattern. The PD simulation method for SHPB test can be used to analyze the dynamic response of solids suffering impact load.
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The force density vector in the Non-Ordinary State-Based (NOSB) PeriDynamics (PD) replaces the internal force vector derived from the divergence of the stress tensor in the classical (local) stress equilibrium equations. It involves only the non-local form of the first order derivatives of stress and displacement components. Inherent in the NOSB PD formulation is the presence of oscillations especially in the regions of steep displacement gradients. This study introduces an alternative form of the force density vector by considering the internal force vector derived directly from the displacement equilibrium equations. It involves only the non-local form of the second-order derivatives of the displacement components. The numerical results from this form of the force density vector do not present any oscillations. Therefore, it is referred to as the Refined NOSB (RNOSB) PD. The simulations concern the comparisons of NOSB and RNOSB PD predictions for an isotropic plate with or without a notch or a crack under quasi-static and dynamic tensile loading. The RNOSB PD proves to be effective and accurate for cracking and fracture analysis without any numerical instability.
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Irregular non-uniform discretization of the solution domain in models based on peridynamic theory can improve computational efficiency by allowing local refinement and remove mesh bias effects on crack initiation and propagation. However, the use of such discretizations generally requires adjustment of the classical peridynamic material parameters and usage of a variable horizon which results in the so-called ghost force effect in the interactions between differing horizons. This study presents a generalization of the original bond-based and ordinary state-based peridynamic models to permit the use of irregular non-uniform domain discretizations, in which the strain energy and thermal potential associated with a bond between two material points is split into two parts based on volumetric ratios. This division is potentially different for each bond due to the presence of irregular non-uniform discretization. The validity and accuracy of this proposed approach is established using several benchmark examples, and its applicability to real engineering problems is demonstrated by modeling thermally induced cracking in a three-dimensional nuclear fuel pellet.
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61Hydrogen leakage accompany with tank wall crack propagation was newly analyzed.61Coupled particle and Eulerian methods has been developed for hydrogen safety problems.61Peridynamics model was verified as effective method for hydrogen tank crack analysis.61Volume concentration of gaseous hydrogen leakage from crack was computationally predicted.
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This paper presents a nonlocal, derivative free model for transient flow in unsaturated, heterogeneous, and anisotropic soils. The formulation is based on the peridynamic model for solid mechanics. In the proposed model, flow and changes in moisture content are driven by pairwise interactions with other points across finite distances, and are expressed as functional integrals of the hydraulic potential field. Peridynamic expressions of the rate of change in moisture content, moisture flux, and flow power are derived, as are relationships between the peridynamic and the classic hydraulic conductivities; in addition, the model is validated. The absence of spacial derivatives makes the model a good candidate for flow simulations in fractured soils and lends itself to coupling with peridynamic mechanical models for simulating crack formation triggered by shrinkage and swelling, and assessing their potential impact on a wide range of processes, such as infiltration, contaminant transport, and slope stability.
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react-text: 379 The kernel in a peridynamic diffusion model represents the detailed interaction between points inside the nonlocal region around each material point. Several versions of the kernel function have been proposed. Although solutions associated with different kernels may all converge, under the appropriate discretization scheme, to the classical model when the horizon goes to zero, their... /react-text react-text: 380 /react-text [Show full abstract]
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In the context of flow and transport in porous and fractured media, Darcy-based continuum models, while computationally inexpensive, are of limited use when the scale of interest is of similar size or smaller than the characteristic network connection length. Recently, we have outlined a non-local Darcy model that bridges the gap between network and Darcy-based descriptions. This formulation is able to account for non-local pressure effects that are not accounted for in a classical Darcy description. At the heart of this non-local flow formulation is a conductivity distribution or kernel that is related to the scalar permeability in the classical Darcy law. In this paper, ensembles of flow networks are considered, of which the necessary statistical information is assumed to be known. In order to relate the conductivity distribution with the flow statistics, a stochastic transport model for fluid particles, termed generalized continuous time random walk (g-CTRW), which is a generalization of correlated continuous time random walk, is introduced. Note that similar assumptions as for correlated CTRW are made, i.e., that lengths and velocities of connections between successive nodes along the trajectories can be described by Markov processes. In order to proceed with a theoretical analysis, a Boltzmann equation is presented, which is consistent with the particle time marching algorithm based on g-CTRW. An important outcome of the analysis is an expression relating the joint probability density function of velocity and connection length in the networks with the conductivity kernel. A numerical, stationary flow example demonstrates how the kernel can be extracted. Further, an algorithm is proposed to compute consistent velocity statistics, mean pressure distribution, and spatially varying conductivity kernel in the case of non-stationary flow. This coupled iterative approach is an attempt to consistently compute stochastic flow and transport in large network ensembles.
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Peridynamic simulations of brittle structures with thermal residual deformation: Strengthening and structural reactivity of glasses under impacts//Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences
In glass research, the effect and influence of pre-deformation by thermal or chemical treatment is of great importance when configuring different mechanical properties or scratch resistance on the surface of glasses. In particular, such pre-deformation affects dynamic fracture or damage evolution when glass structures are under impact or collision conditions. Peridynamics provides a seamless approach for the simulation of dynamic damage evolution of the system under aggressive environments. Revising the pair interaction of each material point, the effect of pre-deformation is implemented, and the corresponding damage evolution can be simulated conveniently. Our approach is composed of two steps: first, a static solution is found via energy minimization with thermal boundary conditions in the peridynamic platform. Second, comparing the initial and the pre-deformed structures from the energy minimization, the effect of residual deformation, strengthening and reactive behaviour of brittle structures are seamlessly simulated. The developed methods are applied to the Prince Rupert drop and Bologna vial, which are classic examples of strengthened glasses. This study reports the first complete and successful simulation of dynamic behaviour of strengthened glasses, and a significant contribution in simulating residual stress behaviour in any material.
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A general state-based peridynamic formulation is presented for convective single-phase flow of a liquid of small and constant compressibility in heterogeneous porous media. In addition to local fluid transport, possible anomalous diffusion due to non-local fluid transport is considered and simulated. The governing integral equations of the peridynamic formulation are computationally easier to solve in domains with discontinuities than the traditional conservation models containing spatial derivatives. A bond-based peridynamic formulation is also developed and demonstrated to be a special case of the state-based formulation. The non-local model does not assume continuity in the field variables, satisfies mass conservation over an arbitrary bounded body and approaches the corresponding local model as the non-local region goes to zero. The exact solution of the local model closely matches the non-local model for a classical two-dimensional flow problem with fluid sources and sinks and for both Neumann and Dirichlet boundary conditions. The model is shown to capture arbitrary flow discontinuities/heterogeneities without any fundamental changes to the model and with small incremental computational costs.
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Prediction of crack paths in a quenched glass plate by using peridynamic theory
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International Journal of Fracture,
156
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165
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177
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The peridynamic theory is employed to predict crack growth patterns in quenched glass plates previously considered for an experimental investigation. The plates containing single and multiple pre-existing initial cracks are simulated to investigate the effects of peridynamic and experimental parameters on the crack paths. The critical stretch value in the peridynamic theory and the gap size between the heat reservoirs are determined to be the most significant parameters. The simulation results are in good agreement with the experimental observations published in the literature.
Thermomechanical modeling for interconnects and electronic packages is a difficult challenge, especially for material interfaces and films under 1 mum dimension. Understanding and prediction of their mechanical failure require the simulation of material behavior in the presence of multiple length scales. However, the classical continuum theory is not capable of predicting failure without a posterior analysis with an external crack growth criteria and treats the interfaces having zero thickness. A new nonlocal continuum theory referred to as peridynamic theory offers the ability to predict failure at these length scales. This study presents a new response function as part of the peridynamic theory to include thermal loading. After validating this response function by comparing against the displacement predictions in benchmark problems against those of finite element method, the peridynamic theory is used to predict damage initiation and propagation in regions having dissimilar materials due to thermomechanical loading.
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SUMMARYPeridynamics is a non-local mechanics theory that uses integral equations to include discontinuities directly in the constitutive equations. A three-dimensional, state-based peridynamics model has been developed previously for linearly elastic solids with a customizable Poisson's ratio. For plane stress and plane strain conditions, however, a two-dimensional model is more efficient computationally. Here, such a two-dimensional state-based peridynamics model is presented. For verification, a 2D rectangular plate with a round hole in the middle is simulated under constant tensile stress. Dynamic relaxation and energy minimization methods are used to find the steady-state solution. The model shows m-convergence and -convergence behaviors when m increases and decreases. Simulation results show a close quantitative matching of the displacement and stress obtained from the 2D peridynamics and a finite element model used for comparison. Copyright 2014 John Wiley & Sons, Ltd.
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Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks
(2017). Peridynamic simulation of transient heat conduction problems in functionally gradient materials with cracks. Journal of Thermal Stresses: Vol. 40, No. 12, pp. 1484-1501.
Abstract This book presents the peridynamic theory, which provides the capability for improved modeling of progressive failure in materials and structures, and paves the way for addressing multi-physics and multi-scale problems. The book provides students and researchers with a theoretical and practical knowledge of the peridynamic theory and the skills required to analyze engineering problems. The text may be used in courses such as Multi-physics and Multi-scale Analysis, Nonlocal Computational Mechanics, and Computational Damage Prediction. Sample algorithms for the solution of benchmark problems are available so that the reader can modify these algorithms, and develop their own solution algorithms for specific problems. Students and researchers will find this book an essential and invaluable reference on the topic. 2014 Springer Science+Business Media New York. All rights are reserved.
[59]
Madenci
E
,
Oterkus
S
.
2017
a. Ordinary state-based peridynamics for thermoviscoelastic deformation
This study presents the ordinary state-based peridynamic (PD) constitutive relations for viscoelastic deformation under mechanical and thermal loads. The behavior of the viscous material is modeled in terms of Prony series. The constitutive constants are the same as those of the classical history-integral model, and they are also readily available from relaxation tests. The state variables are conjugate to the PD elastic stretch measures; hence, they are consistent with the kinematic assumptions of the elastic deformation. The PD viscoelastic deformation analysis successfully captures the relaxation behavior of the material. The numerical results concern first the verification problems, and subsequently, a double-lap joint with a viscoelastic adhesive where failure nucleates and grows.
[60]
Madenci
E
,
Oterkus
S
.
2017
b. Peridynamic modeling of thermo-oxidative damage evolution in a composite lamina
//
In 58th AIAA/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,
2017
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0197
.
Three dimensional models of explicit cracking of nuclear fuel pellets for a variety of power ratings have been explored with peridynamics, a non-local, mesh free, fracture mechanics method. These models were implemented in the explicitly integrated molecular dynamics code LAMMPS, which was modified to include thermal strains in solid bodies. The models of fuel fracture, during initial power transients, are shown to correlate with the mean number of cracks observed on the inner and outer edges of the pellet, by experimental post irradiation examination of fuel, for power ratings of 10 and 15 W g 1UO2. The models of the pellet show the ability to predict expected features such as the mid-height pellet crack, the correct number of radial cracks and initiation and coalescence of radial cracks. This work presents a modelling alternative to empirical fracture data found in many fuel performance codes and requires just one parameter of fracture strain. Weibull distributions of crack numbers were fitted to both numerical and experimental data using maximum likelihood estimation so that statistical comparison could be made. The findings show P-values of less than 0.5% suggesting an excellent agreement between model and experimental distributions.
[62]
Multiphysics
CO
.
2012
.
Comsol Multiphysics User's Guide, Version 4.3a
.
[63]
Nadimi
S
,
Miscovic
I
,
McLennan
J
.
2016
.
A 3D peridynamic simulation of hydraulic fracture process in a heterogeneous medium
.
Journal of Petroleum Science and Engineering,
145
:
444
-
452
.
61Utility of the peridynamics theory (PD) in modeling the hydraulic fracture phenomenon.61Suitable for modeling coupled complex thermal-hydrologic-geomechanical process without using additional failure criteria or crack growth laws.61Using of integral equations, instead of the partial differential equations used in classical formulations.61Application of the peridynamic theory in simulation hydraulic fracturing in a heterogeneous medium.
[64]
Oterkus
S
.
2015
.
Peridynamics for the solution of multiphysics problems. [PhD Thesis]
This study presents peridynamic field equations for mechanical deformation, thermal diffusion, moisture diffusion, electric potential distribution, porous flow and atomic diffusion in either an uncoupled or a coupled manner. It is a nonlocal theory with an internal length parameter. Therefore, it can capture physical phenomenon for the problems which include non-local effects and are not suitable for classical theories. Moreover, governing equations of peridynamics are based on integro-differential equations which permits the determination of the field variable in spite of discontinuities. Inherent with the nonlocal formulations, the imposition of the boundary conditions requires volume constraints. This study also describes the implementation of the essential and natural boundary conditions, and demonstrates the accuracy of their implementation. Solutions coupled field problems concerning plastic deformations, thermomechanics, hygrothermomechanics, hydraulic fracturing, thermal cracking of fuel pellet and electromigration are constructed. Their comparisons with the finite element predictions establish the validity of the PD field equations for coupled field analysis.
[65]
Oterkus
S
,
Fox
J
,
Madenci
E
.
2013
.
Simulation of electro-migration through peridynamics//IEEE 63rd Electronic Components and Technology Conference (ECTC),
2013
:
1488
-
1493
.
This study presents a fully coupled peridynamic thermomechanical equations, and their nondimensional form for two-dimensional analysis. Peridynamic solutions to both the thermal and deformation field equations are verified against the known solutions. Subsequently, the coupled thermal and structural response of a plate with a pre-existing crack is investigated. It accurately predicts the temperature rise at the crack tips as the crack propagates.
[67]
Oterkus
S
,
Madenci
E
.
2014
.
Fully coupled thermomechanical analysis of fiber reinforced composites using peridynamics
//
In 55th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference,
2014
-
0694
.
This study presents the peridynamic simulation of thermal cracking behaviour in uranium dioxide, UO2fuel pellets that are used in light water reactors (LWR). The performance of the reactor is influenced by the thermo-mechanical behaviour of the pellets. During the fission process, the pellets are subjected to high temperature gradients, and the oxygen diffusion significantly affects the temperature distribution. Therefore, a coupled analysis of temperature and oxygen diffusions and deformation is unavoidable in order to predict accurate cracking behavior in a fuel pellet. The accuracy of the predictions is verified qualitatively by comparing with the previous studies.
This study concerns the derivation of the coupled peridynamic (PD) thermomechanics equations based on thermodynamic considerations. The generalized peridynamic model for fully coupled thermomechanics is derived using the conservation of energy and the free-energy function. Subsequently, the bond-based coupled PD thermomechanics equations are obtained by reducing the generalized formulation. These equations are also cast into their nondimensional forms. After describing the numerical solution scheme, solutions to certain coupled thermomechanical problems with known previous solutions are presented.
[71]
Oterkus
S
,
Madenci
E
,
Oterkus
E
,
Hwang
Y
,
Bae
J
,
Han
S
.
2014c
.
Hygro-thermo-mechanical analysis and failure prediction in electronic packages by using peridynamics//IEEE 64th Electronic Components and Technology Conference (ECTC),
2014
:
973
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982
.
61A new fully coupled poroelastic peridynamic formulation is developed.61The formulation is also suitable for fluid-filled fractures.61The formulation is validated by considering one- and two-dimensional consolidation problems.61A hydraulically pressurized crack case is analyzed.
[73]
Ouchi
H
,
Katiyar
A
,
York
J
,
Foster
J T
,
Sharma
M M
.
2015
a. A fully coupled porous flow and geomechanics model for fluid driven cracks: A peridynamics approach
A state-based non-local peridynamic formulation is presented for simulating fluid driven fractures in an arbitrary heterogeneous poroelastic medium. A recently developed peridynamic formulation of porous flow has been coupled with the existing peridynamic formulation of solid and fracture mechanics resulting in a peridynamic model that for the first time simulates poroelasticity and fluid-driven fracture propagation. This coupling is achieved by modeling the role of pore pressure on the deformation of porous media and vice versa through porosity variation with medium deformation, pore pressure and total mean stress. The poroelastic model is verified by simulating the one-dimensional consolidation of fluid saturated rock. An additional porous flow equation with material permeability dependent on fracture width is solved to simulate fluid flow in the fractured region. Finally, single fluid-driven fracture propagation with a two-dimensional plane strain assumption is simulated and verified against the corresponding classical analytical solution.
[74]
Ouchi
H
,
Katiyar
A
,
Foster
J T
,
Sharma
M M
.
2015
b.
A peridynamics model for the propagation of hydraulic fractures in heterogeneous, naturally fractured reservoirs//In SPE Hydraulic Fracturing Technology Conference 2015
The effect of different types of vertical reservoir heterogeneities on fracture propagation was systematically investigated. A fully 3-D, poroelastic model that does not prescribe the crack propagation path is used to estimate the fracture geometry in vertically heterogeneous rocks. Complex fracture trajectories are shown to occur and this limits fracture height growth. It is shown that the presence of bedding planes, layer interfaces and even smaller scale heterogeneities can lead to fracture turning, kinking or branching. The mechanisms that control these characteristic fracture propagation behaviors (“turning”, “kinking”, and “branching”) near the layer interface are explored in detail. In layered systems, the mechanical property contrast between layers, the dip angle, the stress contrast and the mechanical properties of the layer interface all play an important role in controlling the fracture trajectory. Conditions under which each type of behavior is expected to occur are clearly delineated.
[78]
Prakash
N
,
Seidel
G D
.
2016
.
Electromechanical peridynamics modeling of piezoresistive response of carbon nanotube nanocomposites
In this work, a coupled electromechanical peridynamics formulation is presented which is used to study the electrical and piezoresistive response of a carbon nanotube (CNT) reinforced polymer nanocomposite material. CNT nanocomposites are multiscale materials which have unique piezoresistive properties arising from mechanisms operating from the nanoscale to the macroscale. The origin of piezoresistivity in CNT nanocomposites is a nanoscale phenomenon known as electron hopping or the electrical tunneling effect which allows an electric current to flow between neighboring CNTs even when not in contact, thereby forming a conductive network. A nanoscale representative volume element of a CNT bundle is chosen, i.e. a local region of high CNT volume fraction within the polymer matrix, wherein coupled electromechanical peridynamic equations are solved to evaluate the effective electrical and piezoresistive properties. The peridynamics formulation is used to introduce electron hopping in a unique way, through electron hopping bonds which have a horizon distance and conductivity dictated by the appropriate physics operating at the nanoscale. The effective electromechanical response depends on parameters such as CNT volume fraction, properties of the polymer matrix between CNTs and applied strain which are investigated in detail. Both quasistatic and dynamic loading conditions are considered where the effective electromechanical response is found to depend on variations in the local conductivity of intertube regions.
[79]
Prakash
N
,
Seidel
G D
.
2017
.
Computational electromechanical peridynamics modeling of strain and damage sensing in nanocomposite bonded explosive materials (ncbx)
Polymer bonded explosives can sustain microstructural damage due to accidental impact, which may reduce their operational reliability or even cause unwanted ignition leading to detonation of the explosive. Therefore a nanocomposite piezoresistivity based sensing solution is discussed here that employs a carbon nanotube based nanocomposite binder in the explosive material by which in situ... [Show full abstract]
[80]
Read
D T
,
Tewary
V K
,
Gerstle
W H
.
2011
.
Modeling electromigration using the peridynamics approach
.
In Electromigration in Thin Films and Electronic Devices,
45
-
69
.
This chapter presents a summary of the information and reasoning needed to justify learning about peridynamics for the purpose of analyzing electromigration and provides guidance for the development of a complete peridynamics analysis. The additions needed to convert the original peridynamics model as developed for mechanics problems to a multiphysics model capable of treating electromigration are reviewed. Experimental data on void drift by electromigration are introduced to provide a specific target for a demonstration of the peridynamical approach. Model results for the basic phenomena of this experiment are presented. The peridynamics approach appears capable of simultaneously accommodating both constitutive laws and explicit treatment of multibody interactions, for handling different aspects of the behavior of the material system to be modeled.
A flexoelectric peridynamic (PD) theory is proposed. Using the PD framework, the formulation introduces, perhaps for the first time, a nanoscale flexoelectric coupling that entails non-uniform strain in centrosymmetric dielectrics. This potentially enables PD modeling of a large class of phenomena in solid dielectrics involving cracks, discontinuities etc. wherein large strain gradients are present and the classical electromechanical theory based on partial differential equations do not directly apply. Derived from Hamilton's principle, PD electromechanical equations are shown to satisfy the global balance requirements. Linear PD constitutive equations reflect the electromechanical coupling effect, with the mechanical force state affected by the polarization state and the electrical force state in turn by the displacement state. An analytical solution of the PD electromechanical equations in the integral form is presented for the static case when a point mechanical force and a point electric force act in a three dimensional infinite solid dielectric. A parametric study on how the different length scales influence the response is also undertaken.
[82]
Silling
S A
.
2000
.
Reformulation of elasticity theory for discontinuities and long-range forces
.
Journal of the Mechanics and Physics of Solids,
48
:
175
-
209
.
Some materials may naturally form discontinuities such as cracks as a result of deformation. As an aid to the modeling of such materials, a new framework for the basic equations of continuum mechanics, called the ‘peridynamic’ formulation, is proposed. The propagation of linear stress waves in the new theory is discussed, and wave dispersion relations are derived. Material stability and its connection with wave propagation is investigated. It is demonstrated by an example that the reformulated approach permits the solution of fracture problems using the same equations either on or off the crack surface or crack tip. This is an advantage for modeling problems in which the location of a crack is not known in advance.
[83]
Silling
S A
,
Askari
E
.
2005
.
A meshfree method based on the peridynamic model of solid mechanics
An alternative theory of solid mechanics, known as the peridynamic theory, formulates problems in terms of integral equations rather than partial differential equations. This theory assumes that particles in a continuum interact with each other across a finite distance, as in molecular dynamics. Damage is incorporated in the theory at the level of these two-particle interactions, so localization and fracture occur as a natural outgrowth of the equation of motion and constitutive models. A numerical method for solving dynamic problems within the peridynamic theory is described. Accuracy and numerical stability are discussed. Examples illustrate the properties of the method for modeling brittle dynamic crack growth. [All rights reserved Elsevier]
[84]
Silling
S A
,
Epton
M
,
Weckner
O
,
Xu
J F
,
Askari
E
.
2007
.
A generalization of the original peridynamic framework for solid mechanics is proposed. This generalization permits the response of a material at a point to depend collectively on the deformation of all bonds connected to the point. This extends the types of material response that can be reproduced by peridynamic theory to include an explicit dependence on such collectively determined quantities as volume change or shear angle. To accomplish this generalization, a mathematical object called a deformation state is defined, a function that maps any bond onto its image under the deformation. A similar object called a force state is defined, which contains the forces within bonds of all lengths and orientation. The relation between the deformation state and force state is the constitutive model for the material. In addition to providing a more general capability for reproducing material response, the new framework provides a means to incorporate a constitutive model from the conventional theory of solid mechanics directly into a peridynamic model. It also allows the condition of plastic incompressibility to be enforced in a peridynamic material model for permanent deformation analogous to conventional plasticity theory.
Modeling important engineering problems related to flow-induced damage (in the context of hydraulic fracturing among others) depends critically on characterizing the interaction of porous media and interstitial fluid flow. This work presents a new formulation for incorporating the effects of pore pressure in a non-local representation of solid mechanics. The result is a framework for modeling fluid-structure interaction problems with the discontinuity capturing advantages of an integral-based formulation. A number of numerical examples are used to show that the proposed formulation can be applied to measure the effect of leak-off during hydraulic fracturing as well as modeling consolidation of fluid-saturated rock and surface subsidence caused by fluid extraction from a geologic reservoir. The formulation incorporates the effect of pore pressure in the constitutive description of the porous material in a way that is appropriate for nonlinear materials, easily implemented in existing codes, straightforward in its evaluation (no history dependence), and justifiable from first principles. A mixture theory approach is used (deviating only slightly where necessary) to motivate an alteration to the peridynamic pressure term based on the fluid pore pressure. The resulting formulation has a number of similarities to the effective stress principle developed by Terzaghi and Biot and close correspondence is shown between the proposed method and the classical effective stress principle.
[87]
Wang
H L
,
Oterkus
E
,
Oterkus
S
.
2018
.
Predicting fracture evolution during lithiation process using peridynamics
Silicon is regarded as one of the most promising anode materials for lithium-ion batteries due to its large electric capacity. However, silicon experiences large volumetric change during battery cycling which can lead to fracture and failure of lithium-ion batteries. The lithium concentration and anode material phase change have direct influence on hydrostatic stress and damage evolution. High pressure gradient around crack tips causes mass flux of lithium ions which increases the lithium-ion concentration in these regions. Therefore, it is essential to describe the physics of the problem by solving fully coupled mechanical-diffusion equations. In this study, these equations are solved using peridynamics in conjunction with newly introduced peridynamic differential operator concept used to convert partial differential equation into peridynamic form for the diffusion equation. After validating the developed framework, the capability of the current approach is demonstrated by considering a thin electrode plate with multiple pre-existing cracks oriented in different directions. It is shown that peridynamics can successfully predict the crack propagation process during the lithiation process.
[88]
Wang
L J
,
Xu
J F
,
Wang
J X
.
2016
.
The Green's functions for peridynamic non-local diffusion//Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences
A coupled thermo-mechanical bond-based peridynamical (TM-BB-PD) method is developed to simulate thermal cracking processes in rocks. The coupled thermo-mechanical model consists of two parts. In the...
[91]
Wildman
R A
,
Gazonas
G A
.
2015
.
A dynamic electro-thermo-mechanical model of dielectric breakdown in solids using peridynamics
.
Journal of Mechanics of Materials and Structures,
10
:
613
-
630
.
A multiscale implementation of hybrid continuous/discontinuous finite element discretizations of nonlocal models for mechanics and diffusion in two dimensions is developed. The implementation features adaptive mesh refinement based on the detection of defects and results in an abrupt transition between refined elements that contain defects and unrefined elements free of defects. An additional difficulty overcome in the implementation is the design of accurate quadrature rules for stiffness matrix construction that are valid for any combination of the grid size and horizon parameter, the latter being the extent of nonlocal interactions. As a result, the methodology developed can attain optimal accuracy at very modest additional costs relative to situations for which the solution is smooth. Portions of the methodology can also be used for the optimal approximation, by piecewise linear polynomials, of given functions containing discontinuities. Several numerical examples are provided to illustrate the efficacy of the multiscale methodology.
[94]
Xu
Z P
,
Zhang
G F
,
Chen
Z G
,
Bobaru
F
.
2018
.
Elastic vortices and thermally-driven cracks in brittle materials with peridynamics
.
International Journal of Fracture,
209
:
203
--
222
.
Instabilities in thermally-driven crack growth in thin glass plates have been observed in experiments that slowly immerse a hot, pre-notched glass slide into a cold bath. We show that a nonlocal model
[95]
Zhang
H
,
Qiao
P Z
.
2018
.
An extended state-based peridynamic model for damage growth prediction of bimaterial structures under thermomechanical loading
A MOOSE-based implicit peridynamic thermomechanical model
2016
Peridynamics using irregular domain discretization with moose-based implementation//ASME 2017 International Mechanical Engineering Congress and Exposition
Peridynamic simulations of brittle structures with thermal residual deformation: Strengthening and structural reactivity of glasses under impacts//Proceedings of the Royal Society A: Mathematical Physical and Engineering Sciences
A peridynamics model for the propagation of hydraulic fractures in heterogeneous, naturally fractured reservoirs//In SPE Hydraulic Fracturing Technology Conference 2015
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