Journal articles: 'Peridynamics'

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Huang, Zaixing. "Noether’s theorem in peridynamics." Mathematics and Mechanics of Solids 24, no. 11 (November 12, 2018): 3394–402. http://dx.doi.org/10.1177/1081286518812931.

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By introducing a new nonlocal argument, the Lagrangian formulation of peridynamics is investigated. The peridynamic Euler–Lagrange equation is derived from Hamilton’s principle, and Noether’s theorem is extended into peridynamics. With the help of the peridynamic Noether’s theorem, the conservation laws relevant to energy, linear momentum, angular momentum and the Eshelby integral are determined. The results show that the peridynamic conservation laws exist only in a spatial integral form rather than in a pointwise form due to nonlocality. In bond-based peridynamics, energy conservation requires that the influence function is independent of the relative displacement field, or energy dissipation will occur. In state-based peridynamics, the angular momentum conservation causes a constraint on the constitutive relation between the force vector-state and the deformation vector-state. The Eshelby integral of peridynamics is given, which can be used to judge nucleation of defects and to calculate the energy release rates caused by damage, fracture and phase transition.

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Yu, Ming, Zeyuan Zhou, and Zaixing Huang. "Traction-Associated Peridynamic Motion Equation and Its Verification in the Plane Stress and Fracture Problems." Materials 16, no. 6 (March 10, 2023): 2252. http://dx.doi.org/10.3390/ma16062252.

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How to prescribe traction on boundary surface is still an open question in peridynamics. This problem is investigated in this paper. Through introducing the induced body force defined by boundary traction, the Silling’s peridynamic motion equation is extended to a new formulation called the traction-associated peridynamic motion equation, which is verified to be compatible with the conservation laws of linear momentum and angular momentum. The energy conservation equation derived from the traction-associated peridynamic motion equation has the same form as that in the original peridynamics advanced by Silling. Therefore, the constitutive models of the original peridynamics can be directly applied to the traction-associated peridynamic motion equation. Some benchmark examples in the plane stress problems are calculated. The numerical solutions agree well with the classical elasticity solutions, and the volume correction and the surface correction are no longer needed in the numerical algorithm. These results show that the traction-associated peridynamic motion equation not only retains all advantages of the original peridynamics, but also can conveniently deal with the complex traction boundary conditions.

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Yaghoobi, Amin, and Mi G. Chorzepa. "Formulation of symmetry boundary modeling in non-ordinary state-based peridynamics and coupling with finite element analysis." Mathematics and Mechanics of Solids 23, no. 8 (June 12, 2017): 1156–76. http://dx.doi.org/10.1177/1081286517711495.

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Peridynamics is an effective method in computational solid mechanics for dealing with discontinuities. However, its computational cost limits its applications, especially when used in the most general form, non-ordinary state-based peridynamics. This paper considers two approaches which decrease the computational cost. The first approach accounts for symmetry boundary conditions in a peridynamic body. In nonlocal peridynamics, boundary conditions are applied to an area. Therefore, when modeling the symmetry boundary condition, assuming fixed particles around the symmetry axis yields incorrect results. The present formulation introduces constraints which allow modeling of local symmetry conditions. Second, the finite-element–peridynamic coupling method is adopted for non-ordinary state-based peridynamics. The coupling method enables the use of peridynamics around discontinuities like cracks, and the faster finite element for the surrounding body. These two methods effectively reduce the solution time with an acceptable accuracy. The validity of these approaches is studied through various examples. Also, ductile crack growth in a compact tension specimen is studied, applying the presented methods. Good agreement is found when comparing experimental results with corresponding numerical results obtained using either fully peridynamic or coupled models.

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Nishawala, Vinesh V., and Martin Ostoja-Starzewski. "Peristatic solutions for finite one- and two-dimensional systems." Mathematics and Mechanics of Solids 22, no. 8 (April 21, 2016): 1639–53. http://dx.doi.org/10.1177/1081286516641180.

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Peridynamics is a nonlocal continuum mechanics theory where its governing equation has an integro-differential form. This paper specifically uses bond-based peridynamics. Typically, peridynamic problems are solved via numerical means, and analytical solutions are not as common. This paper analytically evaluates peristatics, the static version of peridynamics, for a finite one-dimensional rod as well as a special case for two dimensions. A numerical method is also implemented to confirm the analytical results.

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Shen, Feng, Qing Zhang, and Dan Huang. "Damage and Failure Process of Concrete Structure under Uniaxial Compression Based on Peridynamics Modeling." Mathematical Problems in Engineering 2013 (2013): 1–5. http://dx.doi.org/10.1155/2013/631074.

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Peridynamics is a nonlocal formulation of continuum mechanics, which uses integral formulation rather than the spatial partial differential equations. The peridynamic approach avoids using any spatial derivatives, which arise naturally in the classical local theory. It has shown effectiveness and advantage in solving discontinuous problems at both macro- and microscales. In this paper, the peridynamic theory is used to analyze damage and progressive failure of concrete structures. A nonlocal peridynamic model for concrete columns under uniaxial compression is developed. Numerical example illustrates that cracks in a peridynamic body of concrete form spontaneously. The result of the example clarifies the unique advantage of modeling damage accumulation and progressive failure of concrete based on peridynamic theory. This study provides a new promising alternative for analyzing complicated discontinuity problems. Finally, some open problems and future research trends in peridynamics are discussed.

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Friebertshäuser, Kai, Christian Wieners, and Kerstin Weinberg. "Dynamic fracture with continuum-kinematics-based peridynamics." AIMS Materials Science 9, no. 6 (2022): 791–807. http://dx.doi.org/10.3934/matersci.2022049.

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<abstract>This contribution presents a concept to dynamic fracture with continuum-kinematics-based peridynamics. Continuum-kinematics-based peridynamics is a geometrically exact formulation of peridynamics, which adds surface- or volume-based interactions to the classical peridynamic bonds, thus capturing the finite deformation kinematics correctly. The surfaces and volumes considered for these non-local interactions are constructed using the point families derived from the material points' horizon. For fracture, the classical bond-stretch damage approach is not sufficient in continuum-kinematics-based peridynamics. Therefore it is here extended to the surface- and volume-based interactions by additional failure variables considering the loss of strength in the material points' internal force densities. By numerical examples, it is shown that the presented approach can correctly handle crack growth, impact damage, and spontaneous crack initiation under dynamic loading conditions with large deformations.</abstract>

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Chen, Jingkai, Ye Tian, and Xuezheng Cui. "Free and Forced Vibration Analysis of Peridynamic Finite Bar." International Journal of Applied Mechanics 10, no. 01 (January 2018): 1850003. http://dx.doi.org/10.1142/s1758825118500035.

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Peridynamics is a reformulated nonlocal elasticity theory. Unlike the local elasticity theory, the peridynamics is proposed with no continuum assumption. In this paper, a new analytical approach to analyze the vibration of peridynamic finite bar with specified boundary condition is proposed. It is proved that the nonlocal dispersive relation of the peridynamic bar is nonlinear and can be reduced to the local dispersive relation when the peridynamic horizon goes to zero. The phase velocity, as a function of the wave frequency, is proved to be positive and asymptotically decreasing. The homogenous and the nonhomogeneous solutions of the peridynamic bar vibration equation are derived analytically by using the separation of variables. The mode shape characteristic equation of peridynamic bar, which is a second kind Fredholm integral equation, is expanded with a Taylor series expansion up to the infinite order; the corresponding mode shape is derived by solving a differential equation up to the infinite order. The peridynamic boundary condition is analyzed and compared with the local boundary condition. The numerical modeling based on mesh-free method verifies the analytical results for both free vibration and forced vibration cases.

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Liu, Shankun, Fei Han, Xiaoliang Deng, and Ye Lin. "Thermomechanical Peridynamic Modeling for Ductile Fracture." Materials 16, no. 11 (May 30, 2023): 4074. http://dx.doi.org/10.3390/ma16114074.

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In this paper, we propose a modeling method based on peridynamics for ductile fracture at high temperatures. We use a thermoelastic coupling model combining peridynamics and classical continuum mechanics to limit peridynamics calculations to the failure region of a given structure, thereby reducing computational costs. Additionally, we develop a plastic constitutive model of peridynamic bonds to capture the process of ductile fracture in the structure. Furthermore, we introduce an iterative algorithm for ductile-fracture calculations. We present several numerical examples illustrating the performance of our approach. More specifically, we simulated the fracture processes of a superalloy structure in 800 ℃ and 900 ℃ environments and compared the results with experimental data. Our comparisons show that the crack modes captured by the proposed model are similar to the experimental observations, verfying the validity of the proposed model.

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Freimanis, Andris, and Sakdirat Kaewunruen. "Peridynamic Analysis of Rail Squats." Applied Sciences 8, no. 11 (November 19, 2018): 2299. http://dx.doi.org/10.3390/app8112299.

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Rail surface defects are a serious concern for railway infrastructure managers all around the world. They lead to poor ride quality due to excess vibration and noise; in rare cases, they can result in a broken rail and a train derailment. Defects are typically classified as ‘rail studs’ when they initiate from the white etching layer, and ‘rail squats’ when they initiate from rolling contact fatigue. This paper presents a novel investigation into rail squat initiation and growth simulations using peridynamic theory. To the best of the authors’ knowledge, no other comprehensive study of rail squats has been carried out using this approach. Peridynamics are well-suited for fracture problems, because, contrary to continuum mechanics, they do not use partial-differential equations. Instead, peridynamics use integral equations that are defined even when discontinuities (cracks, etc.) are present in the displacement field. In this study, a novel application of peridynamics to rail squats is verified against a finite element solution, and the obtained simulation results are compared with in situ rail squat measurements. Some new insights can be drawn from the results. The outcome exhibits that the simulated cracks initiate and grow unsymmetrically, as expected and reported in the field. Based on this new insight, it is apparent that peridynamic modelling is well-applicable to fatigue crack modeling in rails. Surprisingly, limitations to the peridynamic analysis code have also been discovered. Future work requires finding an adequate solution to the matter-interpenetration problem.

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Altenbach, Holm, Oleksiy Larin, Konstantin Naumenko, Olha Sukhanova, and Mathias Würkner. "Elastic plate under low velocity impact: Classical continuum mechanics vs peridynamics analysis." AIMS Materials Science 9, no. 5 (2022): 702–18. http://dx.doi.org/10.3934/matersci.2022043.

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<abstract>The aim of this paper is to compare the classical continuum mechanics and the peridynamic models in the structural analysis of a monolithic glass plate subjected to ball drop. Governing equations are recalled in order to highlight the differences and basic features of both approaches. In this study the behavior of glass is assumed to be linear-elastic and damage processes are ignored. The generalized Hooke's law is assumed within the classical theory, while the linear peridynamic solid constitutive model is applied within the peridynamic analysis. Mechanical models for the ball drop simulation are discussed in detail. An emphasis is placed on the discretization including finite element mesh, peridynamic node lattice and time stepping, as well as appropriate constraints and contact conditions in both finite element and non-local peridynamics models. Deflections of the plate after the ball drop are presented as functions of time and the results based on the finite element and peridynamic analysis are compared. Good agreements between the deflection values in selected points of the plate as well as deflection fields at several time points indicate, that the model assumptions for the non-local peridynamic analysis including the horizon size, the short-range force contact settings and the support conditions are well suited. The developed peridynamics models can be applied in the future to analyze damage patterns in glass plates.</abstract>

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Karpenko, Olena, Selda Oterkus, and Erkan Oterkus. "An in-depth investigation of critical stretch based failure criterion in ordinary state-based peridynamics." International Journal of Fracture 226, no. 1 (October 2, 2020): 97–119. http://dx.doi.org/10.1007/s10704-020-00481-z.

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AbstractThis study presents an in-depth investigation of the critical stretch based failure criterion in ordinary state-based peridynamics for both static and dynamic conditions. Seven different cases are investigated to determine the effect of the failure parameter on peridynamic forces between material points and dilatation. Based on crack opening displacement (COD) results from both peridynamics and finite element analysis, it was found that one of the seven cases provides the best agreement between the two approaches. This particular case is further investigated by considering the influence of the discretisation and the horizon sizes on COD and crack propagation speeds. Moreover, PD predictions of COD for PMMA material is analysed with the theory of dynamic fracture mechanics and compared with the fracture experiments. It is shown that the peridynamic model can correctly model, simulate and predict the behaviour of the crack under different loading conditions. Furthermore, the presented PD models capture accurate fracture phenomena, specifically the crack path, branching angles and crack propagation speeds, which are in good agreement with experimental results.

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Vazic, Bozo, Erkan Oterkus, and Selda Oterkus. "In-Plane and Out-of Plane Failure of an Ice Sheet using Peridynamics." Journal of Mechanics 36, no. 2 (January 17, 2020): 265–71. http://dx.doi.org/10.1017/jmech.2019.65.

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ABSTRACTWhen dealing with ice structure interaction modeling, such as designs for offshore structures/icebreakers or predicting ice cover’s bearing capacity for transportation, it is essential to determine the most important failure modes of ice. Structural properties, ice material properties, ice-structure interaction processes, and ice sheet geometries have significant effect on failure modes. In this paper two most frequently observed failure modes are studied; splitting failure mode for in-plane failure of finite ice sheet and out-of-plane failure of semi-infinite ice sheet. Peridynamic theory was used to determine the load necessary for inplane failure of a finite ice sheet. Moreover, the relationship between radial crack initiation load and measured out-of-plane failure load for a semi-infinite ice sheet is established. To achieve this, two peridynamic models are developed. First model is a 2 dimensional bond based peridynamic model of a plate with initial crack used for the in-plane case. Second model is based on a Mindlin plate resting on a Winkler elastic foundation formulation for out-of-plane case. Numerical results obtained using peridynamics are compared against experimental results and a good agreement between the two approaches is obtained confirming capability of peridynamics for predicting in-plane and out-of-plane failure of ice sheets.

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Ahadi, Aylin, Per Hansson, and Solveig Melin. "Simulating Nanoindentation of Thin Cu Films Using Molecular Dynamics and Peridynamics." Solid State Phenomena 258 (December 2016): 25–28. http://dx.doi.org/10.4028/www.scientific.net/ssp.258.25.

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Nanoindentation is a useful experimental method to characterize the micromechanical properties of materials. In this study molecular dynamics and peridynamics are used to simulate nanoindentation, with a spherical indenter targeting a thin single crystal Cu film, resting on an infinitely stiff substrate. The objective is to compare the results obtained from molecular dynamic simulations to those obtained using a peridynamic approach as regards the force-displacement curves and the deformation patterns after that the material parameters in the peridynamic model have been fitted to the force displacement curve from the molecular dynamic simulation.

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Stenström, Christer, and Kjell Eriksson. "The J-area integral applied in peridynamics." International Journal of Fracture 228, no. 2 (January 8, 2021): 127–42. http://dx.doi.org/10.1007/s10704-020-00505-8.

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AbstractThe J-integral is in its original formulation expressed as a contour integral. The contour formulation was, however, found cumbersome early on to apply in the finite element analysis, for which method the more directly applicable J-area integral formulation was later developed. In a previous study, we expressed the J-contour integral as a function of displacements only, to make the integral directly applicable in peridynamics (Stenström and Eriksson in Int J Fract 216:173–183, 2019). In this article we extend the work to include the J-area integral by deriving it as a function of displacements only, to obtain the alternative method of calculating the J-integral in peridynamics as well. The properties of the area formulation are then compared with those of the contour formulation, using an exact analytical solution for an infinite plate with a central crack in Mode I loading. The results show that the J-area integral is less sensitive to local disturbances compared to the contour counterpart. However, peridynamic implementation is straightforward and of similar scope for both formulations. In addition, discretization, effects of boundaries, both crack surfaces and other boundaries, and integration contour corners in peridynamics are considered.

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Mikeš, Karel, Milan Jirásek, Jan Zeman, Ondřej Rokoš, and Ron H. J. Peerlings. "LOCALIZATION ANALYSIS OF DAMAGE FOR ONE-DIMENSIONAL PERIDYNAMIC MODEL." Acta Polytechnica CTU Proceedings 30 (April 22, 2021): 47–52. http://dx.doi.org/10.14311/app.2021.30.0047.

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Peridynamics is a recently developed extension of continuum mechanics, which replaces the traditional concept of stress by force interactions between material points at a finite distance. The peridynamic continuum is thus intrinsically nonlocal. In this contribution, a bond-based peridynamic model with elastic-brittle interactions is considered and the critical strain is defined for each bond as a function of its length. Various forms of length functions are employed to achieve a variety of macroscopic responses. A detailed study of three different localization mechanisms is performed for a one-dimensional periodic unit cell. Furthermore, a convergence study of the adopted finite element discretization of the peridynamic model is provided and an effective event-driven numerical algorithm is described.

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Yang, Zhenghao, Bozo Vazic, Cagan Diyaroglu, Erkan Oterkus, and Selda Oterkus. "A Kirchhoff plate formulation in a state-based peridynamic framework." Mathematics and Mechanics of Solids 25, no. 3 (November 17, 2019): 727–38. http://dx.doi.org/10.1177/1081286519887523.

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In recent years, there has been rapid progress on peridynamics. It has been applied to many different material systems, used for coupled field analysis and is suitable for multi-scale analysis. This study mainly focuses on peridynamic analysis for plate-type structures. For this purpose, a new peridynamic Kirchhoff plate is developed. The new formulation is computationally efficient by having only one degree of freedom for each material point. Moreover, it is based on the state-based peridynamic formulation, which does not impose any limitation on material constants. After presenting how to impose simply supported and clamped boundary conditions in this new formulation, several numerical studies are considered to demonstrate the accuracy and capability of the proposed formulation.

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Li, Tianyi, Xin Gu, Qing Zhang, and Xiaozhou Xia. "Elastoplastic Constitutive Modeling for Reinforced Concrete in Ordinary State-Based Peridynamics." Journal of Mechanics 36, no. 6 (October 23, 2020): 799–811. http://dx.doi.org/10.1017/jmech.2020.50.

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ABSTRACTA non-local, ordinary state-based, peridynamic elastoplastic model is formulated to numerically simulate the fracture of reinforced concrete materials. Several basic definitions are first discussed to avoid confusion; and then, a detailed derivation of the force vector state is presented, leading to a unified expression of force state for one-, two- and three-dimensional elasticity problems. Furthermore, an ordinary state-based peridynamic (OSB PD) elastoplastic analysis approach is developed for both plastic compressible and incompressible materials, including the constitutive relationship, the yield function, the consistency condition and the plasticity flow rule. The peridynamic predictions of a quasi-static deformation of the steel rods are in good agreement with the analytical solution. Moreover, the OSB PD plasticity is verified by analyzing a square plate with or without a central hole suffering different loading-unloading paths. Finally, a two dimensional reinforced concrete clamped beam subjected to impact loading is simulated with the proposed OSB PD elastoplasticity, demonstrating its capability in capturing the damage characteristics and structural failure behavior. Simulation results show good accuracy of the peridynamics in simulating elastoplastic problems.

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Fallah, Arash S., Ilias N. Giannakeas, Rizgar Mella, Mark R. Wenman, Yasser Safa, and Hamid Bahai. "On the Computational Derivation of Bond-Based Peridynamic Stress Tensor." Journal of Peridynamics and Nonlocal Modeling 2, no. 4 (July 2, 2020): 352–78. http://dx.doi.org/10.1007/s42102-020-00036-9.

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Abstract The concept of ‘contact stress’, as introduced by Cauchy, is a special case of a nonlocal stress tensor. In this work, the nonlocal stress tensor is derived through implementation of the bond-based formulation of peridynamics that uses an idealised model of interaction between points as bonds. The method is sufficiently general and can be implemented to study stress states in problems containing stress concentration, singularity, or discontinuities. Two case studies are presented, to study stress concentration around a circular hole in a square plate and conventionally singular stress fields in the vicinity of a sharp crack tip. The peridynamic stress tensor is compared with finite element approximations and available analytical solutions. It is shown that peridynamics is capable of capturing both shear and direct stresses and the results obtained correlate well with those obtained using analytical solutions and finite element approximations. A built-in MATLAB code is developed and used to construct a 2D peridynamic grid and subsequently approximate the solution of the peridynamic equation of motion. The stress tensor is then obtained using the tensorial product of bond force projections for bonds that geometrically pass through the point. To evaluate the accuracy of the predicted stresses near a crack tip, the J-integral value is computed using both a direct contour approximation and the equivalent domain integral method. In the formulation of the contour approximation, bond forces are used directly while the proposed peridynamic stress tensor is used for the domain method. The J-integral values computed are compared with those obtained by the commercial finite element package Abaqus 2018. The comparison provides an indication on the accurate prediction of the state of stress near the crack tip.

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Seleson, Pablo, Michael L. Parks, and Max Gunzburger. "Peridynamic State-Based Models and the Embedded-Atom Model." Communications in Computational Physics 15, no. 1 (January 2014): 179–205. http://dx.doi.org/10.4208/cicp.081211.300413a.

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AbstractWe investigate connections between nonlocal continuum models and molecular dynamics. A continuous upscaling of molecular dynamics models of the form of the embedded-atom model is presented, providing means for simulating molecular dynamics systems at greatly reduced cost. Results are presented for structured and structureless material models, supported by computational experiments. The nonlocal continuum models are shown to be instances of the state-based peridynamics theory. Connections relating multibody peridynamic models and upscaled nonlocal continuum models are derived.

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Zhao, Jinhai, Hesheng Tang, and Songtao Xue. "A new fracture criterion for peridynamic and dual-horizon peridynamics." Frontiers of Structural and Civil Engineering 12, no. 4 (December 13, 2017): 629–41. http://dx.doi.org/10.1007/s11709-017-0447-1.

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Weckner, Olaf, Gerd Brunk, Michael A. Epton, Stewart A. Silling, and Ebrahim Askari. "Green’s functions in non-local three-dimensional linear elasticity." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 465, no. 2111 (August 21, 2009): 3463–87. http://dx.doi.org/10.1098/rspa.2009.0234.

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In this paper, we compare small deformations in an infinite linear elastic body due to the presence of point loads within the classical, local formulation to the corresponding deformations in the peridynamic, non-local formulation. Owing to the linearity of the problem, the response to a point load can be used to obtain the response to general body force loading functions by superposition. Using Laplace and Fourier transforms, we thus obtain an integral representation for the three-dimensional peridynamic solution with the help of Green’s functions. We illustrate this new theoretical result by dynamic and static examples in one and three dimensions. In addition to this main result, we also derive the non-local three-dimensional jump conditions, as well as the weak formulation of peridynamics together with the associated finite element discretization.

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Dipasquale, Daniele, Arman Shojaei, and Soemsak Yooyen. "A Novel Stress Tensor-Based Failure Criterion for Peridynamics." Proceedings 39, no. 1 (January 9, 2020): 23. http://dx.doi.org/10.3390/proceedings2019039023.

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Peridynamic theory has recently shown to be a versatile tool for simulating complex phenomena related to the fracture and fragmentation of structural and composite materials. We introduce a novel failure criterion based on the classic stress tensor which takes inspiration from an approach proposed in the literature. Differently from the classic critical stretch-based failure criterion used in peridynamics, our approach takes into account the total elastic energy stored in the bond allowing to predict with more accuracy problems that involve mixed-mode I-II fracture. In order to show the effectiveness of the proposed failure criterion, a benchmark fracture problem is analyzed showing a good agreement with the experimental results and the numerical results obtained with other numerical methods.

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Ji, Songsong, Gang Pang, Jiwei Zhang, Yibo Yang, and Paris Perdikaris. "Accurate artificial boundary conditions for semi-discretized one-dimensional peridynamics." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 477, no. 2250 (June 2021): 20210229. http://dx.doi.org/10.1098/rspa.2021.0229.

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The peridynamic theory reformulates the equations of continuum mechanics in terms of integro-differential equations instead of partial differential equations. In this paper, we consider the construction of artificial boundary conditions (ABCs) for semi-discretized peridynamics using Green functions. Especially, the Green functions that represent the response to the single wave source are used to construct the accu2rate boundary conditions. The recursive relationships between the Green functions are proposed, therefore the Green functions can be computed through a differential and integral system with high precision. The numerical results demonstrate the accuracy of the proposed ABCs. The proposed method can be applied to modelling of wave propagation for other non-local theories and high-dimensional cases.

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Javili, Ali, Rico Morasata, Erkan Oterkus, and Selda Oterkus. "Peridynamics review." Mathematics and Mechanics of Solids 24, no. 11 (October 11, 2018): 3714–39. http://dx.doi.org/10.1177/1081286518803411.

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Peridynamics (PD) is a novel continuum mechanics theory established by Stewart Silling in 2000. The roots of PD can be traced back to the early works of Gabrio Piola according to dell’Isola et al. PD has been attractive to researchers as it is a non-local 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 PD 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.

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Liu, Shuo, Guodong Fang, Jun Liang, Maoqing Fu, and Bing Wang. "A new type of peridynamics: Element-based peridynamics." Computer Methods in Applied Mechanics and Engineering 366 (July 2020): 113098. http://dx.doi.org/10.1016/j.cma.2020.113098.

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Lu, Wei, Mingyang Li, Bozo Vazic, Selda Oterkus, Erkan Oterkus, and Qing Wang. "Peridynamic Modelling of Fracture in Polycrystalline Ice." Journal of Mechanics 36, no. 2 (February 21, 2020): 223–34. http://dx.doi.org/10.1017/jmech.2019.61.

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ABSTRACTIn this study, a peridynamic material model for a polycrystalline ice is utilised to investigate its fracture behaviour under dynamic loading condition. First, the material model was validated by considering a single grain, double grains and polycrystalline structure under tension loading condition. Peridynamic results are compared against finite element analysis results without allowing failure. After validating the material model, dynamic analysis of a polycrystalline ice material with two pre-existing cracks under tension loading is performed by considering weak and strong grain boundaries with respect to grain interiors. Numerical results show that the effect of microstructure is significant for weak grain boundaries. On the other hand, for strong grain boundaries, the effect of microstructure is insignificant. The evaluated results have demonstrated that peridynamics can be a very good alternative numerical tool for fracture analysis of polycrystalline ice material.

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Vazic, Bozo, Erkan Oterkus, and Selda Oterkus. "Peridynamic Model for a Mindlin Plate Resting on a Winkler Elastic Foundation." Journal of Peridynamics and Nonlocal Modeling 2, no. 3 (January 10, 2020): 229–42. http://dx.doi.org/10.1007/s42102-019-00019-5.

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AbstractIn this study, a peridynamic model is presented for a Mindlin plate resting on a Winkler elastic foundation. In order to achieve static and quasi-static loading conditions, direct solution of the peridynamic equations is utilised by directly assigning inertia terms to zero rather than using widely adapted adaptive dynamic relaxation approach. The formulation is verified by comparing against a finite element solution for transverse loading condition without considering damage and comparing against a previous study for pure bending of a Mindlin plate with a central crack made of polymethyl methacrylate material having negligibly small elastic foundation stiffness. Finally, the fracture behaviour of a pre-cracked Mindlin plate rested on a Winkler foundation subjected to transverse loading representing a floating ice floe interacting with sloping structures. Similar fracture patterns observed in field observations were successfully captured by peridynamics.

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Shen, Feng, Zihan Chen, Jia Zheng, and Qing Zhang. "Numerical Simulation of Failure Behavior of Reinforced Concrete Shear Walls by a Micropolar Peridynamic Model." Materials 16, no. 8 (April 18, 2023): 3199. http://dx.doi.org/10.3390/ma16083199.

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A reinforced concrete shear wall is an important building structure. Once damage occurs, it not only causes great losses to various properties but also seriously endangers people’s lives. It is difficult to achieve an accurate description of the damage process using the traditional numerical calculation method, which is based on the continuous medium theory. Its bottleneck lies in the crack-induced discontinuity, whereas the adopted numerical analysis method has the continuity requirement. The peridynamic theory can solve discontinuity problems and analyze material damage processes during crack expansion. In this paper, the quasi-static failure and impact failure of shear walls are simulated by improved micropolar peridynamics, which provides the whole process of microdefect growth, damage accumulation, crack initiation, and propagation. The peridynamic predictions are in good match with the current experiment observations, filling the gap of shear wall failure behavior in existing research.

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Fan, Lin, Song Rong Qian, and Teng Fei Ma. "The Numerical Methods of Peridynamics Theory Used in Failure Analysis of Materials." Advanced Materials Research 1030-1032 (September 2014): 223–27. http://dx.doi.org/10.4028/www.scientific.net/amr.1030-1032.223.

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In order to analysis the force situation of the material which is discontinuity,we can used the new theory called peridynamics to slove it.Peridynamics theory is a new method of molecular dynamics that develops very quickly.Peridynamics theory used the volume integral equation to constructed the model,used the volume integral equation to calculated the PD force in the horizon.So It doesn’t need to assumed the material’s continuity which must assumed that use partial differential equation to formulates the equation of motion. Destruction and the expend of crack which have been included in the peridynamics’ equation of motion.Do not need other additional conditions.In this paper,we introduce the peridynamics theory modeling method and introduce the relations between peridynamics and classic theory of mechanics.We also introduce the numerical integration method of peridynamics.Finally implementation the numerical integration in prototype microelastic brittle material.Through these work to show the advantage of peridynamics to analysis the force situation of the material.

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Ren, Huilong, Xiaoying Zhuang, and Timon Rabczuk. "A new peridynamic formulation with shear deformation for elastic solid." Journal of Micromechanics and Molecular Physics 01, no. 02 (July 2016): 1650009. http://dx.doi.org/10.1142/s2424913016500090.

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We propose a new peridynamic formulation with shear deformation for linear elastic solid. The key idea lies in subtracting the rigid body rotation part from the total deformation. Based on the strain energy equivalence between classic local model and non-local model, the bond force vector is derived. A new damage rule of maximal deviatoric bond strain for elastic brittle fracture is proposed in order to account for both the tensile damage and shear damage. 2D and 3D numerical examples are tested to verify the accuracy of the current peridynamics. The new damage rule is applied to simulate the propagation of Mode I, II and III cracks.

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Shojaei, Arman, Mirco Zaccariotto, and Ugo Galvanetto. "Coupling of 2D discretized Peridynamics with a meshless method based on classical elasticity using switching of nodal behaviour." Engineering Computations 34, no. 5 (July 3, 2017): 1334–66. http://dx.doi.org/10.1108/ec-03-2016-0078.

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Purpose The paper aims to use a switching technique which allows to couple a nonlocal bond-based Peridynamic approach to the Meshless Local Exponential Basis Functions (MLEBF) method, based on classical continuum mechanics, to solve planar problems. Design/methodology/approach 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 gradual transition between the two approaches takes place. Findings The new coupling technique generates overall grids that are not affected by ghost forces. Moreover, the use of the meshless approach can be limited to a narrow boundary region of the domain, and in this way, it can be used to remove the “surface effect” from the Peridynamic solution applied to all internal points. Originality/value The current study paves the road for future studies on dynamic and static crack propagation problems.

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Zhang, Feng, Xinting Hou, Pihua Ji, Cheng Han, Lei Cheng, and Xiaoxiao Liu. "Dynamic simulation of aircraft electro-impulse de-icing using bond-based peridynamics." Advances in Mechanical Engineering 14, no. 11 (November 2022): 168781322211302. http://dx.doi.org/10.1177/16878132221130218.

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Electro-impulse de-icing is a lightweight type of mechanical de-icing system that consumes little energy, provides high reliability, and offers wide application prospects. In this study, the bond-based peridynamics theory was used to simulate the dynamic damage process and evolution law of aircraft electro-impulse de-icing. The ice layer was simplified as a brittle material that satisfies the linear elastic constitutive relation. A numerical analysis model of the ice layer, aircraft skin substrate, and their interface was established to simulate the dynamic responses under a high strain rate. Considering the anisotropy of ice adhesion on the substrate, the interfacial bonds were described as shear bonds and tensile bonds, and the critical elongations of the two were derived. The tensile and shear capacities of the ice on the substrate were then simulated using these critical elongations, and the peeling rates of single ice particles and interfacial ice layers were taken as indexes describing the efficacy of electro-impulse de-icing. The process of electro-impulse de-icing was then analyzed for an aluminum substrate with two adjacent clamped edges. Finally, the results of the peridynamic simulation were compared with those of existing experiments and finite element models to verify the effectiveness of the peridynamic approach.

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Li, Shuang, Yanli Jin, Xiaohua Huang, and Lianjun Zhai. "An Extended Bond-Based Peridynamic Approach for Analysis on Fracture in Brittle Materials." Mathematical Problems in Engineering 2020 (April 7, 2020): 1–12. http://dx.doi.org/10.1155/2020/9568015.

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To enhance the calculation accuracy of bond-based peridynamics (BPD), a novel attenuation function is introduced to describe the effect of internal length on nonlocal long-range forces. Furthermore, the expression of the micromodulus function is deduced, and the corresponding fracture criteria are established. The validity and accuracy of the extended bond-based peridynamic approach are illustrated by three numerical examples: 2D isotropic plate under uniaxial loading; plate with a circular cutout under quasi-static loading; and a diagonally loaded square plate with a center pre-existing crack. Finally, the influence of the width and the angle of the pre-existing crack on the fracture initiation time and the crack propagation paths are studied by applying the proposed approach.

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Ren, Huilong, Xiaoying Zhuang, Yongchang Cai, and Timon Rabczuk. "Dual-horizon peridynamics." International Journal for Numerical Methods in Engineering 108, no. 12 (July 29, 2016): 1451–76. http://dx.doi.org/10.1002/nme.5257.

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Foster, J. T., S. A. Silling, and W. W. Chen. "Viscoplasticity using peridynamics." International Journal for Numerical Methods in Engineering 81, no. 10 (August 17, 2009): 1242–58. http://dx.doi.org/10.1002/nme.2725.

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Dogra, Vaishally. "Optimization of Compatible Meshfree Quadrature Rule for Nonlocal Problems with Applications to Peri Dynamics Study." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 9, no. 2 (December 30, 2018): 703–13. http://dx.doi.org/10.17762/turcomat.v9i2.13872.

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The optimization of a compatible meshfree quadrature rule for nonlocal problems with applications to Peridynamics is investigated in this study. Peridynamics is a nonlocal continuum mechanics theory that models material behavior based on interactions between material points within a finite neighbourhood. In Peridynamics, the accurate evaluation of nonlocal integral equations is crucial for obtaining reliable and efficient solutions. Traditional numerical integration methods, such as Gaussian quadrature, are not directly applicable to nonlocal problems due to their local nature. Hence, the development of a compatible meshfree quadrature rule that can effectively handle nonlocal interactions is of great importance. The objective it to optimize a quadrature rule that accurately captures the nonlocal interactions in Peridynamics while maintaining computational efficiency. The optimization process involves the selection of appropriate quadrature points and weights that minimize the quadrature error and maximize the computational efficiency. Various optimization techniques, such as genetic algorithms, particle swarm optimization, or machine learning algorithms, are explored to search for an optimal quadrature rule. The optimized quadrature rule is then applied to several Peridynamics problems, including fracture mechanics, material failure, and dynamic response analysis. The performance of the optimized quadrature rule is evaluated by comparing the results with those obtained using traditional quadrature methods and analytical solutions, when available. The accuracy, stability, and computational efficiency of the optimized quadrature rule are analysed and discussed. The findings provide valuable insights into the development and optimization of compatible meshfree quadrature rules for nonlocal problems, particularly in the context of Peridynamics. The optimized quadrature rule offers improved accuracy in capturing nonlocal interactions and reduces the computational cost compared to traditional methods. The applications of the optimized quadrature rule to various Peridynamics problems demonstrate its effectiveness and reliability. The implications of it extend beyond Peridynamics and can be applicable to other nonlocal models in the field of computational mechanics. The optimized quadrature rule has the potential to enhance the accuracy and efficiency of numerical simulations involving nonlocal phenomena, enabling more reliable predictions of material behavior and structural response. The findings of this contribute to the advancement of numerical methods for nonlocal problems and provide a foundation for further research and developments in the field of compatible meshfree quadrature rules for nonlocal problems.

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Nayak, Sumeru, R. Ravinder, N. M. Anoop Krishnan, and Sumanta Das. "A Peridynamics-Based Micromechanical Modeling Approach for Random Heterogeneous Structural Materials." Materials 13, no. 6 (March 13, 2020): 1298. http://dx.doi.org/10.3390/ma13061298.

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This paper presents a peridynamics-based micromechanical analysis framework that can efficiently handle material failure for random heterogeneous structural materials. In contrast to conventional continuum-based approaches, this method can handle discontinuities such as fracture without requiring supplemental mathematical relations. The framework presented here generates representative unit cells based on microstructural information on the material and assigns distinct material behavior to the constituent phases in the random heterogenous microstructures. The framework incorporates spontaneous failure initiation/propagation based on the critical stretch criterion in peridynamics and predicts effective constitutive response of the material. The current framework is applied to a metallic particulate-reinforced cementitious composite. The simulated mechanical responses show excellent match with experimental observations signifying efficacy of the peridynamics-based micromechanical framework for heterogenous composites. Thus, the multiscale peridynamics-based framework can efficiently facilitate microstructure guided material design for a large class of inclusion-modified random heterogenous materials.

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Emmrich, Etienne, and Dimitri Puhst. "Survey of Existence Results in Nonlinear Peridynamics in Comparison with Local Elastodynamics." Computational Methods in Applied Mathematics 15, no. 4 (October 1, 2015): 483–96. http://dx.doi.org/10.1515/cmam-2015-0020.

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AbstractWe give an overview on existence results in nonlinear peridynamics, a nonlocal elasticity theory, and compare these results with those known for classical nonlinear elastodynamics. In peridynamics, it is possible to obtain stronger results under equal or even less restrictive assumptions on the stress.

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Han, Junzhao, Guozhong Wang, Xiaoyu Zhao, Rong Chen, and Wenhua Chen. "Modeling of Multiple Fatigue Cracks for the Aircraft Wing Corner Box Based on Non-Ordinary State-Based Peridynamics." Metals 12, no. 8 (July 30, 2022): 1286. http://dx.doi.org/10.3390/met12081286.

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In the current research, we propose a novel non-ordinary state-based peridynamics (PD) fatigue model for multiple cracks’ initiation and growth under tension–tension fatigue load. In each loading cycle, the fatigue loading is redistributed throughout the peridynamic solid body, leading to progressive fatigue damage formation and expansion in an autonomous fashion. The proposed fatigue model parameters are first verified by a 3D numerical solution, and then, the novel model is used to depict the widespread fatigue damage evolution of the aircraft wing corner box. The modified constitutive damage model has been implemented into the peridynamic framework. Furthermore, the criteria and processes from multiple initiations to propagation are discussed in detail. It was found that the computational results obtained from the PD fatigue model were consistent with those from the test data. The angular errors of multiple cracks are within 2.66% and the number of cycles errors are within 15%. A comparison of test data and computational results indicates that the fatigue model can successfully capture multiple crack formations and propagation, and other behaviors of aluminum alloy material.

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Buryachenko, Valeriy A. "Variational principles and generalized Hill’s bounds in micromechanics of linear peridynamic random structure composites." Mathematics and Mechanics of Solids 25, no. 3 (November 25, 2019): 682–704. http://dx.doi.org/10.1177/1081286519887222.

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We consider a static problem for statistically homogeneous matrix linear peridynamic composite materials (CMs). The basic feature of the peridynamic model considered is a continuum description of a material behavior as the integrated non-local force interactions between infinitesimal particles. In contrast to these classical local and non-local theories, the peridynamic equation of motion introduced by Silling ( J Mech Phys Solids 2000; 48: 175–209) is free of any spatial derivatives of displacement. Estimation of effective moduli of peridynamic CMs is performed by generalization of some methods used in locally elastic micromechanics. Namely, the admissible displacement and force fields are defined. The theorem of work and energy, Betti’s reciprocal theorem, and the theorem of virtual work are proved. Principles of minimum of both potential energy and complimentary energy are generalized. The strain energy bounds are estimated for both the displacement and force homogeneous volumetric boundary conditions. The classical representations of effective elastic moduli through the mechanical influence functions for elastic CM are generalized to the case of peridynamics, and the energetic definition of effective elastic moduli is proposed. Generalized Hill’s bounds on the effective elastic moduli of peridynamic random structure composites are obtained. In contrast to the classical Hill’s bounds, in the new bounds, comparable scales of the inclusion size and horizon are taken into account that lead to dependance of the bounds on both the size and shape of the inclusions. The numerical examples are considered for the 1D case.

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Wang, Fei, Yu’e Ma, Yanning Guo, and Wei Huang. "Studies on Quasi-Static and Fatigue Crack Propagation Behaviours in Friction Stir Welded Joints Using Peridynamic Theory." Advances in Materials Science and Engineering 2019 (October 31, 2019): 1–16. http://dx.doi.org/10.1155/2019/5105612.

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The friction stir welding (FSW) technology has been widely applied in aircraft structures. The heterogeneity of mechanical properties in weld and the hole in structure will lead the crack to turn. Peridynamics (PD) has inherent advantages in calculating crack turning. The peridynamic theory is applied to study the crack turning behaviour of FSW joints in this work. The compact tension (CT) samples with and without a hole are designed. The crack propagation testing under quasistatic and fatigue loads are performed. The peridynamic microplastic model is used and a three-stage fatigue calculation model is developed to simulate the quasistatic fracture and the fatigue crack growth. The results predicted by the peridynamic models are compared with the experimental ones. The effects of welding direction on quasistatic and fatigue crack propagation behaviours are investigated and the effect of hole position on crack path geometry is also studied. It is shown that the crack turning in FSWed CT samples can be captured by the peridynamic microplastic and the three-stage fatigue calculation models. The peridynamic crack growth rates agree with the experimental results. For CT specimen without a hole, the crack turns into the weld zone where the material is softer. The effect of welding direction on crack growth rates is not obvious. For CT sample with a hole, the crack propagation direction has been mainly controlled by the hole location and the welding direction has a slight effect on crack path.

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Sarego, Giulia, Mirco Zaccariotto, and Ugo Galvanetto. "Mixed-Mode Crack Patterns in Ordinary State-Based Peridynamics." Key Engineering Materials 665 (September 2015): 53–56. http://dx.doi.org/10.4028/www.scientific.net/kem.665.53.

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A new nonlocal theory of continuum, called Peridynamics, was introduced in 2000. While the classical theory of solid mechanics employs spatial derivatives in order to solve the motion equation and consequently requires the derivability of the displacement field, Peridynamics employs an integral formulation of the equation of motion which leads to the possibility to analyze structures without specific techniques whenever discontinuities, such as cracks or inhomogeneities, are involved. Peridynamics has proven to be able to handle several phenomena concerning crack propagation. There are two variants of the theory, bond-based and state-based. The former is a particular case of the latter, which can also be found in two versions, the ordinary, in which the interaction force between two nodes is aligned with their current relative position, and the non-ordinary, in which interaction forces can have different directions and classical models can be directly introduced in the formulation, even though in this variant numerical integration problems arise (spurious mode deformation). In this study, fracture patterns for mixed-mode crack propagation cases are investigated while varying two fundamental parameters of Peridynamics, the maximum length of interaction, called horizon, and the ratio between the grid spacing and the horizon, called m-ratio. An ordinary state-based Peridynamics formulation is adopted and numerical results are compared with experimental evidences.

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Aguiar, Adair R., and Alan B. Seitenfuss. "Determination of material properties of a linearly elastic peridynamic material." Mathematics and Mechanics of Solids 27, no. 6 (November 23, 2021): 1069–91. http://dx.doi.org/10.1177/10812865211051406.

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We investigate the properties of an isotropic linear elastic peridynamic material in the context of a three-dimensional state-based peridynamic theory, which considers both length and relative angle changes, and is based on a free energy function proposed in previous work that contains four material constants. To this end, we consider a class of equilibrium problems in mechanics to show that, in interior points of the body where deformations are smooth, the corresponding solutions in classical linear elasticity are also equilibrium solutions in peridynamics. More generally, we show that the equations of equilibrium are satisfied even when two of the four peridynamic constants are arbitrary. Pure torsion of a cylindrical shaft and pure bending of a cylindrical beam are particular cases of this class of problems and are used together with a correspondence argument proposed elsewhere to determine these two constants in terms of the elasticity constants of an isotropic material from the classical linear elasticity. One of the constants has a singularity in the Poisson ratio, which needs further investigation. Two additional experiments concerning bending of cylindrical beam by terminal load and anti-plane shear of a hollow cylinder, which do not belong to the previous class of problems, are used to validate these results.

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Sanfilippo, Danilo, Bahman Ghiassi, Alessio Alexiadis, and Alvaro Garcia Hernandez. "Combined Peridynamics and Discrete Multiphysics to Study the Effects of Air Voids and Freeze-Thaw on the Mechanical Properties of Asphalt." Materials 14, no. 7 (March 24, 2021): 1579. http://dx.doi.org/10.3390/ma14071579.

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This paper demonstrates the use of peridynamics and discrete multiphysics to assess micro crack formation and propagation in asphalt at low temperatures and under freezing conditions. Three scenarios are investigated: (a) asphalt without air voids under compressive load, (b) asphalt with air voids and (c) voids filled with freezing water. The first two are computed with Peridynamics, the third with peridynamics combined with discrete multiphysics. The results show that the presence of voids changes the way cracks propagate in the material. In asphalt without voids, cracks tend to propagate at the interface between the mastic and the aggregate. In the presence of voids, they ‘jump’ from one void to the closest void. Water expansion is modelled by coupling Peridynamics with repulsive forces in the context of Discrete Multiphysics. Freezing water expands against the voids’ internal surface, building tension in the material. A network of cracks forms in the asphalt, weakening its mechanical properties. The proposed methodology provides a computational tool for generating samples of ‘digital asphalt’ that can be tested to assess the asphalt properties under different operating conditions.

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Han, Duanfeng, Yiheng Zhang, Qing Wang, Wei Lu, and Bin Jia. "The review of the bond-based peridynamics modeling." Journal of Micromechanics and Molecular Physics 04, no. 01 (March 2019): 1830001. http://dx.doi.org/10.1142/s2424913018300013.

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Peridynamics theory is a nonlocal meshless method that replaces differential equations with spatial integral equations, and has shown good applicability and reliability in the analysis of discontinuities. Further, with characteristics of clear physical meaning and simple and reliable numerical calculation, the bond-based peridynamics method has been widely applied in the field. However, this method describes the interaction between material points simply using a single elastic “spring”, and thus leads to a fixed Poisson’s ratio, relatively low computational efficiency and other inherent problems. As such, the goal of this review paper is to provide a summary of the various methods of bond-based peridynamics modeling, particularly those that have overcome the limitations of the Poisson’s ratio, considered the shear deformation and modeling of two-dimensional thin plates for bending and three-dimensional anisotropic composites, as well as explored coupling with finite element methods. This review will determine the advantages and disadvantages of such methods and serve as a starting point for researchers in the development of peridynamics theory.

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Yakin, H. N., M. R. M. Rejab, Nur Hashim, and N. Nikabdullah. "A new quasi-brittle damage model implemented under quasi-static condition using bond-based peridynamics theory for progressive failure." Theoretical and Applied Mechanics, no. 00 (2023): 6. http://dx.doi.org/10.2298/tam230404006y.

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A novel quasi-brittle damage model implemented under quasistatic loading condition using bond-based peridynamics theory for progressive failure is proposed to better predict damage initiation and propagation in solid materials. Since peridynamics equation of motion was invented in dynamic configuration, this paper applies the adaptive dynamic relaxation equation to achieve steady-state in peridynamics formulation. To accurately characterise the progressive failure process in cohesive materials, we incorporate the dynamic equation with the novel damage model for quasi-brittle materials. Computational examples of 2D compressive and tensile problems using the proposed model are presented. This paper presents advancement by incorporating the adaptive dynamic equation approach into a new damage model for quasi-brittle materials. This amalgamation allows for a more accurate representation of the behavior of damaged materials, particularly in static or quasi-static loading situations, bringing the framework closer to reality. This research paves the way for the peridynamics formulation to be employed for a far broader class of loading condition behaviour than it is now able to.

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Cluni, Federico, Vittorio Gusella, Dimitri Mugnai, Edoardo Proietti Lippi, and Patrizia Pucci. "A mixed operator approach to peridynamics." Mathematics in Engineering 5, no. 5 (2023): 1–22. http://dx.doi.org/10.3934/mine.2023082.

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<abstract>In the present paper we propose a model describing the nonlocal behavior of an elastic body using a peridynamical approach. Indeed, peridynamics is a suitable framework for problems where discontinuities appear naturally, such as fractures, dislocations, or, in general, multiscale materials. In particular, the regional fractional Laplacian is used as the nonlocal operator. Moreover, a combination of the fractional and classical Laplacian operators is used to obtain a better description of the phenomenological response in elasticity. We consider models with linear and nonlinear perturbations. In the linear case, we prove the existence and uniqueness of the solution, while in the nonlinear case the existence of at least two nontrivial solutions of opposite sign is proved. The linear and nonlinear problems are also solved by a numerical approach which estimates the regional fractional Laplacian by means of its singular integral representation. In both cases, a numerical estimation of the solutions is obtained, using in the nonlinear case an approach involving a random variation of an initial guess of the solution. Moreover, in the linear case a parametric analysis is made in order to study the effects of the parameters involved in the model, such as the order of the fractional Laplacian and the mixture law between local and nonlocal behavior.</abstract>

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Jenabidehkordi, Ali, and Timon Rabczuk. "The Multi-Horizon Peridynamics." Computer Modeling in Engineering & Sciences 121, no. 2 (2019): 493–500. http://dx.doi.org/10.32604/cmes.2019.07942.

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Bobaru, Florin. "PERIDYNAMICS AND MULTISCALE MODELING." International Journal for Multiscale Computational Engineering 9, no. 6 (2011): vii—ix. http://dx.doi.org/10.1615/intjmultcompeng.2011002816.

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Alali, Bacim, and Max Gunzburger. "Peridynamics and Material Interfaces." Journal of Elasticity 120, no. 2 (January 22, 2015): 225–48. http://dx.doi.org/10.1007/s10659-014-9512-3.

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Journal articles on the topic 'Peridynamics'

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