Two-dimensional implementation of the coarsening method for linear peridynamics

Engineering with Computers

et al. 2022

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Peridynamic theory has been shown to possess the capabilities of describing phenomena that theories based on partial differential equations are not capable of describing. These phenomena include nonlocal interactions and presence of singularities in system responses. To exploit the capabilities offered by peridynamics in the homogenization of heterogenous media, a nonlocal computational homogenization theory based on peridynamic correspondence model (non-ordinary state-based peridynamics) is proposed. To set the development of the theory on a rigorous mathematical framework and to ensure consistency with the nonlocal nature of the peridynamic theory, a nonlocal vector calculus was used in the analysis of the nonlocal homogenization theory. The proposed theory is a two-scale micro–macro-homogenization strategy in which the constitutive relation at the macroscale is derived from explicit solution of a nonlocal volume constraint problem at the microscale. To justify the coupling between the two scales, nonlocal analogues of the stress and strain average theorems as well as the Hill–Mandel macrohomogeneity condition were derived. Validation of the proposed theory is achieved via numerical solution of Representative Volume Elements (RVE) from composite materials and comparing the results with those obtained by means of established methodologies.

Section: Introductionmentioning

confidence: 99%

A computational homogenization framework for non-ordinary state-based peridynamics

Galadima

Xia

Engineering with Computers

et al. 2022

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“…It is also applied for composite and polycrystalline materials [32][33][34][35][36]. The theory is also suitable for multi-physics [37,38], coarsening and multiscale modeling [39][40][41].…”

Section: Introductionmentioning

confidence: 99%

An energy-based peridynamic model for fatigue cracking

Nguyen

Oterkus

Engineering Fracture Mechanics

2021

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Fatigue design assessment is a crucial step in the design process of ships and offshore structures. To date, the stochastic approach is commonly used to calculate the total lifetime accumulated fatigue damage and the probability of fatigue failure for the structures. Meanwhile, the details of damage initiation and propagation are infrequently investigated. In terms of predicting crack growth, the traditional approaches face conceptual and mathematical difficulties in terms of predicting crack nucleation and growth, especially for multiple crack paths because the equations in classical continuum mechanics are derived by using spatial derivatives. Peridynamics is a non-local theory using the integral equations rather than differential equations which makes it suitable for damage prediction. In this study, a novel energy-based peridynamic model for fatigue cracking is proposed. The definition of cyclic bond energy release rate range and the energy-based peridynamic fatigue equations for both phases crack initiation and crack growth phases are introduced. For validation, first, a problem of mode-I fatigue crack growth is investigated. Next, different mixed-mode fatigue damages are also investigated and the peridynamic results are compared with the experimental results.

“…The objective of the coarsening algorithm was to derive a simplified model from a detailed and more complex model by retaining fewer DoFs, on the one hand, and preserving the effect of the excluded DoFs in the response of the model, on the other hand. Numerical investigation conducted in both works [18, 19] demonstrated the robustness of the technique in reducing the order of the peridynamic model for a range of problems, without compromising on predictive capability.…”

Section: Introductionmentioning

confidence: 99%

“…A coarsening method for linear peridynamics was proposed and implemented for one dimension [18] and was extended [19] for two-dimensional applications. The objective of the coarsening algorithm was to derive a simplified model from a detailed and more complex model by retaining fewer DoFs, on the one hand, and preserving the effect of the excluded DoFs in the response of the model, on the other hand.…”

Section: Introductionmentioning

confidence: 99%

Model order reduction of linear peridynamic systems using static condensation

Galadima

Mathematics and Mechanics of Solids

Oterkus

2020

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Static condensation is widely used as a model order reduction technique to reduce the computational effort and complexity of classical continuum-based computational models, such as finite-element models. Peridynamic theory is a nonlocal theory developed primarily to overcome the shortcoming of classical continuum-based models in handling discontinuous system responses. In this study, a model order reduction algorithm is developed based on the static condensation technique to reduce the order of peridynamic models. Numerical examples are considered to demonstrate the robustness of the proposed reduction algorithm in reproducing the static and dynamic response and the eigenresponse of the full peridynamic models.