Model order reduction of linear peridynamic systems using static condensation

Oterkus

Engineering with Computers

et al. 2023

Self Cite

This paper proposes a framework for implementing viscoelastic constitutive model from the classical continuum mechanics (CCM) theory within non-ordinary state-based peridynamics (NOSBPD). The motivation stems from the inadequacy of CCM to model very complex material behaviours such as initiation and propagation of cracks and nonlocal behaviour due to size effects. The proposed formulation leverages on the constitutive correspondence between NOSBPD and CCM to incorporate a CCM viscoelastic constitutive model based on hereditary integral into NOSBPD. The combination of hereditary constitutive model and NOSBPD effectively makes this formulation a nonlocal time–space viscoelastic framework where temporal nonlocality is incorporated by a hereditary viscoelastic model which stipulates that the behaviour of a material at any point in time depends on both the present action and the complete history of previous actions on the material, and spatial nonlocality on the other hand is incorporated via the nonlocal mechanism provided by the NOSBPD. For model validation, three benchmark problems were solved using the proposed framework. Results obtained were compared to results from analytical solution and solutions from referenced literature. In addition, parametric study was conducted to determine the influence of nonlocality on numerical prediction. Conclusions drawn from the validation studies presented are that the proposed framework is able to predict viscoelastic responses that agree well with local macro models as well as nonlocal micromodels/nanomodels as reported in the literature.

Section: Introductionmentioning

confidence: 99%

Modelling of viscoelastic materials using non-ordinary state-based peridynamics

Oterkus

Engineering with Computers

et al. 2023

Self Cite

“…The coarsening method was extended in [97] for two-dimensional applications. A related method of model reduction based on static condensation of a highfidelity peridynamic model was proposed in [98].…”

Section: Introductionmentioning

confidence: 99%

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

Xia

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.

“…This therefore necessitates the definition of alternative forms of the nonlocal operators defined in equations (4)–(7). The alternative forms of the nonlocal divergence and gradient operators were given in Galadima et al [61] to be the pairs D , − G * and G , − D * . For example, in the alternative pair D , − G * , while D operates on tensor two-point functions, − G * operates on tensor point functions.…”

Section: Introductionmentioning

confidence: 99%

“…The coarsening methodology has the limitation of not being suitable for reducing dynamic models. A model order reduction for peridynamics based on static condensation was proposed in Galadima et al [61] like the coarsening method but with extended capability of reducing the order of dynamic models. All the frameworks above have been applied only to solid mechanics problems.…”

Section: Introductionmentioning

confidence: 99%

Static condensation of peridynamic heat conduction model

Mathematics and Mechanics of Solids

Oterkus

2022

A model order reduction methodology for reducing the order of the peridynamic transient heat model is proposed. This methodology is based on the static condensation procedure. To set the development of the model reduction procedure on a sound mathematical setting, a nonlocal vector calculus was employed in the formulation of the heat transport problem. The model order reduction framework proposed in this study provides a technique to reduce the dimensionality of a peridynamic transport model while still maintaining accurate prediction of the model response. Moreover, the methodology can be adaptively applied to accommodate different resolution requirements for different sections of the model. Using numerical experiments, the proposed methodology is shown to be capable of accurately reproducing results of the full peridynamic transient heat transport problem.