Implicit implementation of the stabilized non‐ordinary state‐based peridynamic model

Pan, Li; Hao, Zulong; Yu, Shengwen; Zhen, W.Q.

doi:10.1002/nme.6234

Cited by 16 publications

(8 citation statements)

References 26 publications

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“…The NOSBPD theory is known to be susceptible to spurious or zero‐energy modes due to weak coupling of each point to its own family. Herein, the control method without control parameters proposed by Li 40 is employed to suppress the zero‐energy modes

{\underset{̲}{T}}^{s} false[bold-italicx false] false⟨ ξ false⟩ = - ω false((||, bold-italicξ) false) σ^{s} B bold-italicξ + \frac{1}{2} ω false((||, bold-italicξ) false) {bold-italicC}^{s} false⟨ ξ false⟩ \underset{̲}{bold-italicz} false⟨ ξ false⟩

…”

Section: Discretization and Numerical Implementationmentioning

confidence: 99%

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

Sun

Fish

2021

Num Anal Meth Geomechanics

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In the present manuscript fracture propagation in a saturated porous medium is modeled based on the classical Biot theory, where solid skeleton and fluid flow are represented by separate two layers. The non‐ordinary state‐based peridynamics (NOSBPD) layer is employed to capture deformation including fracturing of the solid skeleton, while the fluid flow is controlled by the finite element method (FEM) layer. The interaction between the layers is realized by considering the effects of pore pressure from the FEM layer on the NOSBPD layer and, vice versa, the effect of the volumetric strain, porosity, and permeability variations from the NOSBPD layer on the FEM layer. The coupling terms retain their parent characteristics, that is, the interaction term in the momentum balance equation is approximated by the local FEM formulation whereas the interaction term in the mass balance equation is approximated by the nonlocal NOSBPD formulation. By doing so, the model retains the flexibility of coupling two independent discretizations. The coupled system is solved by a fully implicit solution scheme. The accuracy of the proposed method has been verified against available closed‐form solutions and published numerical approaches for the pressure‐ and fluid‐driven facture propagation problems.

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{\underset{̲}{T}}^{s} false[bold-italicx false] false⟨ ξ false⟩ = - ω false((||, bold-italicξ) false) σ^{s} B bold-italicξ + \frac{1}{2} ω false((||, bold-italicξ) false) {bold-italicC}^{s} false⟨ ξ false⟩ \underset{̲}{bold-italicz} false⟨ ξ false⟩

…”

Section: Discretization and Numerical Implementationmentioning

confidence: 99%

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

Sun

Fish

2021

Num Anal Meth Geomechanics

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“…Breitenfeld considered three methods of zero-energy mode control [32], including supplemental interconnected springs, average displacement state, and penalty approach. Stability method is then suggested by Silling [33] and extended by Li et al [34,35], where additional term is added to the strain energy density that resists zero-energy mode of deformation. Other treatment includes sub-horizon or bond-associated deformation gradient [36,37], stress-point method [38,39], etc.…”

Section: Numerical Instability Controlmentioning

confidence: 99%

Peridynamic Modeling of Brittle Fracture in Mindlin-Reissner Shell Theory

Li¹,

Lai²,

Liu³

2022

Computer Modeling in Engineering &Amp; Sciences

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In this work, we modeled the brittle fracture of shell structure in the framework of Peridynamics Mindlin-Reissener shell theory, in which the shell is described by material points in the mean-plane with its drilling rotation neglected in kinematic assumption. To improve the numerical accuracy, the stress-point method is utilized to eliminate the numerical instability induced by the zero-energy mode and rank-deficiency. The crack surface is represented explicitly by stress points, and a novel general crack criterion is proposed based on that. Instead of the critical stretch used in common peridynamic solid, it is convenient to describe the material failure by using the classic constitutive model in continuum mechanics. In this work, a concise crack simulation algorithm is also provided to describe the crack path and its development, in order to simulate the brittle fracture of the shell structure. Numerical examples are presented to validate and demonstrate our proposed model. Results reveal that our model has good accuracy and capability to represent crack propagation and branch spontaneously.

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“…Similarly, Li et al calculated

Ŵ^{s} false(\underset{bold_}{Y} false)

using the linearized bond‐based peridynamics as References 47,69

\begin{align} Ŵ^{s} (\underline{boldY}) & = \frac{1}{2} ((), \frac{1}{2} \underline{boldT} \underline{boldC} \underline{boldz}) ⊙ \underline{boldz} \end{align}

\begin{align} = \frac{1}{2} \int_{ℋ_{x}} \frac{1}{2} \underline{boldT} (bold-italicξ) \underline{boldC} (bold-italicξ) \cdot \underline{boldz} (boldx, t, bold-italicξ) \cdot \underline{boldz} (boldx, t, bold-italicξ) {dV}_{x^{'}}, \end{align}

where

\underset{bold_}{C} false(ξ false) = c ξ \otimes ξ / {| ξ |}^{3}

is the elastic coefficient state of order 2, and c is the linearized micromodulus in the bond‐based peridynamics, particularly with the values of

\frac{12 E}{π δ^{4}}

for 3D problems and

\frac{9 E}{π δ^{3}}

…”

Section: Formulation Of Peridynamicsmentioning

confidence: 99%

Coupling of non‐ordinary state‐based peridynamics and finite element method with reduced boundary effect

Jin

Hwang

Hong

2021

Numerical Meth Engineering

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In this study, we propose a new numerical technique to couple non-ordinary state-based peridynamics (NOSB-PD) and the finite element method (FEM), and improve the scheme by implementing an effective boundary imposition method and a stabilization method. This coupling scheme takes the mutual advantages of peridynamics and the FEM to solve fracture problems without the imposition of additional criteria, yielding enhanced computational efficiency. In addition, the coupling model brings the reduction of the boundary effect using the boundary imposition method. To combine the peridynamics and FEM, the model is partitioned into the peridynamic subregion and finite element subregion. Subsequently, the two subregions are bridged by interface elements, where peridynamic nodes are embedded. Two types of coupling schemes are developed and the boundary effect of the peridynamic subregion is analyzed in each coupling approach. Moreover, stabilization of the coupling method is implemented to control the zero-energy mode inherent in NOSB-PD to simulate fracture problems. The proposed methodology is verified by solving several quasi-static problems in the one-and two-dimensional domains, and fracture problems are solved. The crack paths predicted by the proposed coupling method are in good agreement with the results of the exact solution and the experiments. K E Y W O R D Sboundary effect, crack propagation, finite element method, non-ordinary state-based peridynamics INTRODUCTIONThe finite element method (FEM), which requires continuous displacement fields to calculate stress fields, is the most popular technique for numerical analysis. 1 However, the displacement fields are usually discontinuous if a crack tip or crack surfaces exist. A remeshing technique might be required to prevent discontinuous fields within meshes, 2,3 notably in nonlinear analysis. In the framework of the FEM, the extended FEM (XFEM) was proposed to deal with the spatial derivatives on either the crack tip or crack surfaces. 4,5 In the XFEM, tip enrichment functions based on analytical solutions are used to describe crack tip fields. 6 Moreover, to circumvent the expensive mesh generation and the problems that arise in the treatment of discontinuities, various meshless techniques, such as smoothed particle hydrodynamics (SPH), 7 finite spheres, 8 and the element-free Galerkin method (EFGM), 9 have been proposed. The SPH method is a nonlocal meshless technique that calculates the field quantities at each node using a smoothing function and is used extensively to solve fluid dynamics and fractures. 10,11 The finite sphere method replaces finite elements with overlapping spheres and the

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Implicit implementation of the stabilized non‐ordinary state‐based peridynamic model

Cited by 16 publications

References 26 publications

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

Coupling of non‐ordinary state‐based peridynamics and finite element method for fracture propagation in saturated porous media

Peridynamic Modeling of Brittle Fracture in Mindlin-Reissner Shell Theory

Coupling of non‐ordinary state‐based peridynamics and finite element method with reduced boundary effect

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