An asymptotically compatible treatment of traction loading in linearly elastic peridynamic fracture

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Abstract

Meshfree discretizations of state-based peridynamic models are attractive due to their ability to naturally describe fracture of general materials. However, two factors conspire to prevent meshfree discretizations of state-based peridynamics from converging to corresponding local solutions as resolution is increased: quadrature error prevents an accurate prediction of bulk mechanics, and the lack of an explicit boundary representation presents challenges when applying traction loads. In this paper, we develop a reformulation of the linear peridynamic solid (LPS) model to address these shortcomings, using improved meshfree quadrature, a reformulation of the nonlocal dilatation, and a consistent handling of the nonlocal traction condition to construct a model with rigorous accuracy guarantees. In particular, these improvements are designed to enforce discrete consistency in the presence of evolving fractures, whose a priori unknown location render consistent treatment difficult. In the absence of fracture, when a corresponding classical continuum mechanics model exists, our improvements provide asymptotically compatible convergence to corresponding local solutions, eliminating surface effects and issues with traction loading which have historically plagued peridynamic discretizations. When fracture occurs, our formulation automatically provides a sharp representation of the fracture surface by breaking bonds, avoiding the loss of mass. We provide rigorous error analysis and demonstrate convergence for a number of benchmarks, including manufactured solutions, free-surface, nonhomogeneous traction loading, and composite material problems. Finally, we validate simulations of brittle fracture against a recent experiment of dynamic crack branching in soda-lime glass, providing evidence that the scheme yields accurate predictions for practical engineering problems.

Introduction

Peridynamics provides a description of continuum mechanics in terms of integral operators rather than classical differential operators [1], [2], [3], [4], [5], [6], [7]. These nonlocal models are defined in terms of a lengthscale δ, referred to as a horizon, which denotes the extent of nonlocal interaction. The nonlocal viewpoint allows a natural description of processes requiring reduced regularity in the relevant solution, such as fracture mechanics [8], [9]. An important feature of such models is that when classical continuum models still apply, they revert back to classical continuum models as δ0. Discretizations which preserve this limit under refinement h0 are termed asymptotically compatible (AC) [10], and there has been significant work in recent years toward establishing such discretizations — for an incomplete list see [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20]. Broadly, strategies either involve adopting traditional finite element shape functions and carefully performing geometric calculations to integrate over relevant horizon/element subdomains, or adopt a strong-form meshfree discretization where particles are associated with abstract measure. The former is more amenable to mathematical analysis due to a better variational setting, while the latter is simple to implement and generally faster [21], [22]. In this paper we pursue the meshfree viewpoint.

For fracture mechanics problems one often refines both δ and h at the same rate under so-called M-convergence, δ=Mh, for M>0 [23]. In this setting, one obtains banded stiffness matrices allowing scalable implementations. Typically in the literature a scheme is termed AC if it recovers the solution in both the finite δ and M-convergence limit — in this work we abuse the definition slightly and only require the M-convergence case for asymptotic compatibility as the relevant limit for problems with a corresponding local limit. This AC property is only one necessary ingredient in achieving a convergent simulation, and our recent work focused upon establishing convergence in this setting for boundary value problems [18], [19]. To achieve similar convergence for problems involving fracture, one must also consider the interplay between consistency of quadrature for discrete operators and the imposition of traction loads as fracture surfaces open up and evolve [24]. For peridynamic fracture problems where the free surface evolves implicitly via the breaking of bonds [17], [25], one lacks an explicit boundary representation over the course of a simulation. In addition to providing challenges regarding accurate imposition of traction loads, the breaking of bonds also renders higher-order numerical quadrature inaccurate, as consistent AC quadrature weights are typically derived in the absence of damage.

Our goal is to provide a comprehensive treatment of fracture, nonlocal quadrature, and traction loading which is able to perform more accurate state-based peridynamic fracture simulations free of spurious surface effects. In particular, when no fracture occurs and therefore the classical continuum theory applies, the formulation should preserve the AC limit under M-convergence. When fracture occurs, the formulation should be able to capture the material damage and the evolving fracture surfaces via bond breaking. This practically means that one is able to incorporate all of the necessary ingredients to perform non-trivial simulations of fracture mechanics while maintaining a scalable implementation and guaranteeing convergence. Such a capability is elusive in the peridynamic literature; while peridynamics has been shown to provide a powerful modeling platform for a broad range of applications [26], [27], the development of efficient discretizations with rigorous underpinnings has lagged behind until the last few years.

The challenge in incorporating traction loading into a peridynamic framework stems from the fact that, in contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the surface [9], [20], [28]. Theoretical and numerical challenges arise in how to mathematically impose nonhomogeneous Neumann boundary conditions properly in the nonlocal model. In peridynamic models, careless imposition of traction loads leads to a smaller effective material stiffness close to the boundary, since the integral on those material points is over a smaller region. Therefore, an unphysical strain energy concentration is induced, leading in turn to an artificial softening of the material near the boundary. Such undesirable phenomena are referred to in the literature as a “surface” or “skin” effects [29], [30]. We propose a novel treatment of nonlocal traction-type boundary conditions which avoid the surface effect by designing a loading aimed to recover the corresponding local traction boundary condition as δ0. The approach requires no explicit representation of the boundary, imposing the traction volumetrically using the same information that would be available during a traditional meshfree bond-based peridynamics simulation. Although the Neumann-constrained nonlocal problem and its AC limit were investigated in nonlocal diffusion models [18], [19], [20], [28], [31], [32], to the authors’ best knowledge, the development of AC peridynamic formulations with traction-type boundary conditions remains restricted to weak formulations, simple traction loadings and/or simple geometries. Several modeling and numerical approaches have been proposed to correct the surface effect [7], [33], [34], [35], [36], [37], [38], [39] but mostly restricted to free surfaces. For nonzero loadings, the tractions are often applied as prescribed body forces through a layer of finite thickness at the material boundary [27], [33], [40], as a surface integral through a weak form [41], or by modifying the nonlocal operator through eigenvalues analysis [42]. Therefore, developing an AC meshfree discretization method for peridynamics which is capable to handle nonhomogeneous traction loadings on complex boundaries is critical for the general practice of peridynamics in realistic engineering applications.

We consider the linear peridynamic solid (LPS) model [43] as a prototypical state-based model appropriate for brittle fracture. The LPS model may be interpreted as a nonlocal generalization of the mixed form of linear elasticity, evolving both displacements and a dilatation. We will show that consistent treatment evolving traction loading will require a modification to the definition of dilatation to guarantee consistency in the presence of fractures; conceptually this corresponds to the fact that dilatation is a kinematic variable without associated boundary conditions, and should be estimated consistently independently of whether a fracture is occurring in the vicinity of a given point. Based on the modified nonlocal dilatation, we further propose a new nonlocal generalization of classical traction loads in the LPS model. Particularly, we convert the local traction loads to a correction term in the momentum balance equation, which provides an estimate for the nonlocal interactions of each material point with points outside the domain. Based on this traction-type boundary condition, a meshfree formulation is developed for the LPS model based on the optimization-based quadrature rule [17], which preserves the AC limit under M-convergence and naturally represents the evolving free surfaces in dynamic fracture problems. We note that asymptotic compatibility is not well-defined for dynamic fracture, as there is no known corresponding local theory for peridynamics with bond breaking.1 However, our modified LPS formulation preserves the AC limit for the linear elastic model with traction loading on the evolving fracture surfaces. This fact, together with the consistent discretization introduced here, provides the opportunity for efficient and accurate peridynamic fracture simulations.

We remark that the paper is organized to first establish the rigorous mathematical underpinnings of the approach, while the second half focuses on a more engineering-oriented exploration of its application. Readers with more applied interests may skip many of the proofs in the work without issue. The work is organized as follows. We recall first the linear peridynamic solid (LPS) model definition in Section 2. In Section 3.1, we introduce a novel approach to apply classical traction loads on the LPS model. After establishing the continuous limits of the scheme, we next pursue a consistent discretization. In Section 4 we introduce meshfree quadrature which preserves asymptotic compatibility in the δ0 limit, and establish the discrete scheme for boundary value problems in the absence of fracture. We proceed to investigate a number of two-dimensional statics problems with analytic solutions for the local limit in Section 5. These test cases include: linear patch tests (Section 5.1); manufactured local limits to illustrate asymptotic convergence rates (Section 5.2); homogeneous materials with free-surfaces or non-zero traction loading on curvilinear surfaces (Section 5.3); composite materials with internal interfaces (Section 5.4). In Section 6, we further extend the proposed formulation to handle dynamic brittle fracture, and provide preliminary validation results by comparing our numerical results with available numerical simulations and experimental measurements on three benchmark problems. Section 7 summarizes our findings and discusses future research.

Section snippets

A linear state-based peridynamic model

We consider the state-based linear peridynamic solid (LPS) model in a body occupying the domain ΩRd, d=2 or 3. Let θ be the nonlocal dilatation, generalizing the local divergence of displacement, and K(r) denote a positive radial function with compactly supported on the δ-ball Bδ(x). The momentum balance and nonlocal dilatation are then given by the following, LδuCαm(δ)Bδ(x)λμK(|yx|)yxθ(x)+θ(y)dyCβm(δ)Bδ(x)μK(|yx|)yxyx|yx|2u(y)u(x)dy=f(x),θ(x)dm(δ)Bδ(x)K(|yx|)(yx)u(y)u(x)dy,

Neumann and mixed-type constraint problems

We now consider a state-based peridynamic problem with general mixed boundary conditions: Ω=ΩDΩN and (ΩD)o(ΩN)o=. Here ΩD and ΩN are both 1D curves. We denote the regions near the boundary Ω as IΩ{xΩ|dist(x,Ω)<δ},BΩ{xΩ|dist(x,Ω)<δ},BBΩ{xΩ|dist(x,Ω)<2δ}.Note that to apply the nonlocal Dirichlet-type boundary condition, u(x)=uD(x) is required in a layer with non-zero volume outside Ω, while the proposed traction load is applied as a Neumann boundary condition on the sharp

Optimization-based meshfree quadrature rules

In this section, we introduce a strong-form particle discretization of the state-based peridynamics introduced above. Discretizing the whole interaction region ΩBBΩ by a collection of points Xh={xi}{i=1,2,,Np}ΩBBΩ, we aim to solve for the displacement uiu(xi) and the nonlocal dilatation θiθ(xi) on all xiXh. We first characterize the distribution of collocation points as follows. Recall the definitions [52] of fill distance hχh,Ω=supuiΩBΩminxiχhxixj2,and separation distance qχh=12min

Numerical verification and asymptotic compatibility

In this section we numerically verify the approach by investigating accuracy when recovering analytic solutions in the M-convergence limit with mixed boundary conditions. We consider: linear patch tests, smooth manufactured solutions, analytical solutions to curvilinear surface loading problems, and analytical solutions to linearly elastic composites. For each case, we consider various combinations of Dirichlet and traction-type boundary conditions, exploring also the effect of reduced

Fracture dynamics for brittle fracture experiments

The previous sections have established the ability of the scheme to recover local solutions of boundary value problems in elasticity governed by traction loadings and ensured that the breaking bonds treatment does not impair the AC convergence of the quadrature treatment. Of course, the main appeal of peridynamic discretizations is to handle fracture problems, and we devote the remainder of the paper to demonstrating how the scheme prescribed previously adapts to practical engineering settings,

Conclusion and future work

Peridynamics presents a flexible framework for modeling fracture mechanics. In particular, bond-based fracture models admit a sharp representation of fracture surfaces while avoiding the loss of mass associated with damage models and element death [73]. This flexibility comes with a cost however, as the free-surface introduced during fracture compounds traditional challenges in peridynamic models related to nonlocal boundary conditions. This work has presented a complete workflow demonstrating

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgments

Sandia National Laboratories is a multi-mission laboratory managed and operated by National Technology and Engineering Solutions of Sandia, LLC., a wholly owned subsidiary of Honeywell International, Inc., for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S.

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      The LPS model may be interpreted as a nonlocal generalization of the mixed form of linear elasticity, evolving both displacements and a dilatation. We begin with a review of the deterministic LPS model for heterogeneous materials [34] in Section 2.1, then extend the formulation to the stochastic LPS problem with random parameters in Section 2.2. Finally, we discuss the treatment of material fracture, including the damage criteria and the handling of free surfaces created by the evolving fractures, in Section 2.3.

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