Computationally efficient and effective peridynamic model for cracks and fractures in homogeneous and heterogeneous materials

https://doi.org/10.1016/j.cma.2022.115318Get rights and content

Highlights

  • We propose a peridynamic-based algorithm for cracking behavior analysis.

  • The crack evolutions for problems under different materials are studied.

  • Experimental and numerical studies are used in the verification processes.

  • The efficiency and applicability of the algorithm are studied.

  • The effects of horizon size and number of particles on the peridynamic modeling are examined.

Abstract

Efficiency has long been a bottleneck for peridynamics simulation as compared to the grid-based methods. This paper proposes a super-fast peridynamic (SFPD) model for crack and fracture simulations in homogeneous and heterogeneous materials in a most efficient manner. The philosophy behind this SFPD algorithm is to decrease the computational time needed at every simulation step rather than to increase the time step, since the peridynamic stability and convergence depend primarily on a small time step. The proposed SFPD algorithm surpasses the classical models and algorithms as follows: (1) the SFPD is hundreds of times faster than the regular peridynamic approach for the same computational task; (2) the peridynamic convergency is dramatically enhanced despite using a smaller number of particles by adopting a new strategy of time step calculation; (3) the SFPD algorithm can simulate the cracks in a wide range of scales including the subatomic scale; (4) the prediction of crack velocity using the SFPD algorithm is much better than other models’ predictions when compared with experimental measurements. Various cracking problems are investigated in both homogeneous and heterogeneous materials under different boundary conditions. The newly proposed algorithm is well-validated to have huge potential in boosting computational efficiency. Moreover, we recommend adopting this algorithm in the software development of crack simulation packages.

Introduction

The prediction of cracking behaviors in different types of solids is attracting great interest in the solid mechanics field due to its importance in engineering applications [1]. Some analytical approaches were presented to study the fatigue crack growth [2], [3], where the stress intensity at the crack tip was calculated to predict the fatigue crack propagation. In another study [4], fracture analysis based on scaling assumptions was utilized to examine the cracking behavior in glass-ceramic slabs under cooling or drying conditions. However, the analytical solutions can only be implemented on a limited variety of geometry and boundary conditions. Therefore, many numerical approaches were presented to simulate the cracks, since the experimental approach is expensive and cannot be practically implemented under different boundary conditions [5], [6]. The finite element method (FEM) is one of the most popular numerical methods and has been widely used to predict the cracking behavior in different types of solids [7], [8]. However, the computational efforts are increased dramatically when more mesh elements are reproduced around the crack path [9], [10]. Moreover, mesh elements may suffer from ill-conditioning due to excessive remeshing, which leads to the termination of the analysis [11]. Alternatively, the extended finite element method (XFEM) has been adopted to study the crack initiation and propagation in different materials and boundary conditions [12], [13]. However, XFEM adds more complexity to the FEM and may increase the computational efforts tremendously. In [14], [15], a new mesoscopic network mechanics method based on FEM was developed to study the crack mechanism in cross-linked elastomers such as rubbers and gels. The crack is simulated by deleting the overstretched chains and elements; thus, a careful check must be performed at each deletion operation, which may increase the complexity. Coupling between the interaction integral method and FEM based on the criterion of maximum tangential stress (MTS) was also adopted to investigate the crack pattern in central cracked Brazilian disc (CCBD) specimens [16].

The phase-field method has been adopted by many researchers to study the crack mechanism in different materials, such as quasi-brittle materials [17], [18], and in different loading types such as fatigue [19], tensile [20] and thermal loadings [21]. The capability of the phase-field method to represent the cracks with a diffusive region outperforms other crack treatments such as the crack treatments in Griffith [22] and Irwin [23]. When FEM is used to solve the phase-field differential equations, there are more requirements such as mesh-refinement, a small time step and a large number of mesh elements, which in turn increases the computational cost and, in some cases, limits its usage [24]. To overcome these limitations, the phase-field method was coupled with other methods to study the crack mechanism in composites, like the coupling scheme with the cohesive zone method (CZM) [25], [26]. It is also expected to evaluate the mechanical properties of composite materials using machine learning techniques to reduce the computational cost [27]. However, the computational efficiency of the previous coupling schemes is still an issue since the coupling strategy will add more computational steps and complexity to the problem under investigation.

The CZM has been implemented to predict the crack shape in many materials, particularly concrete, cementitious composites and polymers [28]. According to CZM, elements with zero thickness are added in the possible crack paths, where the crack is recorded when these elements are damaged [29]. Therefore, CZM requires possible predictions of crack paths. Moreover, very fine mesh is required near the cohesive zone elements, which in turn increases the computational cost significantly [30]. Alternatively, the discrete element method (DEM) was employed by many researchers to simulate cracks in rocks [31], cementitious samples [32], and laminated glass [33]. DEM assumes that particles are rigid and can overlap or detach based on the contact forces governed by force–displacement law [34]. The main limitation of DEM is the vast number of particles needed in the problem discretization, particularly in large or complex domains [35], [36]. A lattice model has also been employed to study the fracture mechanics in different brittle materials such as concrete and mortar [37]. The material in the lattice model is discretized into a finite number of elastic beams or spring mesh elements, where the fracture is simulated by conducting linear elastic analysis [38]. The mesh elements that exceed the maximum strength or energy are removed from the lattice model. Similar to DEM, the lattice model requires a large number of mesh elements to produce accurate results, therefore, a huge amount of computational resources are needed [39].

Recently, meshfree methods were used to overcome the difficulties and deficiencies encountered in the mesh-based methods. The peridynamics-based method is considered one of the most popular approaches used to simulate cracks. Peridynamics was first presented by Silling [40], [41] to solve the deformation problems of materials with a discontinuity (crack). Later, peridynamics has been employed to simulate the cracks in a wide range of materials like concrete [42], glass [43], PMB material [44] and steel plate [45], [46]. Peridynamics surpasses the phase-field method by its ability to simulate the cracks in 3D, while the phase-field method is best suited for 2D problems [47]. Moreover, the phase-field method requires extensive mesh refinement around the crack path and leads eventually to huge computational costs [48]. Furthermore, the crack path must be previously redefined to conduct the refinement. Due to these drawbacks of the phase-field method, peridynamics is considered a preferable method to simulate fracture. However, peridynamics is a nonlocal extension of continuum mechanics, so it suffers from a huge computational cost. Moreover, the stability and convergence of the peridynamic method depend primarily on using a very small time step (from nanoseconds to microseconds) [49]. The non-local nature of peridynamics requires more attention to the discretization parameters such as the horizon radius (m) because the damage and displacement of every particle depend on its neighboring particles (non-local). Previous attempts were devoted to enhancing the computational efficiency and the stability of peridynamics, such as by introducing several types of micro-modules functions [50], [51] or using different kernel formulations with the peridynamic models [52]. Most recent attempts have used coupling schemes with FEM [53], [54]. However, the previous attempts did not give a solid solution to how to increase the computational efficiency of the peridynamics (PD). Software packages and codes used to simulate the cracking problems are still suffering from the huge computational cost issue, whether these software packages and codes are using FEM, the cohesive zone model, or the peridynamic approach [55], [56].

Several peridynamic algorithms were presented in an attempt to decrease the computational cost for crack simulation problems. Most of these algorithms use the neighbor list technique to decrease the computational steps. For instance, searching for neighboring particles can be conducted only in the nearby regions of the particle instead of the whole domain [57] or by creating a neighbor list for each particle [58]. However, more efficient enhancements are needed since the computational cost is still large. For example, the computational costs for algorithms 1 and 2 presented in [58] for each time step are around 13 and 21 s, respectively, for only 4725 particles and when m equals only 2. If similar algorithms are used in problems that need more particles and time steps, huge computational efforts are required.

Measurement of the crack velocity in different materials is considered an interesting and challenging topic. Several experiments were conducted to measure the crack velocity in different materials such as steel [59], [60] and Polymethylmethacrylate (PMMA) [61], [62], [63]. The published numerical models did not give a solid value for the crack velocity measured in the experiments. For instance, the crack velocity in PMMA plate, numerically calculated using peridynamics [64], is about 538 m/s even for the crack propagation without oscillation or branching. However, the experimental measurement for the critical crack velocity in PMMA is 330 m/s [65], [66], where the branching occurs immediately after that velocity. Another challenging topic is to simulate the fracture at an atomic scale due to its important applications and also due to the lack of clear analytical solutions or experimental data. Several numerical models were developed in an attempt to study the crack mechanics at the atomic scale [67], [68], [69], [70] or to simulate multiple cracks in composites [71], [72], [73]. However, several difficulties were encountered in the previous models such as the stability and the computational efforts.

In this paper, we present a new algorithm based on the peridynamic method, which is capable of simulating the cracks in different materials (homogeneous and heterogeneous materials) and under varying boundary conditions in less than 2 min when using 10 000 particles and 2000 time steps, where every particle interacts with up to 28 of its neighboring particles (m=3). Moreover, our SFPD algorithm is capable of measuring the crack velocity in different materials efficiently. Furthermore, crack simulation in atomic and even subatomic scales is now feasible using our SFPD algorithm. The proposed algorithm is straightforward and is very convenient to implement in most software packages. The remainder of this paper is structured as follows. Section 2 presents the basic peridynamic formulations. Section 3 illustrates the details of the super-fast peridynamic algorithm (SFPD). Several numerical examples are presented in Section 4 to solve the crack simulation in both homogeneous and heterogeneous materials in different scales and conditions. Also, convergence analysis for peridynamic parameters is examined in detail. Finally, our conclusions are stated in Section 5.

Section snippets

Peridynamic formulation

In the peridynamic theory, the solid body is represented by a set of particles interacting with each other within a domain (Hx) by a radius of δ (horizon size) as depicted in Fig. 1.

The main advantage of the peridynamic theory is using the integral form to solve the equations of motion according to Eq. (1) [44], [49], [53]: ρüx,t=Hxfux,tux,t,xxdv+bx,twhere ρ is the density, ü(x,t) is the acceleration of particle x at time t, f is the force per volume square, dv is the partial volume

Super-fast peridynamic algorithm (SFPD)

In this section, the super-fast peridynamic algorithm (SFPD) is presented to solve the crack simulation in both homogeneous and heterogeneous materials based on the peridynamic formulation. Table 1 shows the SFPD algorithm:

The following points are concluded from the SFPD algorithm (Table 1):

  • The size of looping is minimized by including only the nearby particles within the domain Hx for each particle x. This is done by building the interaction matrix only once in the initialization stage (before

Results and discussion

In this section, the SFPD algorithm is utilized to simulate the cracks in homogeneous and heterogeneous materials in different cases and boundary conditions. The proposed algorithm is well validated by comparing its results with experiments and numerical models. MATLAB code is developed based on the SFPD algorithm and executed on the HP Laptop with Intel ® Core® i7-8550U [email protected] GHz, 1992 MHz, 4 Core(s), 8 Logical Processor and 8 GB of RAM.

Conclusions

A new robust algorithm based on peridynamic modeling is presented in this work. The mesh-free nature of peridynamics is well utilized in a manner that eliminates the slow simulation drawback of the peridynamic method. We focus here on decreasing the computational cost at every simulation step; therefore, we allow the peridynamic model to simulate the cracks with the suitable time step needed for the required stability.

The proposed algorithm is capable of predicting the crack behavior in

CRediT authorship contribution statement

D.A. Abdoh: Methodology, Investigation, Formal analysis, Writing – original draft. B.B. Yin: Formal analysis, Writing – review & editing. V.K.R. Kodur: Resources, Writing – review & editing. K.M. Liew: Conceptualization, Methodology, Project administration, Funding acquisition, Supervision, Writing – review & editing.

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

The authors gratefully acknowledge the supports provided by the Research Grants Council of the Hong Kong Special Administrative Region, China (Project No. 9043135, CityU 11202721). The first author, D.A. Abdoh under the supervision of Professor K.M. Liew, acknowledges the UGC-Postgraduate Studentship awarded by the Hong Kong Government to support his Ph.D program in the Department of Architecture and Civil Engineering, City University of Hong Kong , Hong Kong, China.

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