A state-based peridynamic analysis in a finite element framework
Introduction
The finite element method is computationally robust and very effective for modeling structures with complex geometries and different materials under general loading conditions. Its computational model requires explicit representation of cracks for the prediction of crack growth and propagation. However, the stresses at the crack tips are mathematically singular because of the undefined spatial derivatives of displacements. Techniques such as the eXtended Finite Element Method (XFEM) by Moës et al. [1] or its extensions can facilitate the prediction of onset of crack initiation and its propagation path in conjunction with multiple external criteria. However, the choice of such criteria for injection of discontinuous displacement enrichment functions introduces uncertainties in crack topology especially in the presence of multiple interacting cracks and non-planar 3D crack surfaces.
The peridynamics (PD) introduced by Silling [2] and Silling et al. [3] is extremely suitable for failure analysis of structures because its governing equations are always defined at the fracture surfaces. In the PD theory, internal forces are expressed through nonlocal interactions between pairs of material points within a continuous body, and damage is introduced through the removal of interactions. This feature allows crack nucleation and propagation at multiple sites with arbitrary paths in the structure.
The bond-based PD analysis to model isotropic materials employs a single PD material parameter referred to as the “micromodulus” and the stretch between two material points. Thus, it results in a reduction of independent material constants while imposing a fixed value on the Poisson’s ratio [4]. Furthermore, it does not distinguish between the dilatation and distortional deformation modes. Therefore, Silling et al. [3] introduced the ordinary state-based PD analysis to remove such limitations. The applications of the PD theory to investigate material failure can be found in recent books by Madenci and Oterkus [5] and Bobaru et al. [6].
Because the basic PD equations are entirely consistent with the Classical Continuum Mechanics (CCM), Macek and Silling [7] showed that the rod elements already existing in the FEA framework can be employed to represent the bond-based PD interactions. Specifically, they generated a mesh that resembled the nonlocal interactions. The classical element stiffness properties are related to their PD counterparts through a correspondence table. They demonstrated such capability by considering complex impact and penetration type loading conditions within the ABAQUS framework, a commercially available FEA software. Their approach was adopted by Lall et al. [8] to investigate damage in electronic components due to drop-shock and later by Beckmann et al. [9] to determine thermomechanical strains. Analogous to the correspondence table for the structural rod element, Diyaroglu et al. [10] provided the correspondence tables for thermal link and surface elements available in ANSYS to perform analysis of heat and electrical conduction and moisture (wetness) concentration. Similarly, De Meo and Oterkus [11] presented a finite element implementation of a PD modeling of pitting corrosion damage model. Also, Dorduncu et al. [12] extended the concept introduced by Silling and Macek [7], and developed a truss element for ordinary state-based PD analysis. The stiffness matrix for this element involves the PD material parameters which depend on the nature of discretization, the size of interaction domain between the two points as well as the engineering material constants.
In all of these previous studies, the PD material parameters are determined by equating the classical strain energy density (SED) under simple loading conditions to that of PD at a point with a complete domain of interaction [5] in a uniformly discretized computational domain. However, these PD parameters are not valid if the point is located near a surface because of its incomplete domain of interaction. Therefore, they require a correction procedure due to surface effects.
In addition, the PD analysis requires volume integration over the interaction domain which must conform to the geometry of the integration domain for determining the PD material parameters. However, the numerical integration performed as part of the PD analysis introduces inaccuracy arising from the approximation of circular or spherical analytical integration of interaction domains. Therefore, it requires a volumetric correction procedure to improve the integration accuracy [13]. Such corrections rely on regular geometry for the integration domain and the volume of PD points. They cannot be directly applied to the arbitrary domains of interactions that are present in non-uniform discretization. Also, a nonuniform discretization often results in irregular shapes for the interaction domain in which the points can have different volumes. The interaction domains associated with each point in nonuniform discretization can be different with varying number of family members.
Within the context of state-based PD, this study extends the PD truss element developed by Dorduncu et al. [12] for nonuniform PD domain discretization with irregular interaction domain without requiring a surface and a volume correction procedure. The uniform or nonuniform PD discretization can be generated using standard finite element (FE) mesh generation software that enables the construction of computational models to represent specific geometries. The applicability of this element is demonstrated by considering a structured and an unstructured nonuniform domain discretization by using an in-house program. Its accuracy is established by considering a plate with or without a defect in the form of a hole or a crack under tension. In the presence of a defect, the solution is achieved by employing an implicit algorithm until crack initiation or growth, and then continuing with an explicit algorithm.
Section snippets
Peridynamic equation of motion
Peridynamics introduced by Silling [2] and Silling et al. [3] is a nonlocal continuum theory. Silling et al. [3] derived the PD equation of motion aswhere is the mass density, and and represent the acceleration and the body force vector, respectively, at point . The internal force vector, is defined aswhere and are the displacement vectors at points and , respectively. Their positions in the deformed
PD element stiffness matrix
A PD element accounts for the PD interactions among the nodes. The nodes in the family of node are connected through PD elements as shown in Fig. 3.
The internal force vector at node can be expressed asin which represents the force in the element between nodes and , and is the number of PD elements connected at node . Therefore, the internal work arising from the deformation of elements connected node can be expressed asor
PD governing equations in finite element framework
For a two-dimensional analysis, the domain is split into the interior (bulk) region, and the boundary layer region, as shown in Fig. 4. The width of the boundary layer is defined by which is much smaller than the characteristic dimension of the domain. Also, the boundary of the domain defined by is split into the segments of and which are subjected to external loads and displacement constraints, respectively. In the context of finite element analysis, the domain is discretized
Failure criteria
The onset of cracking and its propagation can be simulated by breaking the PD elements. Element breakage criteria can be based on critical stretch, critical energy release rate, maximum tangential stress or maximum principle stress, etc. Although not a limitation, in this study the onset of cracking and propagation occurs when the maximum principal stress, in an element between node k and j exceeds the uniaxial tensile strength of the material, [18], and the elements connected to
Numerical results
In order to demonstrate the capability of this new element for PD analysis, the numerical results concern four distinct geometric configurations and loading conditions. Unlike the previous PD predictions as discussed by Madenci and Oterkus [5], these PD predictions are free of surface effects because the present approach involves the direct use of the engineering constants without any calibration, and enables the direct imposition of the essential and natural boundary conditions.
First, a square
Conclusions
This study describes the development of a PD truss element based on the state-based peridynamic interactions and PD LSM for approximating the spatial derivatives. It presents the element stiffness matrices for internal and boundary elements. This new element provides computational efficiency because of the ability to have non-uniform discretization in the analysis of deformation and cracking within a finite element framework. Its capability is demonstrated by considering a plate with and
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2022, Composite StructuresCitation Excerpt :They demonstrated that the material variations and boundary/loading conditions played a major role on the crack onset and propagation. The BB PD does not distinguish between dilatation and distortional parts of deformation with a constraint on the Poisson’s ratio (1/3 for 2D and 1/4 for 3D) [33–37]. Later, Silling et al. [30] introduced a more general formulation called “ordinary state-based (OSB)” PD theory to eliminate the limitations of the BB PD theory.
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2021, Theoretical and Applied Fracture MechanicsCitation Excerpt :Both proposed methods are free of the PD surface effect using variable horizons. 2D problems were only taken into account in the reference studies mentioned above [26–29]. For the surface effect correction with arbitrary horizon domains, out-of-plane deformations for thin-walled structures still have been not discussed.
Peridynamics modelling of dynamic tensile failure in concrete
2021, International Journal of Impact EngineeringCitation Excerpt :Moreover, PD theory permits the cracks to spontaneously occur at multiple sites and freely extend along an arbitrary path without the need for external remedial techniques. Besides, one can easily combine the the PD theory and the FEM when performing the numerical analysis [30]. Furthermore, the PD differential operator for numerical analysis with detailed mathematical formulae can be approached through [31].
Numerical study on dynamic icebreaking process of an icebreaker by ordinary state-based peridynamics and continuous contact detection algorithm
2021, Ocean EngineeringCitation Excerpt :Moreover, PD can also be applied to subjects other than fracture mechanics (Javili et al., 2019). Many updated mathematical models based on PD theory have been developed over the years: dual-horizon PD was proposed to solve the issue of spurious wave reflections when variable horizons are adopted (Ren et al., 2017); Dual-support SPH was developed in solid within the framework of variational principle (Ren et al., 2019); the PD differential operator provided a differential form for numerical analysis (Gao and Oterkus, 2020); as well as the combination of PD and FEM model (Madenci et al., 2018) and PD least squares minimization (Madenci et al., 2019). Recently, PD was applied to predict ice-structure interaction for a simple structure interaction with ice (Liu et al., 2017; Lu et al., 2018), which showed PD's potential to study ice fracture.