Overcoming the limitations of distinct element method for multiscale modeling of materials with multimodal internal structure

https://doi.org/10.1016/j.commatsci.2015.02.026Get rights and content

Highlights

  • We propose general expressions describing interaction between simply deformable elements.

  • Realization of various rheological and fracture models is discussed.

  • Models accounting for grain and phase boundaries within the DEM framework are proposed.

  • Examples of DEM application to materials with multiscale internal structure are shown.

Abstract

This paper develops an approach to model the deformation and fracture of heterogeneous materials at different scales (including multiscale modeling) within a discrete representation of the medium. Within this approach, molecular dynamics is used for the atomic-scale simulation. The simply deformable distinct element method is applied for simulating at higher length scales. This approach is proposed to be implemented using a general way to derive relations for interaction forces between distinct elements in a many-body approximation similar to that of the embedded atom method. This makes it possible to overcome limitations of the distinct element method which are related to difficulties in implementing complex rheological and fracture models of solids at different length scales. For an adequate description of the mechanical behavior features of materials at the micro- and mesoscales, two kinds of models that consider grain and phase boundaries within the discrete element framework are proposed. Examples are given to illustrate the application of the developed formalism to the study of the mechanical response (including fracture) of materials with multiscale internal structure. The examples show that the simply deformable distinct element method is a correct and efficient tool for analyzing complex problems in solid mechanics (including mechanics of discontinua) at different scales.

Introduction

Currently, the design of advanced materials is in many respects based on theoretical results obtained with computer simulation. The necessity for computer simulation in modern materials science and mechanical engineering has promoted a rapid development of numerical methods and generated a need for the development of methodologies and approaches of multiscale modeling of material deformation and fracture. In the last years, multiscale modeling has evolved into an independent line of computer-aided research and design of materials, which has greatly influenced the development of computational methods and led to the development of various combined numerical techniques [1], [2], [3], [4]. Despite a significant difference in various multiscale modeling approaches, all of them are similar in that they take into account (explicitly and implicitly) the contribution of deformation mechanisms of spatial and structural scales, lower with respect to the considered one, to the mechanical response of the material at the considered scale (this assumption is called the structure–property paradigm [1]). This actually means the requirement for taking into account the internal structural and rheological features of basic structural elements at different scales, first of all, at the nano- and microscale. The basic elements of the internal structure are not only grains and inclusions of other phases but also interfaces between them and discontinuities whose deformation ability and evolution play a crucial role for a wide range of materials, primarily, nanostructured materials.

The whole family of various methods that take into account the hierarchy of structural scales on the mechanical response of the material can be arbitrarily divided into two types (methodologies). The first one is based on decomposing a problem into fine and coarse scales [2]. The practical implementations of this methodology as a rule reduce to two-scale methods and models. These are, particularly, the variational multiscale method proposed by Hughes et al. [5] and different multigrid methods including combined techniques in which a system is divided into domains with different discretization step modeled with the use of different numerical methods (finite element method, molecular dynamics method, discrete element method, etc.) [6], [7], [8], [9]. The second, most widely adopted methodology implies the determination of a hierarchy of basic structural scales in the modeled material and determination of a representative volume or a system of representative volumes with qualitatively different structural and phase composition on each scale [10], [11], [12], [13], [14]. The representative volume of a considered scale must contain a sufficient number of structural elements of this scale for correct subsequent homogenization. Numerical results on the mechanical response of the representative volume are used to determine its integral mechanical properties, including the equation of state and its parameters, the fracture criterion and fracture parameters, and so on. These parameters are taken as input parameters (response parameters of discretization units) on a higher scale. Consistent implementation of the given procedure starting from the atomic or nanoscale yields the construction of micro-, meso- and macroscale models of solids with multiscale internal structure. It should be noted that this methodology is based on two essential approximations. The first one implies a unique relation between characteristic sizes of the internal structure elements on a considered scale and characteristic times of accommodation/relaxation processes of this scale. The second approximation implies that energy thresholds of activation of deformation mechanisms depend uniquely on their spatial scales. Today, there are different practical implementations of the discussed methodology. The most rigorous and advanced implementations take into account the kinetics of deformation, including relaxation, processes on lower scales through the introduction of additional input parameters (internal variables) governed by special evolution equations [13], [15].

Despite different formalisms of different approaches, a common problem of multiscale modeling is the different representation of the medium in numerical methods applied for modeling at different scales. For example, the integral properties of the atomic and nanoscale representative volumes are determined using molecular dynamics and molecular mechanics methods, Monte-Carlo method and others. These methods are based on a discrete description of solids. At the same time, multiscale modeling of materials on higher spatial and structural scales is conventionally performed using continuum mechanics methods (finite element and finite difference methods). Different concepts of the medium representation result in qualitatively different formalisms of these numerical methods, particularly, different formulation of equations of motion and state. The question naturally arises whether it is possible and efficient to perform numerical modeling of material deformation and fracture on all scales using numerical methods that refer to a unified (discrete) concept of the medium representation. The importance of this question is particularly related to the necessity of taking into account the evolution of different-scale initial discontinuities (cracks and damages) and new discontinuities formed under deformation. A large group of problems in which a direct description of discontinuous (discrete) structure of a considered medium at different scales is crucial for obtaining an adequate solution has been called the mechanics of discontinua [16]. The necessity of describing a solid as a multiscale discontinuous (discrete) system led to the development of a broad class of “discrete” numerical methods based on the representation of the medium as an ensemble of interacting particles (currently these methods are called particle-based) [17], [18], [19], [20], [21], [22], [23], [24]. The principal difference of these methods from computational methods in continuum mechanics is the replacement of a continuous representation of the material or medium by an ensemble of interacting point masses (at the atomic scale within the framework of molecular dynamics or Monte-Carlo method) or by an ensemble of interacting particles of finite size (at higher spatial scales). The difference in the medium representation within continuum and discrete concepts determines differences in governing and balance equations. In particular, in explicit formulation conventional equations of motion of continuum are replaced by ordinary differential equations (for translations and rotations) governing the evolution of a particle ensemble. Relations between local stresses and strains or their time derivatives are replaced by expressions for potentials/forces of particle–particle interaction. One of the most important consequences of these features of particle-based methods is an inherent ability of discrete domains (particles) to change surroundings. This makes “discrete” numerical methods extremely attractive for direct modeling of complicated fracture-related processes at different spatial and structural scales up to the macroscale [16], [18], [22], [25], [26], [27].

Well-known numerical methods widely applied for problem solving in the mechanics of discontinua are discrete element methods (DEM). The term DEM implies a large group of modeling techniques that treat a solid as an ensemble of deformable or rigid bodies of arbitrary shape [16], [17], [18], [19], [20], [21], [24], [27], [28], [29], [30], [31]. The main differences between various representatives of the DEM group are in the principles of formulation of motion equations and in the approximations to the description of element deformability. There are two approaches to formulate the motion equations of discrete elements: implicit (represented by discontinuous deformation analysis) and explicit [29], [31]. In the framework of the first approach motion equations are written for all the elements in matrix form and solved simultaneously using corresponding methods for solving of sets of linear/nonlinear algebraic equations. In the second approach, which is used in this paper, motion equations are written and solved “individually” for each element.

Various representatives of the group of explicit discrete element methods differ in the approximations used to describe (i) strain distribution in the bulk of discrete element, and (ii) influence of shape/geometry of the element on its kinematics and interaction with surrounding [31]. In particular, for discrete elements of complex shape, conventionally approximated by generally shaped polygons or polyhedrons, the motion equations for rotational degrees of freedom has a complex form. Besides, presence of vertices and edges for such elements lead to separation of the interaction with surrounding into several types (vertex–vertex, vertex–face, vertex–edge, edge–edge, etc.) [31]. Such details are of principle for modeling blocky and granular materials and media. At the same time, use of polygonal shape of discrete elements in modeling consolidated materials with complex internal structure (polycrystals, composites) at the mesoscopic scale results in some limitations for spatial discretization. In particular, a polygonal element is correct structural model for separate grain or phase inclusion. However, modeling grain/inclusion by ensemble of interacting polygonal elements may lead to generation of artificial effects. That is why simplified elements are mainly used in DEM-based modeling of consolidated polycrystal and multiphase materials at mesoscopic scale. In the framework of this approach it is assumed that interaction between discrete elements occurs on plane faces (face–face interaction). The size of face is chosen based on the local packing of elements so that imaginary polygons (obtained by connecting the vertices of the plane faces) fill the space without voids (or produce required fraction and shape of the voids). The effect of vertices and edges is usually neglected here. One representative of such simplification of element geometry is its interpretation as equivalent circular disc (in 2D problem statement) or sphere [29], [30], [31] (obviously, that is applicable for equiaxed or nearly equiaxed elements). As a consequence of such approximation there is a simple form of motion equations for rotational degrees of freedom of distinct element; namely the Euler equation becomes analogous to Newton equation for translational degrees of freedom. In many recent papers such simplification of discrete element is associated with well known distinct element method proposed by Cundall [18], [29]. Hereinafter we will keep the meaning of the term “distinct element” as a particular case of wider term “discrete element” (note, that in some papers the term “distinct element method” has more general sense and is used for denoting all the group of explicit DEMs [31]).

Geometrical peculiarities of discrete element substantially define the details of strain distribution in the bulk of element. For the problems where integral response of the system is determined by slipping of structural elements (blocks), and deformation of the blocks (modeled by discrete elements) is assumed to be small, the approximation of rigid bodies connected by springs of given stiffness and viscosity is valid [18], [29], [30], [32]. Another limiting case (especially actual for discrete elements of complex polygonal/polyhedral shape) is internal discretization of the discrete element connected with dividing of its space into a finite number of internal finite elements or finite difference zones for detail calculation of displacement, strain and stress distributions. An intermediate approximation is the model of simply deformable discrete element that takes into account the possibility of specimen deformation in a linear approximation of displacement distribution in the element volume [17], [20], [30], [31], [33], [34], [35]. This approximation actually means that stresses and strains are homogeneously distributed in the element volume. Homogeneously distributed (or, in other words, averaged) internal stresses and strains are calculated on the basis of superposing parameters of interaction between the element and its neighbors. The calculated stresses are in turn used to calculate element–element interaction forces [30], [33]. The described approximation suits well, in particular, for distinct elements, which have equiaxed or nearly equiaxed shape. As was discussed above, such elements may constitute material grains, inclusions of different phases and interface zones between them at different scales, starting from the nanoscale. Therefore, the development of the formalism of simply deformable distinct elements is relevant to a broad class of problems in multiscale modeling of heterogeneous consolidated materials.

Despite the fact that distinct element method has simple mathematical formulation and apparent advantages in modeling deformation and fracture at various scales, its application to materials with complex internal structure is limited mainly to the brittle and weakly bonded media. These limitations are due to insufficient development of mathematical models of interaction between distinct elements. A conventional approach is based on the use of pair-wise elastic interaction forces (resistance to compression/tension/shear in bonded pairs of elements and resistance to compression in pairs of contacting particles). Corresponding spring stiffness values are derived on the assumption that the strain energy stored in a unit cell of a deformed ensemble of elements is equivalent to the associated strain energy of an equivalently deformed continuum [36], [37]. It is necessary to note that distinct element method has its origin in the well-known molecular dynamics method [23] applied for studying deformation and fracture of materials at the atomic scale (explicit DEM is often considered as an extension of molecular dynamics method to the micro-, meso- and macroscale). The modern stage of development of distinct element method is in many respects similar to that of molecular dynamics method in the 1980s. That was the period of using mainly pair-wise interatomic potentials (Lennard-Jones, Morse, pseudopotential). The use of such simplified interactions made it difficult to study some complex phenomena related to the nucleation of plastic deformation, nanocrack propagation and surface effects. These difficulties have been overcome owing to the development of many-body potentials of interatomic interaction (embedded atom method [38], electron density functional theory [39], etc.). Fundamentally similar problems exist in distinct element method. The use of conventional models of pair-wise interaction between discrete elements often makes it difficult to correctly simulate irreversible strain accumulation in ductile consolidated materials. Correct modeling of the material’s plastic behavior (stress relaxation) is critically important even in one-scale problems. In multiscale modeling, a correct account for material ductility due to mechanisms belonging to lower structural scales becomes determinative.

This paper proposes the mathematical formalism of simply deformable distinct element method for adequate modeling mechanical behavior of elastic–plastic materials with taking into account internal interfaces. This formalism is based on previous achievements of the authors concerning the development of generalized relations for interacting forces between movable cellular automata. The movable cellular automaton (MCA) method is a hybrid numerical technique which unites distinct element method and cellular automaton method [40], [41]. The main difference between МСА and conventional implementations of distinct element method is the postulated general structure of expressions for the interaction force between automata. These expressions have a many-body form [41], [42]. Based on them, here the authors suggest a general and advanced approach to deriving relations for interaction forces between distinct elements in a many-body approximation similar to that of the embedded atom method. It is shown for isotropic elastic–plastic materials that the use of the derived relations makes possible the implementation of different models and criteria of elasticity, plasticity and fracture (including that for anisotropic materials and crystallites) within the mathematical formalism of simply deformable distinct element method. The mechanical response of solids (including materials with submicro- and nanophases, nanostructured materials) at the meso- and microscale is described using physically substantiated models that take into account grain and phase boundaries. The proposed approach to formulating element–element interactions allows one to overcome the above-described limitations of distinct element method and to greatly expand its application area to a broad range of spatial and structural scales as well as to materials with different rheology. The possibilities and efficiency of multiscale numerical study of the mechanical response (including fracture) of heterogeneous materials by distinct element method are illustrated by the examples of constructing a multiscale model of sintered solids that contain different-scale (from nano- to mesoscale) structural constituents.

Section snippets

General statements

In the framework of distinct element method the consolidated material under consideration is treated as an ensemble of interacting particles (elements) having finite size. Interacting pairs of neighboring elements are considered as being in contact. To model a consolidated fragment of solid these contacts are assumed to be bonded initially. For damage/crack faces corresponding contacts are considered as unbonded [18], [43]. Switching between these two states of the contact is based on assigned

Fracture model and criteria

A fundamental feature of the DEM formalism is an inherent ability of discrete elements to change surroundings (interacting neighbors). This allows considering DEM as an attractive numerical technique to be used for direct modeling multiple fracture accompanied by formation and mixing of large number of fragments. Coupling (adhesion) of fragments can be included in the model as well. These abilities are taken into account by means of change of the contact type from bonded to unbonded or vice

Direct and indirect models of interfaces between structural elements

The key point in describing the mechanical response of solids (including materials with submicro- and nanoscopic phases as well as nanostructured materials) at the meso-, micro- and nanoscale is to take into account the role of grain boundaries and phase interfaces. The geometric and physical and mechanical properties of such interfaces depend on the characteristic size of the structural elements separated by interfaces as well as on the conditions of material production. Particularly, the

Application results and discussion

The developed formalism of the method of simply deformable distinct elements has allowed us to overcome the essential difficulties in describing the mechanical response of solids (particularly, inelastic deformation of elements). It made possible the application of the method at different scales (including nano- and macroscale) and, moreover, implementation of different multiscale approaches and models within its framework. This section gives two examples illustrating the possibilities and

Conclusions

A common problem of multiscale numerical modeling is the different representation of the medium in numerical methods applied for modeling at different scales. Different concepts of the medium representation result in qualitatively different formalisms of these numerical methods, particularly, in a different formulation of equations of motion and state. The paper proposes a unified multiscale approach to the modeling of materials with multimodal internal structure based on the concept of a

Acknowledgements

This work was supported by the Russian Science Foundation (Grant 14-19-00718). S.G. Psakhie acknowledges financial support from the Program of Basic scientific research of the State academies of sciences for 2013-2020 (Russia).

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