eXtended Variational Quasicontinuum Methodology for Lattice Networks with Damage and Crack Propagation

Lattice networks with dissipative interactions are often employed to analyze materials with discrete micro- or meso-structures, or for a description of heterogeneous materials which can be modelled discretely. They are, however, computationally prohibitive for engineering-scale applications. The (variational) QuasiContinuum (QC) method is a concurrent multiscale approach that reduces their computational cost by fully resolving the (dissipative) lattice network in small regions of interest while coarsening elsewhere. When applied to damageable lattices, moving crack tips can be captured by adaptive mesh refinement schemes, whereas fully-resolved trails in crack wakes can be removed by mesh coarsening. In order to address crack propagation efficiently and accurately, we develop in this contribution the necessary generalizations of the variational QC methodology. First, a suitable definition of crack paths in discrete systems is introduced, which allows for their geometrical representation in terms of the signed distance function. Second, special function enrichments based on the partition of unity concept are adopted, in order to capture kinematics in the wakes of crack tips. Third, a summation rule that reflects the adopted enrichment functions with sufficient degree of accuracy is developed. Finally, as our standpoint is variational, we discuss implications of the mesh refinement and coarsening from an energy-consistency point of view. All theoretical considerations are demonstrated using two numerical examples for which the resulting reaction forces, energy evolutions, and crack paths are compared to those of the direct numerical simulations.Comment: 36 pages, 23 figures, 1 table, 2 algorithms; small changes after review, paper title change

Multiscale quasicontinuum modelling of fibrous materials

Structural lattice models and discrete networks of trusses or beams are regularly used to describe the mechanics of fibrous materials. The discrete elements naturally represent individual fibers and yarns present at the mesoscale. Consequently, relevant mesoscale phenomena, e.g. individual fiber failure and bond failure, culminating in macroscopic fracture can be captured adequately. Even macroscopic phenomena, such as large rotations of yarns and the resulting evolving anisotropy, are automatically incorporated in lattice models, whereas they are not trivially established in continuum models of fibrous materials. Another advantage is that by relatively straightforward means, lattice models can be altered such that each family of discrete elements describes the mechanical response in one characteristic direction of a fibrous material. This ensures for a straightforward experimental identification of the elements’ parameters. In this thesis such an approach is adopted for a lattice model of electronic textile. A lattice model for interfiber bond failure and subsequent fiber sliding is also formulated. The thermodynamical basis of this lattice model ensures that it can be used in a consistent manner to investigate the effects of mesoscale parameters, such as the bond strength and the fiber length, on the macroscopic response. Large-scale (physically relevant) lattice computations are computationally expensive because lattice models are constructed at the mesoscale. Consequently, large-scale computations involve a large number of degrees of freedom (DOFs) and extensive effort to construct the governing equations. Principles of the quasicontinuum (QC) method are employed in this thesis to reduce the computational cost of large-scale lattice computations. The advantage is that the QC method allows the direct and accurate incorporation of local mesoscale phenomena in regions of interest, whereas substantial computational savings are made in regions of less interest. Another advantage is that the QC method completely relies on the lattice model and does not require the formulation of an equivalent continuum description. The QC method uses interpolation to reduce the number of DOFs and summation rules to reduce the computational cost needed to establish the governing equations. Large interpolation triangles are used in regions with small displacement fluctuations. In fully resolved regions the dimensions of the interpolation triangles are such that the exact lattice model is captured. Summation rules are used to sample the contribution of all nodes to the governing equations using a small number of sampling nodes. In this thesis, one summation rule is proposed that determines the governing equations exactly, even though a large reduction of the number of sampling points is obtained. This summation rule is efficient for structural lattice models with solely nearest neighbor interactions, but it is inefficient for atomistic lattice computations that include interactions over longer ranges. Therefore, a second ’central’ summation rule is proposed, in which significantly fewer sampling points are selected to increase the computational efficiency, at the price of the quality of the approximation. The QC method was originally proposed for (conservative) atomistic lattice models and is based on energy-minimization. Lattice models for fibrous materials however, are often non-conservative and energy-based QC methods can thus not straightforwardly be used. Examples are the lattice model proposed for woven fabrics and the lattice model to describe interfiber bond failure and subsequent frictional fiber sliding proposed in this thesis. A QC framework is therefore proposed that is based on the virtual-power statement of a non-conservative lattice model. Using the virtual-power statement, dissipative mechanisms can be included in the QC framework while the same summation rules suffice. Its validity is shown for a lattice model with elastoplastic trusses. The virtual-power-based QC method is also adopted to deal with the lattice model for bond failure and subsequent fiber sliding presented in this thesis. In contrast to elastoplastic interactions that are intrinsically local dissipative mechanisms, bond failure and subsequent fiber sliding entail nonlocal dissipative mechanisms. Therefore, the virtualpower-based QC method is also equipped with a mixed formulation in which not only the displacements are interpolated, but also the internal variables associated with dissipation

A Variational Formulation of Dissipative Quasicontinuum Methods

Lattice systems and discrete networks with dissipative interactions are successfully employed as meso-scale models of heterogeneous solids. As the application scale generally is much larger than that of the discrete links, physically relevant simulations are computationally expensive. The QuasiContinuum (QC) method is a multiscale approach that reduces the computational cost of direct numerical simulations by fully resolving complex phenomena only in regions of interest while coarsening elsewhere. In previous work (Beex et al., J. Mech. Phys. Solids 64, 154-169, 2014), the originally conservative QC methodology was generalized to a virtual-power-based QC approach that includes local dissipative mechanisms. In this contribution, the virtual-power-based QC method is reformulated from a variational point of view, by employing the energy-based variational framework for rate-independent processes (Mielke and Roub\'i\v{c}ek, Rate-Independent Systems: Theory and Application, Springer-Verlag, 2015). By construction it is shown that the QC method with dissipative interactions can be expressed as a minimization problem of a properly built energy potential, providing solutions equivalent to those of the virtual-power-based QC formulation. The theoretical considerations are demonstrated on three simple examples. For them we verify energy consistency, quantify relative errors in energies, and discuss errors in internal variables obtained for different meshes and two summation rules.Comment: 38 pages, 21 figures, 4 tables; moderate revision after review, one example in Section 5.3 adde

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

ADAPTIVE REFINEMENT AND MULTISCALE MODELING IN 2D PERIDYNAMICS

The original peridynamics formulation uses a constant nonlocal region, the horizon, over the entire domain. We propose here adaptive refinement algorithms for the bond-based peridynamic model for solving statics problems in two dimensions that involve a variable horizon size. Adaptive refinement is an essential ingredient in concurrent multiscale modeling, and in peridynamics changing the horizon is directly related to multiscale modeling. We do not use any special conditions for the “coupling” of the large and small horizon regions, in contrast with other multiscale coupling methods like atomistic-to-continuum coupling, which require special conditions at the interface to eliminate ghost forces in equilibrium problems. We formulate, and implement in two dimensions, the peridynamic theory with a variable horizon size and we show convergence results (to the solutions of problems solved via the classical partial differential equations theories of solid mechanics in the limit of the horizon going to zero) for a number of test cases. Our refinement is triggered by the value of the nonlocal strain energy density. We apply the boundary conditions in a manner similar to the way these conditions are enforced in, for example, the finite-element method, only on the nodes on the boundary. This, in addition to the peridynamic material being effectively “softer” near the boundary (the so-called “skin effect”) leads to strain energy concentration zones on the loaded boundaries. Because of this, refinement is also triggered around the loaded boundaries, in contrast to what happens in, for example, adaptive finite-element methods

Multiscale Modelling of Molecules and Continuum Mechanics Using Bridging Scale Method

his PhD dissertation is about developing a multiscale methodology for coupling two different time/length scales in order to improve properties of new space materials. Since the traditional continuum mechanics models cannot describe the influence of the nanostructured upon the mechanical properties of materials and full atomistic description is still computationally too expensive, millions of degrees of freedom are needed just for modeling few hundred cubic nanometers, this leads to a coupled system of equations of finite element (FE) in continuum and molecular dynamics (MD) in atomistic domain. Coupling efficiently and accurately two dissimilar domains presents challenges especially in handshaking area where the two domains interact and transfer information. The objective of this study is (i) develop a novel nodal position FE method that can couple with the MD easily, (ii) develop a proper methodology to couple the FE with MD for FE/MD multi-scale modeling and let the information transfer in a seamless manner between the two domains, and (iii) implement complicated cases to confirm accuracy and validity of the proposed model

The role of the patch test in 2D atomistic-to-continuum coupling methods

For a general class of atomistic-to-continuum coupling methods, coupling multi-body interatomic potentials with a P1-finite element discretisation of Cauchy--Born nonlinear elasticity, this paper adresses the question whether patch test consistency (or, absence of ghost forces) implies a first-order error estimate. In two dimensions it is shown that this is indeed true under the following additional technical assumptions: (i) an energy consistency condition, (ii) locality of the interface correction, (iii) volumetric scaling of the interface correction, and (iv) connectedness of the atomistic region. The extent to which these assumptions are necessary is discussed in detail.Comment: Version 2: correction of some minor mistakes, added discussion of multiple connected atomistic region, minor improvements of styl

Computational Multiscale Methods

Computational Multiscale Methods play an important role in many modern computer simulations in material sciences with different time scales and different scales in space. Besides various computational challenges, the meeting brought together various applications from many disciplines and scientists from various scientific communities