An Analysis of the Quasicontinuum Method

The aim of this paper is to present a streamlined and fully three-dimensional version of the quasicontinuum (QC) theory of Tadmor et al. and to analyze its accuracy and convergence characteristics. Specifically, we assess the effect of the summation rules on accuracy; we determine the rate of convergence of the method in the presence of strong singularities, such as point loads; and we assess the effect of the refinement tolerance, which controls the rate at which new nodes are inserted in the model, on the development of dislocation microstructures.Comment: 30 pages, 16 figures. To appear in Jornal of the Mechanics and Physics of Solid

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

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

Multi-scale simulation of the nano-metric cutting process

Molecular dynamics (MD) simulation and the finite element (FE) method are two popular numerical techniques for the simulation of machining processes. The two methods have their own strengths and limitations. MD simulation can cover the phenomena occurring at nano-metric scale but is limited by the computational cost and capacity, whilst the FE method is suitable for modelling meso- to macro-scale machining and for simulating macro-parameters, such as the temperature in a cutting zone, the stress/strain distribution and cutting forces, etc. With the successful application of multi-scale simulations in many research fields, the application of simulation to the machining processes is emerging, particularly in relation to machined surface generation and integrity formation, i.e. the machined surface roughness, residual stress, micro-hardness, microstructure and fatigue. Based on the quasi-continuum (QC) method, the multi-scale simulation of nano-metric cutting has been proposed. Cutting simulations are performed on single-crystal aluminium to investigate the chip formation, generation and propagation of the material dislocation during the cutting process. In addition, the effect of the tool rake angle on the cutting force and internal stress under the workpiece surface is investigated: The cutting force and internal stress in the workpiece material decrease with the increase of the rake angle. Finally, to ease multi-scale modelling and its simulation steps and to increase their speed, a computationally efficient MATLAB-based programme has been developed, which facilitates the geometrical modelling of cutting, the simulation conditions, the implementation of simulation and the analysis of results within a unified integrated virtual-simulation environment

An analysis of the field theoretic approach to the quasi-continuum method

Using the orbital-free density functional theory as a model theory, we present an analysis of the field theoretic approach to quasi-continuum method. In particular, by perturbation method and multiple scale analysis, we provide a formal justification for the validity of the coarse-graining of various fields, which is central to the quasi-continuum reduction of field theories. Further, we derive the homogenized equations that govern the behavior of electronic fields in regions of smooth deformations. Using Fourier analysis, we determine the far-field solutions for these fields in the presence of local defects, and subsequently estimate cell-size effects in computed defect energies.Comment: 26 pages, 1 figur

MULTIRESOLUTION MOLECULAR MECHANICS: THEORY AND APPLICATIONS

A general mathematical framework, Multiresolution Molecular Mechanics (MMM), is proposed to consistently coarse-grain molecular mechanics at different resolutions in order to extend the length scale of nanoscale modeling of crystalline materials. MMM is consistent with molecular mechanics in the sense that the constitutive description such as energy and force calculations is exactly the same as molecular mechanics and no empirical and phenomenological constitutive relationships in continuum mechanics are employed. As such, MMM can converge to full molecular mechanics naturally. As many coarse-graining approaches, MMM is based on approximating the total potential energy of a full atomistic model. Analogous to quadrature rules employed to evaluate energy integrals in finite element method (FEM), a summation rule is required to evaluate finite energy summations. Most existing summation rules are specifically designed for the linear interpolation shape function and their extensions to high order shape functions are currently not clear. What distinguishes MMM from existing works is that MMM proposes a novel summation rule framework SRMMM that is valid and consistent for general shape functions. The key idea is to analytically derive the energy distribution of the coarse-grained atomistic model and then choose some quadrature-type (sampling) atoms to accurately represent the derived energy distribution for a given shape function. The optimal number, weight and position of sampling atoms are also determined accordingly, similar to the Gauss quadrature in FEM. The governing equations are then derived following the variational principle. The proposed SRMMM is verified and validated numerically and compared against many other summation rules such as Gauss-quadrature-like rule. And SRMMM demonstrates better performance in terms of accuracy and computational cost. The convergence property of MMM is also studied numerically and MMM shows FEM-like behavior under certain circumstance. In addition, MMM has been applied to solve problems such as crack propagation, atomic sheet shear, beam bending and surface relaxations by employing high order interpolation shape functions in one (1D), two (2D) and three dimensions (3D)