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

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

Improving the initial guess for the Newton-Raphson protocol in time-dependent simulations

A general linearisation procedure for the consistent tangent of a small-strain visco-plastic material model is presented in this note. The procedure is based on multi-variable linearisation around a so-called 'reference state'. In particular, the linerarisation of the time integration scheme is found to yield an extra term compared to classical expressions, which only appears because the material response is time-dependent. It has the effect of yielding a very accurate initial guess for the Newton-Raphson protocol based on the ongoing viscous flow. It is shown, using a modern variational FFT-based solver, that the extra term reduces both the CPU time and the number of Newton-Raphson iterations by around a factor two.Comment: Journal of Computational Physics, 202

LOCALIZATION ANALYSIS OF DAMAGE FOR ONE-DIMENSIONAL PERIDYNAMIC MODEL

Peridynamics is a recently developed extension of continuum mechanics, which replaces the traditional concept of stress by force interactions between material points at a finite distance. The peridynamic continuum is thus intrinsically nonlocal. In this contribution, a bond-based peridynamic model with elastic-brittle interactions is considered and the critical strain is defined for each bond as a function of its length. Various forms of length functions are employed to achieve a variety of macroscopic responses. A detailed study of three different localization mechanisms is performed for a one-dimensional periodic unit cell. Furthermore, a convergence study of the adopted finite element discretization of the peridynamic model is provided and an effective event-driven numerical algorithm is described

MOLECULAR STATICS SIMULATION OF NANOINDENTATION USING ADAPTIVE QUASICONTINUUM METHOD

In this work, molecular statics is used to model a nanoindentation test on a two-dimensional hexagonal lattice. To this end, the QuasiContinuum (QC) method with adaptive propagation of the fully resolved domain is used to reduce the computational cost required by the full atomistic model. Three different adaptive mesh refinement criteria are introduced and tested, based on: (i) the Zienkiewicz–Zhu criterion (used for the deformation gradient), (ii) local atoms’ site energy, and (iii) local lattice disregistry. Accuracy and efficiency of individual refinement schemes are compared against the full atomistic model and obtained results are discussed

Sampling point selection for energy estimation in the quasicontinuum method

The quasicontinuum (QC) method reduces computational costs of atomistic calculations by using interpolation between a small number of so-called repatoms to represent the displacements of the complete lattice and by selecting a small number of sampling atoms to estimate the total potential energy of the interpolated problem. In this contribution two new sampling point selections are introduced for the QC method. The first selection determines the total potential energy of the lattice exactly in correspondence with the interpolation. Since no error due to summation occurs, the fully resolved regions around lattice defects can remain small. However, in this case many sampling atoms must be used. Therefore a second sampling point selection is derived from the first selection that uses only one sampling atom to represent all atoms within interpolation together with the repatoms. This ensures that the exact lattice model is recovered in the fully resolved regions while a smooth transition is achieved towards coarse regions in which the method becomes very close to the local QC method