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

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

The skeletons of free distributive lattices

AbstractThe skeletons of free distributive lattices are studied by methods of formal concept analysis; in particular, a specific closure system of sublattices is elaborated to clarify the structure of the skeletons. Up to five generators, the skeletons are completely described

On the efficient use of lattice models

No abstract

Fatigue phase-field damage modeling of rubber using viscous dissipation: Crack nucleation and propagation

By regularizing sharp cracks within a pure continuum setting, phase-damage models offer the ability to capture crack nucleation as well as crack propagation. Crack branching and coalescence can furthermore be described without any additional efforts, as geometrical descriptions of the cracks are not required. In this contribution, we extend our previous phase-field model for rate-dependent fracture of rubbers in a finite strain setting (Loew et al., 2019) to describe damage under cyclic loading. The model is derived from the balance of mechanical energy and introduces a fatigue damage source as a function of the accumulated viscous dissipation under cyclic loading. We use uniaxial cyclic tension to present the influence of the fatigue material parameters and to confirm the model’s energy balance. The parameters are subsequently identified using monotonic and cyclic experiments of a plane stress nature. Finally, the model is validated by separate experiments, which demonstrate that the model accurately predicts (fatigue) crack nucleation as well as propagation

Fusing the Seth-Hill strain tensors to fit compressible elastic material responses in the nonlinear regime

Strain energy densities based on the Seth-Hill strain tensors are often used to describe the hyperelastic mechanical behaviours of isotropic, transversely isotropic and orthotropic materials for relatively large deformations. Since one parameter distinguishes which strain tensor of the Seth-Hill family is used, one has in theory the possibility to t the material response in the nonlinear regime. Most often for compressible deformations however, this parameter is selected such that the Hencky strain tensor is recovered, because it yields rather physical stress-strain responses. Hence, the response in the nonlinear regime is in practise not often tailored to match experimental data. To ensure that elastic responses in the nonlinear regime can more accurately be controlled, this contribution proposes three generalisations that combine several Seth-Hill strain tensors. The generalisations are formulated such that the stress-strain responses for in finitesimal deformations remain unchanged. Consequently, the identifi cation of the Young's moduli, Poisson's ratios and shear moduli is not a ffected. 3D fi nite element simulations are performed for isotropy and orthotropy, with an emphasis on the identifi cation of the new material parameters

Maximally Concentrated Sequences after Half-sample Shifts

It is well known that index (discrete-time)-limited sampled sequences leak outside the support set when a band-limiting operation is applied. Similarly, a fractional shift causes an index-limited sequence to be infinite in extent due to the inherent band-limiting. Index-limited versions of discrete prolate spheroidal sequences (DPSS) are known to experience minimum leakage after band-limiting. In this work, we consider the effect of a half-sample shift and provide upper bounds on the resulting leakage energy for arbitrary sequences. Furthermore, we find an orthonormal basis derived from DPSS with members ordered according to energy concentration after half sample shifts; the primary (first) member being the global optimum