Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

Beex, Lars; Rokoš, Ondřej; Zeman, Jan; Bordas, Stéphane

doi:10.1002/gamm.201510018

Cited by 19 publications

(18 citation statements)

References 39 publications

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“…Consequently, the summation error outweighs the interpolation error. These two reasons and the example presented in Beex et al (2015b), Sections 4.2 and 5.2 motivate the following test, in which the deformation field due to bending is linear rather than constant and more pronounced interpolation errors are expected.…”

Section: Uniaxial Loading Testmentioning

confidence: 94%

“…Instead of refining the triangulation in the coarse part of the domain, an alternative approach for decreasing the interpolation error would be to use higher-order approximations, as shown e.g. in Beex et al (2015b). Furthermore, the total error due to interpolation and summation (Fig.…”

Section: Pure Bending Testmentioning

confidence: 99%

“…Before closing this contribution by a summary and conclusions in Section 6, we perform numerical tests on three benchmark examples presented in Section 5, two of which have been adopted from Beex et al (2014c), Section 4, and Beex et al (2015b), Section 4.2. We demonstrate that both approaches, represented by Eqs.…”

Section: Introductionmentioning

confidence: 99%

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A variational formulation of dissipative quasicontinuum methods

Rokoš

Beex

Zeman

et al. 2016

International Journal of Solids and Structures

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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 Roubíček, 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.

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Section: Uniaxial Loading Testmentioning

confidence: 94%

Section: Pure Bending Testmentioning

confidence: 99%

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A variational formulation of dissipative quasicontinuum methods

Rokoš

Beex

Zeman

et al. 2016

International Journal of Solids and Structures

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“…Note that higher-order polynomial approximations inside elements can be adopted as well, see e.g. Beex et al (2014a), Yang and To (2015), or Beex et al (2015b).…”

Section: General Frameworkmentioning

confidence: 99%

eXtended variational quasicontinuum methodology for lattice networks with damage and crack propagation

Rokoš

Peerlings

Zeman

2017

Computer Methods in Applied Mechanics and Engineering

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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.

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“…Paper materials (wood fibres [1,2,3,4,5]), fabrics (yarns [6,7,8,9,10]), and metal foams (struts [11,12,13,14,15]) are examples of materials with slender components in their microstructure. Micromechanical models of such materials often represent each slender constituent as a beam, which yields a string of beam finite elements when discretized [16] (or springs [17,18]). In most cases, contact between the slender constituents is essential to be incorporated.…”

Section: Introductionmentioning

confidence: 99%

Contact between shear-deformable beams with elliptical cross sections

2019

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Slender constituents are present in many structures and materials. In associated mechanical models, each slender constituent is often described with a beam. Contact between beams is essential to incorporate in mechanical models, but associated contact frameworks are only demonstrated to work for beams with circular cross-sections. Only two studies have shown the ability to treat contact between beams with elliptical cross-sections, but those frameworks are limited to point-wise contact, which narrows their applicability. This contribution presents initial results of a framework for shear-deformable beams with elliptical cross-sections if contact occurs along a line or at an area (instead of at a point). This is achieved by integrating a penalty potential over one of the beams' surfaces. Simo-Reissner Geometrically Exact Beam (GEB) elements are employed to discretise each beam. As the surface of an assembly of such beam elements is discontinuous, a smoothed surface is introduced to formulate the contact kinematics. This enables the treatment of contact for large sliding displacements and substantial deformations.

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Higher‐order quasicontinuum methods for elastic and dissipative lattice models: uniaxial deformation and pure bending

Cited by 19 publications

References 39 publications

A variational formulation of dissipative quasicontinuum methods

A variational formulation of dissipative quasicontinuum methods

eXtended variational quasicontinuum methodology for lattice networks with damage and crack propagation

Contact between shear-deformable beams with elliptical cross sections

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