Localization of deformation and loss of macroscopic ellipticity in microstructured solids

d’Avila, Maria Paola Santisi; Triantafyllidis, Nicolas; Wen, Guilin

doi:10.1016/j.jmps.2016.07.009

Cited by 20 publications

(15 citation statements)

References 24 publications

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“…This material would be ideal not only to theoretically analyze instabilities, but also to practically realize the 'architected materials' which are preconized to yield extreme mechanical properties such as foldability, channeled response, and surface effects [1][2][3]. The crucial step towards the definition of a class of these materials was made by Triantafyllidis [4][5][6][7][8][9] and Ponte Castañeda [10][11][12][13][14][15][16][17][18], who laid down a general framework for the homogenization of elastic composites and for the analysis of bifurcation and strain localization in these materials. In particular, they showed how to realize an elastic material displaying a prestress-sensitive incremental response, exactly how it is postulated for nonlinear elastic solids subject to incremental deformation.…”

Section: Introductionmentioning

confidence: 99%

“…The results that will be presented also demonstrate how lattice models of heterogeneous materials can be highly effective to obtain analytical expressions for homogenized properties, thus allowing an efficient analysis of the influence of the Published in Journal of the Mechanics and Physics of Solids (2021), 146, 104198, DOI: doi.org/10.1016/j.jmps.2020.104198 microstructural parameters. This is a clear advantage over continuum formulations for composites, where analytical results can only be obtained for simple geometries and loading configurations (as for instance in the case of laminated solids [8,9,48]). Several new features are found, including a 'super-sensitivity' of the localization direction to the preload state and the conditions in which a perfect correspondence between the lattice and the continuum occurs (so that the discrete system and the equivalent solid share all the same bifurcation modes).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Bordiga

Cabras

Bigoni

2021

Journal of the Mechanics and Physics of Solids

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A lattice (or 'grillage') of elastic Rayleigh rods (possessing a distributed mass density, together with rotational inertia) organized in a parallelepiped geometry can be axially loaded up to an arbitrary amount without distortion and then be subject to incremental time-harmonic dynamic motion. At certain threshold levels of axial load, the grillage manifests instabilities and displays non-trivial axial and flexural incremental vibrations. Including every possible structural geometry and for an arbitrary amount of axial stretching, Floquet-Bloch wave asymptotics is used to homogenize the in-plane mechanical response, so to obtain an equivalent prestressed elastic solid subject to incremental time-harmonic vibration, which includes, as a particular case, the incremental quasi-static response. The equivalent elastic solid is obtained from its acoustic tensor, directly derived from homogenization and shown to be independent of the rods' rotational inertia. Loss of strong ellipticity in the equivalent continuum coincides with macrobifurcation in the lattice, while micro-bifurcation remains undetected in the continuum and corresponds to a vibration of vanishing frequency of the lowest dispersion branch of the lattice, occurring at finite wavelength. Dynamic homogenization reveals the structure of the acoustic branches close to ellipticity loss and the analysis of forced vibrations (both in physical space and Fourier space) shows low-frequency wave localizations. A perturbative approach based on dynamic Green's function is applied to both the lattice and its equivalent continuum. This shows that only macro-instability corresponds to localization of incremental strain, while micro-instabilities occur in modes which spread throughout the whole lattice with an 'explosive' character. In particular, extremely localized mechanical responses are found both in the lattice and in the solid, with the advantage that the former can be easily realized, for instance via 3D printing. In this way, features such as shear band inclination, or the emergence of a single shear band, or competition between micro and macro instabilities become all designable features. The comparison between the mechanics of the lattice and its equivalent solid shows that the homogenization technique allows an almost perfect representation, except when micro-bifurcation is the first manifestation of instability. Therefore, the presented results pave the way for the design of architected cellular materials to be used in applications where extreme deformations are involved.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Bordiga

Cabras

Bigoni

2021

Journal of the Mechanics and Physics of Solids

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“…Future work will focus on reducing this cost, as well as more thoroughly investigating the effects of microscopic buckling on macroscopic material stability. 70 How to cite this article: Alberdi R, Zhang G, Khandelwal K. A framework for implementation of RVE-based multiscale models in computational homogenization using isogeometric analysis. Int J Numer Methods Eng.…”

Section: Discussionmentioning

confidence: 99%

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

Alberdi

Zhang

Khandelwal

2018

Numerical Meth Engineering

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Summary This study presents an isogeometric framework for incorporating representative volume element–based multiscale models into computational homogenization. First‐order finite deformation homogenization theory is derived within the framework of the method of multiscale virtual power, and Lagrange multipliers are used to illustrate the effects of considering different kinematical constraints. Using a Lagrange multiplier approach in the numerical implementation of the discrete system naturally leads to a consolidated treatment of the commonly employed representative volume element boundary conditions. Implementation of finite deformation computational strain‐driven, stress‐driven, and mixed homogenization is detailed in the context of isogeometric analysis (IGA), and performance is compared to standard finite element analysis. As finite deformations are considered, a numerical multiscale stability analysis procedure is also detailed for use with IGA. Unique implementation aspects that arise when computational homogenization is performed using IGA are discussed, and the developed framework is applied to a complex curved microstructure representing an architectured material.

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“…A recent study have also shown that upon the loss of strict rank-one convexity there is not always a discontinuous/localized deformation pattern on the bifurcated branch [51]. The presence or absence of the localized deformation depends on the stability of the bifurcated branch [51]. When there is a discontinuous deformation corresponds to the loss of strict rank-one convexity, i.e.…”

Section: Macroscale Stabilitymentioning

confidence: 98%

Computational design of finite strain auxetic metamaterials via topology optimization and nonlinear homogenization

Zhang

Khandelwal

2019

Computer Methods in Applied Mechanics and Engineering

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A novel computational framework for designing metamaterials with negative Poisson's ratio over a large strain range is presented in this work by combining the density-based topology optimization together with a mixed stress/deformation driven nonlinear homogenization method. A measure of Poisson's ratio based on the macro deformations is proposed, which is further validated through direct numerical simulations. With the consistent optimization formulations based on nonlinear homogenization, auxetic metamaterial designs with respect to different loading orientations and with different unit cell domains are systematically explored. In addition, the extension to multimaterial auxetic metamaterial designs is also considered, and stable optimization formulations are presented to obtain discrete metamaterial topologies under finite strains. Various new auxetic designs are obtained based on the proposed framework. To validate the performance of optimized designs, a multiscale stability analysis is carried out using the Bloch wave method and rank-one convexity check. As demonstrated, short and/or long wavelength instabilities can occur during the loading process, leading to a change of periodicity of the microstructure, which can affect the performance of an optimized design.

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Localization of deformation and loss of macroscopic ellipticity in microstructured solids

Cited by 20 publications

References 24 publications

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

Dynamics of prestressed elastic lattices: Homogenization, instabilities, and strain localization

A framework for implementation of RVE‐based multiscale models in computational homogenization using isogeometric analysis

Computational design of finite strain auxetic metamaterials via topology optimization and nonlinear homogenization

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