Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

Czarnecki, S.; Lewiński, Tomasz

doi:10.1002/pssb.201600821

Cited by 17 publications

(14 citation statements)

References 44 publications

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“…A new challenge is to develop effective methods for recovering the microstructures based on the results of the vector topology optimization, due to the need to consider (usually very many) Pareto optimal solutions-see ref. [78].…”

Section: Resultsmentioning

confidence: 99%

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Czarnecki

Łukasiak

2020

Physica Status Solidi (b)

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The article is focused on recovering of two-phase microstructures of effective properties for 2D elasticity (see e.g., ref. [1]) predicted by two, stress-based topology optimization methods: isotropic material design (IMD) and cubic material design (CMD)-see e.g., refs. [2-5]. In these articles, the reconstruction of microstructures in subdomains where the auxetic properties [6-13] of the optimal, nonhomogeneous isotropic or cubic elastic material appear has been undertaken. In the present article, being an extension of the previous work, [14] the homogenization method based on parametric description of the shape of the domain filled by the given isotropic material inside the representative volume elements (RVEs), instead of the combinatorial search among the admissible microstructures, is implemented. Within the 2D setting discussed, the local cubic symmetry of the material reduces to the local symmetry of a square. The main goal is to recover isotropic microstructures or microstructures of symmetry of a square at arbitrary points of the body in a relatively simple way, considering the heterogeneity of the distribution of optimal elastic moduli. In the case of an isotropic material, the results obtained in ref. [14] are used as a starting point to the further search of the auxetic microstructures, realizing the Cherkaev-Gibiansky bounds. [15] In the case of 2D models of materials showing auxetic properties, a series of works devoted to their research at the level of regular atomic structures (e.g., hard cyclic hexamers [HCHs] or hard cyclic tetramers and many others) were particularly important in the development of the theory of such materials-see refs. [16-24]. Especially important in this respect is the ref. [16], in which a constant thermodynamic tension Monte Carlo (MC) simulations revealed the first planar, isotropic (and chiral) thermodynamically stable structure of negative Poisson's ratio in tilted phase formed by HCHs. In ref. [17], the exact solution for negative Poisson's ratio at high densities and in the static limit (i.e., at zero temperature) was obtained for planar, isotropic (and chiral) structure of cyclic hexamers interacting through nearest-neighbouring site-site inversepower potential. In ref. [18], MC simulations performed on hard, cyclic tetramers revealed an interesting feature of such a very simple model of molecules composed of four identical hard disks: depending on its shape, all possible material behavioursregarding negative Poisson's ratio could arise and thermodynamically stable auxetic (or partially auxetic) chiral structures of square symmetry could be spontaneously formed. Analyzing in ref. [19], other 2D cyclic trimers model of triatomic molecules in which "atoms" are distributed on vertices of equilateral triangles, the authors have found the exact solution at zero temperature for the planar auxetic structure which is isotropic but not chiral (soft cyclic trimers). The both chiral and nonchiral molecular models were studied in ref. [20], where the influence of molecular geome...

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Section: Resultsmentioning

confidence: 99%

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Czarnecki

Łukasiak

2020

Physica Status Solidi (b)

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“…The recovery and approximation of the Pareto optimal microstructures appear independently in the case of vector optimization, e.g., in problem of minimizing the compliances of the elastic body for the multiple loading conditions, see ref. .…”

Section: Final Remarksmentioning

confidence: 99%

An Explicit Construction of the Underlying Laminated Microstructure of the Least Compliant Elastic Bodies

Czarnecki

2018

Physica Status Solidi (b)

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The paper discusses the problem of recovery of the microstructure of the least compliant bodies of non-homogeneous optimal isotropic properties predicted by the Isotropic Material Design method. The three-dimensional microstructure is assumed as constructed by a subsequent lamination process in which two isotropic materials of ordered properties are used, the process being repeated three or six times, subsequently. The Hooke tensors closest to the optimal ones are found by making use of the analytical Francfort-Murat formulae for effective moduli of third and sixth rank sequential laminates. The ability to model an auxetic behavior within the subdomains where the optimal Poisson's ratio assumes negative values is also shown.

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“…Deformation of the optimal least compliant elastic bodies made of isotropic or cubic material of spatially varying properties turns out to be point-wise bounded if the unit cost of the designs is assumed as proportional to the trace of the Hooke tensor (see [ 50 , 67 ]). Consequently, the magnitudes of the optimal moduli follow the values of the stress characteristics.…”

Section: Final Remarksmentioning

confidence: 99%

The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies

2017

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The paper discusses the problem of manufacturability of the minimum compliance designs of the structural elements made of two kinds of inhomogeneous materials: the isotropic and cubic. In both the cases the unit cost of the design is assumed as equal to the trace of the Hooke tensor. The Isotropic Material Design (IMD) delivers the optimal distribution of the bulk and shear moduli within the design domain. The Cubic Material Design (CMD) leads to the optimal material orientation and optimal distribution of the invariant moduli in the body made of the material of cubic symmetry. The present paper proves that the varying underlying microstructures (i.e., the representative volume elements (RVE) constructed of one or two isotropic materials) corresponding to the optimal designs constructed by IMD and CMD methods can be recovered by matching the values of the optimal moduli with the values of the effective moduli of the RVE computed by the theory of homogenization. The CMD method leads to a larger set of results, i.e., the set of pairs of optimal moduli. Moreover, special attention is focused on proper recovery of the microstructures in the auxetic sub-domains of the optimal designs.

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Pareto Optimal Design of Non‐Homogeneous Isotropic Material Properties for the Multiple Loading Conditions

Cited by 17 publications

References 44 publications

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

Recovery of the Auxetic Microstructures Appearing in the Least Compliant Continuum Two‐Dimensional Bodies

An Explicit Construction of the Underlying Laminated Microstructure of the Least Compliant Elastic Bodies

The Isotropic and Cubic Material Designs. Recovery of the Underlying Microstructures Appearing in the Least Compliant Continuum Bodies

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