A stress-based formulation of the free material design problem with the trace constraint and multiple load conditions

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Using both mathematical and numerical methods, the optimal distributions of material characterized by the Young modulus and Poisson ratio (as well as other moduli of isotropy) maximizing the overall stiffness of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support are constructed. The overall stiffness of the body is defined as the inverse of the work of external forces on displacements, called here the compliance of the structure. The isoperimetric condition bounds the integral of the trace of the Hooke tensor. It is proved that isotropic composite materials forming the bodies of extremely high stiffness exhibit negative Poisson ratio in large subdomains, which points at the significance of the auxetic material in modern structural design. The obtained results show that the whole range of possible variation of the Poisson ratio is used, from −1 to 1/2, which proves usefulness of the auxetic materials.

“…The optimal Kelvin modulus

λ_{1}^{*} (x)

defining the optimal, non‐homogeneous, anisotropic Hooke tensor (see Refs. ) is shown in Fig. b.…”

Section: Case Studiesmentioning

confidence: 90%

Section: Formulation Of An Optimal Designmentioning

confidence: 99%

Section: Formulation Of An Optimal Designmentioning

confidence: 99%

“…The optimal modulus

λ_{1}^{*} = λ_{1}^{*} true(x true)

x \in Ω

defines the optimal, non‐homogeneous, anisotropic Hooke tensor

C^{*}

because all remaining optimal Kelvin moduli

λ_{2}^{*}, λ_{3}^{*}, λ_{4}^{*}, λ_{5}^{*}, λ_{6}^{*}

λ_{1}^{*}

have been added mainly in order to compare FMD solutions with the solutions obtained by the IMD method.…”

Section: Case Studiesmentioning

confidence: 99%

“…The optimal plots of Kelvin modulus

λ_{1}^{*} true(x true)

defining the optimal, non‐homogeneous, anisotropic Hooke tensor (see Refs. ), optimal bulk, and shear moduli

k^{*} true(x true)

μ^{*} true(x true)

, as well as three fields

H_{1111}^{*} true(x true)

H_{1122}^{*} true(x true)

H_{2323}^{*} true(x true)

x \in Ω

of the optimal, non‐homogeneous, isotropic Hooke tensor are shown in Fig. b where the “bottom,” left visible side is clamped.…”

Section: Case Studiesmentioning

confidence: 99%

See 3 more Smart Citations

The emergence of auxetic material as a result of optimal isotropic design

Czarnecki

Wawruch

2015

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

Czarnecki

Lewiński

2017

The present paper concerns two methods: IMD (isotropic material design) and YMD (Young modulus material design) concerning Pareto optimal distributions of the material and its isotropic properties. The merit function is the weighted sum of compliances corresponding to n independent loading conditions. The unit cost of the design is assumed as equal to the trace of the elastic moduli tensor C. In the IMD method, the design variables are the bulk and shear moduli. In the YMD method, the Poisson ratio is assumed as given, possibly non‐uniformly distributed in the design domain, the Young modulus being the only design variable. Both the methods reduce to the auxiliary minimization problem involving statically admissible stress fields corresponding to the subsequent loading conditions. The integrand is expressed by a norm of the collection of these stresses, hence has linear growth. The numerical method proposed refers to the primal formulation: it is based on piece‐wise polynomial approximation of the set of statically admissible stresses. The exemplary Pareto optimal solutions show that admitting negative values of Poisson's ratio may contribute to a decrease of the total compliances.

Auxetic Properties of the Stiffest Elastic Bodies as a Result of Topology Optimization and Microstructures Recovery Based on Homogenization Method

Czarnecki,

Łukasiak

2024

Herein, several semianalytical formulae for the optimal distribution of elastic moduli in 2D or 3D domains occupied by nonhomogeneous, least compliant bodies are presented and three different methods for recovering microstructures predicted by the isotropic material design method are proposed, which is a stress‐based variant of topology optimization for isotropic bodies. In all five presented variants of the topology optimization of elastic bodies, the numerical implementation of these formulae requires only the use of any algorithm of searching for the minimum of functional in multidimensions without any constraints, even box constraints on design parameters. For each variant considered, the formulae defining this functional are defined analytically. All results concern the design of composite structures that maximize their stiffness (equivalently minimize their compliance) while satisfying a certain isoperimetric condition introducing the upper limit of the available material resource. The article pays special attention to the fact that the stiffest, heterogeneous isotropic elastic material has almost always auxetic properties. Based on the asymptotic homogenization method and adopted bending‐free lattice model, an algorithm for the numerical recovering of nonhomogeneous isotropic microstructure in the entire domain occupied by the plate is proposed, with a fairly easy‐to‐implement ability to model an auxetic behavior.