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Isotropic Material Design PDF Print E-mail
Isotropic Material Design
S. Czarnecki
cmstComputational Methods in Science and Technology, CMST 21(2) 49-64 (2015)
DOI: 10.12921/cmst.2015.21.02.001
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Abstract:
The paper deals with optimal distribution of the bulk and shear moduli minimizing the compliance of an inhomogeneous isotropic elastic 3D body transmitting a given surface loading to a given support. The isoperimetric condition is expressed by the integral of the trace of the Hooke tensor being a linear combination of both moduli. The problem thus formulated is reduced to an auxiliary 3D problem of minimization of a certain stress functional over the stresses being statically admissible. The integrand of the auxiliary functional is a linear combination of the absolute value of the trace and norm of the deviator of the stress field. Thus the integrand is of linear growth. The auxiliary problem is solved numerically by introducing element-wise polynomial approximations of the components of the trial stress fields and imposing satisfaction of the variational equilibrium equations. The under-determinate system of these equations is solved numerically thus reducing the auxiliary problem to an unconstrained problem of nonlinear programming.
 
The emergence of auxetic material as a result of optimal isotropic design PDF Print E-mail
The emergence of auxetic material as a result of optimal isotropic design
S. Czarnecki, P. Wawruch
czar wawPhys. Status Solidi B, 1–11 (2015) / DOI 10.1002/pssb.201451733
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Abstract:
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.


 
Recovery of two-phase microstructures of planar isotropic elastic composites PDF Print E-mail
10th World Congress on Structural and Multidisciplinary Optimization
May 19-24,2013, Orlando,Florida, USA

Recovery of two-phase microstructures of planar isotropic elastic composites
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Abstract
The isotropic elastic mixtures composed of two isotropic materials of the bulk moduli and shear moduli are characterized by the effective bulk and shear moduli. In the planar problems the theoretically
admissible pairs, for given volume fraction ρ 0 of material, lie within a rectangular domain of vertices determined by the Hashin-Shtrikman bounds. The tightest bounds in 2D known up till now are due to
Cherkaev and Gibiansky (CG). The microstructures corresponding to the interior of the CG area can be of arbitrary rank, in the meaning of the hierarchical homogenization. In the present paper a family of composites is constructed of the underlying microstructures of rank 1. The consideration is confined to the microstructures possessing rotational symmetry of angle 1200. To find the effective moduli the homogenization method is used: the local basic cell problems are set on a cell Y of the shape of a hexagonal domain. The periodicity conditions refer to the opposite sides of Y. Such a non-conventional basic cell choice generates automatically the family of isotropic mixtures. The subsequent points are found by solving the inverse homogenization problems with the isoperimetric condition expressing the amounts of both the materials within the cell. The isotropy conditions, usually explicitly introduced into the inverse homogenization formulation, do not appear, as being fulfilled by the microstructure construction. The method put forward makes it possible to localize each admissible pair by appropriate choice of the layout of both the constituents within the repetitive sub-domain of Y.
Keywords: isotropic composites, inverse homogenization, recovery of microstructures.
 
Theoretical Foundations of Civil Engineering, Polish-Ukrainian Transactions, 2014 PDF Print E-mail
tfoce 2014Uderzenie punktu materialnego w belkę, Jemielita G., Kozyra Z., Theoretical Foundations of Civil Engineering, Polish-Ukrainian Transactions. Ed. by W. Szcześniak, pp. 51-58, Warsaw 2014.
Streszczenie: W pracy wyznaczono prędkość masy i prędkość punktów materialnych belki w trakcie zderzenia.  Wyznaczono też przemieszczenia dynamiczne belki przy zderzeniu elastycznym i nieelastycznym w zależności od przemieszczenia statycznego.

Izotropowe kompozyty periodyczne - homogenizacja struktur I rzedu, Łukasiak T., Theoretical Foundations of Civil Engineering, Polish-Ukrainian Transactions. Ed. by W. Szcześniak, pp. 77-84, Warsaw 2014.
Streszczenie:  Materiały kompozytowe w 2D charakteryzuja się z reguły ortotropowymi cechami. Interesującymi, nie tylko z teoretycznego punktu widzenia, są takie kompozyty których makro-właściwości są izotropowe np. stal posiadająca wewnętrzną mikro-strukturę traktowana jest makroskopowo jako ciało izotropowe. W pracy przedstawiono homogenizację kompozytów wykorzystując metodę elementów skończonych.

Parametric Modeling of Space Frame Structures, Kacprzyk Z., Ostapska–Łuczkowska K., Theoretical Foundations of Civil Engineering, Polish-Ukrainian Transactions. Ed. by W. Szcześniak, pp. 167-172, Warsaw 2014.
Abstract: The subject of this paper is a study of different mathematical models of curves and surfaces applied in CAD systems with an aim to identify best modelling approach for a certain design problems. The study is supported by practical examples performed with the use of commercial software. Examples show how modelling approach differs or should differ for different space frames types. For better model efficiency, irrelevant elements are neglected in certain models.
 
Two-phase isotropic composites with prescribed bulk and shear moduli PDF Print E-mail
Two-phase isotropic composites with prescribed bulk and shear moduli.
T. Łukasiak
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ABSTRACT: The paper deals with the inverse homogenization problem: to reconstruct the layout of two elastic and isotropic materials given by bulk and shear  moduli within a hexagonal (2D) pe-
riodicity cell, corresponding to the predefined values of the bulk and shear moduli, of the effective isotropic composite and to the given isoperimetric condition concerning the volume fractions. The effective
isotropic moduli are computed according to the homogenization algorithm, with using appropriate Finite Elements (FE) techniques along with periodicity assumptions. The inverse problem thus formulated can be ef-
fectively solved numerically by the Sequential Linear Programming (SLP) method. The isotropy conditions, usually explicitly introduced into the inverse homogenization formulation, do not appear in the algorithm, as
being fulfilled by the microstructure construconstruction. The rotational symmetry of angle 120o of the resulting representative volume element is assumed.
 
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