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1

On Cherkaev–Lurie–Milton theorem in the plane problems of linear elasticity

Lewiński T.

Archives of Mechanics

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2022

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Vol. 74, nr 4

319--339

EN

The paper delivers a full justification of the Cherkaev–Lurie–Milton theorem in application to the elasticity problem of in-plane loaded plates, 2D periodic elastic composites, elasticity of thin plates subjected to transverse loads as well as in-plane periodic thin plates in bending. The theorem is treated as natural extension of Michell’s result on 2D elasticity and the Gauss–Bonnet formula applied to the deflection surface of a thin plate subject to bending.

2

Optimum design of elastic moduli for the multiple load problems

Lewiński T.

Archives of Mechanics

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2021

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Vol. 73, nr 1

27--66

EN

The paper deals with minimization of the weighted average of compliances of structures, made of an elastic material of spatially varying elasticity moduli, subjected to n load variants acting non-simultaneously. The trace of the Hooke tensor is assumed as the unit cost of the design. Three versions of the free material design are discussed: designing the moduli of arbitrary anisotropy (AMD), designing the moduli of an isotropic material (IMD), designing of Young’s modulus for a fixed Poisson ratio (YMD). The problem is in all cases reduced to the Linear Constrained Problem (LCP) of Bouchitté and Fragalà consisting of two mutually dual problems: stress based and strain based, the former one being characterized by the integrand of linear growth depending on the trial statically admissible stresses. The paper shows equivalence of the stress fields solving the (LCP) problem and those appearing in the optimal body subjected to subsequent load cases.

3

Zagadnienia kształtowania konstrukcji w ujęciu współczesnej mechaniki

Bołbotowski K., Czarnecki S., Czubacki R., Dzierżanowski G., Lewiński T., Łukasiak T., Sokół T.

Inżynieria i Budownictwo

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2017

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R. 73, nr 8

437--441

PL

Przedstawiono współczesne kierunki rozwoju teorii optymalizacji topologii konstrukcyjnej dźwigarów płaskich, przestrzennych oraz rusztów. Omówiono metody konstrukcji bazowych prowadzące do kratownicowych lub belkowych aproksymacji dźwigarów Michella, metodę strut and tie w wersji wspomaganej przez metodę konstrukcji bazowych, metodę interpolacji materiałowej oraz metodę projektowania materiału. Każda z tych metod łączy w sobie optymalizację kształtu i materiału.

EN

The paper concerns the contemporary trends of the theory of optimization of structural topology of planar and spatial girders as well as grillages. The following methods discussed are the ground structure method leading to truss-like or beam-like approximations of Michell structures, the strut and tie method supported by the ground structure method, the method of material interpolation schemes and the free material design method. Each of these methods combines the shape design and the material design.

4

Simultaneous design of optimal shape and local cubic material characteristics

Czubacki R., Lewiński T., Ganghoffer J. F.

Engineering Transactions

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2017

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Vol. 65, iss. 1

11--17

EN

This paper deals with the minimum compliance problem of the femur bone made of a nonhomogeneous elastic material with cubic symmetry. The elastic moduli as well as the trajectories of anisotropy directions are design variables. The isoperimetric condition determines the value of the cost of the design expressed as the integral of the trace of the Hooke’s tensor. The optimum design is found for a selected design domain and a single load case. The optimal cubic material characteristics are reflected by the properties of the underlying microstructure. Admissible microstructures are reconstructed, thus delivering a deeper insight into the optimum design. The obtained microstructures are second- rank laminates composed of an isotropic material and voids. To eliminate the degeneracy of the design at least three load cases should be considered.

5

On shape and material optimization of isotropic bodies

Czarnecki S., Lewiński T., Wawruch P.

Engineering Transactions

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2017

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Vol. 65, iss. 1

3--9

EN

This paper deals with the free material design and its two constrained versions constructed by imposing isotropy with (i) independent bulk and shear moduli, and (ii) fixed Poisson’s ratio. In the latter case, the Young modulus is the only design variable. The moduli are viewed as non-negative, thus allowing for the appearance of void domains within the design domain. The paper shows that all these methods reduce to one stress-based problem in which the norm involved reflects the type of the constraints imposed.

6

Solution of the three forces problem in the case of two forces being mutually orthogonal

Sokół T., Lewiński T.

Engineering Transactions

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2016

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Vol. 64, iss. 4

485--491

EN

The present paper delivers some new numerical and exact solutions of the three forces problem that is one of fundamental problems of Michell’s truss theory. The present problem is to find the lightest fully stressed truss transmitting three self-equilibrated co-planar forces. In this study, we limit our considerations to the case of two forces being mutually orthogonal. The aim of this paper is to classify possible layouts of optimal trusses depending on the position of the applied lateral point load (the positions of the other two forces are fixed, which however does not restrict the scope of the study). The exact analytical solutions are obtained with a great help of numerical solutions that enable proper prediction of the optimal layouts.

7

Topology optimization in structural mechanics

Lewiński T., Czarnecki S., Dzierżanowski G., Sokół T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2013

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Vol. 61, nr 1

23-37

EN

Optimization of structural topology, called briefly: topology optimization, is a relatively new branch of structural optimization. Its aim is to create optimal structures, instead of correcting the dimensions or changing the shapes of initial designs. For being able to create the structure, one should have a possibility to handle the members of zero stiffness or admit the material of singular constitutive properties, i.e. void. In the present paper, four fundamental problems of topology optimization are discussed: Michell’s structures, two-material layout problem in light of the relaxation by homogenization theory, optimal shape design and the free material design. Their features are disclosed by presenting results for selected problems concerning the same feasible domain, boundary conditions and applied loading. This discussion provides a short introduction into current topics of topology optimization.

8

A stress-based formulation of the free material design problem with the trace constraint and single loading condition

Czarnecki S., Lewiński T.

Bulletin of the Polish Academy of Sciences. Technical Sciences

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2012

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Vol. 60, nr 2

191-204

EN

The problem to find an optimal distribution of elastic moduli within a given plane domain to make the compliance minimal under the condition of a prescribed value of the integral of the trace of the elastic moduli tensor is called the free material design with the trace constraint. The present paper shows that this problem can be reduced to a new problem of minimization of the integral of the stress tensor norm over stresses being statically admissible. The eigenstates and Kelvin’s moduli of the optimal Hooke tensor are determined by the stress state being the minimizer of this problem. This new problem can be directly treated numerically by using the Singular Value Decomposition (SVD) method to represent the statically admissible stress fields, along with any unconstrained optimization tool, e.g.: Conjugate Gradient (CG) or Variable Metric (VM) method in multidimensions.

9

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part I. Solving the local optimization problem

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 1

21-21

EN

The paper deals with compliance minimization of a transversely homogeneous plate, subjected to the in-plane and transverse loadings acting simultaneously. The set of design variables includes the eigenstates of Hooke’s tensor whose eigenvalues, i.e. Kelvin moduli fields, are assumed to be fixed on the middle plane of the plate, but no isoperimetric condition is imposed. The optimization task reduces to an equilibrium problem of an effective hyperelastic plate. The effective potential is explicitly expressed in terms of the invariants of both the strain fields involved.

10

Compliance minimization of thin plates made of material with predefined Kelvin moduli. Part II. The effective boundary value problem and exemplary solutions

Dzierżanowski G., Lewiński T.

Archives of Mechanics

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2012

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Vol. 64, nr 2

137-137

EN

The compliance minimization of transversely homogeneous plates with predefined Kelvin moduli leads to the equilibrium problem of an effective hyperelastic plate with the hyperelastic potential expressed explicitly in terms of both the membrane and bending strain measures, as derived in Part I of the present paper. The aim of this second part of the paper is to show convexity of this potential and, consequently, uniqueness of solutions of the minimum compliance problem considered. Theoretical results are illustrated by numerically calculated optimal trajectories of the eigenstate corresponding to the largest Kelvin modulus.

11

On the fundamental problem of optimum design of anisotropic plates

Lewiński T.

Foundations of Civil and Environmental Engineering

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2006

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No. 7

215-226

EN

The paper deals with the free material design problem of minimum compliance of an anisotropic clastic plate loaded in-plane. All the characteristics of the plate stiffness tensor or of the form of the Hookc tensor for plane case, are treated as design variables. The cost function is expressed in terms of the Kelvin moduli. The necessary conditions of optimality are discussed. They imply that the deformation state within the optimal plate must satisfy the condition of colinearity of stress and strain tensors.

12

Shaping the stiffest three-dimensional structures from two given isotropic materials.

Czarnecki S., Lewiński T.

Computer Assisted Mechanics and Engineering Sciences

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2006

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Vol. 13, No. 1

53-83

EN

The paper concerns layout optimization of elastic three dimensional bodies composed of two isotropic materials of given amount. Optimal distribution of the materials corresponds to minimization of the total compliance or the work of the given design-independent loading. The problem is discussed in its relaxed form admitting composite domains, according to the known theoretical results on making the minimum compliance problems well posed. The approach is based upon explicit formulae for the components of Hooke's tensor of the third rank stiff two material composites. An appropriate derivation of these formulae is provided. The numerical algorithm is based on COC concept, the equilibrium problems being solved by the ABAQUS system. Some of the optimal layouts presented compare favourably with the known benchmark solutions. The paper shows how to use commercial FEM codes to find optimal composite designs within linear elasticity theory.

13

The lightest plane structures of a bounded stress level transmitting a point load to a circular support

Graczykowski C., Lewiński T.

Control and Cybernetics

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2005

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Vol. 34, no 1

227-253

EN

The paper refers to the problem of Michell (1904) of finding the lightest fully stressed structures, composed of possibly infinite number of members, trasmitting a given load to a support forming a circle. The point load can be located within or outside the circle. The known analysis by Hemp (1973) is enhanced here by disclosing the explicit formulae for the weights of the ribs and the interior (fibrous domain). The optimal weight can be found by two manners: by applying the primal integral formula involving the density of the reinforcement or by computing the work of the load on the adjoint displacement. One of the aims of the paper is to show that both these formulae are equivalent. This identity is essential since in the case of point loads the equivalence of the primal and dual formulations has not been proved till now. Tlie analytically found layouts are confirmed by analysis of trusses (of finite number of joints) approximating the exact Michell-like solutions.

14

Optimal design of membrane shells. Homogenization-based relaxation of the two-phase layout problem

Lewiński T, Telega J. J.

Journal of Theoretical and Applied Mechanics

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2003

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Vol. 41 nr 3

545-560

EN

The aim of using shell structures instead of plates is to avoid bending, hence the vital role of the membrane theory. Within this theory the classical optimum design problem is formulated: lay out two isotropic materials such that the shell becomes the stiffest possible. The amount of both the materials is fixed. The aim of the present paper is to reformulate this problem in a form assuring its well-posedness. The membrane approximation can be introduced from the very beginning or be imposed upon the relaxation. In the present paper it is shown that the latter modeling leads to a better formulation. It does not lose its stability even if one material degenerates to a void, thus leading to a well-posed shape design problem.

PL

Odpowiednie kształtowanie konstrukcji powłokowych zezwala na minimalizację efektów zginania. Konstrukcje zaprojektowane idealnie powinny pracować bezmomentowo, co podkreśla szczególną rolę teorii powłok błonowych, czyli powłok nie podlegających zginaniu.W pracy rozpatrujemy klasyczne zadanie optymalizacji rozmieszczenia dwu materiałów izotropowych w powłoce przcującej bezmomentowo w celu maksymalizacji jej sztywności. Ilość obu materiałów jest z góry ustalona. Celem pracy jest przeformułowanie tego zagadnienia do postaci dobrze postawionej. Założenie bezmomentowej pracy powłoki może być narzucone od początku lub przyjęte już po procesie relaksacji (w sensie rachunku wariacyjnego). W tej pracy wykazujemy, że ta ostatnia metoda modelowania jest bardziej korzystna.Otrzymuje się sformułowanie, które zachowuje się stabilnie nawet wtedy, gdy jeden z materiałów degeneruje się do pustek, co zezwala na otrzymanie dobrze sformułowanego zadania optymalizacji kształtu.

15

Layout optimization of two isotropic materials in elastic shells

Dzierżanowski G., Lewiński T

Journal of Theoretical and Applied Mechanics

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2003

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Vol. 41 nr 3

459-472

EN

The two-phase layout problem within the plate theory was solved by Gibiansky and Cherkaev in 1984. The same problem in the plane stress formulation was solved by the same authors in eventually cleared up by Allaire and Kohn in 1993. In the thin shell theory both these formulations are coupled, which is clearly seen in the homogenization formulae found by Lewiński and Telega in 1988, Telega and Lewiński in 1998, and a general setting of the layout problem presented in the book by the same authors. The aim of the present paper is to set this problem within the Mushtari-Donnell-Vlasov approximation. The main result of the present examination is the lower bound of the complementary energy found by using the translation method. The translation matrix involves off-diagonal components, which leads to the effective complementary potential of a specific coupled form, expressible in terms of invariants of the stress and couple results.

PL

Zagadnienie optymalnego rozkładu dwóch materiałów izotropowych w sprężystych plytach cienkich rozwiązali Gibianskij i Czerkajew w roku 1984. Minimalizacji podlegała podatność płyty. Analogiczne zadanie dotyczące teorii tarcz rozwiązali ci sami autorzy w 1987r. Sformułowanie to uzupełnili i uściślili Allaire i Kohn w roku 1993. W zadaniu dotyczącym powłok cienkich oba te sformułowania są ze sobą sprzężone, co jasno jest widoczne w formułach homogenizacji znalezionych w pracach Lewińskiego i Telegi z roku 1988 oraz pracach Telegi i Lewińskiego z roku 1998; ogólne, niejawne sformułowanie tego zadania optymalizacji omówiono w książce tych samych autorów. Celem niniejszej pracy jest sformułowanie tego zadania w sposób jawny w zakresie technicznej teorii powłok Musztariego-Donnella-Własowa. W pracy wyprowadzamy w sposób jawny dolne oszacowanie energii komplementarnej z wykorzystaniem metody translacji. Macierz traslacji zawiera tutaj składniki pozadiagonalne. Ta postać macierzy translacji prowadzi do zastępczego potencjału o specyficznej postaci sprzężonej, wyrażalnej za pomocą niezmienników sił wewnętrznych w powłoce.

16

Circular and annular two-phase plates of minimal compliance

Kolanek K., Lewiński T.

Computer Assisted Mechanics and Engineering Sciences

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2003

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Vol. 10, No. 2

177-199

EN

The paper deals with optimal design of thin plates. The plate thickness assumes two possible values: h1 and h2 and the plate volume is given. The problem of minimizing the plate compliance needs relaxation. The relaxed formulation was found by Gibiansky and Cherkaev in 1984. In the present paper a finite element approximation of this problem is presented in the framework of rotationally symmetric bending of circular and annular plates. The problem is composed of a nonlinear equilibrium problem coupled with a minimum compliance problem. The aim of the present paper is to analyze the forms of the optimal solutions, in particular, to look into the underlying microstructures. It is proved that in some solutions a ribbed microstructure occurs with ribs non-coinciding with both the radial and circumferential directions. Due to non-uniqueness of the sign of an angle of inclination of ribs the appearance of this microstructure does not contrasts with the radial symmetry of the problem. In the degenerated problem when the smallest thickness h1 vanishes the above interpretation of the inclined ribbed microstructure becomes incorrect; in these regions one can assume that the plate is solid but with a varying thickness. The degenerated case of h1=0 was considered in the papers by Rozvany et al. and Ong et al. but there such a microstructure was not taken into account. One of the aims of the paper is to re-examine these classical and frequently cited results.

17

On algebraic equations of elastic trusses, frames and grillages

Lewiński T

Journal of Theoretical and Applied Mechanics

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2001

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Vol. 39 nr 2

307-322

EN

The algebraic equations of elastic frames are obtained by imposing natural constraints on the trial displacement fields. The obtained set of equations describe deformations of trusses, frames made of both compressible and incompressible bars, grillages of rigid joints as well as pinjointed grillages. A nwe form of the set of algebraic equations of frames with a diagonal constitutive matrix is put forward. The presented approach is well-suited for educational purposes.

PL

O algebraicznych równaniach sprężystych kratownic, ram i rusztów. Algebraiczne równania ram srężystych otrzymano na drodze nałożenia więzów na próbne pola przemieszczeń. Znaleziony układ równań opisuje odkształcenia kratownic, ram wykonanych z prętów ściśliwych bądź nieściśliwych, rusztów o węzłach sztywnych oraz rusztów przegubowych. Wyprowadzono nową postać równań algebraicznych z diagonalną macierzą kostytutywną. Przedstawione podejście dobrze pasuje do celów dydaktycznych.

18

Michell-like grillages and structures with locking

Lewiński T., Telega J. J.

Archives of Mechanics

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2001

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Vol. 53, nr 4-5

457-485

EN

The paper generalizes the Michell theory of plane pseudo-continua to the anti-plane problems in which the loading is perpendicular to the plane of the structure. The starting point is the minimum compliance problem for a two-phase Kirchhoff plate. Upon relaxation, one of the materials can be degenerated to a void (or microvoids) and by imposing the condition of the volume being small, one arrives at the Michell-like problem for a locking plate. The locking locus B is determined explicitly; it tends to a square if the Poisson ratio tends to 1. In the last case the locking locus coincides with that used in the Rozvany-Prager theory of optimal grillages. A theory of perfectly-locking and elastic-locking plates and shells, not necessarily isotropic, is formulated. Dual extremum and existence theorems are also given.

19

Static analysis of closed cylindrical shells subjected to asymmetric loading

Bielecki T., Lewiński T.

Archives of Civil Engineering

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1999

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Vol. 45, nr 2

147-162

EN

The paper deals with stress and deformation analysis of closed cylindrical shells made from a homogenous lineary elastic material. Numerous available analyses prove that the classical Vlasov's approximate theory can lead to essential errors. On the other hand the known equations of Flügge are complicated, hence inconvenient. In 1968 Koiter derived equations with degree of complexity similar of those of Vlasov and leading to the results not worse than those predicted by Flügge's approach. Just this approach of Koiter is taken here (after Niordson as the analytical method. The present paper recalls briefly the derivation of these equations and general representation of their solutions. The analysis is confined to the case of transverse loading and bending moments prescribed along the boundaries. This class of problems comprises such important examples like the response of the vessels to the wind loading and deformation of tubes filled up with liquid. The present paper gives a brief account of such analyses.

PL

Przedmiotem rozważań w tej pracy są zagadnienia statyczne zamkniętych powłok walcowych o przekroju kołowym. Powłoki walcowe znajdują szerokie zastosowanie w technice, a w szczególności w budownictwie jako zbiorniki, silosy, naczynia ciśnieniowe lub kominy. Teoria powłok walcowych jest dobrze opracowana, zarówno w zakresie powłok cienkich jak i powłok średniej grubości, por. Flügge, Mazurkiewicz i Nagórski. Algorytmy analityczne wynikające z tych teorii nie są jednak proste i dlatego wydaje się, że każda próba ich uproszczenia - bez utraty dokładności — jest godna uwagi. Liczne badania dowiodły, że stosowanie uproszczeń teorii technicznej może prowadzić do znacznych błędów. Dopiero w 1959 r. S.D. Morley zproponował pewne uproszczenia równań Flüggego, które prowadziły do ułatwień w algorytmie analizy statycznej i jednocześnie nie powodowały istotnych błędów. Jednak rozważania Morleya zupełnie nie przekonują, robią wrażenie formalnych. Powtórnego wyprowadzenia równań Morleya dokonał W.T. Koiter w 1968 r., korzystając z zasady opcjonalności związków konstytutywnych Love'a I przybliżenia. Wyprowadzenie Koitera spopularyzował Niordson. Jednym z celów pracy jest dokładne omówienie teorii Morleya-Koitera. Punktem wyjścia jest teoria Love'a I przybliżenia w ujęciu Sandersa - Leonarda. Następnie, korzystając z zasady opcjonalności, postulujemy związki konstytutywne jak w teorii II przybliżenia, przyjmując jednak tensory sprzęgające jako zależne od pewnych parametrów sterujących (stałe di). Parametry di dobieramy tak, aby wyzerowały się wszystkie operatory różniczkowe trzeciego rzędu w równaniach przemieszczeniowych. Otrzymujemy najprostsze możliwe równania opisujące pracę powłoki walcowej, bez dodatkowych założeń dotyczących np. małej wyniosłości. Równania te rozsprzęgamy metodą przedstawień Galerkina, a następnie rozdzielamy zmienne. Po zmiennej v rozwijamy poszukiwane funkcje w szeregi Fouriera. Względem zmiennej E dostajemy równania różniczkowe zwyczajne zależne od n-tego numeru harmoniki. Znajdujemy pierwiastki równań charakterystycznych. Postać tych równań tłumaczy zabiegi Morleya i Koitera: mamy do czynienia z równaniami, które dają się rozwiązać analitycznie. Niejako przy okazji udało się znaleźć pewne błędy ideowe w książce Niordsona, a następnie poprawić je. Opracowano algorytmy analizy zbiorników walcowych obciążonych obwodowo i powierzchniowo. Szczegółowe omówienie stosowanych metod można znaleźć w pracy Bieleckiego.

20

Selected equilibrium problems of thin elastic helicoidal shells

Knabel J., Lewiński T.

Archives of Civil Engineering

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1999

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Vol. 45, nr 2

245-257

EN

The paper deals with selected, rotationally-symmetric, statical problems of a thin elestic helicoidal shell within the framworks of two versions of the thin shell theory. The main results are found by solving the equations of the Budiansky-Sanders-Koiter (BSK) theory, commonly viewed as the best version of all first-order Love-type shell approximations. The results are obtained by imposing some simplifications in the compatibility equation, silimarly as it was proposed by Mathuna in 1963. These results are compared to the predictions of the Vlasov-Mushtari-Donnell (VMD) theory.

PL

Przedmiotem pracy są wybrane, obrotowo-symetryczne zagadnienia statyki powłok helikoidalnych w ramach dwu wersji teorii powłok cienkich. Dokładniejsze wyniki otrzymano na drodze całkowania równań Budiansky'ego-Sandersa-Koitera (BSK), tworzacych tzw. "najlepszą" wersję teorii Love'a pierwszego przybliżenia. Wyniki te otrzymano na drodze pewnych uproszczeń w równaniach nierozdzielności, podobnych do uproszczeń proponowanych w pracy Mathuny (1963). Przewidywania teorii BSK porównano z wynikami otrzymanymi w ramach teorii Własova-Mushtari'ego-Donnella.