Tomasz Lewiński

Plates, laminates and shells : asymptotic analysis and homogenization

Mathematical preliminaries: function spaces, convex analysis, variational convergence. Elastic plates: three-dimensional analysis and effective models of composite plates thin plates in bending and stretching non-linear behaviour of plates moderately thick transversely symmetric plates sandwich plates with soft core. Elastic plates with cracks: unilateral cracks in thin plates unilateral cracks in plates with transverse shear deformation part-through the thickness cracks stiffness loss of cracked laminates comments and bibliographical notes. Elastic-perfectly plastic plates: mathematical complements homogenisation of functional with linear growth homogenisation of plates loaded by forces and moments comments and bibliographical notes. Elastic and plastic shells: linear and non-linear models of elastic shells homogenisation and stiffnesses of thin periodic elastic shells linear approach homogenised properties of thin periodic elastic shells undergoing moderately large relations around tangents perfectly plastic shells. Application of homogenisation methods in optimum design of plates and shells: mathematical complements two-phase plate in bending, Hashin-Shtrikamn bounds two-phase plate Hashin-Shtrikman bounds for the in-plane problem explicit formulae for effective bending stiffnesses and compliances of ribbed plates explicit formulae for effective membrane stiffnesses and compliance s of ribbed plates thin bending two-phase plates of minimum compliance minimum compliance problem for thin plates of varying thickness -application of young measures thin shells of minimum compliances Truss-like Michell continua.

Effective models of composite periodic plates—I. Asymptotic solution

Abstract This paper is concerned with the statical problem of plates with oscillating material properties and rapidly varying face shapes. Both the material moduli and the geometrical data vary periodically with the same period. By using the asymptotic method of Caillerie (1982, C. R. Acad. Sci. Paris 294(II) , 159), the higher-order terms of the expansion for deformations and stresses are arrived at. These terms are determined by a subsequent homogenized problem. For new auxiliary functions of a local variable, appropriate basic cell problems are established. Proof that the conventional asymptotic expansion does not comprise boundary effects at the clamped edge is also given.

Energy change due to the appearance of cavities in elastic solids

Abstract The paper presents an overview of the problem of assessing an increment of strain energy due to the appearance of small cavities in elastic solids. The following approaches are discussed: the compound asymptotic method by Mazja et al., the Eshelby-like method used in the classical works on the mechanics of composites, the homogenization method, and the topological derivative method proposed by Sokolowski and Żochowski. The increment of energy is expressed by a quadratic form with respect to strains referring to the virgin solid. All the methods lead to the same formula for the increment of energy. It is expressed by a quadratic form with respect to strains referring to the virgin solid. This quadratic form turns out to be unconditionally positive definite. Explicit formulae are derived for an elliptical hole and for a spherical cavity. The results derived determine the characteristic function of the bubble method of the optimal shape design of elastic 2D and 3D structures.

Effective models of composite periodic plates—III. Two-dimensional approaches

Abstract This paper is concerned with two-dimensional approximations of the three-dimensional local problems of Caillerie [(1984). Math. Meth. Appl. Sci. 6 , 159–191] and Kohn and Vogelius [(1984). Int. J. Solids Structures 20 , 333–350]. The solutions to these problems make it possible to evaluate the effective stiffnesses of periodic plates. The Kirchhoff-type approximation results in the formulae shown by Duvaut [1976. In Theoretical and Applied Mechanics (Edited by W. T. Koiter), pp. 119–132. North-Holland, Amsterdam]. By imposing Hencky-Reissner-type constraints, one is led to new formulae which have a wider range of applicability. The paper also discusses the formulae which result from homogeneizing two-dimensional Kirchhoffs, Reissner-Henckys and Reddys equations of plates with periodic structure.

Topology optimization in structural and continuum mechanics

Structural topology optimization.- On basic properties of Michells structures.- Validation of numerical method by analytical benchmarks and verification of exact solutions by numerical methods.- Introduction to shape and topology optimization.- Homogenization method for shape and topology optimization.- Level set method for shape and topology optimization.- Compliance minimization of two-material elastic structures.- Some notes on topology optimization of vibrating continuum structures.- Topology optimization of vibrating continuum structures for maximum values of simple and multiple eigenfrequencies and frequency gaps.- On optimum shape design and periodicity of band-gape beam structures.- Topological design for minimum dynamic compliance of continuum structures subjected to forced vibration.- Topological design for minimum sound emission from bi-material structures subjected to forced vibration.- Discrete material optimization of vibrating laminated composite plates for minimum sound emission.- Optimization of diffusive transport problems.- Fluid topology optimization: stokes and Navier-stokes models.- Topology optimization of coupled multi-physics problems.- Topology optimization based on the extended finite element method.- Topology optimization of meso- and nano-scale problems.- Topology optimization under uncertainty.- A brief review of numerical methods of structural topology optimization.- The free material design in planar elasticity.

Effective models of composite periodic plates—II. Simplifications due to symmetries

Abstract This paper is aimed at investigating how material and geometrical symmetries of a composite plate simplify the formulae derived in Part I by the asymptotic method. It occurs that the symmetry with respect to the middle plane results in splitting the subsequent overall and local problems into membrane and bending problems. Additional assumptions of orthotropy of the material and of certain symmetries of the periodicity cells imply far-reaching simplifications, e.g. the vanishing of some terms in the first-order correctors for displacements, and the cancelation of discrepancies in the formulation of some boundary conditions. In the last section, a computational algorithm for evaluating the effective stiffnesses is suggested.

On the twelfth-order theory of elastic plates

The aim of the present contribution is to find the primal formulation of the latter theory thus enabling one to compare this approach with the displacement-based approach of the same order and with other relevant plate theories of order twelve, eight and six

Effective stiffnesses of transversely non-homogeneous plates with unidirectional periodic structure

Abstract This paper presents a derivation and analysis of closed formulae for the membrane, bending and reciprocal effective stiffnesses of elastic orthotropic and transversely asymmetric plates with microstructures periodic in one direction. The derivation is based on the concept of imposing Hencky-type displacement constraints on the solutions to the Caillerie-Kohn-Vogelius local homogenization problems. Reduction of the dimension of the problems by one makes it possible to solve the new approximate local problems exactly, thus enabling one to find the relevant formulae in closed form, ready for engineering applications as well as for optimization and sensitivity studies. This paper generalizes the previous results of the author [ Int. J. Solids Structures 29 , 309–326 (1992)], concerning the bending of transversely symmetric periodic plates.

On Basic Properties of Michell’s Structures

The paper is an introduction towards Michell’s optimum design problems of pin-jointed frameworks. Starting from the optimum design problem in its discrete setting we show the passage to the discrete-continuous setting in the kinematic form and then the primal stress-based setting. The classic properties of Hencky nets are consistently derived from the kinematic optimality conditions. The Riemann method of the net construction is briefly recalled. This is the basis for finding the analytical solutions to the optimum design problems. The paper refers mainly to the well known solutions (e.g. the cantilevers) yet discusses also the open problems concerning those classes of solutions in which the kinematic approach cannot precede the static analysis.

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

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.