Begell House
International Journal for Multiscale Computational Engineering
International Journal for Multiscale Computational Engineering
1543-1649
9
5
2011
MULTISCALE MECHANICAL MODELLING OF COMPLEX MATERIALS AND ENGINEERING APPLICATIONS 2
FOREWORD of the special issue edited by Patrizia Trovalusci and Martin Ostoja-Starzewski
Patrizia
Trovalusci
Department of Structural Engineering and Geotechnics, Sapienza University of Rome, 00184 Roma, Italy
Martin
Ostoja-Starzewski
Department of Mechanical Science & Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801,
vii-ix
HOMOGENIZATION OF RANDOM PLATES
The homogenization of elastic periodic plates is as follows: The three-dimensional (3D) heterogeneous body is replaced by a homogeneous Love-Kirchhoff plate whose stiffness constants are computed by solving an auxiliary boundary problem on a 3D unit cell that generates the plate by periodicity in the in-plane directions. In the present study, a generalization of the above-mentioned approach is presented for the case of a plate cut from a block of linear elastic composite material considered to be statistically uniform random in the in-plane directions. The homogenized bending stiffness and the moduli for in-plane deformation of the random plate are defined in four equivalent manners: (1) the first definition considers statistically invariant stress and strain fields in the infinite plate. In the other definitions, a finite representative volume element of the plate is submitted on its lateral boundary to suitable (2) kinematically uniform conditions, (3) statically uniform conditions, and (4) periodic conditions. The relationships between these four definitions are studied and hierarchical bounds are derived.
Karam
Sab
Université Paris-Est, Laboratoire Navier, Ecole des Ponts ParisTech, IFSTTAR, CNRS
503-513
DYNAMICS OF RETICULATED STRUCTURES: EVIDENCE OF ATYPICAL GYRATION MODES
This paper deals with the dynamic behavior of periodic reticulated beams made of symmetric unbraced framed cells. Such archetypical cells can present a high contrast between shear and compression deformabilities that opens the possibility of enriched local kinematics. Through the homogenization method of periodic discrete media associated with a systematic use of scaling, the existence of atypical gyration modes is established theoretically. These latter modes appear when the elastic moment is balanced by the rotation inertia, conversely to "natural" modes where the elastic force is balanced by the translation inertia. A generalized beam modeling including both "natural" and gyration modes is proposed and discussed through a dimensional analysis. The results are confirmed on numerical examples.
Celine
Chesnais
École Nationale des Travaux Publics de l'État, Université de Lyon, DGCB, FRE CNRS 3237, Lyon, France
S.
Hans
École Nationale des Travaux Publics de l'État, Université de Lyon, DGCB, FRE CNRS 3237, Lyon, France
Claude
Boutin
École Nationale des Travaux Publics de l'État, Université de Lyon, DGCB, FRE CNRS 3237, Lyon, France
515-528
THE MATCHED ASYMPTOTIC EXPANSION FOR THE COMPUTATION OF THE EFFECTIVE BEHAVIOR OF AN ELASTIC STRUCTURE WITH A THIN LAYER OF HOLES
In the framework of matched asymptotic expansions we introduce a new efficient and robust method to approximate the behavior of a structure containing a thin layer with periodically distributed micro-holes. A surface (in three dimensions) or a line (in two dimensions), on which particular jumping conditions are defined, substitutes for the initial problem.
Giuseppe
Geymonat
Université Montpellier II, LMGC, UMR-CNRS 5508, Case Courier 048, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
Francoise
Krasucki
Université Montpellier 2, I3M, UMR-CNRS 5149, Case Courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5, France
Sofiane
Hendili
Université Montpellier 2, I3M, UMR-CNRS 5149, Case Courier 051, Place Eugène Bataillon, 34095 Montpellier Cedex 5; EPIMACS, INRIA Rocquencourt
Marina
Vidrascu
EPIMACS, INRIA Rocquencourt
529-542
A COSSERAT BASED MULTI-SCALE MODEL FOR MASONRY STRUCTURES
This paper presents a multi-scale model for the analysis of the in-plane structural response of regular masonry. It is based on a computational periodic homogenization technique and is characterized by the adoption of the Cosserat continuum model at the macroscopic structural level, taking into account the influence of the microstructure on the global response and correctly describing the localization phenomena; at the microscopic representative volume element (RVE) level, where the nonlinear constitutive behavior, geometry, and arrangement of the masonry constituents are modeled in detail, a standard Cauchy model is employed. An isotropic nonsymmetric damage model is adopted for the bricks and mortar joints. The solution algorithm is based on a parallelization strategy and on the finite-element method. Some numerical applications on typical masonry structures are reported, showing both the global response curves and the stress and damage distributions on the RVEs.
Maria Laura
De Bellis
Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma "La Sapienza," Via Eudossiana, 18, 00184 Roma, Italy
Daniela
Addessi
Dipartimento di Ingegneria Strutturale e Geotecnica, Università di Roma "La Sapienza," Via Eudossiana, 18, 00184 Roma, Italy
543-563
NON-LOCAL COMPUTATIONAL HOMOGENIZATION OF PERIODIC MASONRY
Micro-polar and second-order homogenization procedures for periodic elastic masonry have been implemented to include geometric and material length scales in the constitutive equation. From the evaluation of the numerical response of the unit cell representative of the masonry to properly prescribed displacement boundary conditions related to homogeneous macro-strain fields, the elastic moduli of the higher-order continua are obtained on the basis of an extended Hill-Mandel macro-homogeneity condition. Elastic moduli and internal lengths for the running bond masonry are obtained in the case of Cosserat and second-order homogenization. To evaluate these results, a shear layer problem representative of a masonry wall subjected to a uniform horizontal displacement at points on the top is analyzed as a micro-polar and a second-order continuum and the results are compared to those corresponding with the reference heterogeneous model. From this analysis the second-order homogenization appears to provide better results in comparison with the micro-polar homogenization.
Andrea
Bacigalupo
Department of Civil, Environmental and Architectural Engineering, University of Genova, via Montallegro, 1-16145 Genova, Italy
Luigi
Gambarotta
Department of Civil, Environmental and Architectural Engineering, University of Genova, via Montallegro, 1-16145 Genova, Italy
565-578
FRACTIONAL DIFFERENTIAL CALCULUS FOR 3D MECHANICALLY BASED NON-LOCAL ELASTICITY
This paper aims to formulate the three-dimensional (3D) problem of non-local elasticity in terms of fractional differential operators. The non-local continuum is framed in the context of the mechanically based non-local elasticity established by the authors in a previous study; Non-local interactions are expressed in terms of central body forces depending on the relative displacement between non-adjacent volume elements as well as on the product of interacting volumes. The non-local, long-range interactions are assumed to be proportional to a power-law decaying function of the interaction distance. It is shown that, as far as an unbounded domain is considered, the elastic equilibrium problem is ruled by a vector fractional differential operator that corresponds to a new generalized expression of a fractional operator referred to as the central Marchaud fractional derivative (CMFD). It is also shown that for bounded solids the corresponding integral operators contain only the integral term of the CMFD and no divergent terms on the boundary appear for a one-dimensional solid case. This aspect is crucial since the mechanical boundary conditions may be easily enforced as in classical local elasticity theory.
Mario
Di Paola
Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica, Università degli Studi di Palermo, Viale delle Scienze, I-90128, Palermo, Italy
Massimiliano
Zingales
Dipartimento di Ingegneria Strutturale, Aerospaziale e Geotecnica, Università degli Studi di Palermo, Italy
579-597
MULTIFIELD CONTINUUM SIMULATIONS FOR DAMAGED MATERIALS: A BAR WITH VOIDS
This work is based on the formulation of a continuum model with microstructure for the study of the mechanical behavior of microcracked materials. Such a continuum is named multifield continuum because it is characterized by field descriptors accounting for the presence of material internal structure. In particular, the disturbance due to the presence of distributed microcracks in the material is revealed by an additional kinematical field representing the smeared displacement jump over the microcracks. According to the approach of the classical molecular theory of elasticity, the constitutive multifield continuum (macromodel) has been obtained by requiring the energy equivalence with an appropriate discrete micromodel. The stress-strain relations of the continuum have been explicitly identified by selecting the response functions of the interactions of the discrete model and depend on the geometry of the material's internal phases. Attention is here focused on theoretical and numerical investigations on a one-dimensional microcracked bar by varying the microcrack density and size. The effectiveness of the multi-field model, in representing the gross mechanical behavior of such materials with internal structure, is ascertained by comparing the multifield solutions with the numerical solutions obtained by using finite-element simulations for a linear elastic strip having different distributions of voids.
Patrizia
Trovalusci
Department of Structural Engineering and Geotechnics, Sapienza University of Rome, 00184 Roma, Italy
Valerio
Varano
Department of Structures, Roma Tre University of Rome, 00146 Roma, Italy
599-608