Patrizia Trovalusci

Non-linear micropolar and classical continua for anisotropic discontinuous materials

Non-linear Cosserat and Cauchy anisotropic continua equivalent to masonry-like materials, like brick/block masonry, jointed rocks, granular materials or matrix/particle composites, are presented. An integral procedure of equivalence in terms of mechanical power has been adopted to identify the effective elastic moduli of the two continuous models starting from a Lagrangian system of interacting rigid elements. Non-linear constitutive functions for the interactions in the Lagrangian system are defined in order to take into account both the low capability to carry tension and the friction at the interfaces between elements. The non-linear problem is solved through a finite element procedure based on the iterative adjustment of the continuum constitutive tensor due to the occurrence of some limit situation involving the contact actions of the discrete model. Differences between the classical and the micropolar model are investigated with the aid of numerical analyses carried out on masonry walls made of blocks of different size. The capability of the micropolar continuum to discern, unlike the classical continuum, the behaviour of systems made of elements of different size is pointed out. It is also shown that for anisotropic materials, even in the elastic case, the micropolar solution in general does not tend to the classical solution when the size of the elements vanishes. 2002 Elsevier Science Ltd. All rights reserved.

Limit Analysis for No-tension and Frictional Three-Dimensional Discrete Systems

ABSTRACT Masonry-like materials, such as brick-block masonry or rocky materials, are modelled as discrete systems of rigid blocks in dry contact. Compressive and frictional interactions are considered. A computer procedure to determine the collapse load for two and three dimensional structures using a non-standard limit analysis approach is implemented. The mechanisms of hingeing, sliding, and twisting at the block interfaces are accounted for. The solution of the problem of non linear programming, corresponding to limit analysis in the presence of friction at interfaces, is obtained by solving a preliminary problem of linear programming, corresponding to a linearized limit analysis in the presence of dilatancy at the interfaces. Numerical analyses are performed for two and three dimensional systems with many degrees of freedom. The non linear and linearized solutions are compared and discussed. Experimental results are also presented and compared with the numerical solutions.

Masonry as structured continuum

A structured continuum model is formulated to describe the behaviour of block masonry modelled as distinct rigid body systems with elastic interfaces. A correspondence between the two motions is obtained by postulating a relationship between the displacement fields of the continuum and the discrete models. The constitutive functions for the dynamic actions of the continuum are derived by equating the power of the two models.SommarioViene presentato un modello di continuo con struttura atto a descrivere il comportamento meccanico di murature a blocchi, pensate come un sistema di corpi rigidi con contatti puntuali elastici. Il moto del sistema discreto e di quello continuo sono messi in relazione postulando una corrispondenzatra i campi di spostamento. Le funzioni costitutive delle azioni dinamiche del modello continuo sono ricavate uguagliando la potenza meccanica spesa in moti corrispondenti.

Material symmetries of micropolar continua equivalent to lattices

Some aspects concerning the identification of continuum coarse models from fine discrete models are discussed. The preservation of the mechanical power, in the transition from the microscopic to the macroscopic description, is required. A procedure based on the equivalence of the virtual power provides a natural way to select the continuum satisfying this requirement. Having the advantages of an integral procedure, it gives good results if the coarse model is a multifield continuum with strain fields compatible with those of the fine model. In this situation both models share the same material symmetry group. This is shown with reference to rigid particle systems. In particular, the symmetry group of the discrete material is defined and its transformation into that of an equivalent micropolar continuum is studied in detail. Numerical analyses are also performed to investigate the effect of change in the material symmetries.

Constitutive Relations for Elastic Microcracked Bodies: From a Lattice Model to a Multifield Continuum Description

A continuum model suitable for the description of microcracked bodies is shown. The influence of microcracks on the mechanical behavior of the body is estimated through a microstructural field added to the displacement one. This field represents the perturbation to the regular displacement field due to the presence of microcracks. It is an observable quantity; its rate must satisfy appropriate balance equations. The problem of deriving constitutive relations for such a model at least in the linear elastic case is dealt with. Constitutive equations are deduced from a lattice model using an integral identification procedure based on the equivalence in terms of virtual work, without resorting to limit processes. The discrete model considered is made of two superposed lattices; the first one is constituted of material points connected by elastic links; the second one is made of empty closed shells interacting between themselves and with the first lattice. As sample test, a one-dimensional problem is shown.

A Generalized Continuum Formulation for Composite Microcracked Materials and Wave Propagation in a Bar

A multifield continuum is adopted to grossly describe the dynamical behavior of composite microcracked solids. The constitutive relations for the internal and external (inertial) actions are obtained using a multiscale modeling based on the hypotheses of the classical molecular theory of elasticity and the ensuing overall elastodynamic properries allow us to take properly into account the microscopic features of these materials. Referring to a one-dimensional microcracked bar, the ability of such a continuum to reveal the presence of internal heterogeneities is investigated by analyzing the relevant dispersive wave propagation properties. Scattering of traveling waves is shown to be associated with the microcrack density in the bar.

Molecular Approaches for Multifield Continua: origins and current developments

The mechanical behaviour of complex materials, characterised at finer scales by the presence of heterogeneities of significant size and texture, strongly depends on their microstructural features. Attention is centred on multiscale approaches which aim to deduce properties and relations at a given macroscale by bridging information at proper underlying microlevel via energy equivalence criteria. Focus is on physically–based corpuscular–continuous models originated by the molecular models developed in the 19th century to give an explanation per causas of elasticity. In particular, the ‘mechanistic–energetistic’ approach by Voigt and Poincare who, when dealing with the paradoxes deriving from the search of the exact number of elastic constants in linear elasticity, respectively introduced molecular models with moment and multi–body interactions is examined. Thus overcoming the experimental discrepancies related to the so–called central–force scheme, originally adopted by Navier, Cauchy and Poisson.

Coupling Continuum and Discrete Models of Materials with Microstructure: a Multiscale Algorithm

The importance of a multiscale modeling to describe the behavior of materials with microstructure is commonly recognized. In general, at the different scales the material may be described by means of different models. In this paper we focus on a specific class of materials for which it is possible to identify (at the least) two relevant scales: a macroscopic scale, where continuum mechanics applies; and a microscale, where a discrete model is adopted. The conceptual framework and the theoretical model were discussed in previous work. This approach is well suited to study multifield and multiphysics problems. We present here the multiscale algorithm and the computer code that we developed to implement this strategy. The solution of the problem is searched for at the macroscale using nonlinear FEM. During the construction of the FE solution, the material behavior needs to be described at Gauss points. This step is performed numerically, formulating an equivalent problem at the microscale where the inner structure of the material is described through a lattice-like model. The two scales are conceptually independent and bridged together by means of a suitable localization-homogenization procedure. We show how different macroscopic models (e.g. Cauchy vs. Cosserat continuum) can be easily recovered starting from the same discrete system but using different bridges. The interest of this approach is shown discussing its application to few examples of engineering interest (composite materials, masonry structures, bone tissue).

Strain Rates of Micropolar Continua Equivalent to Discrete Systems

Assemblies of rigid blocks can be perceived as fine models of rocky materials, masonry materials and any other composite material made of elements essentially undeformable with respect to the matrix. However, it is often convenient to model such materials as continua endowed with structure in the sense specified in [ ?]. In an earlier work, a continuum constitutive model of a discrete system has been derived in the framework of micropolar linear elasticity [ ?]. In particular, the equivalence of the mechanical power of the continuum and the discrete model for any virtual velocity fields led to the identification of the constitutive functions of the continuum stress measures. In the present note, this integral criterion of equivalence is adopted to identify the strain rates of a micropolar elastic‐perfectly plastic continuum equivalent to rigid particles systems with no-tension constraints; the correspondence between the power of the continuum and of the discrete model is required for any virtual stress field. 2. Identification of the Plastic Strain Rates of the Equivalent Continuum Let w and W be, respectively, the velocity vector and the angular velocity skewsymmetric tensor of a material point of a Cosserat continuum. The linearized strain measures are defined as U D grad w W and U D grad W. The mechanical density power of a linear elastic‐plastic Cosserat continuum can be written (e.g. [?]) as P D S .U e C U p / C 1 S .U e C U p /; (1) where the second-order tensors U e and U p and the third-order tensors U e and U p represent the elastic (‘e’) and the plastic (‘p’) part of the strain and of the micro-strain rates, while S is the second-order linearized stress tensor and S the third-order linearized micro-stress tensor. Considering a representative part (‘module’) M, of volume V, of the discrete system, the mean power formula can be written as D 1

Some Novel Numerical Applications of Cosserat Continua

Cosserat continua demonstrated to have peculiar mechanical properties, with respect to classic Cauchy continua, due to the fact that they are able to more accurately model heterogeneous materials, as particle composites, masonry-like materials and others, taking into account, besides the disposition, the size and the orientation of the heterogeneities [1-2]. On the contrary, classical Cauchy elasticity fails in the modeling of problems in which the characteristic internal length is comparable with the structural length (e.g. [3]). For this reason, many studies are devoted to the numerical implementation of the Cosserat model for practical engineering purposes.Due to the complexity of real physical systems, generally it is not convenient to use semi- analytical approaches for solving such problems, due to limitations related to the boundary conditions and constitutive material models. Thus, numerical approaches are considered as a better choice in these studies. In particular, the authors in the present paper are studying some reference benchmarks, well known from the literature of Cosserat continua, by comparing the solution provided by strong and weak formulations. On one hand, the high accuracy of the so- called Strong Formulation Finite Element Method (SFEM) [4-6] is compared to the numerical models provided by a Finite Element (FE) modeling given by COMSOL® Multiphysics. Convergence, stability and reliability of both modelling will be discussed. The aim is to investigate the more convenient numerical strategies for solving the Cosserat elastic problem. Moreover, differences between classical (Cauchy) and micropolar (Cosserat) model will be discussed in order to emphasize the capabilities of the Cosserat continua in modelling the aforementioned class of problems.