Scale effects in orthotropic composite assemblies as micropolar continua: A comparison between weak-and strong-form finite element solutions

The aim of the present work was to investigate the mechanical behavior of orthotropic composites, such as masonry assemblies, subjected to localized loads described as micropolar materials. Micropolar models are known to be effective in modeling the actual behavior of microstructured solids in the presence of localized loads or geometrical discontinuities. This is due to the introduction of an additional degree of freedom (the micro-rotation) in the kinematic model, if compared to the classical continuum and the related strain and stress measures. In particular, it was shown in the literature that brick/block masonry can be satisfactorily modeled as a micropolar continuum, and here it is assumed as a reference orthotropic composite material. The in-plane elastic response of panels made of orthotropic arrangements of bricks of different sizes is analyzed herein. Numerical simulations are provided by comparing weak and strong finite element formulations. The scale effect is investigated, as well as the significant role played by the relative rotation, which is a peculiar strain measure of micropolar continua related to the non-symmetry of strain and work-conjugated stress. In particular, the anisotropic effects accounting for the micropolar moduli, related to the variation of microstructure internal sizes, are highlighted

Material Symmetries in Homogenized Hexagonal-Shaped Composites as Cosserat Continua

In this work, material symmetries in homogenized composites are analyzed. Composite materials are described as materials made of rigid particles and elastic interfaces. Rigid particles of arbitrary hexagonal shape are considered and their geometry described by a limited set of parameters. The purpose of this study is to analyze different geometrical configurations of the assemblies corresponding to various material symmetries such as orthotetragonal, auxetic and chiral. The problem is investigated through a homogenization technique which is able to carry out constitutive parameters using a principle of energetic equivalence. The constitutive law of the homogenized continuum has been derived within the framework of Cosserat elasticity, wherein the continuum has additional degrees of freedom with respect to classical elasticity. A panel composed of material with various symmetries, corresponding to some particular hexagonal geometries defined, is analyzed under the effect of localized loads. The results obtained show the difference of the micropolar response for the considered material symmetries, which depends on the non-symmetries of the strain and stress tensor as well as on the additional kinematical and work-conjugated statical descriptors. This work underlines the importance of resorting to the Cosserat theory when analyzing anisotropic materials

Inherent strength and stiffness anisotropy of laminated rocks

The variation of rock strength and stiffness, known as mechanical anisotropy, is expected at different scales: large (rock mass) - or small (intact rock) - scales. It is always mandatory for engineering applications built either on or in anisotropic rock masses to investigate the strength and deformation behavior of those masses. To achieve this goal, continuum-based constitutive models are presented to analyze the mechanical anisotropy. One of both implemented models is named ‘Transubi model’ which considers the transverse isotropic elasticity into bi-linear Mohr-Coulomb strain hardening/softening plastic framework. Experimental investigations and numerical simulations focused mainly on the influence of the mechanical anisotropy on the plastic zoning around excavated openings in laminated rocks. Later, the Transubi model was applied to a tunnel excavated in a shaly facies formation of bedded argillaceous Opalinus clay in an URL (FE-tunnel) to obtain the short-term stability insights. Overall, the research outcomes may have a prospective impact regarding the understanding of anisotropy of laminated, bedded and foliated rocks which improves the deformation behaviour predictability using continuum-based numerical modeling tools

Multi-scale techniques for mansonry structures

The aim of this work is, hence, to adopt the computational homogenization techniques to obtain the global response of masonry structures. Since the experimental global response curves, obtained in typical shear tests on masonry panels, show stiffness and resistance degradation, damage is the fundamental ingredients which must be taken into account in such problems. Moreover, as it is well known, due to the aforementioned softening behavior, regularization techniques are required in order to avoid spurious mesh dependencies when a numerical solution is sought in the framework of finite element method. The first step of this work is the adoption of the standard first order computational homogenization, where Cauchy continuum is used both at the macro and micro-level. This approach is well known in literature and several authors applied it to different engineering problems. An example of the adoption of regularization techniques in the context of multi-scale approaches is found in Massart (2003). Hence a regularization based on the imposition of the macroscopical length scale at the micro-level, in the framework of the fracture energy regularization, is proposed. However, as previously stated, many authors have pointed out the inner limits of first order computational homogenization. Such a formulation, in fact, may be adopted only if 1)the microstructure is very small with respect to the characteristic size at the macro-scale; 2)the absolute size of the constituents does not affect the mechanical properties of the homogenized medium and in presence of low macroscopic gradients of stresses and strains. As a consequence no localization phenomena typically exhibited by masonry can be analyzed. For masonry structures, instead, microstructural typical sizes are comparable with the macro-structural sizes; shape, size and arrangement of the constituents strongly affect the mechanical global response and high deformation gradients typically appear. An enriched formulation is then proposed in order to overcome these problems, based on the adoption of a Cosserat medium at the macro-level and a Cauchy medium at the micro-level. The theoretical and computational schemes remain the same as before but for the fact that the two media present different variables. In particular in the Cosserat medium additional strain and stress variables appear, with respect to the Cauchy continuum, as a consequence of the independent rotational degree of freedom assigned to every material point. Thus, a more sophisticated kinematic map, containing higher order polynomial expansions, is needed to state proper bridging conditions between the two levels. The innovative contribution of this work concerns the adoption of an enhanced multi-scale computational homogenization technique for studying the masonry response, together with the employment of damage models for the constituents description. Thus, by exploiting the inner regularization properties of the Cosserat continuum at the macro-level and by adopting a classical fracture energy regularization at the micro-level, localization phenomena, typically exhibited by masonry structures, are analyzed. Since this material shows a typical strain softening behavior, an ad hoc regularization technique has been developed at both levels in order to obtain objective numerical responses. To the knowledge of the author, no previous examples of Cosserat-Cauchy computational homogenization techniques, taking into account localization effects, have been presented. A possible objection to the use of a fully-coupled multi-scale technique could be related to the high computational efforts required, but here the use of parallel computing brings them down. In this context, these procedures strike a good balance between the achievement of detailed information at the scale of the constituents and the requirement of holding the computational costs down

New insights on homogenization for hexagonal-shaped composites as Cosserat continua

In this work, particle composite materials with different kind of microstructures are analyzed. Such materials are described as made of rigid particles and elastic interfaces. Rigid particles of arbitrary hexagonal shape are considered and their geometry is described by a limited set of parameters. Three different textures are analyzed and static analyses are performed for a comparison among the solutions of discrete, micropolar (Cosserat) and classical models. In particular, the displacements of the discrete model are compared to the displacement fields of equivalent micropolar and classical continua realized through a homogenization technique, starting from the representative elementary volume detected with a numeric approach. The performed analyses show the effectiveness of adopting the micropolar continuum theory for describing such materials

Numerical simulations of materials with micro-structure : limit analysis and homogenization techniques

Continuum-based numerical methods have played a leading role in the numerical solution of problems in soil and rock mechanics. However, for stratified soils and fractured rocks, a continuum assumption often leads to difficult parameters to define and over-simplified geometry to be realistic. In such cases, approaches that consider the micro-structure of the material can be adopted. In this paper, two of such approaches are detailed, namely limit analysis incorporating fractures and individual blocks, and elastoplastic homogenization of layered soils

Masonry nonlinear response: modeling and analysis of the effects of damaging mechanisms

Over the last decades, many efforts were devoted to develop efficient and accurate numerical procedures for the assessment of the structural capacity of masonry constructions. The main difficulties in modeling this type of material are due to its heterogeneous nature. Indeed, masonry is composed by blocks, stones or bricks, connected with or without mortar, whose geometry, mechanical properties and arrangement strongly affect the overall response. Among the available modeling strategies, finite element models appear to be suitable tools to describe the evolution of the nonlinear mechanisms developing in the material under typical loading conditions. Within this framework, macromechanical models, which consider masonry as an equivalent homogeneous, isotropic or anisotropic medium, are a fair compromise between accuracy and computational burden. Stemming on the above considerations, this work focuses on the development of constitutive laws involving damage and plasticity inner variables, tailored to the macromechanical analysis of 2D masonry structures. Herein, a new isotropic damage-plastic model, which is an enhanced version of that presented by Addessi et al. (2002), is proposed. This model is able to capture the degrading mechanisms due to propagation of microcracks and accumulation of irreversible strains, as well as the stiffness recovery related to cracks re-closure. Moreover, to account for the variation of the mechanical properties in the different material directions, a novel orthotropic damage model is developed to deal with regular masonry textures. The proposed models are implemented in finite element procedures, where the mesh-dependency problem is efficiently overcome by adopting nonlocal integral formulations. Numerical applications are performed to assess the models capacity of describing the material inelastic behavior and comparisons of numerically and experimentally evaluated responses are also provided for some masonry panels. Finally, the effects of degrading mechanisms on masonry dynamic behavior are investigated. For this purpose a systematic approach is adopted, based on the evaluation of the frequency response curves of masonry walls. The obtained curves show peculiar characteristics due to the irreversible effect of damage, which leads to degradation of the structural mechanical properties and the related variation of the natural frequencies, which in turn significantly influence the dynamic amplification of the response. The numerical results are also confirmed by shaking table tests performed on tuff masonry walls loaded out-of-plane

Derivation of the out-of-plane behaviour of masonry through homogenization strategies: Micro-scale level

Two simple and reliable homogenized models are presented for the characterization of the masonry behaviour via a representative volume element (RVE) defined at a structural level. An FE micro modelling approach within a plate formulation assumption (Kirchhoff-Love and Mindlin-Reissner theory) using Cauchy continuum hypotheses and first-order homogenization theory is adopted. Brick units are considered elastic and modelled through quadrilateral finite elements (FEs) with linear interpolation. Mortar joints are assumed to be inelastic and reduced to zero-thickness interface FEs. A multi-surface plasticity model governs the strength envelope of mortar joints. It can reproduce fracture, frictional slip and crushing along the interface elements, hence making possible the prediction of a stepped, toothed or de-bonding failure pattern of masonry.Validation tests on the homogenized procedures are undertaken to conclude on the correct identification of the elastic stiffness properties, in the ability to reproduce the masonry orthotropic behaviour and the effect of potential pre-compressive states. Furthermore, the approaches are extended to characterize a case study of an English-bond masonry wall. Both the validation and application steps provide excellent results when compared with available experimental and numerical data from the literature. Conclusions on the influence of three-dimensional shear stresses and the effect of potential discontinuities along the thickness direction are also outlined.The two homogenized approaches are, for the running- and English-bond masonry cases, integrated within a FE code. By providing reliable and low computational cost solutions', these are particularly suitable to be combined within multi-scale approaches.This work was supported by FCT (Portuguese Foundation for Science and Technology), within ISISE, scholarship SFRH/BD/95086/2013. This work was also partly financed by FEDER funds through the Competitivity Factors Operational Programme - COMPETE and by national funds through FCT - Foundation for Science and Technology within the scope of the project POCI-01-0145-FEDER-007633

Mechanics of Materials

[no abstract available

Anisotropic Continuum-Molecular Models: A Unified Framework Based on Pair Potentials for Elasticity, Fracture and Diffusion-Type Problems

This paper presents a unified framework for continuum-molecular modeling of anisotropic elasticity, fracture and diffusion-based problems within a generalized two-dimensional peridynamic theory. A variational procedure is proposed to derive the governing equations of the model, that postulates oriented material points interacting through pair potentials from which pairwise generalized actions are computed as energy conjugates to properly defined pairwise measures of primary field variables. While mass is considered as continuous function of volume, we define constitutive laws for long-range interactions such that the overall anisotropic behavior of the material is the result of the assigned elastic, conductive and failure micro-interaction properties. The non-central force assumption in elasticity, together with the definition of specific orientation-dependent micromoduli functions respecting material symmetries, allow to obtain a fully anisotropic non-local continuum using a purely pairwise description of deformation and constitutive properties. A general and consistent micro-macro moduli correspondence principle is also established, based on the formal analogy with the classic elastic and conductivity tensors. The main concepts presented in this work can be used for further developments of anisotropic continuum-molecular formulations to include other mechanical behaviors and coupled phenomena involving different physics