Masonry-like Materials Research Articles

Mesoscopic simulation of uniaxial compression fracture of concrete via the nonlocal macro–meso-scale consistent damage model

To capture the uniaxial compression fracture of concrete composite is still a great challenging problem. To this end, the newly proposed nonlocal macro–meso-scale consistent damage (NMMD) model is extended to simulate uniaxial compression fracture of concrete composites. In the NMMD model, a mesoscopic structure composed of material point pairs is attached to each material point. A decomposition scheme suitable for masonry-like materials is adopted to decompose the elongation quantity of point pairs. The topologic damage is defined as the weighted summation of mesoscopic damage in neighboring material point pairs driven by the decomposed elongation quantity. The physically-based energetic degradation function is employed to bridge the geometric discontinuity in solids and the energy dissipation, and thus the topologic damage is integrated into the framework of continuum damage mechanics. The Monte-Carlo method and two-phase random field model are employed to generate stochastic circular and irregular aggregates representing the meso structure of concrete, respectively. The interface transition zone is prescribed by reducing the critical value of interface point pairs. Numerical results indicate that both the crack patterns and stress–strain curves of the uniaxial compression experiments on concrete panels can be well captured by the proposed NMMD model without mesh size sensitivity. The influence of horizontal constraints on the top and bottom surfaces is investigated and shown to have an effect on the crack patterns, strength and deformation capability of the specimen. The necessity of decomposition is also illustrated.

Sensitivity analysis on mechanical parameters of continuum models for simulating in-plane tests on existing masonry panels

Understanding and modeling the structural behavior of existing masonry buildings is a fundamental task in view of their management and conservation, especially in seismic-prone areas. Among the different modeling approaches for the computational analysis of unreinforced masonry structures, this work investigates the ability of two material continuum models, the modified masonry-like material and Mazars’ material, to simulate real in-plane failure modes of masonry panels, considering the influence of the variability of mechanical parameters on the results. As the first part of a broader benchmark, this work focuses on the study of two recurring test patterns for masonry panels subject to both horizontal and vertical loads, given the availability of experimental data. To account for the parameter variability that typically affects masonry materials, uncertainty propagation is performed in a probabilistic framework, and a gPCE-based proxy model makes it possible to turn time-consuming numerical runs into significantly faster analytical evaluations. Sensitivity analyses are carried out in terms of Sobol’ indices to evaluate the influence of input variability on the ultimate horizontal load. The most relevant masonry parameters for each model are thus highlighted and the issue of model calibration is finally addressed by relying on Bayesian inverse methods. This work also underlines the pros and cons of the different hypotheses, which underlie the two nonlinear constitutive laws.

A semidefinite programming approach for the projection onto the cone of negative semidefinite symmetric tensors with applications to solid mechanics

We propose an algorithm for computing the projection of a symmetric second-order tensor onto the cone of negative semidefinite symmetric tensors with respect to the inner product defined by an assigned positive definite symmetric fourth-order tensor {mathcal {C}}. The projection problem is written as a semidefinite programming problem and an algorithm based on a primal-dual path-following interior point method coupled with a Mehrotra’s predictor-corrector approach is proposed. Implementations based on well-known symmetrization schemes and on direct methods are theoretically and numerically investigated taking into account tensors {mathcal {C}} arising in the modelling of masonry-like materials. For these special cases, indications on the preferable symmetrization scheme that take into account the conditioning of the arising linear systems are given.

An orthotropic damage model with internal sliding and friction for masonry-like material

Quasi-brittle materials like masonry exhibit mechanical properties and develop nonlinearities that are mainly driven by their joints. In the present work, a Continuum Damage Mechanics point of view is considered to describe the macroscopic behavior of masonry. Based on damage models developed for ceramic matrix composite materials, a fixed directional damage approach is proposed. From this formulation, unilateral effect, as well as internal sliding and friction coupled with damage, are introduced. Numerical examples of the response of the model for different loading cases involving cyclic and non-proportional loadings are carried out and compared to experimental results.

A quasi-monolithic phase-field description for mixed-mode fracture using predictor–corrector mesh adaptivity

In this work, we develop a mixed-mode phase-field fracture model employing a parallel-adaptive quasi-monolithic framework. In nature, failure of rocks and rock-like materials is usually accompanied by the propagation of mixed-mode fractures. To address this aspect, some recent studies have incorporated mixed-mode fracture propagation criteria to classical phase-field fracture models, and new energy splitting methods were proposed to split the total crack driving energy into mode-I and mode-II parts. As extension in this work, a splitting method for masonry-like materials is modified and incorporated into the mixed-mode phase-field fracture model. A robust, accurate and efficient parallel-adaptive quasi-monolithic framework serves as basis for the implementation of our new model. Three numerical tests are carried out, and the results of the new model are compared to those of existing models, demonstrating the numerical robustness and physical soundness of the new model. In total, six models are computationally analyzed and compared.

Thermodynamically consistent non-local continualization for masonry-like systems

Masonry-like systems composed by modular stiff units bonded by soft connections represent an efficient, versatile and ultimately successful strategy for natural and artificial macro-scale architectures. The static and dynamic behavior of masonry-like materials characterized by a running bond periodic pattern of rigid rectangular blocks and elastic interfaces is described by formulating equivalent nonlocal continuum models. The paper discusses the thermodynamic restrictions limiting the consistency of the standard continualization strategies. The inherent pathologies recognized in the macroscopic quasi-static response and/or in the dynamic dispersion properties of different continuum models motivate the original proposal of an enhanced continualization strategy. Based on the series expansion of the pseudo-differential functions accounting for shift operators and proper downscaling laws, the enhanced continualization scheme allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall elastic moduli, as well as qualitative agreement and convergent matching of the frequency dispersion functions. The theoretical findings are successfully verified though the solution of representative static and dynamic benchmark problems.

The natural frequencies of masonry beams

The present paper aims at analytically evaluating the natural frequencies of cracked slender masonry elements. The problem is dealt with in the framework of linear perturbation, and the small oscillations of the structure are studied under loaded conditions, after the equilibrium for permanent loads has been achieved. A masonry beam element made of masonry-like material is considered, and some explicit expressions of the beam's fundamental frequency as a function of the external loads and the amplitude of imposed deformations are derived. The analytical results are validated via finite-element analysis.

Enfermedad meningocócica invasiva C en la era posvacunal en un hospital terciario y fallos vacunales

Masonry-like systems composed by modular stiff units bonded by soft connections represent an efficient, versatile and ultimately successful strategy for natural and artificial macro-scale architectures. The static and dynamic behavior of masonry-like materials characterized by a running bond periodic pattern of rigid rectangular blocks and elastic interfaces is described by formulating equivalent nonlocal continuum models. The paper discusses the thermodynamic restrictions limiting the consistency of the standard continualization strategies. The inherent pathologies recognized in the macroscopic quasi-static response and/or in the dynamic dispersion properties of different continuum models motivate the original proposal of an enhanced continualization strategy. Based on the series expansion of the pseudo-differential functions accounting for shift operators and proper downscaling laws, the enhanced continualization scheme allows formulating homogeneous non-local continuum models of increasing orders, analytically featured by characteristic non-local constitutive and inertial terms. The enhanced continualization shows thermodynamic consistency in the definition of the overall elastic moduli, as well as qualitative agreement and convergent matching of the frequency dispersion functions. The theoretical findings are successfully verified though the solution of representative static and dynamic benchmark problems.

Some Novel Numerical Applications of Cosserat Continua

Cosserat continua demonstrated to have peculiar mechanical properties, with respect to classic Cauchy continua, because they are able to more accurately describe heterogeneous materials, as particle composites and masonry-like material, taking into account size effects. Many studies have been devoted to their numerical implementation. In this paper, some reference benchmarks, referred to an isotropic heterogeneous sample, are shown by comparing the solutions provided by strong and weak formulations. The strong formulation finite element method (SFEM), implemented in MATLAB®, is compared to the finite element method (FEM), given by COMSOL® Multiphysics, and the advantages of the two approaches are highlighted and discussed.

Masonry-like material with bounded shear stress

Abstract The constitutive equation of masonry-like materials has been generalized in order to account for a limit to the tangential component of the stress, that is proportional to the normal component. This result has been achieved by suitably modifying the convex set made up of all the admissible stresses. The constitutive equation obtained in this manner retains all the properties of normal elastic materials. The model developed has been implemented in the FEM computer code MADY, and applied to the study of masonry shear-walls and arches under monotonic in-plane loads. The reliability of the proposed model is checked through comparisons with some experimental results.

A continuum model of interlocking structural systems

Masonry systems made of interlocking square-cut stones have long been studied by mathematicians and architects. It so happens that, under appropriate boundary conditions, ashlars interlock, and a stable structure results. Recently, the idea that new masonry-like materials can be designed on the basis of this archetypal principle has been put forward and applications have been proposed under the name of topological interlocking materials. In this paper, a mathematical model is proposed, that describes these materials as a special class of continua with microstructure. The system is viewed as a continuous body, the material elements of which are ashlar blocks endowed with an interaction structure based, to within certain approximations, on their interlocking geometry. The case of rigid ashlars with purely plastic interactions is developed in detail.

Bounded shear stress in masonry-like bodies

In the paper, a hyperelastic material with limited tensile and compressive strength is considered. In addition, the maximum value of the tangential component of the stress that can be attained on each plane depends on the intensity of the normal component. Previously formulated for the plane case, the constitutive equation of this material is now extended to three-dimensional bodies, constituting henceforth a generalization of the well-known model of masonry-like materials. The proposed constitutive law has been implemented in the finite element code MADY and has been applied to the study of masonry panels under different load conditions.

A 3D Masonry-Like Model with Bounded Shear Stress

This paper deals with non linear elastic materials for which not all the stresses are admis-sible but only those which belong to the stress range, i.e. a closed and convex subset of the spaceof all symmetric tensors. The constitutive equation that has been formulated and explicitly solved issufficiently general to include, besides the so-called masonry-like materials, many others whose stressrange is obtained experimentally or is theoretically defined. The model, implemented into the finiteelement code MADY, has been used to analyze a masonry panel under a bi-directional monotonicallyincremental load and the obtained numerical results have been discussed.

Continuous and discrete models for masonry like material: A critical comparative study

Abstract The aim of this paper is to present a critical comparative review of different models that may be adopted for modelling the mechanical behaviour of masonry, with particular attention to microstructured models. Several continuous and discrete models are discussed. Such models are based on the following assumptions: i) the structure is composed of rigid blocks; ii) the mortar is modelled as an elastic material or an elastic interface. The rigid block hypothesis is particularly suitable for historical masonry, in which stone blocks may be assumed as rigid bodies. For this type of masonry, mortar thickness is negligible if compared with block size, hence it can be modelled as an interface. Masonry-like materials may be modelled taking into account their heterogeneity by adopting a heterogeneous Finite Element Model (FEM) or a Discrete Element Model (DEM). The former seems to be more representative of masonry, but it is computationally onerous and results interpretation may be difficult; the latter is limited to rigid block assumption and mortar joints modelled as interfaces. For this reason, continuous equivalent models may be suitable to investigate masonry behaviour. Continuum equivalent models provide, in an analytical form, constitutive functions, but Cauchy model may be not suitable to describe masonry behaviour due to not negligible size of heterogeneity (block size) with respect to masonry panel size. For this reason, micropolar equivalent continuum may be adopted. By reference to the existing literature, a simple and effective DEM is adopted, in which masonry is modelled as a ‘skeleton’ having a behaviour depending on forces and moments transferred between blocks through the interfaces (mortar joints). Moreover for the micropolar equivalent continuum, an ad hoc enriched homogenised FEM is formulated by means of triangular elements. The proposed numerical models represent two possible simple approaches for solving heterogeneous problems. Such models are developed both by means of fast numerical routines and do not require specific computer codes, whereas the heterogeneous FEM may be studied by adopting a traditional FE code. DEM and heterogeneous FEM are adopted to verify reliability and application field of Cauchy and micropolar continua. Moreover, sensitivity of micropolar model to the Representative Elementary Volume (REV) chosen is discussed. For these purposes, ad hoc FE models are adopted, with constitutive functions obtained from an identification procedure (both for Cauchy and micropolar continua). An extensive comparison between DEM, heterogeneous FEM and equivalent homogenous FEM is presented in some meaningful cases, taking into account also the effect of heterogeneity size on models behaviour.

Multi-scale and multi-physics modelling for complex materials

Complex materials, used in many engineering and technology fields or present in nature, are characterized by an internal structure, made of very different constituents arranged according to various geometries, and by generally non-linear and non-elastic constitutive behaviour. It is widely recognized that important macroscopic properties, such as the macroscopic stiffness and strength, are governed by processes that occur at one to several scales below the level of observation. A thorough understanding of how these processes influence the gross behaviour is key to the analysis of existing and the design of performance improved composite materials. The present volume is a special issue of invited papers focused on multiscale and multiphysics methodologies for the simulation the of complex materials behaviour. A primary aim of this collection is to give an overview of the interdisciplinary research oriented to the modelling of materials with complex microstructures and complex behaviour, thus providing a theoretical, computational and experimental framework for bridging the gap between mechanical engineering and material science. The papers represent a survey of several approaches proposed for studying materials with internal structure (at the nano, micro, meso-scale), ranging from fibre or particle composites up to porous, cementitious and masonry-like materials. Laminated, functionally graded materials and materials with random microstructure are covered. The basic multiscale modelling strategies, towards modern complex multiphase materials subjected to various types of mechanical, thermal loadings as well as environmental effects are included. While highlighting the underlying conceptual, theoretical and experimental bases concerning a wide range of complex constitutive models (plasticity, damage, fracture, etc.), attention is focused on the computational issues. In all the contributions mechanics is coupled with other concurrent physical T. Sadowski (&) Faculty of Civil Engineering and Architecture, Lublin University of Technology, Nadbystrzycka 40, 20-618, Lublin, Poland e-mail: t.sadowski@pollub.pl

Particulate random composites homogenized as micropolar materials

Many composite materials, widely used in different engineering fields, are characterized by random distributions of the constituents. Examples range from polycrystals to concrete and masonry-like materials. In this work we propose a statistically-based scale-dependent multiscale procedure aimed at the simulation of the mechanical behavior of a two-phase particle random medium and at the estimation of the elastic moduli of the energy-equivalent homogeneous micropolar continuum. The key idea of the procedure is to approach the so-called Representative Volume Element (RVE) using finite-size scaling of Statistical Volume Elements (SVEs). To this end properly defined Dirichlet, Neumann, and periodic-type non-classical boundary value problems are numerically solved on the SVEs defining hierarchies of constitutive bounds. The results of the performed numerical simulations point out the importance of accounting for spatial randomness as well as the additional degrees of freedom of the continuum with rigid local structure.

Derivation of microstructured continua from lattice systems via principle of virtual works: the case of masonry-like materials as micropolar, second gradient and classical continua

The description of the mechanical behaviour of brick/block masonry through equivalent continua is presented here as a paradigmatic example of the problem of gross modelling of discontinuous and heterogeneous materials as continua with microstructure. The approaches reported in the literature differ for the way identification of the continuum is carried out or the nature of the continuum itself. In this paper, continuous models equivalent to rigid particle systems with free or constrained rotations are derived within the general framework of the principle of virtual work. In particular, an integral equivalence procedure is used to derive micropolar, second gradient and classical models. The non-classical models have in the field equations non-standard kinematic and static descriptors accounting for the presence of the material internal structure. The differences in the material responses of the various continua are identified referring to their internal work formulas. For the reference material, it is shown that, unlike the Cauchy continuum, both micropolar and second gradient models are effective in the presence of load and geometrical singularities, which involve significant scale effects on the material response. On the other hand, the second gradient model, as well as the classical model, disregards the role of relative rotation between the local rigid rotation (macrorotation) and the microrotation, which is related to the presence of non-symmetric strains. This circumstance, significant in strongly anisotropic systems, allow us to point out the advantages of the micropolar modelling especially for orthotropic masonry assemblies made of elements of any size. These statements are discussed by means of selected numerical examples of masonry panels differing in size, shape and arrangement, under shear loading conditions.

Singular stress fields for masonry-like vaults

In the present paper, we apply the theorems of limit analysis to vaults modeled as masonry-like materials, that is, unilateral continuous bodies. On allowing for singular stresses, we consider statically admissible stress field concentrated on surfaces lying inside the masonry. Such structures are unilateral membranes, whose geometry is described a la Monge, and the equilibrium of them, under vertical loads, is formulated in the Pucher form. The problem is reduced to a singular partial differential equation of the second order where the shape f and the stress function F appear symmetrically. The unilateral restrictions require that the membrane surface lies in between the extrados and intrados surfaces of the vault and that the stress function be concave. Such a constraint is, in general, not satisfied on a given shape for given loads: in such a case, the shape has to be modified to fit the constraint. In a sense, the unilateral assumption renders the membrane an underdetermined structure that must adapt its shape in order to satisfy the unilateral restrictions. A number of simple examples are presented to illustrate how the method works.

Variational fracture mechanics to model compressive splitting of masonry-like materials

A regularized variational model of fracture mechanics for masonry-like materials has been recently proposed: this is based upon the competition between bulk-energy release and surface-energy production due to the nucleation and/or progression of cracks, assumed they can open in mode I only. This model is applied here to derive a theory of strength in confined masonry-like materials, where an inhomogeneous state of stress is due to heterogeneous inclusions or boundary constraints. The theory accords the phenomenon of rupture an energetic interpretation. Under tension, opening of mode I fractures at right angle to the axis of loading is clearly energetically favorable; under compression, the solid splits because in doing so the stress is released so to reduce the total energy. Numerical experiments have been performed for prismatic solids under fixed lateral confinement and increasing uniaxial tension or compression up to failure. Representative domains for the strength under biaxial stress are thus deduced.

Genesis of the multiscale approach for materials with microstructure

This paper presents an overview of the origin of multiscale approaches in mechanics. While the pioneer molecular models of linear elastic bodies by Navier, Cauchy and Poisson were contradicted by experiments, the phenomenological energetic approach by Green still seems suitable for simple materials only. Voigt’s molecular model, here reinterpreted in the light of contemporary mechanics, reconciled the two approaches providing a conceptual guideline for developing constitutive models based on a direct link between continuum and discrete solid mechanics. Such a theoretical background proves to be especially suitable for new complex materials. An example referred to masonry-like materials is given.