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In the present work, we present a linear poroelastic model for a quasi-static oscillating bilayer beam made by two thick strata in perfect bond at their interface, characterized by different elastic moduli and permeability, both directly influenced by porosity. The solution is built up by considering the structure as a three-dimensional object and is obtained analytically for the full coupling of mass balance and mechanical equations, by imposing weak boundary conditions on the sole forces emerging at the lateral outermost surfaces, neglecting the inertial effects. After verifying the accuracy of the solution by comparing theoretical outcomes and numerical results from Finite Element simulations, ad hoc sensitivity analyses were then carried out in order to investigate the effects on the stress distribution and fluid flow due to prescribed inhomogeneities related to variations of material and geometrical parameters across the beam thickness, by keeping constant the total volume of the porous solid skeleton and applying self-equilibrated and oscillating in time axial forces and bending moments at two opposite sides of the structure. Some dynamic analyses were also performed to highlight proper periods and frequencies ranges where pulse loadings ensured the quasi-static response of the system, however discussing interferences due to transient regimes for selected cases on the basis of numerical analyses. Finally, we show that, by playing with ratios between elastic moduli, thickness and porosity of the two layers, a number of counterintuitive or even non-conventional behaviors can be observed, which give rise to unexpected back-flow phenomena and anomalous fluid content distributions, as well as to stress peaks kindled by moving the layer interface and overall stiffening that could be all exploited for the design of new multi-objective poroelastic metamaterials.

In the present paper five beam models are presented to reproduce the experimental response of the End Notch Flexure (ENF) test. In particular, a specific ENF scheme is considered; it is made of two symmetric laminates partially bonded by a adhesive layer, whose thickness is not negligible. The failure occurs at the interface between one laminate and the adhesive, leading to an unsymmetrical failure; moreover, because of the presence of the adhesive layer, the two arms representing the initial defect of the ENF are not in contact. The experimental setup is described and five beam models are introduced to simulate the nonlinear response of the ENF scheme, characterized by an increasing improved kinematics. Failure is studied considering both linear and nonlinear elastic fracture mechanics, introducing in the latter case a very simple cohesive interface model. Solutions obtained by the five beam models are compared with the ones recovered with finite element (FE) analyzes and with the results of the experimental tests, remarking the reliability, the simplicity and the computational efficiency of the beam schemes. It is found that the improved kinematics and the cohesive fracture evolution play an important role in accurately reproducing the response of the ENF scheme.

In this paper, an enhanced Virtual Element Method (VEM) formulation is proposed for plane elasticity. It is based on the improvement of the strain representation within the element, without altering the degree of the displacement interpolating functions on the element boundary. The idea is to fully exploit polygonal elements with a high number of sides, a peculiar VEM feature, characterized by many displacement degrees of freedom on the element boundary, even if a low interpolation order is assumed over each side. The proposed approach is framed within a generalization of the classic VEM formulation, obtained by introducing an energy norm in the projection operator definition. Although such generalization may mainly appear to have a formal value, it allows to effectively point out the mechanical meaning of the quantities involved in the projection operator definition and to drive the selection of the enhanced representations. Various enhancements are proposed and tested through several numerical examples. Numerical results successfully show the capability of the enhanced VEM formulation to (i) considerably increase accuracy (with respect to standard VEM) while keeping the optimal convergence rate, (ii) bypass the need of stabilization terms in many practical cases, (iii) obtain natural serendipity elements in many practical cases, and (vi) effectively treat also nearly incompressible materials., 27 pages, 6 figures

In this paper, a new beam model based on a 5-parameter displacement field, accounting for an enhanced kinematics and able to reproduce the Poisson effect, is proposed. The displacement field enriches the classical three-parameter Timoshenko beam with two new parameters capable to simulate the shortening effect over the thickness. The related differential equations, derived from a variational formulation, are analytically solved and implemented in a deformation method approach, capable of solving structural problems of beams with general geometry, boundary and load conditions. Several numerical applications are developed, highlighting the characteristic of the proposed 5-parameters model and comparing the results with those obtained using the classical Bernoulli, Timoshenko and Reddy beam models. Numerical results show that, although for homogeneous beam the differences in terms of generalized stress and displacement are generally very small, the proposed model returns local stress enriched by the new parameters, with more significant differences where the Poisson effect is more pronounced.

In this paper, the Fiber Reinforced Cementitious Matrix (FRCM) material is considered as a composite material obtained embedding a fiber grid into the mortar matrix. The mechanical response of the FRCM is, hence, derived through an homogenization procedure for periodic composite materials, considering the nonlinear behavior of the constituents. In particular, the mortar is modeled introducing a nonlocal damage constitutive law, characterized by two different damage parameters, in tension and in compression, and by a plastic response in compression. A linear response is considered for the fiber grid, while the possible decohesion of the fiber from the matrix is accounted for introducing suitable interfaces. The numerical procedure is detailed and implemented in a finite element code. Several numerical applications are presented, investigating the tensile and shear response of the FRCM. For the tensile test, a comparison with experimental evidences is illustrated; moreover, a sensitivity analysis is performed investigating the influence of the nonlocal radius, the fracture energy and the fiber stiffness on the overall tensile response of the FRCM. Then, the shear response is reproduced, remarking the importance of the damage and plasticity in compression. Finally, the effect of the confinement on the shear response of the FRCM is also investigated.

A novel two-scale modeling approach, linking different structural models at macro and microscale, is proposed to describe response of masonry walls with periodic texture. At the higher macroscopic scale, the real heterogeneous material is modeled as a homogenized medium, considering the classical Mindlin-Reissner theory for flat shells. At the lower microscopic scale, a representative masonry Unit Cell (UC), accounting for the actual geometry, arrangement and nonlinear behavior of constituent materials, is analyzed in detail by resorting to a three-dimensional Cauchy model. The UC is modeled as the assembly of elastic bricks and nonlinear zero-thickness interfaces, in which the sliding frictional and damaging mechanisms are concentrated. To perform the macro–micro information transition a proper kinematic map is defined, whereas the upscaling process is performed via a homogenization procedure based on the Transformation Field Analysis (TFA), properly extended to the case of interfaces. The developed homogenization procedure invokes a generalized Hill-Mandel principle and requires to satisfy ‘non-standard’ constraints at the microlevel, for which the perturbed Lagrangian method is employed. Numerical applications are performed to prove the model efficiency in describing the response of a running bond UC subjected to in-plane and out-of-plane loads. Special attention is devoted to the analysis of shell bending and shear behavior, comparing the results obtained with the proposed model with those recovered by detailed micromechanical analyses.

In the present paper a mathematical model able to reproduce the mechanical response of a shape memory alloy (SMA) helical spring is presented. The proposed model is based on a simplified but effective theory of the helix that assumes small strains and large displacements. The kinematics of the helix and the related generalized strain measures are introduced, with stress resultants conjugated to them and the corresponding equilibrium equations of the curved beam. A specific constitutive law for this structural model is then presented for the SMA material, which accounts for the phase transformation due to the coupling of the shear and axial strains and to the effect of the temperature variation as well. Then, a numerical procedure, based on the backward Euler time integration algorithm and the prediction-correction technique for solving the nonlinear time step, is developed. Several numerical applications are presented to assess the reliability of the proposed SMA helix model and the implemented numerical procedure, and to investigate the response of the helix. Initially, the behavior of a circular SMA cross-section is studied, comparing the results obtained by the proposed modeling approach with the ones carried out by finite element analyses. Then, the helix structural element is considered and results are again compared with finite element outcomes. Finally, a sensitivity analysis is performed varying the pitch of the helix.

The paper presents two reduced order homogenization techniques for studying the response of nonlinear composite materials. The first approach is based on Transformation Field Analysis, which considers the presence of eigenstrains to account for inelastic strains, while the second approach is derived from Hashin–Shtrikman variational principle, which introduces eigenstresses, namely polarization stresses, on a homogeneous elastic reference material to account for the heterogeneity and the inelastic response. In particular, the case of elasto-plastic periodic composites is investigated. A Unit Cell is identified and divided in subsets. In each subset, the eigenstrains or eigenstresses are assumed uniform. A very effective technique is derived for the Hashin–Shtrikman approach updating the elastic reference material properties during the inelastic strain evolution. Numerical procedures are implemented to derive the nonlinear response of composites with plastic constituents. Several numerical applications are carried out to assess the effectiveness of the two presented reduced order models. A deep investigation on the differences and similarities of the two approaches is presented, proving their equivalence under particular circumstances. Simple and complex loading histories are considered, comparing the results with a finite element solution, considered as the reference solution.

In this paper, a first-order virtual element method for Reissner–Mindlin plates is presented. A standard displacement-based variational formulation is employed, assuming transverse displacement and rotations as independent variables. In the framework of the first-order virtual element, a piecewise linear approximation is assumed for both displacement and rotations on the boundary of the element. The consistent term of the stiffness matrix is determined assuming uncoupled polynomial approximations for the generalized strains, with different polynomial degrees for bending and shear parts. In order to mitigate shear locking in the thin-plate limit while keeping the element formulation as simple as possible, a selective scheme for the stabilization term of the stiffness matrix is introduced, to indirectly enrich the approximation of the transverse displacement with respect to that of the rotations. Element performance is tested on various numerical examples involving both thin and thick plates and different polygonal meshes.

Recently, many studies showed the opportunity to use a CO2 laser for treating the surfaces for structural bonding. In this work, a study on the effectiveness of a laser texturing on ENF bonded joints made with CFRP and epoxy adhesive has been developed. The experimental results showed that the use of this technology allows obtaining mechanical resistances of the bonded joints higher than 80% compared to untreated ones. Furthermore, a numerical model is proposed to forecast the response of the ENF specimens, and the numerical results are in good agreement with the experimental ones. This work represents a basis for the development of an innovative approach to design the laser texturing as a function of the applied stress.

Optimization of materials and structures is a crucial step in the design of man-made mechanical components for a wide field of engineering applications. It also plays a key role in mechanobiology of living systems, being involved by nature across the scales, from single-cell to tissues and organs, as a strategy to minimize metabolic cost and maximize biomechanical performances. The synergy between thecontinuously increasing developmentof high-resolution 3D printing technologies andthe possibility to predict chemical and physical properties through molecular dynamics-based numerical analyses hasrecently contributed to boost the use of both design and topology optimization procedures. They are employed in ab initio simulations as key strategies for deciding microstructures to improve mechanical performances and, concretely, to achieve prototypes of new material components. With this in mind, we here proposeto abandon the classical approach of using a single scalar objective function employed in the classical design and topology optimization strategies, to introduce multiple quantities to be minimized, identified as the differences between material yield stress and the maximum von Mises stress. After mathematically justifying the well-posedness of thisunconventionalchoicefor the case at hand, it is highlightedthat the proposedstrategy is based on the concept of "equalizing" a proper stress measure at any point of the body and, for this reason, it is baptized as Galilei’s optimization, in honor of the Italian scholar who somehow first wondered about the possibility of changing sizes of beams to have uniform internal forces and, in turn, minimum weight. By exploiting analytical solutions and ad hoc implementing a parametric finite element algorithm to be applied to a wide variety of solids with arbitrary complex structural geometries, including nested or hierarchically organized architectures, it is firstdemonstrated that the proposed optimization strategy roughlyretraces principles invoked by nature to guide growth, remodeling and shaping of biomaterials. More importantly, by means of several benchmark examples, we finally show the proposed procedure might be also helpfully employed to conceive a new class of micro-structured, eventually 3D-printed materials exhibiting surprising post-elastic properties, such as high overall resilience and toughness, in particular obtaining a decrease of stress concentration and a slowing down ofcrack propagation as direct effects of the optimization, which de facto minimizesstress gradients wherever in the solid domain.

Masonry structures, although classically suitable to withstand gravitational loads, are sensibly vulnerable if subjected to extraordinary actions such as earthquakes, exhibiting cracks even for events of moderate intensity compared to other structural typologies like as reinforced concrete or steel buildings. In the last half-century, the scientific community devoted a consistent effort to the computational analysis of masonry structures in order to develop tools for the prediction (and the assessment) of their structural behavior. Given the complexity of the mechanics of masonry, different approaches and scales of representation of the mechanical behavior of masonry, as well as different strategies of analysis, have been proposed. In this paper, a comprehensive review of the existing modeling strategies for masonry structures, as well as a novel classification of these strategies are presented. Although a fully coherent collocation of all the modeling approaches is substantially impossible due to the peculiar features of each solution proposed, this classification attempts to make some order on the wide scientific production on this field. The modeling strategies are herein classified into four main categories: block-based models, continuum models, geometry-based models, and macroelement models. Each category is comprehensively reviewed. The future challenges of computational analysis of masonry structures are also discussed.

Fracture is one of the most commonly encountered failure modes of engineering materials and structures. Prevention of cracking-induced failure is, therefore, essential to save lives and contain costs and should be considered a social commitment. However, the understanding of nucleation and propagation of complex crack patterns in real structures is still an open problem, for which traditional techniques are unable to provide satisfactory results. In this field, the variational formulation of fracture mechanics as an energy minimisation problem opened new perspectives; in particular, the phase field approach to fracture revealed to be a versatile and powerful tool for the investigation of crack problems. Despite the widespread use of plates and shells in engineering applications, the failure mechanisms of these kind of structures are quite poorly understood. As a matter of fact, few attempts have been devoted to the formulation and applications of the phase field approach to plates and shells. In this contribution an alternative phase field formulation is proposed, which relies on the subdivision of the thin (or slender) solid into several layers. In this way, while the mechanical behaviour of the solid is governed by classical theories of plates and shells, the phase field equation has to be satisfied within each layer, that is, in a domain with reduced dimension. The proposed procedure is validated and critically examined.

In this paper, the tensile and shear constitutive behavior of Fiber Reinforced Cementitious Matrix (FRCM) composites is addressed. A nonlinear finite-element computational approach is developed, by accounting for the micromechanical mechanisms via the introduction of interface elements inside and between the FRCM constituents, i.e. fibers and mortar. The damage, the friction and the unilateral contact are considered for reproducing the mortar cracking and the slippage between the fibers and the mortar. The numerical approach is illustrated and it is validated with an available experimental result, highlighting the effectiveness of such proposed modeling approach for reproducing the constitutive tensile behavior of FRCM materials. Finally, the shear behavior is investigated, also considering cyclic loading patterns under different compressive load levels.

Aim of the paper is to present a one dimensional simple model for the study of the bond behavior of Fabric Reinforced Cementitious Matrix (FRCM) strengthening systems externally applied to structural substrates. The equilibrium of an infinitesimal portion of the reinforcement and the mortar layers composing the strengthening systems allows to derive the governing equations. An analytical solution is determined solving the system of differential equations. In particular, in the first part of the paper a nonlinear shear-stress slip law characterized by a brittle post-peak behavior with a residual shear strength in the post peak phase is introduced for either the lower reinforcement-mortar interface (approach 1) or both the lower and the upper interface (approach 2). In the latter approach, a calibration of the shear strength of the upper interface is proposed in order to implicitly account for the effect of the damage of the mortar on the bond behavior. In the second part of the paper it is presented the solution of the problem in the case of softening behavior by approximating the shear-stress slip law throughout a step function. Comparisons with experimental data, available in literature, are presented in order to assess the reliability of the proposed approach.

In this paper a new numerical approach for determining admissible thrust curves for masonry arches is proposed. Arbitrary loading conditions, including distributed loads applied to the extrados and to the intrados of the arch, but also horizontal inertial forces simulating the effects of seismic actions are considered for arches of any geometry. The admissible solutions, corresponding to equilibrium thrust curves entirely contained in the thickness of the arch, are consistent with the lower bound theorem of Limit Analysis and, thus, are “safe” solutions from a structural point of view. The well-established Milankovitch's theory for the equilibrium of masonry arches is reviewed and generalized. Then, a specific formulation of the theory is presented, allowing the construction of an effective and efficient numerical procedure based on the Point Collocation Method and enriched by a constrained optimization routine. The latter is aimed at determining, among all the admissible equilibrium solutions, the optimal solution matching specific requirements of interest for applications, as the solution corresponding to the maximum or minimum thrust. The proposed procedure is discussed and validated with reference to the cases of circular, parabolic and pointed arches. In particular, maximum and minimum thrust solutions have been determined for all the examined cases.

This paper presents a kinematic enriched finite element model for nonlinear analysis of brick masonry walls loaded in their plane. The finite element accounts for the transverse deformation of the wall and permits to reproduce mortar-brick interaction in wall thickness direction. Non-local constitutive relationships are considered both for mortar and bricks, adopting a damage-friction law for the mortar and an isotropic damage model for the bricks, both accounting for tensile failure mechanisms. A numerical procedure is developed for evaluation of damage and friction in mortar and brick materials. Numerical applications are presented, comparing results obtained by the proposed finite element with experimental outcomes.

Several tools for the prediction and the assessment of the structural behavior of masonry buildings have been developed in recent decades. Numerical tools have been favorably developed and preferred over analytical approaches, given the complex mechanical response of masonry and the irregular geometries of historic masonry buildings. In this chapter, a thorough review of numerical strategies for the analysis of masonry structures is presented. Additionally, classification of these strategies is also suggested to logically organize the extensive literature on this topic. Even though a wholly congruent categorization of all the numerical tools is essentially unrealistic given the specific aspects of each solution developed, the existing numerical strategies are subdivided into four classes: block-based models, continuum models, geometry-based models, and macroelement models. Each class is thoroughly reviewed and the open challenges in numerical modeling of masonry structures are critically examined.

Aim of the present study is to propose an enhanced method for the domain decomposition (clustering) of the representative volume element (RVE) of composite materials to be used with homogenization techniques, based on the PieceWise Uniform Transformation Field Analysis (PWUTFA). With PWUTFA, both constitutive and evolutive equations for the constituents of the composite material are written in terms of averages in each cluster; moreover, it is not required to solve via FEM the nonlinear micro-mechanical problem, allowing to drastically reduce the number of internal variables. PWUTFA is founded on the idea that it is possible to divide the RVE into large subdomains (clusters) that should group together finite elements having, under any applied loading condition, the most similar values of strain. Accordingly, in this study a multi-objective optimization-based approach is proposed with the aim to simultaneously reduce both the error in the approximation of the strain fields and the number of clusters in which the domain is decomposed. Different clustering solutions, obtained through the proposed optimization approach are analyzed, and the corresponding mechanical responses are compared with the ones obtained by the finite element analysis and by uniform transformation field analysis.

This chapter presents a survey of the homogenization and multiscale analysis of periodic, masonry walls subjected to in-plane loading conditions. The scale of the structure, macroscale, and that of the material, microscale, are introduced, invoking the assumption of the separation of the scales. Two-dimensional (2D) Cauchy and Cosserat models are adopted at the macroscale to describe the response of the wall. A unit cell (UC) is introduced and analyzed adopting three-dimensional, 2D, and enriched plane state Cauchy models. A damage-plastic model is considered for the mortar joints, accounting for the unilateral effect due to reclosure of the mortar joint and friction occurring when the microcracks are closed. A different damage model is introduced for bricks capable of properly describing their compressive failure. The homogenization for deriving the overall nonlinear response of the UC is performed by developing a piecewise transformation field analysis technique. Several numerical applications are presented to assess the material modeling and the effectiveness of the adopted homogenization technique and that of the multiscale approach. These concern the response of the UC to verify, on the one hand, the ability of the constitutive laws proposed for brick and mortar to describe the response of the masonry and, on the other hand, the performance of the homogenization technique. Then, structural analyses of masonry elements are presented, comparing the results obtained using different modeling approaches among them and with available experimental evidence. Finally, concluding remarks are given.

This paper presents two micromechanical and a multiscale two-dimensional finite element models for the analysis of masonry walls under out-of-plane instability effects. The micromechanical analysis are performed considering elastic beams to model the bricks and either nonlinear beams or interfaces to model the mortar layers. The beam finite elements rely on the force-based formulation and account for large displacements by making use of the corotational approach. This latter is properly formulated and extended to interface elements to include nonlinear geometry effects. The multiscale model is defined by applying a two-scale beam-to-beam homogenization procedure, developed for masonry elements with periodic bricks arrangements. Hence, a Unit Cell made of a single linear elastic brick and a single nonlinear mortar layer is introduced at the microscopic level, which is linked to the macroscale level through a semi-analytical homogenization technique. In all models, a damage formulation with friction plasticity governs the mortar constitutive relationship. Computational details on model implementation and solution procedures are given for all approaches. Correlation studies are performed to assess the proposed numerical micromechanical and multiscale procedures. In particular, the behavior of an unreinforced wall tested under vertically eccentric load is reproduced and advantages and disadvantages of proposed approaches are discussed.

Recent experimental studies involving Fabric Reinforced Cementitious Matrix (FRCM) strengthening systems externally applied to structural supports have underlined important aspects characterizing their response. In particular, failure mechanisms quite different from the ones emerged in case of traditional Fiber Reinforced Polymers (FRP) have been observed. These mechanisms particularly underline the role of additional phenomena to necessarily consider for the study and the development of theoretical models/design formulas specific for FRCMs. Aim of the present paper is to analyze the role of different aspects characterizing the behavior of FRCM strengthening systems externally applied on structural supports. Among these, it is mainly considered the role of the upper mortar layer, and, in particular, the influence of its possible damage state during the interaction mechanism between the reinforcement and the matrix. With this aim, a simple theoretical model is here presented. It accounts for the interaction between the reinforcement and the mortar matrix at the level of interface, by considering both the de-bonding and the tensile failure of mortar phenomena, which generally affect the resistant mechanism of FRCMs. The results presented in the paper underline the role of the different components of the strengthening system and their influence both in terms of local and global response.

International audience; The present paper deals with the modeling of bonded interfaces adopting the asymptotic expansion technique. The equilibrium problem of a composite body made of two adherents issn perfect contact with an elastic interface is considered and a classical rescaling technique is introduced. The asymptotic expansion method is reviewed; in fact, the representation form for the displacement and stress vector fields are introduced and tsshe effect of higher order terms is taken into account. Using the classical scheme of matched asymptotic expansions, the interface conditions are obtained. The cases of hard and soft interfaces are considered: the first is derived assuming the elasticity coefficients independent of the adherent thickness, the second considers the elasticity properties linearly depending on the thickness. Numerical investigations are performed in the framework of the finite element method. In particular, comparisons of the results obtained by modeling the adhesive as a continuum material (discretized in finite elements even in the thickness) with the results carried out using hard, soft interface models at the first and higher order expansion are performed.

Recent experimental studies involving Fabric Reinforced Cementitious Matrix (FRCM) strengthening systems externally applied to structural supports have underlined important aspects characterizing their response. In particular, failure mechanisms quite different from the ones emerged in case of traditional Fiber Reinforced Polymers (FRP) have been observed. These mechanisms particularly underline the role of additional phenomena to necessarily consider for the study and the development of theoretical models/design formulas specific for FRCMs. Aim of the present paper is to analyze the role of different aspects characterizing the behavior of FRCM strengthening systems externally applied on structural supports. Among these, it is mainly considered the role of the upper mortar layer, and, in particular, the influence of its possible damage state during the interaction mechanism between the reinforcement and the matrix. With this aim, a simple theoretical model is here presented. It accounts for the interaction between the reinforcement and the mortar matrix at the level of interface, by considering both the de-bonding and the tensile failure of mortar phenomena, which generally affect the resistant mechanism of FRCMs. The results presented in the paper underline the role of the different components of the strengthening system and their influence both in terms of local and global response.

In this paper a new multiscale approach is presented for the analysis of structures made of composite material characterized by elastoplastic or viscoplastic nonlinear response. Scale separation is assumed so that the homogenization theory can be applied: at the structural scale (macroscale) the material appears as homogeneous, while at the microscale it is characterized by a heterogeneous microstructure that affects the global behavior of the structure. The resolution of the micromechanical problem is performed developing a reduced order homogenization technique based on a mixed variational formulation, named Mixed Transformation Field Analysis (MxTFA). The microscopic reduced internal variables are the stress and plastic multiplier parameters of RVE subsets, whose evolution is computed enforcing the weak form of the governing equations for every subset. Some numerical tests are performed to show the accuracy and efficiency of the proposed multiscale technique.

In this paper the capacity of an innovative composite basalt grid with inorganic matrix (FRCM) has been evaluated both in terms of repairing pre-damaged and strengthening clay brick walls under out-of-plane loads. Experimental tests have been performed on full scale clay brick walls subjected to out-of-plane loads. A wall, damaged after a test, has been repaired by means of basalt FRCM. A similar wall has been tested directly, without pre-damage, after strengthening by means of FRCM. This allowed to remark the effect of retrofitting pre-damaged and new walls. To simulate a non-uniform out-of-plane behaviour of the wall, two adjacent edges of the wall have been constrained and the other two were left free while a pointwise normal force has been applied at the free opposite corner of the wall. The purpose of this work was to assess the potentiality of FRCM to recover the capacity of a wall after significant damage and to increase the global response of strengthened wall not previously damaged. The experimental results demonstrated that the externally bonded strengthening was able to prevent a brittle failure and it was not affected by debonding; ultimate load of the retrofitted wall almost doubled with respect to the unreinforced configuration, despite complex stress state, and that the failure was governed by shear sliding at higher displacement levels.

The present work deals with the formulation of a virtual element method for two dimensional structural problems. The contribution is split in two parts: in part I, the elastic problem is discussed, while in part II (Artioli et al. in Comput Mech, 2017) the method is extended to material nonlinearity, considering different inelastic responses of the material. In particular, in part I a standardized procedure for the construction of all the terms required for the implementation of the method in a computer code is explained. The procedure is initially illustrated for the simplest case of quadrilateral virtual elements with linear approximation of displacement variables on the boundary of the element. Then, the case of polygonal elements with quadratic and, even, higher order interpolation is considered. The construction of the method is detailed, deriving the approximation of the consistent term, the required stabilization term and the loading term for all the considered virtual elements. A wide numerical investigation is performed to assess the performances of the developed virtual elements, considering different number of edges describing the elements and different order of approximations of the unknown field. Numerical results are also compared with the one recovered using the classical finite element method.

In this paper, a numerical method for assessing the safety of masonry arches is presented. The catenary and the thrust curve of existing arch differential equations are revised.The variational formulation of the differential equations of the catenary and of the thrust curve is presented with its finite element approximation. The optimization problem, solving the nonlinear constrained minimization which allows determining the thrust curve in equilibrium with a given load for a prescribed geometry, corresponding to the minimum value of the thrust force is discussed. Numerical examples are presented in order to assess the ability of the presented numerical procedure in solving the problem of the determination of the thrust curve, comparing the analytical, when available, with the numerical solutions. The values of the minimum and maximum allowable thrust forces are also provided.

The present paper deals with the formulation of an innovative interface model characterized by nonlinear response. The interface is assumed to be composed of several thin layers, one of which is characterized by nonlinear response. In this layer, denoted throughout the paper as detachment layer, the nonlinearity is modeled adopting a damage law and the decohesion process occurs. The kinematics of the interface is enriched accounting for the in-plane strain component; in such a way, the presence of the stress component arising in the plane of the interface can be computed and taken into account for the evaluation of the nonlinear response of the detachment layer. In fact, when in-plane compressive stresses are present a confinement effect occurs, slowing down the damage growth. On the contrary, when tensile in-plane stress is present in the detachment layer, the damage evolves more rapidly. The interface model is implemented in a finite element computer code and a numerical procedure is proposed. Two numerical applications are presented to assess the ability of the proposed interface model and of the implemented numerical procedure to reproduce specific decohesion problems. The first application deals with the debonding of a FRP reinforcement from a concrete support; comparison of the numerical results with available experimental evidences are given. The second application concerns the mixed mode decohesion in an edge-notch four-point bending specimen.

The present paper deals with the homogenization problem of periodic composite materials, considering a Cosserat continuum at the macro-level and a Cauchy continuum at the micro-level. Consistently with the strain-driven approach, the two levels are linked by a kinematic map based on a third order polynomial expansion. Because of the assumed regular texture of the composite material, a Unit Cell (UC) is selected; then, the problem of determining the displacement perturbation fields, arising when second or third order polynomial boundary conditions are imposed on the UC, is investigated. A new micromechanical approach, based on the decomposition of the perturbation fields in terms of functions which depend on the macroscopic strain components, is proposed. The identification of the linear elastic 2D Cosserat constitutive parameters is performed, by using the Hill–Mandel technique, based on the macrohomogeneity condition. The influence of the selection of the UC is analyzed and some critical issues are outlined. Numerical examples for a specific composite with cubic symmetry are shown.

The present paper deals with the problem of the detachment of fiber reinforced polymer (FRP) from a masonry support. In particular, a new interphase model simulating the mechanical bond between masonry and FRP is proposed. It is obtained as a system of two interfaces: one reproduces the response of the glue and of the skin-deep layer and it is characterized by a linear elastic behavior whereas the second describes the nonlinear response of the detachment layer. The effect of the possible confinement due to the stress components acting in the plane of the detachment layer is also considered. Two numerical applications are reported. The first illustrates a comparison between the experimental laboratory tests and numerical simulations, showing the good performances of the proposed model. The second application is specifically devoted to the investigation of the effects of the presence of mortar joints in the detachment mechanism.

In this paper the mechanical response of porous Shape Memory Alloy (SMA) is modeled. The porous SMA is considered as a composite medium made of a dense SMA matrix with voids treated as inclusions. The overall response of this very special composite is deduced performing a micromechanical and homogenization analysis. In particular, the incremental Mori---Tanaka averaging scheme is provided; then, the Transformation Field Analysis procedure in its uniform and nonuniform approaches, UTFA and NUTFA respectively, are presented. In particular, the extension of the NUTFA technique proposed by Sepe et al. (Int J Solids Struct 50:725---742, 2013) is presented to investigate the response of porous SMA characterized by closed and open porosity. A detailed comparison between the outcomes provided by the Mori---Tanaka, the UTFA and the proposed NUTFA procedures for porous SMA is presented, through numerical examples for two- and three-dimensional problems. In particular, several values of porosity and different loading conditions, inducing pseudoelastic effect in the SMA matrix, are investigated. The predictions assessed by the Mori---Tanaka, the UTFA and the NUTFA techniques are compared with the results obtained by nonlinear finite element analyses. A comparison with experimental data available in literature is also presented.

The present study deals with the determination of the nonlinear response of the masonry regarded as a regular, i.e. periodic, composite material made of bricks and mortar. A homogenization procedure is applied deriving the masonry overall mechanical response on the basis of the study of a single unit cell. An enriched plane state kinematic model including the effect of the transversal strains of the masonry is presented. This is a simplified form of the full three-dimensional approach. Different cohesive constitutive models are introduced for the brick and mortar; in particular, the frictional effect, playing an important role in the masonry response, is accounted for in the mortar joints. Two main issues are addressed: (a) different structural models are considered at macro- and micro-scale: the macro-model is formulated in the two-dimensional plane state context, while the enriched plane state kinematic approach is adopted at the microlevel; (b) a nonlocal integral strain technique, able to overcome the classical localization drawbacks due to the softening response of the masonry constituents, is developed for the case of periodic media. Numerical applications are presented to assess the effectiveness of the proposed modeling approach.

The present study proposes a homogenization technique for the estimation of the overall behavior of composite materials characterized by a nonlinear behavior. A mixed nonuniform TFA procedure is proposed in order to study the mechanical response of periodic composites characterized by linear isotropic hardening plasticity. As the case of periodic composites is considered, the homogenization is performed on a repetitive unit cell that plays the equivalent role of representative volume element for random media. A numerical application is performed to test the effectiveness of the proposed technique. In particular, the homogenization results are compared with the ones carried out by micromechanical nonlinear finite element analyses.

In this paper, a new mixed nonuniform Transformation Field Analysis (TFA) homogenization technique for deriving the overall mechanical response of heterogeneous materials is proposed. The new TFA approach is based on a mixed stress variational formulation of the elasto-plasticity theory, assuming stresses and plastic multiplier as independent variables. The case of periodic composites is considered, so that the homogenization is performed on a repetitive unit cell that plays the equivalent role of representative volume element for random media. The unit cell of the material is divided into subsets and in each subset the weak form of the compatibility condition and of the consistency condition are enforced, leading to a simultaneous evaluation of the stress and of the plastic strain field. It is assumed that the stress field satisfies a priori the equilibrium equations while the inelastic strain distribution is evaluated as function of the assumed stress resulting nonuniform in each subset. Some numerical examples are performed in the framework of plane stress plasticity in order to assess the effectiveness of the proposed technique. In particular, the homogenization results are compared with the ones carried out by micromechanical nonlinear finite element analyses.

This paper deals with the nonlinear analysis of masonry walls loaded in their plane. The masonry is regarded as a composite material made of bricks joined by mortar. To correctly reproduce the mortar–brick interaction in the direction of the thickness of the wall, an enriched kinematic model is proposed, so that the model is able, in a feasible form, to account for the out-of-plane strains due to the in-plane loading acting on the wall. Nonlocal nonlinear constitutive laws are considered both for the mortar and the bricks. In particular, a damage-friction law is considered for the mortar, while a damage model with two alternative yield functions is proposed for the bricks, both based on a tensile failure mechanism. A 2D finite element (FE) accounting for the three-dimensional kinematic effect is developed. This is implemented in a numerical procedure based on the backward Euler step-by-step time integration of the constitutive evolution laws and on the predictor-corrector algorithm for the solution of the nonlinear problem. Five applications are presented to highlight the effectiveness of the proposed nonlinear model and the implemented FE procedure. The first aims to show the model’s ability to reproduce the failure of the masonry for transversal damage; the others deal with comparisons with classical small scale and structural scale experimental tests.

Aim of the present paper is the development of a homogenization technique able to determine the overall mechanical response of composite materials taking into account the cracking and plastic behavior of its constituents and the decohesion process among them. A representative volume element of a composite material is studied. The plastic effects in the constituents are considered introducing a plastic model with isotropic and kinematic hardening. The debonding between constituents and the cracking process are described introducing a cohesive damage interface model that takes into account also the unilateral contact and frictional effects. In particular, a new procedure based on the nonuniform Transformation Field Analysis is presented. The plastic strains in the constituents and the inelastic relative displacements along the interfaces are approximated as linear combinations of inelastic scalar modes that are functions of the spatial variables. The coefficients of the linear combinations are the internal variables of the problem that are computed solving the evolutive problem. Some numerical applications are carried out to verify the efficiency of the proposed homogenization approach in reproducing the overall mechanical response of composites characterized by cracking and plastic phenomena in the constituents and debonding between them. The homogenization results are compared with the solution obtained by micro-mechanical nonlinear finite element analyses.

A cohesive zone model is formulated to describe the mechanics of initiation and propagation of cracks and the associated asperity degradation and nonlinear dilation along structural interfaces of quasi-brittle materials, such as concrete, rocks and masonry, subjected to monotonic or cyclic loading. Using a two-scale approach, a cohesive-law is determined at each point of a smooth macroscale interface by resolving a problem at the micro-scale for a representative interface area (RIA), where the geometry of the asperities is modelled using three differently inclined microplanes. On each microplane a cohesive-frictional cohesive law is then used. In this paper, the finite depth of the asperities is accounted for by considering the progressive reduction in contact area between each couple of interfacing microplanes for increasing opening (macro-scale) relative-displacement. Furthermore, the rupture of the asperities and associated flattening of the fracture surface is captured by a progressive reduction of the inclination angles of the microplanes in the RIA. Numerical examples are reported to assess the sensitivity of the shear-stress slip curves and of the nonlinear dilation upon the geometry of the asperities in the RIA. Numerical–experimental comparisons are then presented to illustrate the predictive capability of the model in simulating granite rock joints subjected to monotonic and cyclic shear loading and the concrete-bar interaction in a pull-out test.