Cosserat model for periodic masonry deduced by nonlinear homogenization

The paper deals with the problem of the determination of the in-plane behavior of periodic masonry material. The macromechanical equivalent Cosserat medium, which naturally accounts for the absolute size of the constituents, is derived by a rational homogenization procedure based on the Transformation Field Analysis. The micromechanical analysis is developed considering a Cauchy model for masonry components. In particular, a linear elastic constitutive relationship is considered for the blocks, while a nonlinear constitutive law is adopted for the mortar joints, accounting for the damage and friction phenomena occurring during the loading history. Some numerical applications are performed on a Representative Volume Element characterized by a selected commonly used texture, without performing at this stage structural analyses. A comparison between the results obtained adopting the proposed procedure and a nonlinear micromechanical Finite Element Analysis is presented. Moreover, the substantial differences in the nonlinear behavior of the homogenized Cosserat material model with respect to the classical Cauchy one, are illustrated.     

Homogenization of heterogeneous masonry beams

This study presents a two-scale model to describe the out-of-plane masonry response. One-dimensional (1D) structural elements, like masonry columns or strips of long wall characterized by the periodic repetition of bricks and mortar arranged in stack bond, are considered. A damage-friction plasticity law is adopted to model the mortar joint constitutive response, while the bricks are assumed as linear elastic. A 1D beam formulation is introduced at both the structural and micromechanical scale, linking the two levels by means of a kinematic map. This expresses the microscopic beam strains in the masonry unit cell (UC) as function of the macroscopic generalized strains. The kinematic field in the UC is completed by adding an unknown periodic fluctuation term. A nonlinear homogenization procedure is developed, proposing a semi-analytical solution for the micromechanical problem, based on the fiber discretization of the mortar joints. A force-based beam-column finite element procedure is adopted at the structural scale and the solution algorithm for the element state determination is illustrated in details. Some numerical applications, showing the UC constitutive response and the behavior of masonry structural elements, are finally presented.     

Compatible model for herringbone bond masonry: Linear elastic homogenization,failure surfaces and structural implementation

A simplified kinematic procedure at a cell level is proposed to obtain in-plane elastic moduli and macroscopic masonry strength domains in the case of herringbone masonry. The model is constituted by two central bricks interacting with their neighbors by means of either elastic or rigid-plastic interfaces with friction, representing mortar joints. The herringbone pattern is geometrically described and the internal law of composition of the periodic cell is defined.A sub-class of possible elementary deformations is a-priori chosen to describe joints cracking under in-plane loads. Suitable internal macroscopic actions are applied on the Representative Element of Volume (REV) and the power expended within the 3D bricks assemblage is equated to that expended in the macroscopic 2D Cauchy continuum. The elastic and limit analysis problem at a cell level are solved by means of a quadratic and linear programming approach, respectively.To assess elastic results, a standard FEM homogenization is also performed and a sensitivity analysis regarding two different orientations of the pattern, the thickness of the mortar joints and the ratio between block and mortar Young moduli is conducted. In this way, the reliability of the numerical model is critically evaluated under service loads.When dealing with the limit analysis approach, several computations are performed investigating the role played by (1) the direction of the load with respect to herringbone bond orientation, (2) masonry texture and (3) mechanical properties adopted for joints.At a structural level, a FE homogenized limit analysis is performed on a masonry dome built in herringbone bond. In order to assess limit analysis results, additional non-linear FE analyses are performed, including a full 3D numerical expensive heterogeneous approach and models where masonry is substituted with an equivalent macroscopic material with orthotropic behavior and possible softening. Reliable predictions of collapse loads and failure mechanisms are obtained, meaning that the approach proposed may be used by practitioners for a fast evaluation of the effectiveness of herringbone bond orientation.     

Closed-form expressions for the macroscopic flexural rigidity coefficients of periodic brickwork

Approximate expressions for the macroscopic out-of-plane elastic coefficients of brick masonry with a regular pattern are derived in closed form using a homogenization approach for periodic media. Following an approach similar to the Method of Cells for fiber reinforced composites, a (piecewise-)differentiable expression depending on very a limited number of degrees of freedom and fulfilling suitable periodicity conditions is proposed for the microscopic transverse displacement field over any Representative Volume Element (RVE). Some of the equilibrium conditions at the interfaces between bricks and mortar joints are also fulfilled. By averaging the moment and curvature fields over the RVE, the macroscopic bending stiffness coefficients can be explicitly obtained. Using the FE solution of a masonry panel subjected to elementary load conditions as a benchmark, the proposed approach is found to accurately match the numerically obtained stiffness coefficients, for masonry elements of different geometry and different mechanical properties. In several instances, the proposed expressions agree with the numerical predictions better than other analytical expressions available in the literature.     

Masonry walls strengthened with composite materials – dynamic out-of-plane behavior

The dynamic behavior of masonry walls strengthened with composite materials and subjected to dynamic out-of-plane loading is analytically investigated. The analytical model derived in the paper focuses on one-way action through the height of the wall and it is based on dynamic equilibrium and the compatibility conditions between the structural components (masonry units, mortar joints, FRP reinforcement, and adhesive layers). The cracking and the nonlinear and dissipative behavior of the mortar material, the breathing of cracks, the rocking phenomenon, the development of arching forces, the interaction between the existing wall and the composite system, and the formation of debonded zones near the cracked mortar joints are considered in the nonlinear dynamic analysis. A numerical study that examines the capabilities of the model, quantifies the response of the strengthened wall to dynamic loads such as free and forced vibrations and seismic base excitation, and compares it to the response of the unstrengthened wall is presented. A summary and conclusions close the paper.     

A nonlinear homogenization procedure for periodic masonry

The present paper deals with the problem of the determination of the in-plane behavior of masonry material. The masonry is considered as a composite material composed by a regular distribution of blocks connected by horizontal and vertical mortar joints. The overall constitutive relationships of the regular masonry are derived by a rational micromechanical and homogenization procedure. Linear elastic constitutive relationship is considered for the blocks, while a new special nonlinear constitutive law is proposed for the mortar joints. In particular, a mortar constitutive law, which accounts for the coupling of the damage and friction phenomena occurring during the loading history, is proposed; the developed model is based on an original micromechanical analysis of the damage process of the mortar joint. Then, an effective nonlinear homogenization procedure, representing the main novelty of the paper, is proposed; it is based on the transformation field analysis, using the technique of the superposition of the effects and the finite element method. The presented methodology is implemented in a numerical code. Finally, numerical applications are performed in order to assess the performances of the proposed procedure in reproducing the mechanical behavior of masonry material.     

Meshless meso-modeling of masonry in the computational homogenization framework

In the present study a multi-scale computational strategy for the analysis of structures made-up of masonry material is presented. The structural macroscopic behavior is obtained making use of the Computational Homogenization (CH) technique based on the solution of the Boundary Value Problem (BVP) of a detailed Unit Cell (UC) chosen at the mesoscale and representative of the heterogeneous material. The attention is focused on those materials that can be regarded as an assembly of units interfaced by adhesive/cohesive joints. Therefore, the smallest UC is composed by the aggregate and the surrounding joints, the former assumed to behave elastically while the latter show an elastoplastic softening response. The governing equations at the macroscopic level are formulated in the framework of Finite Element Method (FEM) while the Meshless Method (MM) is adopted to solve the BVP at the mesoscopic level. The material tangent stiffness matrix is evaluated at both the mesoscale and macroscale levels for any quadrature point. Macroscopic localization of plastic bands is obtained performing a spectral analysis of the tangent stiffness matrix. Localized plastic bands are embedded into the quadrature points area of the macroscopic finite elements. In order to validate the proposed CH strategy, numerical examples relative to running bond masonry specimens are developed.     

Nonlinear parameter identification of models for masonry

This paper investigates the parameter estimation problem for brick masonry models. An identification procedure is proposed in which the uncertainties of known parameters and/or errors of measurements are its elements of distinction. The minimization process of the discrepancies between experimental data and theoretical measurements takes place by means of a first order iterative method. The identification procedure is applied to two different problems: the calibration of an interface model for brick–mortar joint in its functional form through monotonic experimental tests; to evaluate the unknown parameters of a continuum model for brick masonry walls in its non-holonomic form by means of in-plane cyclic shear–compression test of masonry panels. The general framework of the non-linear estimate methodology, the parameter identification problems and the numerical results are presented.     

Homogenization towards a mechanistic Rigid Body and Spring Model (HRBSM) for the non-linear dynamic analysis of 3D masonry structures

A mechanistic model with rigid elements and interfaces suitable for the non-linear dynamic analysis of full scale 3D masonry buildings is presented. The model relies into two steps: in the first step, a simplified homogenization is performed at the meso-scale to deduce the mechanical properties of a macroscopic material, to be used in structural applications; the second step relies into the implementation of a Rigid Body and Spring Model (RBSM) constituted by rigid elements linked with homogenized interfaces. In the homogenization step, a running bond elementary cell is discretized with 24 three-node plane-stress elastic triangular elements and non-linear interfaces representing mortar joints. It is shown how the mechanical problem in the unit cell is characterized by few displacement variables and how homogenized stress–strain curves can be evaluated by means of a semi-analytical approach. The second step relies on the implementation of the homogenized curves into a RBSM, where an entire masonry structure can be analyzed in the non-linear dynamic range through a discretization with rigid elements and inelastic interfaces. Non-linear structural analyses are conducted on a church façade interconnected with a portion of the perpendicular walls and on a small masonry building, for which experimental and numerical data are available in the literature, in order to show how quite reliable results may be obtained with a limited computational effort.     

Computational multiscale methods for granular materials

The fine-scale heterogeneity of granular material is characterized by its polydisperse microstructure with randomness and no periodicity. To predict the mechanical response of the material as the microstructure evolves, it is demonstrated to develop computational multiscale methods using discrete particle assembly-Cosserat continuum modeling in micro- and macro- scales, respectively. The computational homogenization method and the bridge scale method along the concurrent scale linking approach are briefly introduced. Based on the weak form of the Hu-Washizu variational principle, the mixed finite element procedure of gradient Cosserat continuum in the frame of the second-order homogenization scheme is developed. The meso-mechanically informed anisotropic damage of effective Cosserat continuum is characterized and identified and the microscopic mechanisms of macroscopic damage phenomenon are revealed.     

A Double-Strand Elastic Rod Theory

Motivated by applications in the modeling of deformations of the DNA double helix, we construct a continuum mechanics model of two elastically interacting elastic strands. The two strands are described in terms of averaged, or macroscopic, variables plus an additional small, internal or microscopic, perturbation. We call this composite structure a birod. The balance laws for the macroscopic configuration variables of the birod can be cast in the form of a classic Cosserat rod model with coupling to the internal balance laws through the constitutive relations. The internal balance laws for the microstructure variables also take a mathematical form analogous to that for a Cosserat rod, but with coupling to the macroscopic system through terms corresponding to distributed force and couple loads.     

Micromechanical analysis of heterogeneous materials subjected to overall Cosserat strains

In the framework of the computational homogenization procedures, the problem of coupling a Cosserat continuum at the macroscopic level and a Cauchy medium at the microscopic level, where a heterogeneous periodic material is considered, is addressed. In particular, non-homogeneous higher-order boundary conditions are defined on the basis of a kinematic map, properly formulated for taking into account all the Cosserat deformation components and for satisfying all the governing equations at the micro-level in the case of a homogenized elastic material. Furthermore, the distribution of the perturbation fields, arising when the actual heterogeneous nature of the material is taken into account, is investigated. Contrary to the case of the first-order homogenization where periodic fluctuations arise, in the analyzed problem more complex distributions emerge.     

梯度增强Cosserat连续体的广义Hill定理

基于经典Cauchy连续体的Hill定理,在平均场理论的框架下导出了梯度增强Cosserat连续体细、宏观均匀化方法的广义Hill定理。在梯度增强Cosserat连续体中,不仅宏观样条点上的应变和应力张量,而且它们的梯度均作用于与该样条点相关联的细观表征元（RVE）。依据此广义Hill定理,对梯度增强Cosserat连...     

A simple meso-macro model based on SQP for the non-linear analysis of masonry double curvature structures

A 3D model for the evaluation of the non-linear behavior of masonry double curvature structures is presented. In the model, the heterogeneous assemblage of blocks is substituted with a macroscopically equivalent homogeneous non-linear material. At the meso-scale, a curved running bond representative element of volume (REV) constituted by a central block interconnected with its six neighbors is discretized through of a few six-noded rigid wedge elements and rectangular interfaces. Non linearity is concentrated exclusively on joints reduced to interface, exhibiting a frictional behavior with limited tensile and compressive strength with softening. The macroscopic homogenous masonry behavior is then evaluated on the REV imposing separately increasing internal actions (in-plane membrane actions, meridian and parallel bending, torsion and out-of-plane shear). This simplified approach allows to estimate heuristically the macroscopic stress–strain behavior of masonry at the meso-scale. The non-linear behavior so obtained is then implemented at a structural level in a novel FE non-linear code, relying on an assemblage of rigid infinitely resistant six-noded wedge elements and non-linear interfaces, exhibiting deterioration of the mechanical properties.Several numerical examples are analyzed, consisting of two different typologies of masonry arches (a parabolic vault and an arch in a so-called “skew” disposition), a ribbed cross vault, a hemispherical dome and a cloister vault. To fully assess numerical results, additional non-linear FE analyses are presented. In particular, a simplified model is proposed, which relies in performing at a structural level a preliminary limit analysis – which allows to identify the failure mechanism – and subsequently in modeling masonry through elastic elements and non-linear interfaces placed only in correspondence or near the failure mechanism provided by limit analysis. Simulations performed through an equivalent macroscopic material with orthotropic behavior and possible softening are also presented, along with existing experimental evidences (where available), in order to have a full insight into the capabilities and limitations of the approach proposed.     

Kinematic FE limit analysis homogenization model for masonry walls reinforced with continuous FRP grids

A homogenization model for periodic masonry structures reinforced with continuous FRP grids is presented. Starting from the observation that a continuous grid preserves the periodicity of the internal masonry layer, rigid-plastic homogenization is applied directly on a multi-layer heterogeneous representative element of volume (REV) constituted by bricks, finite thickness mortar joints and external FRP grids. In particular, reinforced masonry homogenized failure surfaces are obtained by means of a compatible identification procedure, where each brick is supposed interacting with its six neighbors by means of finite thickness mortar joints and the FRP grid is applied on the external surfaces of the REV. In the framework of the kinematic theorem of limit analysis, a simple constrained minimization problem is obtained on the unit cell, suitable to estimate – with a very limited computational effort – reinforced masonry homogenized failure surfaces.A FE strategy is adopted at a cell level, modeling joints and bricks with six-noded wedge shaped elements and the FRP grid through rigid infinitely resistant truss elements connected node by node with bricks and mortar. A possible jump of velocities is assumed at the interfaces between contiguous wedge and truss elements, where plastic dissipation occurs. For mortar and bricks interfaces, a frictional behavior with possible limited tensile and compressive strength is assumed, whereas for FRP bars some formulas available in the literature are adopted to reproduce the delamination of the truss from the support.Two meaningful structural examples are considered to show the capabilities of the procedure proposed, namely a reinforced masonry deep beam (0°/90° continuous reinforcement) and a masonry beam in simple flexion for which experimental data are available. Good agreement is found between present model and alternative numerical approaches.     

Damage,plasticity and failure of ceramics and cementitious composites subjected to multi-axial state of stress

Experimental data on mechanical behavior of ceramics and cementitious composites subjected to triaxial state of stress and verification of the theoretical model capable to describe deformability and fracture of brittle rock-like materials are presented in the paper. To check the validity of the theoretical model the stress–strain curves and stresses at material fracture determined experimentally for brick and mortar were compared with the theoretical predictions. The limit surface at material fracture obtained experimentally from triaxial tests was used in numerical analysis of masonry specimens subjected to compressive loading. These numerical results obtained by employing the Finite Element Method software package Mafem3D were compared with experimental data available in the literature. Fairly good agreement of numerical predictions with experimental results for masonry specimens was observed.