A homogenized Love–Kirchhoff model for out-of-plane loaded random 2D lattices: Application to “quasi-periodic” brickwork panels

A homogenization procedure for finding the bending stiffness of a 2D regular lattice with random local interactions is proposed. The kinematic and static methods are used to provide explicit upper and lower bounds for the homogenized moduli. The proposed homogenization procedure is applied to a masonry obtained by a random perturbation of the periodic running bond masonry [Cecchi, A., Sab, K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A 28, 610–625].A numerical evaluation of the scatter between the discrete models and the 2D Love–Kirchhoff model is performed on a test case, for various values of the random perturbation parameter and of the parameter that characterizes the heterogeneity of the wall. As expected, when the number of heterogeneities in the structure is large enough, the average response of the random discrete model converges to an asymptotic response. It is shown that this asymptotic response is very close to that of the periodic discrete model which is in turn very close to the response of the deterministic homogenized model. Similarly to the conclusion of Cecchi and Sab [Cecchi A., Sab K., 2009. Discrete and continuous models for in plane loaded random elastic brickwork. Eur. J. Mech. A. 28, 610–625.] dedicated to in-plane loading, the present results concerning out-of-plane loading show (both by means of a discrete model and a homogenized model) that the running bond pattern may be used successfully to analyze historical masonries with blocks having irregular widths in the horizontal direction.     

A homogenized Reissner–Mindlin model for orthotropic periodic plates: Application to brickwork panels

This paper describes a new procedure for the homogenization of orthotropic 3D periodic plates. The theory of Caillerie [Caillerie, D., 1984. Thin elastic and periodic plates. Math. Method Appl. Sci., 6, 159–191.] – which leads to a homogeneous Love–Kirchhoff model – is extended in order to take into account the shear effects for thick plates. A homogenized Reissner–Mindlin plate model is proposed. Hence, the determination of the shear constants requires the resolution of an auxiliary 3D boundary value problem on the unit cell that generates the periodic plate. This homogenization procedure is then applied to periodic brickwork panels.A Love–Kirchhoff plate model for linear elastic periodic brickwork has been already proposed by Cecchi and Sab [Cecchi, A., Sab, K., 2002b. Out-of-plane model for heterogeneous periodic materials: the case of masonry. Eur. J. Mech. A-Solids 21, 249–268 ; Cecchi, A., Sab, K., 2006. Corrigendum to A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork [Int. J. Solids Struct., vol. 41/9–10, pp. 2259–2276], Int. J. Solids Struct., vol. 43/2, pp. 390–392.]. The identification of a Reissner–Mindlin homogenized plate model for infinitely rigid blocks connected by elastic interfaces (the mortar thin joints) has been also developed by the authors Cecchi and Sab [Cecchi A., Sab K., 2004. A comparison between a 3D discrete model and two homogenised plate models for periodic elastic brickwork. Int. J. Solids Struct. 41/9–10, 2259–2276.]. In that case, the identification between the 3D block discrete model and the 2D plate model is based on an identification at the order 1 in the rigid body displacement and at the order 0 in the rigid body rotation.In the present paper, the new identification procedure is implemented taking into account the shear effect when the blocks are deformable bodies. It is proved that the proposed procedure is consistent with the one already used by the authors for rigid blocks. Besides, an analytical approximation for the homogenized shear constants is derived. A finite elements model is then used to evaluate the exact shear homogenized constants and to compare them with the approximated one. Excellent agreement is found. Finally, a structural experimentation is carried out in the case of masonry panel under cylindrical bending conditions. Here, the full 3D finite elements heterogeneous model is compared to the corresponding 2D Reissner–Mindlin and Love–Kirchhoff plate models so as to study the discrepancy between these three models as a function of the length-to-thickness ratio (slenderness) of the panel. It is shown that the proposed Reissner–Mindlin model best fits with the finite elements model.     

Homogénéisation d'un milieu élastique fortement hétérogèneHomogenization of a strongly heteregeneous elastic medium

We consider the homogenization of an elastostatic problem in a strongly heterogeneous periodic medium made of two connected components having comparable tensors of elastic moduli, separated by a third medium (soft layer), the thickness of which is of the same order ε than the basic periodicity cell, and such that its elastic moduli tensor becomes infinitely small following a rate ε^r, r>0. If r?2, we identify the homogenized problem. Otherwise, we have to assume moreover that there are no volume forces in the third medium. To cite this article: M. Mabrouk, A. Boughammoura, C. R. Mecanique 330 (2002) 543–548.     

Multiscale homogenization of n-component composites with semi-elliptical random interface defects

Effective elastic characteristics of periodic multicomponent composite materials with random interface defects are studied in the paper. The defects are assumed to be semi-elliptical and lying with major semi axes along the interfaces, where minor and major semi-axes as well as the defects number are given as input random variables. The homogenization approach has a multiscale character—some algebraic approximation is used first to calculate effective elastic parameters of the interphase including all defects located at the same interface. Equations for interphase random elastic parameters are obtained using MAPLE symbolic mathematics in conjunction with probabilistic generalized perturbation method. A different homogenization method is applied at the micro scale, where the cell problem is solved numerically using the Finite Element Method (FEM) program. Since the composites considered exhibit random variations of both elastic properties and the interface defects, the overall homogenized characteristics must be obtained as random quantities, which is realized on the micro scale by the Monte-Carlo simulation. The proposed interface defects model obeys the porosity effects resulting from the nature of some matrices in engineering composites as well as the interface cracks appearing as a result of composites ageing during static or fatigue fracture.     

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.     

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.     

Non-linear micropolar and classical continua for anisotropic discontinuous materials

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

Second-order homogenization of periodic materials based on asymptotic approximation of the strain energy: formulation and validity limits

In this paper a second-order homogenization approach for periodic material is derived from an appropriate representation of the down-scaling that correlates the micro-displacement field to the macro-displacement field and the macro-strain tensors involving unknown perturbation functions. These functions take into account of the effects of the heterogeneities and are obtained by the solution of properly defined recursive cell problems. Moreover, the perturbation functions and therefore the micro-displacement fields result to be sufficiently regular to guarantee the anti-periodicity of the traction on the periodic unit cell. A generalization of the macro-homogeneity condition is obtained through an asymptotic expansion of the mean strain energy at the micro-scale in terms of the microstructural characteristic size ?; the obtained overall elastic moduli result to be not affected by the choice of periodic cell. The coupling between the macro- and micro-stress tensor in the periodic cell is deduced from an application of the generalised macro-homogeneity condition applied to a representative portion of the heterogeneous material (cluster of periodic cell). The correlation between the proposed asymptotic homogenization approach and the computational second-order homogenization methods (which are based on the so called quadratic ansätze) is obtained through an approximation of the macro-displacement field based on a second-order Taylor expansion. The form of the overall elastic moduli obtained through the two homogenization approaches, here proposed, is analyzed and the differences are highlighted. An evaluation of the developed method in comparison with other recently proposed in literature is carried out in the example where a three-phase orthotropic material is considered. The characteristic lengths of the second-order equivalent continuum are obtained by both the asymptotic and the computational procedures here analyzed. The reliability of the proposed approach is evaluated for the case of shear and extensional deformation of the considered two-dimensional infinite elastic medium subjected to periodic body forces; the results from the second-order model are compared with those of the heterogeneous continuum.     

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.     

A homogenized viscoelastic model for masonry structures

A linear viscous model for evaluating the stresses and strains produced in masonry structures over time is presented. The model is based on rigorous homogenization procedures and the following two assumptions: that the structure is composed of either rigid or elastic blocks, and that the mortar is viscoelastic. The hypothesis of rigid block is particularly suitable for historical masonry, in which stone blocks may be assumed as rigid bodies, while the hypothesis of elastic blocks may be assumed for newly constructed brickwork structures. The hypothesis of viscoelastic mortar is based on the observation that non-linear phenomena may be concentrated in mortar joints. Under these assumptions, constitutive homogenized viscous functions are obtained in an analytical form.Some meaningful cases are discussed: masonry columns subject to minor and major eccentricity, and a masonry panel subject to both horizontal and vertical loads. The major eccentricity case is analysed taking into account both the effect of viscosity and the no-tension hypothesis, whereas the bi-dimensional loading case is analysed to verify the sensitivity of masonry behaviour to viscous function. In the masonry wall considered, the principal stresses are both of compression, and the no-tension assumption may therefore be discounted.     

In plane loaded masonry walls: DEM and FEM/DEM models. A critical review

This work is dedicated to the assessment of the nonlinear behaviour of masonry panels with regular texture and subject to in-plane loads, by means of numerical pushover analysis and an analytical homogenized model. Two numerical models are considered and adopted for performing a set of numerical tests: a discrete model developed by authors and a discrete/finite element model frequently adopted in rock mechanics field and effectively extended to masonry structures. In both models the hypotheses of rigid blocks and elastic–plastic joints following a Mohr–Coulomb yield criterion are adopted. The aim of this work is twofold: (1) a comparison and a calibration of the numerical models, evaluating their effectiveness in determining ultimate loads and collapse mechanisms of masonry panels, by assuming a nonlinear homogenized model for regular masonry as reference solution; (2) the evaluation of sensitivity of masonry behaviour and numerical models to panel dimension ratio and to varying masonry texture. In a first case study, sliding collapse mechanisms changing to overturning collapse mechanisms for increasing panel and block height-to-width ratio are obtained and the results given by the numerical models turn out to be in good agreement. Furthermore, a second case study, dedicated to square panels supported at base ends and vertically loaded, shows different ‘arch mechanisms’ depending on block height-to-width ratio.     

Homogenization of masonry walls with a computational oriented procedure. Rigid or elastic block?

The idea behind this paper is to present a numerical procedure for the analysis of masonry walls, based on the application of an asymptotic homogenization method. In this paper, a masonry wall, obtained by the regular repetition of blocks between which mortar is laid, is modelled as a periodic body in the two plane directions. The local problem is formulated for a base cell tied to the geometry of the body and in a position to generate it entirely through some law of its internal composition. Two homogenized models are formulated: the first envisages that both phases, block and mortar, behave in linear elastic fashion; the other envisages that the mortar behaves in linear elastic fashion, while the block is infinitely stiff. The two models are described theoretically and the construction of the model according to the characteristic module is numerically defined. In the case where the infinitely stiff (rigid) block is assumed, not only is the formulation of the model made extremely simple, but any numerical problems tied to great differences in the numerical values characterizing the constitutive modules of the two phases are overcome. In this regard, the domain of applicability of this model is sought both by comparing the homogenized constitutive functions, while varying the ratios of the elastic coefficients of the mortar and the block, with the rigid solution, and by analysing the structural behaviour that derives from the application, or not, of the rigid model, this being done for two sample problems. It should be underlined that the rigid-block model furnishes qualitatively sound structural answers even for very low ratios between the elastic moduli of the two phases composing the wall, and furnishes answers that are quantitatively sound as well for ratios of the order of 30:1, a realistic ratio in the case of ancient walls. The results obtained can be extended to heterogeneous materials in general, that is, to many of the innovatory materials, the composites, where the constituent phases have stiffness characteristics that are rather different and the condition of regularity of alternation of the phases is adequately plausible.     

Strength domain of non-periodic masonry by homogenization in generalized plane state

This paper deals with the evaluation of the strength domain for non-periodic masonry using a random media micromechanical approach. The generalized plane state formulation is used in order to more accurately describe the masonry behavior and failure criteria proposed in literature. Considering the masonry as a heterogeneous material with random microstructure, the elastic characteristics of the homogenized continuum and the strength domain are evaluated by using the hierarchy theory related to partitions with increasing dimensions through the application of natural and essential boundary conditions with proportionally growing values. An overall failure criterion based on the mean stress state of each phase is introduced. The proposed procedure is validated by comparison with the experimental results obtained with periodic masonry subjected to biaxial stress states recovering the main failure mechanisms. Then the approach is applied to an actual non-periodic masonry introducing peculiar algorithms in order to evaluate strength surfaces and to verify the convergence of the domains obtained through the application of natural and essential boundary conditions with the increasing size of portion dimensions at the mesoscale level.     

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.     

Stochastic representation of the mechanical properties of irregular masonry structures

A procedure for the stochastic characterization of the elastic moduli of plane irregular masonry structures is presented in this paper. It works in the field of the random composite materials by considering the masonry as a mixture of stones (or bricks) and mortars. Once that the elastic properties of each constituent are known (deterministically or stochastically), the definition of the overall masonry elastic properties requires the knowledge of the random field describing the irregular geometry distribution. This last one is obtained by a software, implemented ad hoc, that, starting from a colour digital photo of the masonry and using the instruments of the digital image processing techniques, gives the random features of this field in both the space and frequency domain. The definition of the stochastic properties of masonry structures may be very useful both for the application of the stochastic homogenization techniques and for the direct stochastic analysis of the structures.     

Evaluation of a Statistically Equivalent Periodic Unit Cell for a quasi-periodic masonry

The paper presents a method to estimate the Statistically Equivalent Periodic Unit Cell (SEPUC) corresponding to a masonry with quasi-periodic texture. The identification of the texture and the constituent phases (unit blocks and mortar joints) is achieved by means of digital image processing techniques applied to color image of the masonry wall. A statistical analysis of geometrical parameters (width and height of blocks, thickness and length of mortar joints) allows to estimate their probability distribution and to identify the typology of the texture. Subsequently a Monte Carlo analysis is performed using several tentative SEPUCs generated with different dimensions of blocks and joints according to the estimated distributions. A criterion was eventually proposed to identify, among the numerically generated ones, the SEPUC which is more suitable to model the behavior of masonry wall. The SEPUC is analyzed with techniques available for periodic texture, applying periodic boundary conditions, in order to estimate the equivalent elastic stiffness. The proposed method is validated comparing the results in the elastic range obtained with SEPUC and those obtained imposing essential and natural boundary conditions on the original texture.     

On the microstructure of open-cell foams and its effect on elastic properties

Synthetic open-cell foams have a complex microstructure consisting of an interconnected network of cells resulting from the foaming process. The cells are irregular polyhedra with anywhere from 9 to 17 faces in nearly monodisperse foams. The material is concentrated in the nearly straight ligaments and in the nodes where they intersect. The mechanical properties of such foams are governed by their microstructure and by the properties of the base material. In this study micro-computed X-ray tomography is used to develop 3D images of the morphology of polyester urethane and Duocel aluminum foams with different average cell sizes. The images are used to establish statistically the cell size and ligament length distributions, material distributions along the ligaments, the geometry of the nodes and cell anisotropy. The measurements are then used to build finite element foam models of increasing complexity that are used to estimate the elastic moduli. In the most idealized model the microstructure is represented as a regular Kelvin cell. The most realistic models are based on Surface Evolver simulations of random soap froth with N³ cells in spatially periodic domains. In all models the cells are elongated in one direction, the ligaments are straight but have a nonuniform cross sectional area distribution and are modeled as shear deformable beams. With this input both the Kelvin cell models and the larger random foam models are shown to predict the elastic moduli with good accuracy but the random foams are 5–10% stiffer.     

The influence of finite extensibility on the eigenspectrum of dilute polymeric solutions

Linear stability of plane Couette flow of dilute polymeric solutions modeled using the FENE-P constitutive equation has been examined. Specifically, the spurious, discrete and continuous spectra are identified as a function of the ratio of the solvent to the total viscosity β, maximum chain length L, disturbance wavenumber α and the Weissenberg number (We). It is observed that reducing L shifts the entire spectrum to the left (in the eigenspectrum plane) by a factor directly related to the steady-state value of the Peterlin function f. Additionally, decreasing the value of L causes the splitting of the regular continuous spectrum, initially located at −1/We, into two branches. Overall, the plane Couette flow of a FENE-P model is found to be unconditionally stable. Specifically, decreasing L increases the decay rate of the most dangerous disturbance while increasing the number of discrete and spurious modes. However, increasing both β and α decreases the number of discrete and spurious modes whereas increasing We has no influence on the number of spurious modes but results in an increase in the number of discrete modes.     

Bounds on the non-local effective elastic properties of composites

The paper deals with the effective linear elastic behaviour of random media subjected to inhomogeneous mean fields. The effective constitutive laws are known to be non-local. Therefore, the effective elastic moduli show dispersion, i.e¹ they depend on the “wave vector” k of the mean field. In this paper the well-known Hashin-Shtrikman bounds (1962) for the Lamé parameters of isotropic multi-phase mixtures are generalized to inhomogeneous mean fields k ≠ 0. The bounds involve two-point correlations of random elastic moduli. In the limit k → ∞ the bounds converge to the exact result. The interest is focussed on composites with cell structures and on binary mixtures. To illustrate the results, numerical evaluations are carried out for a binary cell material composed of nearly spherical grains of equal size.     

Effects of cell shape and strut cross-sectional area variations on the elastic properties of three-dimensional open-cell foams

The Voronoi tessellation technique and the finite element (FE) method are utilized to investigate the microstructure-property relations of three-dimensional (3-D) cellular solids (foams) that have irregular cell shapes and non-uniform strut cross-sectional areas (SCSAs). Perturbations are introduced to a regular packing of seeds to generate a spatially periodic Voronoi diagram with different degrees of cell shape irregularity (amplitude a), and to the constant SCSA to generate a uniform distribution of SCSAs with different degrees of SCSA non-uniformity (amplitude b). Twenty FE models are constructed, based on the Voronoi diagrams for twenty foam samples (specimens) having the same pair of a and b, to obtain the mean values and standard deviations of the elastic properties. Spatially periodic boundary conditions are applied to each specimen. The simulation results indicate that for low-density imperfect foams, the elastic moduli increase as cell shapes become more irregular, but decrease as SCSAs get less uniform. When the relative density (R) increases, the elastic moduli of imperfect foams increase substantially, while the Poisson's ratios decrease moderately. The effect of the interaction between the two types of imperfections on foam elastic properties appears to be weak. In addition, it is found that the strut cross-sectional shape has a significant effect on the foam properties. Also, the elastic response of foams with the cell shape and SCSA imperfections appears to be isotropic regardless of changes in a, b and R and the strut cross-sectional shape.