Raimondo Luciano

Homogenization technique and damage model for old masonry material

Abstract Masonry is a composite material realized by the inclusion of bricks into the matrix of mortar. In the present paper, a micromechanical approach for defining the properties of a periodic masonry material is proposed. A damage model for old masonries is presented. In fact, it is assumed that the damage is due to the coalescence and growth of the fractures only in the mortar. A repetitive unit cell is chosen and eight possible undamaged and damaged states for the masonry are identified. The homogenization theory for material with periodic microstructure is used to define the overall moduli of the uncracked and cracked masonry. Variational formulations of the periodic problem are given. A numerical procedure for the computation of the elastic properties of the undamaged and damaged masonry material is developed. Then, the damage evolution of the masonry, which accounts for the exact geometry and for the mechanical properties of the constituents of the composite, is obtained. Energy and local strength criteria for the mortar are proposed. The behavior of a typical masonry material is studied and the results are put in comparison with the ones available in the literature. Finally, a simple structural application is developed.

Formulas for the stiffness of composites with periodic microstructure

Abstract In this paper, the mechanical behavior of composite materials with periodic microstructure is analysed. The corresponding elastic problem is solved by using the Fourier series technique and assuming the homogenization eigenstrain to be piecewise constant. Then, the coefficients of the overall stiffness tensor of the composite material are expressed analytically in terms of the elastic properties of the constituents (fibers and matrix) and as a function of nine triple series which take into account the geometry of the inclusions. In the case of composite materials reinforced by long fibers, simple formulas for evaluating these series are proposed. Close-form expressions for the elastic moduli of the fiber reinforced composite with periodic microstructure and for the equivalent transversely isotropic material are obtained. Finally, several comparisons with experimental results are presented.

Damage of masonry panels reinforced by FRP sheets

Abstract In this paper the mechanical behavior of a masonry wall is studied. The masonry is regarded as a composite realized by a regular arrangement of blocks into a matrix of mortar. Hence, a panel of masonry is a three-dimensional heterogeneous body with a finite thickness and R 2 -periodicity in the plane of the wall. A micromechanical approach is proposed to get the overall properties of the masonry. Then, a case of a wall reinforced by FRP-layered sheets placed on the surfaces of the wall is analyzed. To model the overall behavior of the unreinforced and reinforced masonry, by accounting for the progressive damage of the mortar, of the block and of the FRP sheets, a simple homogenization technique is proposed. Two different damage criteria are adopted for the mortar and the block, within isotropic viscoelastic and elastic damage models. Furthermore, a brittle damage model is used for the reinforcement. Finally, numerical applications are developed by adopting the proposed procedure in order to investigate on the damage of the unreinforced and reinforced masonry panels.

Variational bounds for the overall properties of piezoelectric composites

In this paper, variational bounds for the overall properties of periodic heterogeneous media with nonlinear and nonlocal piezoelectric constitutive relationships are obtained. First, elementary bounds are developed by extending to piezoelectric materials the well-known Voigt and Reuss bounds. Then, by generalizing the two Hashin-Shtrikman principles, eight new variational principles are derived and applied to obtain bounds. In fact, the variational principles developed are based on auxiliary electroelastic equilibrium problems, which can be solved by a transformation from the space domain to the Fourier domain. As an application, fibre-reinforced linear piezoelectric composites are considered, and expressions of upper and lower bounds for the overall properties of these composites are developed in closed form.

Variational methods for the homogenization of periodic heterogeneous media

Abstract In this paper, several variational principles for the evaluation of the overall properties of composite materials with periodic microstructure are introduced. The two classical homogenization problems corresponding to assigned average strain or assigned average stress are considered. The periodicity of the variables governing the problem is enforced either by considering special representations (i.e. Fourier series) of the periodic part of the displacement and stress fields or by adopting appropriate boundary conditions on the unit cell. In particular, the boundary conditions ensuring the periodicity of the governing variable are introduced in the functionals by using Lagrangian multipliers. Once the variational principles are introduced, the Fourier series technique and the finite element method are adopted to obtain rational approximation procedures. Finally, numerical applications are carried out in order to assess the performances of the proposed methods in the computation of estimates or bounds on the overall elastic properties of a composite material, and in the determination of the displacement and stress distribution in the unit cell.

Micromechanical formulas for the relaxation tensor of linear viscoelastic composites with transversely isotropic fibers

Abstract Explicit analytical expressions for the relaxation moduli in the Laplace domain of composites with viscoelastic matrix and transversely isotropic fibers are developed. The correspondence principle in viscoelasticity is applied and the problem in the Laplace domain is studied using the solution of the elastic problem having periodic microstructure. Formulas for the Laplace transform of the relaxation functions of the composite are obtained in terms of the properties of the matrix and the fibers. The inversion to the time domain of the relaxation and the creep functions of composites reinforced by transversely isotropic fibers is carried out numerically when a power law model is applied to represent the viscoelastic behavior of the matrix. Finally, comparisons with experimental results are presented.

A damage model for masonry structures

In this paper a brittle damage model for masonry structures characterized by a unit cell composed of mortar, blocks and a finite number of fractures is proposed. Initially, a micromechanically consistent binary representation of the damage variable is proposed for regular heterogeneous media, and a discrete damage kinetic law is developed. Then, a typical masonry material is considered and a specific damage model is derived from the previous one by employing the classical cohesive Coulomb failure criterion and by taking into account the specific microstructure of the masonry. This damage model is used to formulate the nonlinear quasistatic problem of masonry structures subject to assigned loading paths. Then, a numerical procedure based on the finite element method is implemented in a computer code. Finally, some applications are developed to analyze the nonlinear behavior of simple masonry structures and to show the capability of the proposed damage model to capture the differences of behavior by varying the microstructure and the physical properties of the masonry.

Bounds on non-local effective relations for random composites loaded by configuration-dependent body force

Abstract An infinite domain occupied by composite material is considered. The composite is treated as random. The stress, strain and displacement fields within it are produced by body force with two types of components. One is deterministic while the other depends on the particular realisation of the medium as would be the case, for instance, under gravity loading, so that the distribution of a density (physical or otherwise) enters the problem, as well as the distribution of elastic constants. This feature introduces additional constitutive operators defining the mean response of the body. Besides the usual (non-local) effective modulus operator, there is another operator that provides a contribution to the mean stress from the fluctuating part of the density, and one more that accounts for the non-zero contribution to the mean energy from the interaction of the fluctuating part of the stress field with itself. Bounds of Hashin–Shtrikman type are developed which provide restrictions on all these operators, in real space and in the transform domain. The formalism also provides direct Hashin–Shtrikman approximations for the non-local operators.

Boundary-layer corrections for stress and strain fields in randomly heterogeneous materials

Abstract Boundary-layer effects on the effective response of fibre-reinforced media are analysed. The distribution of the fibres is assumed random. A methodology is presented for obtaining non-local effective constitutive operators in the vicinity of a boundary. These relate ensemble averaged stress to ensemble averaged strain. Operators are also developed which re-construct the local fields from their ensemble averages. These require information on the local configuration of the medium. Complete information is likely not to be available, but averages of these operators conditional upon any given local information generate corresponding conditional averages of the fields. Explicit implementation is performed within the framework of an approximation of Hashin–Shtrikman type. Two types of geometry are considered in examples: a half-space and a crack in an infinite heterogeneous medium. These are representative, asymptotically, of the field in the vicinity of any smooth boundary, and in the vicinity of a crack tip, respectively. Results have been obtained for the case of anti-plane deformation, realized by the imposition of either Dirichlet or Neumann conditions on the boundary; those for the Neumann condition are presented and discussed explicitly. The stresses in both fibre and matrix adjacent to a crack tip are shown to differ substantially from the values that would be predicted by ordinary homogenization.

Non-local constitutive response of a random laminate subjected to configuration-dependent body force

Abstract The non-local constitutive behaviour of an infinite composite laminate is analyzed. The laminate is treated as random and is subjected to a combination of deterministic and configuration-dependent body force. In this case, in addition to the effective non-local elastic operator, other non-local constitutive operators must be considered in order to define the mean response of the body. For a laminate subjected to forces that vary only in the direction of lamination, these operators are obtained explicitly. The Hashin-Shtrikman principles developed by Luciano and Willis (Luciano, R., Willis, J.R., 2001a. Bounds on non-local effective relations for random composites loaded by configuration-dependent body force, to appear in J. Mech. Phys. Solids), which provide bounds for the operators for general composites, are shown to generate exactly the two operators that define the stress, while giving only bounds for the remaining operator that appears in the expression for the total energy. The case of a two-phase laminate with the layers arranged periodically is presented as an example.