Homogenization of metamaterials using full-wave simulations

We present a full-wave homogenization method to determine the effective material parameters of metamaterials by considering a spherical piece of metamaterial. We use a T-matrix approach that is accelerated by a multilevel fast multipole method that is stable at low frequencies. To determine the T-matrix of one inclusion in the metamaterial a Method of Moments surface integral equation is used that is also accelerated using another multilevel fast multipole method that is stable at low frequencies. We also derive a new closed-form expression to extract the effective material parameters from the T-matrix of the spherical piece of material. Examples verify the accuracy and limitations of the method. We show results for metamaterials comprising more than 40,000 particles.     

Homogenization of Carbides in Ni60/WC Composite Coatings Made by Fiber Laser Remelting

In the current study, the effect of laser remelting on homogenization of carbides in WC-reinforced Ni60 composite coatings was investigated. Ni60 + 50 wt% WC composite coatings were fabricated on the surface of Q550 steel by LDF4000-100 fiber laser device. First cladding layer was made by rectangle laser spot and then circular laser spot was utilized to remelt the coatings. Microstructure characteristics were investigated by optical microscope and scanning electron microscopy. Elemental distribution and phase constitution were analyzed by energy disperse spectroscopy and X-ray diffraction. The results showed that accumulation of residual WC particles was sufficiently eliminated under the effect of laser remelting. The irregularly shaped carbides in first cladding layer were transformed into well-distributed polygonal carbides by laser remelting. Statistical analysis indicated median size of reinforcement particles decreased from 35.40 to 5.62 µm. Microhardness of remelted region had a smooth profile and decreased by ?50 HV_0.1 than that of first cladding region. Homogenization of carbides in nickel composite coating was well realized by laser remelting.     

Strain energy-based homogenization of nonlinear elastic particulate composites

The macroscopic constitutive law for a heterogeneous solid containing two dissimilar nonlinear elastic phases undergoing finite deformation is obtained. Attention is restricted to the case of spherical symmetry such that only the materials consisting of an irregular suspension of perfectly spherical particles experiencing all-round uniform loading are considered which leads to a one-dimensional modeling. For the homogenization procedure, a strain-energy based scheme which utilizes Hashin’s composite sphere is employed to obtain the macroscopic stress-deformation relation added by the initial volume fraction of the particles. As applications of the procedure, the closed-form macroscopic stress expression for a generalized Carroll composite material is derived. Then, by choosing carbon black-filled rubbers, unknown bulk modulus of the carbon black particles is calculated. Finally, the particle-reinforced flexible polyurethane foam is studied using the Ritz method. It is shown that the analytical outcome for composites filled by compressible inclusions is applicable for porous materials with the same matrix.     

An extension of the Secant Method for the homogenization of the nonlinear behavior of composite materials

We discuss the consequences of a different application of the principle the Modified Secant Method is [C. R. Acad. Sci. Paris Sér. IIb 320 (1995) 563] based on. In fact, we directly compute the second-order averages of the local fields available from the linear elastic homogenization procedure exploited in order to evaluate the effective elastic moduli. This method, which can be seen either as a simplification of the Modified Secant Method or as an extension of the Secant Method [J. Mech. Phys. Solids 26 (1979) 325], may be useful for any composite whose overall elastic constants need to be estimated by modeling the microstructure through Morphologically Representative Patterns [J. Mech. Phys. Solids 44 (1996) 307], which is for instance the case of syntactic foams [Int. J. Solids and Structures 38/40-41 (2001) 7235]. In order to show the accuracy of the proposed method, we apply it to several examples and compare its results with those obtainable by means of other analytical methods available in the literature, with numerical results of Finite Element simulations, and with experimental results. Closed-form solutions are derived for the effective yield stress of porous metals and incompressible composites reinforced with rigid spheres.     

Corrugated laminate homogenization model

This work addresses the problem of finding a substitute material model for describing the load response of globally flat corrugated sheets made from multidirectional laminates. Exact solutions of the equations governing thin singly curved shells give the displacements, strains and stresses within a unit cell extending over one period of the corrugation pattern. The forces reacting to enforced unit deformations determine the constitutive law of the substitute material model. The analytical results are compared with those of finite-element models.     

A review of homogenization and topology optimization III-topology optimization using optimality criteria

This is the first part of a three-paper review of homogenization and topology optimization, viewed from an engineering standpoint and with the ultimate aim of clarifying the ideas so that interested researchers can easily implement the concepts described. In the first paper we focus on the theory of the homogenization method where we are concerned with the main concepts and derivation of the equations for computation of effective constitutive parameters of complex materials with a periodic micro structure. Such materials are described by the base cell, which is the smallest repetitive unit of material, and the evaluation of the effective constitutive parameters may be carried out by analysing the base cell alone. For simple microstructures this may be achieved analytically, whereas for more complicated systems numerical methods such as the finite element method must be employed. In the second paper, we consider numerical and analytical solutions of the homogenization equations. Topology optimization of structures is a rapidly growing research area, and as opposed to shape optimization allows the introduction of holes in structures, with consequent savings in weight and improved structural characteristics. The homogenization approach, with an emphasis on the optimality criteria method, will be the topic of the third paper in this review.     

Effective stiffness and microscopic deformation of an orthotropic plate containing arbitrary holes

Many engineering materials and structures, such as cellular structures, sandwich core structures and laminated plates with holes, can be modeled by an inclusion problem with anisotropic matrix. The paper studies the effective properties and the microscopic deformation of anisotropic plates with periodic holes by using direct and mathematical homogenization. The effective stiffnesses are calculated by different homogenization methods and the microscopic deformation of a RVE is modeled by the finite element method for the plate with arbitrarily shaped holes. All of the effective stiffness coefficients, especially stretching-shear coupling coefficients are evaluated.     

A multiresolution homogenization of modal analysis with application to layered media

We apply multiresolution techniques to study the effective properties of boundary value problems. Conditions under which boundary values are preserved in the effective (‘homogenized') formulation are developed and discussed. Relations between the eigenfunctions and eigenvalues of the generic formulation and those of the effective formulation are explored. Applications to the construction of effective Green function in a complex lamination are discussed. The analytic results are demonstrated via numerical computations.     

Micro-stress analysis and identification of lightweight aggregate’s failure strength by micromechanical modeling

This work is based on a dual approach of experiments and micromechanical modeling in order to characterize the failure behaviors of lightweight aggregate concretes (LWAC). Many classes of LWAC were tested, based on five families of lightweight aggregates (LWA) and three types of mortar matrices: normal, high performance (HP) and very high performance (VHP). Micromechanical modeling is based on an iterative homogenization approach and associated localization: local stress distributions during the uniaxial compression tests can be predicted in LWAC’s components and at their interface. Experimental compressive strengths were measured on manufactured 16 × 32 cm cylindrical specimens. The confrontations between micromechanical modeling and experiments were used to identify LWA’s failure strengths which are difficult to measure, and to quantify the inaccuracies related to conventional methods. These corrected values of LWA’s failure strength were introduced into a failure criterion modeling: associated predictions of LWAC’s compressive strength are in good agreement with the experimental measurements.