Effective Moduli Method Research Articles

Modeling of ice-matrix composite fracture

This article discusses the results of a study that used the structural approach to conduct multiscale modeling of composite materials. These materials are composed of an ice matrix that has been enhanced with high-strength basalt fibers. The research involved numerical modeling to analyze the fracture process of composite materials made by freezing fresh water with added basalt fibers. The analysis included experimental data from samples of pure ice and composites with different amounts of filler. The effective modulus of elasticity of the composites was determined by considering the quantity of basalt fiber reinforcement. The discrepancy between the strength obtained by finite element complex ANSYS numerical calculation and the experimental data was revealed by a series of macroscopic test calculations of the composite material sample, according to which the effective modulus of elasticity of the composite was recalculated and corrections were introduced into the Voight and Reuss models, taking into account the non-uniform distribution of the fibers and the non-ideal adhesion between the fibers and water ice. A stochastic model of crack growth at the micro level was also used with the data on the diameter and length of basalt fibers and their random distribution in the layers of the composite material. Using the newly acquired effective elastic moduli and the SmartCrackGrowth algorithm, a satisfactory agreement of the crack growth rate with the obtained by stochastic calculation was achieved. The refined method of effective elastic moduli and multilevel model of stress-strain state calculation of composite materials are recommended for assessing the strength of ice cover and designing winter roads with high load-bearing capacity and long operational duration in the Arctic and Subarctic environments.

О предельных переходах в пространственных задачах гомогенизации двухкомпонентных диэлектрических композитов с экстремальными модулями одной из фаз

The spatial problem of calculating the effective permittivity of two-component composite, consisting of a base material filling a spherical layer and one spherical inclusion, is considered. The homogenization problem is solved by effective moduli method with calculation of the energy characteristics in the composite medium and in its individual phases. In the obtained solution, the limit transitions are made for two extreme cases: pores or inclusions with zero dielectric constant and conductive inclusions with infinitely high dielectric constant. The solutions of these problems are compared with the solutions of homogenization problems for a medium with void and for a medium with conductive inclusion boundary. In problems with one basic material, the properties of inclusions were taken into account only by the corresponding boundary conditions on the interface. It is shown that calculations of the effective permittivity by energy criterion give correct results in all the cases considered, while the calculations by the average permittivity for a composite with a conductive inclusion boundary may be erroneous.

Limit transitions in plane homogenization problems for two-phase dielectric composites with extreme material properties of one phase

The plane problem of calculating the effective dielectric constant of two-phase composite consisting of main material with one circular inclusion is considered. To solve the homogenization problem, the method of effective moduli with the support of the energy balance between the composite and the homogeneous comparison medium was used. In the solution obtained, the passage to the limit was made for the case of an inclusion with zero dielectric constant and for the case of a conductive inclusion. The limiting solutions are compared with the solutions of homogenization problems for a medium with a hollow and for a medium with a conducting inclusion boundary.

Homogenization of Porous Piezocomposites with Extreme Properties at Pore Boundaries by Effective Moduli Method

The article deals with problems on determining the effective moduli of porous piezoceramic materials with substances that have extreme properties and are deposited on the pore boundaries. The argumentation for the method of effective moduli is given. The reciprocity relation, from which the properties of effective moduli that are similar to known ones for moduli of piezoelectric materials can be obtained, is proved. It is noted that for homogenization of porous piezoelectric composites with rigid or electrode pore boundaries, it is necessary to use models for materials having extremely large moduli that are located on the pore boundaries. Using numerical calculations, anomalous properties of various effective piezoelectric moduli that are promising for practical applications of the investigated piezocomposites have been revealed.

Effective properties of a porous inhomogeneously polarized by direction piezoceramic material with full metalized pore boundaries: Finite element analysis

This paper concerns the homogenization problems for porous piezocomposites with infinitely thin metalized pore surfaces. To determine the effective properties, we used the effective moduli method and the finite element approaches, realized in the ANSYS package. As a simple model of the representative volume, we applied a unit cell of porous piezoceramic material in the form of a cube with one spherical pore. We modeled metallization by introducing an additional layer of material with very large permittivity coefficients along the pore boundary. Then we simulated the nonuniform polarization field around the pore. For taking this effect into account, we previously solved the electrostatic problem for a porous dielectric material with the same geometric structure. From this problem, we obtained the polarization field in the porous piezomaterial; after that, we modified the material properties of the finite elements from dielectric to piezoelectric with element coordinate systems whose corresponding axes rotated along the polarization vectors. As a result, we obtained the porous unit cell of an inhomogeneously polarized piezoceramic matrix. From the solutions of these homogenization problems, we observed that the examined porous piezoceramics composite with metalized pore boundaries has more extensive effective transverse and shear piezomoduli, and effective dielectric constants compared to the conventional porous piezoceramics. The analysis also showed that the effect of the polarization field inhomogeneity is insignificant on the ordinary porous piezoceramics; however, it is more significant on the porous piezoceramics with metalized pore surfaces.

Analysis of porosity influence on the effective moduli of ceramic matrix PZT composite using the simplified finite element model

The problem of determining the effective moduli of a ceramic matrix piezocomposite with respect to multiscale porosity was considered. To solve the homogenization problem, the method of effective moduli in the standard formulation, the finite element method and the ANSYS computational package were used. Various models of two-phase and three-phase composites consisting of a piezoceramic matrix, elastic inclusions of corundum and pores of various sizes have been investigated. Finite element models of representative volumes of 3–0 and 3–0–0 connectivities were developed. The results of computational experiments showed that effective moduli depend quite significantly not only on the volume fractions of inclusions and pores, but also on the structure and size of pores in comparison with the characteristic sizes of inclusions.

Stresses in inhomogeneous elastic–viscoelastic–elastic sandwich plates via hyperbolic shear deformation theory

A hyperbolic shear deformation theory is proposed to investigate the bending analysis of inhomogeneous elastic/viscoelastic/elastic (EVE) sandwich plates. The sandwich plate is consisting of two elastic material faces and viscoelastic material core. Three kinds of symmetric sandwich plates that are classified depending on thickness of each layer are presented. A fourth type of sandwich plate is considered without viscoelastic core. The equilibrium equations have been solved by using Illyushin’s approximation method as well as the effective moduli method. The deflection and stresses of simply supported EVE sandwich plates have been presented due to a hyperbolic theory and compared with those due to other familiar plate theories. The effect of different parameters on bending analysis of sandwich plates is discussed.

Simulation and Finite Element Analysis of Porous Piezoceramic Disk-Shaped Transducer with Plano-Concave Surface

The paper investigates a disk-shaped piezoelectric transducer with plano-concave surface. Special form of the construction enhances the flexibility of the transducer and improves its effectiveness similar to the known transducers of Cymbal and Moonie types. The analysis of the transducer was performed using the finite element technologies and ANSYS finite element software. The influence of the porosity of the piezoceramic material of the transducer on its performance was studied both in static analysis and steady-state oscillations mode. The porosity of the piezoceramic material was taken into account by the effective properties of porous piezoceramic as a composite structure, which were calculated in numerical experiments on solving the homogenization problems by the effective moduli method and the finite element method.

Elastic and Viscoelastic Stresses of Nonlinear Rotating Functionally Graded Solid and Annular Disks with Gradually Varying Thickness

Abstract Analytical and numerical nonlinear solutions for rotating variable-thickness functionally graded solid and annular disks with viscoelastic orthotropic material properties are presented by using the method of successive approximations.Variable material properties such as Young’s moduli, density and thickness of the disk, are first introduced to obtain the governing equation. As a second step, the method of successive approximations is proposed to get the nonlinear solution of the problem. In the third step, the method of effective moduli is deduced to reduce the problem to the corresponding one of a homogeneous but anisotropic material. The results of viscoelastic stresses and radial displacement are obtained for annular and solid disks of different profiles and graphically illustrated. The calculated results are compared and the effects due to many parameters are discussed.

Finite element simulation of effective properties of microporous piezoceramic material with metallized pore surfaces

ABSTRACTThe paper presents a computer design of effective properties of microporous piezoceramic materials with fully metallized pore surfaces based on the simulation of the representative volume structure, the effective moduli method of the composite mechanics, and the finite element method. ANSYS finite element software was used to obtain numerical solution of the homogenization problems. To simulate the metallization on the boundaries of pores with piezoceramic matrix, the boundary conditions of free electrodes and elastic shell finite elements were used. The influence of the pore surface metallization on the values of effective moduli for microporous piezoceramics PZT-4 was investigated numerically.

Computer design of porous and ceramic piezocomposites in the finite element package ACELAN

The paper is devoted to finite element software ACELAN-COMPOS developed by the authors. It presents the approaches to the determination of the effective properties of piezoelectric two-phase composite materials based on the effective moduli method, the algorithms for generating representative volumes with specified properties, and the finite element techniques. Brief description of two algorithms of representative volume generation are provided. One of these algorithms supports connectivity between two phases of the composite, and the other creates inclusions in granular shape as the second phase. The results of calculating the effective moduli of porous piezoceramics with open and closed porosity and the effective moduli of polycrystalline piezoceramics are given.

Local buckling of moderately thick stepped skew viscoelastic composite plates using the element-free Galerkin method

In this paper, the quasi-static stability of moderately thick skew viscoelastic composite plates subjected to in-plan forces is analyzed using the element-free Galerkin method. The study is based on the first-order shear deformation theory and the method of effective moduli. The shape functions are constructed using the moving least squares approximation and the essential boundary conditions are introduced into the formulation through the Lagrange multiplier method and the orthogonal transformation techniques. The method is programmed and several numerical examples are presented to demonstrate the scope and efficacy of the procedure. Also, local buckling of viscoelastic stepped plates is studied.

Modeling of Magnetoelectric Composites by Effective Moduli and Finite Element Methods. Theoretical Approaches

The work discusses the approaches to the determination of the effective properties of magnetoelectric composite materials based on the effective moduli method, modeling of the representative volumes with account for the microstructure and the finite element technique for the representative volumes.

Finite element modeling of porous thermoelastic composites with account for their microstructure

The paper discusses approaches to the determination of the effective moduli of porous anisotropic thermoelastic composite materials based on the effective moduli method, modeling of the representative volumes with account for microstructure and finite element technologies of solving static thermoelastic problems for heterogeneous bodies. The paper formulates a set of boundary-value problems of thermoelasticity with boundary conditions of the first kind that enables computing all effective stiffness moduli, thermal conductivity moduli and coefficients of thermal stresses for porous anisotropic material. The technology of the finite element method is described for the numerical solution of boundary-value thermoelastic problems in the representative volume of a porous material. A range of the simulation methods for the inner structure of the representative volumes in cubic finite element lattice is represented by the way of giving properties of the porous material to a certain number of finite elements. The methods considered are the simple random method, the method that supports the connectivity of the skeleton for the porosity up to 90%, the Witten-Sander method that enables us to obtain a cluster from the pores of fractal type, and the growing from the plane method that forms several clusters from the pores in the vicinity of one of the edges of the cubic lattice. The example considered is the model of porous silicon that is taken as a material of cubic symmetry. The computation results allow analyzing the influence of various structures of the representative volumes on the effective moduli. The comparison of the computed effective characteristics of porous silicon with a range of known analytical and numerical results has been carried out. The results of the test computations have demonstrated that the effective material properties of porous thermoelastic composites can strongly depend on the models of their representative volumes.

Local Buckling of Rectangular Viscoelastic Composite Plates Using Finite Strip Method

This article presents the stability analysis of reinforced viscoelastic composite plates subjected to in-plane loading using a finite strip method. The equations governing the stiffness and geometry matrices of moderately thick rectangular plates in the time domain are presented using higher-order shear deformation theory and effective moduli method. The critical buckling loads of viscoelastic plates are determined through solving the eigenvalue problems related to the global stiffness and geometry matrices. The accuracy of the developed model is verified against the results reported elsewhere while a comprehensive parametric study is presented to show the effects of different variables on local buckling coefficients.

Local buckling of thin and moderately thick variable thickness viscoelastic composite plates

This paper addresses the finite strip formulations for the stability analysis of viscoelastic composite plates with variable thickness in the transverse direction, which are subjected to in-plane forces. While the finite strip method is fairly well-known in the buckling analysis, hitherto its direct application to the buckling of viscoelastic composite plates with variable thickness has not been investigated. The equations governing the stiffness and the geometry matrices of the composite plate are solved in the time domain using both the higher-order shear deformation theory and the method of effective moduli. These matrices are then assembled so that the global stiffness and geometry matrices of a moderately thick rectangular plate are formed which lead to an eigenvalue problem that is solved to determine the magnitude of critical buckling load for the viscoelastic plate. The accuracy of the proposed model is verified against the results which have been reported elsewhere whilst a comprehensive parametric study is presented to show the effects of viscoelasticity parameters, boundary conditions as well as combined bending and compression loads on the critical buckling load of thin and moderately thick viscoelastic composite plates.

Rotating Variable-Thickness Inhomogeneous Cylinders: Part II—Viscoelastic Solutions and Applications

Analytical solutions for the rotating variable-thickness inhomogeneous, orthotropic, hollow cylinders under plane strain assumption are developed in Part I of this paper. The extensions of these solutions to the viscoelastic case are discussed here. The method of effective moduli and Illyushin's approximation method are used for this purpose. The rotating fiber-reinforced viscoelastic homogeneous isotropic hollow cylinders with uniform thickness are obtained as special cases of the studied problem. Numerical application examples are given for the dimensionless displacement of and stresses in the different cylinders. The influences of time, constitutive parameter and elastic properties on the stresses and displacement are investigated.

Bending response of inhomogeneous fiber-reinforced viscoelastic sandwich plates

The static response of an inhomogeneous fiber-reinforced viscoelastic sandwich plate is investigated by using the first-order shear deformation theory. Several types of sandwich plates are considered taking into account the symmetry of the plate and the thickness of each layer. In addition, two cases are considered depending on the viscoelastic material which are included in the core or the faces of the sandwich plates. The method of effective moduli and Illyushin's approximation method are used to solve the equations governing the bending of simply supported inhomogeneous fiber-reinforced viscoelastic sandwich plates. Numerical computations were carried out to study the effect of the time parameter on deflections and stresses at different values of the aspect ratio, side-to-thickness ratio and constitutive parameter.

Determination of sensitivity coefficients of the elastic effective properties for periodic fiber-reinforced composites using the response function method

Derivation and implementation of the homogenization method including determination of sensitivity gradients of the effective elasticity tensor using combined numerical-analytical approach are addressed in this paper. This is possible thanks to an application of the numerical response function together with the effective moduli method known from classical homogenization theory. Computational procedure is implemented using 4-noded quadrilateral plane strain finite elements (program MCCEFF) and the symbolic computations system MAPLE. The sensitivity coefficients are determined on the basis of partial derivatives of the homogenized elasticity tensor calculated using the response function method with respect to all composite components’ elastic characteristics. They are further separately subjected to normalization procedure for a final comparison with each other. Such an enriched homogenization procedure is tested on the periodic fiber-reinforced two component composites; the results of computational analysis are compared to the results of the central finite difference approach applied before. Computational methodology proposed here may be further successively applied not only in the context of homogenization method but also to extend various discrete computational techniques like boundary/finite element, finite difference and volumes together with various meshless methods.

Stress concentrations in a viscoelastic composite plate weakened by a triangular hole

The problems of determining the stress concentrations around a triangular hole in a fiber-reinforced viscoelastic composite plate under uniform tension or pure bending are presented. Using the method of effective moduli, these problems are solved by reduction to the corresponding problems for an anisotropic elastic material. Illyushin’s approximation method is used to derive the viscoelastic solution. Numerical computations were carried out and the results show how the stress distribution around the hole changes with time, and also how the stress concentration factor depends on the constitutive parameter.