An eigenfunction expansion-variational method for the anti-plane electroelastic behavior of three-phase fiber composites

The anti-plane electroelastic behavior of three-phase piezoelectric composites (fiber/interphase/matrix) with doubly periodic microstructures is dealt with. A new variational functional for a unit cell is constructed by incorporating the periodic boundary conditions into the energy functional. Then, by combining with the eigenfunction expansions of the complex potentials satisfying the fiber-interphase-matrix interfacial conditions, an eigenfunction expansion-variational method based on a unit cell is developed. The numerical results of the effective electroelastic moduli show a rapid convergence of the present method. A unified first-order approximation formula is also provided, where an equivalent parameter matrix reflecting the overall influence of the electroelastic properties of the fiber and interphase on the effective properties, is found. The equivalent parameter matrix can greatly simplify the complicated relation of the effective electroelastic properties to the internal structure of a three-phase fiber composite. Though the equivalent parameter matrix is extracted in the first-order approximation formula, its validity is also verified in the high-order numerical results.     

A complex variable solution of two-dimensional heat conduction of composites reinforced with periodic arrays of cylindrically orthotropic fibers

Problems of two-dimensional steady-state heat conduction for composites with doubly periodic arrays of cylindrically orthotropic fibers are dealt with. A new complex variable method is presented by introducing an appropriate coordinate transformation to convert the governing differential equation into a harmonic one, and the eigenfunction expansions of the field variables in a unit cell are derived. Then by using a generalized variational functional which absorbs the periodicity condition, an eigenfunction expansion–variational method based on a unit cell is developed to solve such problems. A convergence analysis and a comparison with finite element calculations are conducted to demonstrate the correctness and efficiency of the present method. A discussion is made about the effects of the cylindrical orthotropy of the fiber and the existence of the isotropic core in the fiber on the effective conductivity of the composite. An engineering equivalent parameter, which reflects the overall influence of the thermal conductivities of the matrix and fibers as well as the interfacial characteristic on the effective thermal conductivity of the composite, is found. It is shown that the present first-order approximation of the effective thermal conductivity of the composite can be written in a unified formula for different microstructural characteristics and possesses a good engineering accuracy.     

Eigenstrain and Fourier series for evaluation of elastic local fields and effective properties of periodic composites

The elastic stress and strain fields and effective elasticity of periodic composite materials are determined by imposing a periodic eigenstrain on a homogeneous solid, which is constrained to be equivalent to the heterogeneous composite material through the imposition of a consistency condition. To this end, the variables of the problem are represented by Fourier series and the consistency condition is written in the Fourier space providing the system of equations to solve. The proposed method can be considered versatile as it allows determining stress and strain fields in micro-scale and overall properties of composites with different kinds of inclusions and defects. In the present work, the method is applied to multi-phase composites containing long fibers with circular transverse section. Numerical solutions provided by the proposed method are compared with finite element results for both unit cell containing a single fiber and unit cell with multiple fibers of different sizes.     

A unified analysis of various problems relating to circular holes with edge cracks

A unified method of analysis is developed for various problems relating to elastic plates containing circular holes with edge cracks. The method is based on the analysis of a unit rectangular region containing a circular hole with edge cracks, where the boundary conditions of the outer edges are suitably adjusted in order to treat various problems including periodic arrays of holes with edge cracks. The method is applied to five problems, and accurate values of the stress intensity factors are obtained. These analytical values of the stress intensity factors are fitted by polynomials which are convenient for practical applications.     

Doubly-periodic array and zig-zag array of cracks in solids under uniaxial tension

In fatigue of materials, it is often observed that a number of cracks initiate from preexisting defects and propagate to form major cracks causing the final failure. In this paper, we consider a doubly-periodic array and a zig-zag array of cracks in a two-dimensional solid subjected to tension, as simplified models of randomly distributed cracks in materials. The analysis is based on the eigenfunction expansions of the complex stress potentials for properly chosen unit regions. Numerical results for the stress intensity factors, crack opening displacements and effects of cracks on the tensile stiffnesses of these solids are given for various combinations of the parameters. The results are then fitted to reliable polynomial formulae for convenience of engineering applications.     

Collinear periodic cracks in an anisotropic medium

The problem of collinear periodic cracks in an anisotropic medium is examined in this paper. By means of Stroh formalism and the conformal mapping method, we obtain general periodic solutions for collinear cracks. The corresponding stress intensity factors, crack opening displacements and strain energy release rate are found.     

Closed-form effective elastic constants of frame-like periodic cellular solids by a symbolic object-oriented finite element program

In this study, the effective elastic constants of several 2D and 3D frame-like periodic cellular solids with different unit-cell topologies are analytically derived using the homogenization method based on equivalent strain energy. The analytical expressions of strain energy of a unit cell under different strain modes are determined using a generic symbolic object-oriented finite element (FE) program written in MATLAB. The obtained analytical expressions of the strain energy are then used to symbolically compute the effective elastic constants that include Young’s moduli, Poisson’s ratios, and shear moduli. The obtained analytical effective elastic constants are numerically verified using results from an ordinary numerical FE program. The obtained closed-form effective elastic constants are also compared with some existing solutions from the literature. This study demonstrates that symbolic computation platforms can be properly used to provide efficient methodologies for finding useful analytical solutions of mechanical problems. Without the symbolic object-oriented FE program in this study, elaborate and tedious analytical analysis has to be manually performed for each different unit cell. The symbolic object-oriented FE program provides analytical analysis of unit cells that is accurate and fast. The object-oriented programming technique allows the symbolic FE program in this study to be efficiently implemented.     

Sensitivity studies for the effective characteristics of periodic multicomponent composites

The main goal is to present the application of design sensitivity analysis in homogenization of periodic multicomponent composites. The effective modulus approach is used to determine homogenized characteristics of the composite together with some upper and lower bounds estimators for these quantities. The approach related to homogenization problem is presented in a general form for a linear elastic n-component periodic composite and is implemented in the finite element method homogenization-oriented computer program MCCEFF. Sensitivity coefficients are determined numerically for various components of homogenized elasticity tensor and, using symbolic analysis, for their bounds, in both cases with respect to material parameters of the components. Structural response functional for such a composite is proposed as the strain energy in various strain states of the composite cell to detect the most influential parameters of the most popular fiber-reinforced structures as well as in the case of multicomponent superconducting composite device.     

Two analytical models for the study of periodic fibrous elastic composite with different unit cells

In this work, the effective elastic moduli of two-phase fibrous periodic composites are obtained by means of the Asymptotic Homogenization Method (AHM) and eigenfunction expansion-variational method (EEVM), for different types of parallelogram cells. The constituents exhibit transversely isotropic properties. A doubly periodic parallelogram array of cylindrical inclusions under longitudinal shear is considered. The behavior of the shear elastic coefficient for different geometry arrays of the cell related to the angle of the fibers is studied. Some numerical examples and comparisons with other theoretical results demonstrate that both methods (AHM and EEVM) are efficients for the analysis of composites with presence of rhombic cell. The effect of the configuration of the cells on the shear effective property is observed.     

压电复合材料的共线周期性裂纹平面问题的电弹性场分析

利用复变函数知识、半逆解法及待定系数法, 研究了压电复合材料的共线周期性裂纹问题, 给出了在电不可渗透边界条件下的应力、电位移、应力强度因子、电位移强度因子和机械应变能释放率的解析解。当裂纹间距趋于无穷时, 共线周期性裂纹退化为一条单裂纹, 得到了压电复合材料一条单裂纹的结果。通过数值算例讨论了共线周期性裂纹的裂纹长度、裂纹间距和机电载荷对机械应变能释放率的影响规律。结果表明, 机械应变能释放率随着共线周期性裂纹的裂纹长度、共线周期性裂纹的裂纹间距、机械载荷和正电场的增大而增大, 随着负电场的增大而减小。     

A Three-Dimensional Strain Gradient Plasticity Analysis of Particle Size Effect in Composite Materials

Recent experiments on particle-reinforced metal-matrix composite materials have shown particle size effects. Small particles tend to give larger plastic work hardening than large particles at the same particle volume fraction. Prior models used to study the particle size effect are based on the strain gradient plasticity theories, and these models are mainly axisymmetric models with vanishing lateral stress tractions in order to represent the uniaxial tension condition. However, the prior results fall short to agree with the experimental data. A three-dimensional (3D) unit-cell model is adopted in the present study. The periodic boundary conditions are imposed for the 3D unit cell to ensure the compatibility of the unit cell before and after the deformation. The particles are elastic, while the metal matrix is elastic-plastic and is characterized by the conventional theory of mechanism-based strain gradient plasticity, which is established from the Taylor dislocation model but does not involve the higher-order stress. It is shown that the 3D unit-cell model with the periodic boundary conditions gives better agreements with the experimental data than the unit-cell model with the traction-free boundary conditions on the lateral surfaces.     

Doubly periodic arrays of cracks and thin inhomogeneities in an infinite magnetoelectroelastic medium

A two-dimensional boundary element method for the analysis of a magnetoelectroelastic medium containing doubly periodic sets of cracks or thin inclusions is developed in this paper. The integral equations and closed-form expressions for corresponding kernels are obtained. Based on the quasi-periodicity of extended displacement and stress function, the integral representations for average stress, strain, electric displacement, magnetic induction etc. are developed. The algorithm of effective properties determination is given. The numerical examples prove the efficiency and high accuracy of the proposed approach in determination of stress, electric displacement and magnetic induction intensity factors and effective properties of the material containing doubly periodic arrays of cracks or thin inclusions.     

Stiffness reduction of cracked solids

A method for the determination of the effective moduli of elastic solids containing a doubly periodic rectangular array of cracks is given. The derivation is based on the analysis of a unit cell in which the displacement vector is expanded to a second order in the distances from centerlines. The equilibrium equations in conjunction with the continuity conditions for the displacements and tractions, give a system of equations for the elastic field variables. The determination of the elastic internal energy provides the requested effective moduli of the cracked body. The method is applied to predict the loss of stiffness of cracked isotropic solids and unidirectional composites, as well as cracked cross-ply laminates.     

Isogeometric topology optimization of anisotropic metamaterials for controlling high-frequency electromagnetic wave

This study presents a level set–based topology optimization with isogeometric analysis (IGA) for controlling high-frequency electromagnetic wave propagation in a domain with periodic microstructures (unit cells). The high-frequency homogenization method is applied to characterize the macroscopic high-frequency waves in periodic heterogeneous media whose wavelength is comparative to or smaller than the representative length of a unit cell. B-spline basis functions are employed for the IGA discretization procedure to improve the performance of electromagnetic wave analysis in a unit cell and topology optimization. Also, to keep the same order of continuity on the periodic boundaries as on other element edges in the domain, we propose the extended domain approach, while incorporating Floquet periodic boundary condition (FPBC). Two types of optimization problems are taken as examples to demonstrate the effectiveness of the proposed method in comparison with the standard finite element analysis (FEA). The optimization results provide optimized topologies of unit cells qualified as anisotropic metamaterials with hyperbolic and bidirectional dispersion properties at the macroscale.     

Large deformation response of a hyperelastic fibre reinforced composite: Theoretical model and numerical validation

The mechanical behaviour of an incompressible neo-Hookean material directionally reinforced with a generalised neo-Hookean fibre is examined in the finite deformation regime. To consider the interaction between the fibre and the matrix, we use a composite model for this transversely isotropic material based on a multiplicative decomposition of deformation, which decouples the uniaxial deformation along the fibre direction from the remaining shear deformation. The model is then verified numerically by a unit cell model with periodic boundary conditions. The strain energy stored in the unit cell is compared with the energy predicted by the proposed theoretical model and excellent agreement is reported.     

Fractional Talbot effect in a tapered gradient-index medium: unit cell

A generalization of the fractional Talbot effect to the case of a tapered gradient-index medium for uniform illumination is considered. A unit cell of the fractional Talbot image contains the superposition of unit cell images of the periodic object.     

Failure Analysis and Remedy Procedure for Cracking of Fuel Nozzles in a Gas Turbine

A common failure in a certain type of gas turbine, observed during the first periodic inspection, is radial cracks in the tip plate of gas fuel nozzles. Here, each gas turbine has 18 nozzles. In all nozzles and in all similar units, these cracks of lengths ranging from 1 mm to a maximum of 14.5 mm are observed. As prescribed by the manufacturer, the defective part must be removed and replaced by welding and machining of a new one. But this problem is repeated and observed in the next periodic visits, and in all units. Depending on the number of nozzles in each gas turbine unit and the number of units in total, these repairs are very expensive and time-consuming. In this paper, the failure is analyzed and the causes of the cracks in the nozzles are investigated. Studies show that the main causes of nozzle failure are residual stresses caused by welding and thermal stresses caused by the start-up and shutdown processes. According to results, a solution has been proposed to release these residual and thermal stresses. After the implementation of this method in 1998, no more failure has been reported by the repair team, which proves the effectiveness of this solution. Since this paper has been prepared based on technical reports from the years between 1996 and 1998, the cited references of this paper are these technical reports.     

Nucleation of cracks in two-dimensional periodic cellular materials

This paper describes the nucleation and propagation of cracks in brittle cellular material. Four basic patterns with triangular, square, hexagonal and kagome-type cells are considered. The cracks propagate by sequential failure of critical elements. The analysis technique hinges on the combined use of the structural variation method and the representative cell method. While the latter allows for the analysis of periodic structures under arbitrary loads, by means of the discrete Fourier transform, the former analyzes modified structures (the cracked lattices) on the basis of analysis of the pristine structure (the periodic lattices). Within the assumptions of Bernoulli–Euler beam theory the suggested method for the analysis of infinite cracked lattices is exact. Although most cracks follow intuitive paths it was found that the microstructure of cellular materials has a significant influence on the crack pattern.     

Calculation of effective coefficients for piezoelectric fiber composites based on a general numerical homogenization technique

Numerical unit cell models for 1–3 periodic composites made of piezoceramic unidirectional cylindrical fibers embedded in a soft non-piezoelectric matrix are developed. The unit cell is used for prediction of the effective coefficients of the periodic transversely isotropic piezoelectric cylindrical fiber composite. Special emphasis is placed on the formulation of the boundary conditions that allows the simulation of all modes of overall deformation arising from any arbitrary combination of mechanical and electrical loading. The numerical approach is based on the finite element method (FEM) and it allows the extension to composites with arbitrary geometrical inclusion configurations, providing a powerful tool for fast calculation of their effective properties. For verification the effective coefficients are evaluated for square and hexagonal arrangements of unidirectional piezoelectric cylindrical fiber composites. The results obtained from the numerical technique are compared with those obtained by means of the analytical asymptotic homogenization method (AHM) for different fiber volume fractions.     

Accurate and versatile modeling of electromagnetic scattering on periodic nanostructures with a surface integral approach

A surface integral formulation for light scattering on periodic structures is presented. Electric and magnetic field equations are derived on the scatterers' surfaces in the unit cell with periodic boundary conditions. The solution is calculated with the method of moments and relies on the evaluation of the periodic Green's function performed with Ewald's method. The accuracy of this approach is assessed in detail. With this versatile boundary element formulation, a very large variety of geometries can be simulated, including doubly periodic structures on substrates and in multilayered media. The surface discretization shows a high flexibility, allowing the investigation of irregular shapes including fabrication accuracy. Deep insights into the extreme near-field of the scatterers as well as in the corresponding far-field are revealed. This method will find numerous applications for the design of realistic photonic nanostructures, in which light propagation is tailored to produce novel optical effects.