Evaluation of porosity in composite aircraft structures

Conclusion The use of an analytical model for determining the moduli of elasticity of composite laminates made of woven or unidirectional plies with different porosity levels was described. The analysis of aircraft composite parts with different levels of voids (porosity) was based on a method which utilizes the results of state-of-the-art nondestructive testing methods (ultrasonic through transmission, loading, or pulse/echo) as the starting data for the analytical model. The porosity distribution over the volume of the material and correlations for the nondestructive testing methods were determined for epoxy-carbon laminates with standard and stiffened binders and corroborating experiments were conducted. It was shown that the moduli of elasticity of composite laminates decrease with an increase in the porosity levels. The type, thickness, and layup of the laminate are the basic factors that affect the decrease in the elastic properties of porous composite laminates.Translated from Mekhanika Kompozitnykh Materialov, Vol. 30, No. 6, pp. 813–830, November–December, 1994.     

Aerated autoclaved concrete: Stochastic structure model and elastic properties

Aerated autoclaved concrete (AAC) is a modern and important construction material, whose elastic properties are primarily defined by its porosity. The possibility to predict elastic properties of AAC based on the voids distribution is very important. The report describes simulations of the mechanical properties of AAC, based on a stochastic-geometric model of its structure. The model is the well-known “cherry-pit” model, which presents a random system of partially overlapping spheres. In the mechanical analysis the solid phase is approximated by a network model with the help of the so-called radical tessellation with respect to the hard spheres of the “cherry-pit” model. The network edges are modelled in ANSYS as 3D beams. In this approach, the discretized elements (the edges) have in distinction to FE calculations with small polyhedral same dimension as the air voids and so the numerical costs can be drastically reduced. The FE simulations calculate the elastic constants and energy concentrations, which are responsible for the material failures, in large samples. Comparisons with fracture tests showed good matching between simulations and experiments. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A generalized self-consistent method for composites with random elastic properties of inclusions

The generalized self-consistent method is extended to the problems of statistical mechanics of composites with random elastic properties of inclusions. This approach makes it possible to reduce the problem of predicting the effective elastic properties of composites with random structures to a sequence of simpler homogenized boundary-value problems for solitary inclusions with inhomogeneous elastic transition layers in a homogeneous effective elastic medium and with the corresponding boundary conditions. The elastic properties of a solitary inclusion for the gth homogenized problem are found from the solutions of the gth and (g+1)th homogenized problems. The elastic properties and sizes of the transition layers account for the random distribution, random sizes, and random elastic properties of inclusions in the composite. A test problem of predicting the effective elastic properties of a transversely isotropic layer composite with random elastic properties of some layers is solved by using the method proposed. The solution obtained coincides with the known exact solution [1].Perm State Technical University, Perm, Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 35, No. 6, pp. 785–796, November–December, 1999.     

Propagation of elastic waves in certain composites

Conclusions 1. Relationships have been obtained for determining nine elastic characteristics of orthotropic composite materials from the properties of the starting components and the assigned reinforcement scheme.2. Formulas are given for calculating the propagation velocity of three types of elastic flat waves for an arbitrary direction in one of the planes of elastic symmetry of a uniform orthotropic material.3. It has been shown that the velocity of the first arrival of a packet of ultrasonic vibrations which is recorded in an experiment is equal to the velocity of motion of the wave front in a limitless medium even for rather thin (5–10 mm) fiberglass-plastic specimens with unidirectional or cross-reinforced schemes.4. The dependences of elastic properties and rates of propagation of elastic vibrations on direction which are calculated theoretically from the properties of the starting components and the reinforcement scheme agree satisfactorily with experimental results.Translated from Mekhanika Polimerov, No. 3, pp. 531–536, May–June, 1978.     

The influence of imperfections on the creep behavior of woven polymer composites at elevated temperatures

The dynamic behavior of compression molded polymer/woven graphite fiber composites at elevated temperatures is investigated analytically. This is performed with the objective of predicting the initiation of catastrophic failure that may occur after prolonged usage of the material at these temperatures. Special attention is paid to the behavior of the voids present in them where the failure may occur. The polymer matrix is modeled as a 4-parameter model (Maxwell-Voigt combination) (Govindarajan et al., in: Advances in Computer-Aided Engineering, ASME, 1994) while the composite structure is modeled using the fiber undulation model (Ishikawa and Chou, J. Mater. Sci. 17, pp. 3211–3220, 1982). The relation between the polymer properties and the ambient temperature is modeled after Arhenius' relation (Govindarajan et al., 1994; Ferry, Viscoelastic Properties of Polymers, Wiley, New York, 1961). The multiple phases in the matrix are taken into account through Eshelby's theory (Proc. Royal Soc. London A 241, pp. 376–396, 1957) and its extension for multiple occurrences of the same phase (Tanden and Weng, Polymer Composites 5, pp. 327–333, 1984; Weng, Internat. J. Eng. Sci. 22 (7), pp. 845–856, 1984) which assumes an ellipsoidal shape for inclusions. The resulting elastic equations are transformed into the time domain using Laplace transformation and the correspondence principle (Govindarajan et al., 1994; Wang and Weng, ASME J. Appl. Mech, 1992). All the voids are considered to be prolate ellipsoids with the 1-axis being the axis of symmetry. The distribution of voids is assumed to be of a Gaussian form with respect to the aspect ratio. The response of the composite under creep condition (constant load) has been simulated. Relations between the applied stress and the stresses in the matrix/void phase are also supplied, so that the influence of the voids may be characterized. The model is then applied to simulate the behavior of an epoxy/woven graphite composite to obtain the numerical results.     

Polydisperse structures of composites with random elastic properties of inclusions

A generalized self-consistent method [1, 2] is developed and applied to the boundary-value problems of composites with random elastic properties of inclusions. The approach suggested makes it possible to allow for a random mutual arrangement, statistical dispersion of elastic properties and sizes of the inclusions, and their mutual correlation in terms of special homogenized indicator functions. For comparison, the analytical solutions and those obtained from a corresponding sequence of H+1 (H=0,1,…) linked homogenized problems of the self-consistent method for the strain distribution in the inclusions and for the tensor of effective elastic properties of the composite are given. A numerical calculation of the effective transversely isotropic elastic characteristics for a unidirectional polydisperse fibrous composite is also presented. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 1, pp. 33–58, January–February, 2000.     

Structural approach to calculation of the effect of moisture on elastic characteristics of organoplastics

Data have been obtained for the structural calculation of the effect of moisture on the elastic characteristics of organoplastics from the properties of components. The distribution of moisture between the fiber and matrix — the components of a unidirectional composite — is considered. The elastic properties of the fiber are determined by an inverse calculation using the experimental dependences of the composite and matrix on moisture. The moisture effect on the properties of the materials is taken into account with influence functions, which differ by more than 25% for various characteristics. The results can be used for calculating the elastic properties of composites with various reinforcement schemes and at the nonequilibrium distribution of the moisture concentration in an actual environment.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Institute of Polymer Mechanics, Riga, LV-1006, Latvia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 34, No. 4, pp. 525–530, July–August, 1998.     

Computational approach for composite materials with coupled constitutive laws

A numerical scheme based on fast Fourier transform is presented to compute the effective response and the local fields within a heterogeneous material which exhibits a coupled constitutive law. It consists in the iterative resolution of periodic coupled Lippmann–Schwinger equations. This approach is illustrated in the case of electroelastic composite materials. By using an augmented Lagrangian formulation, a simple iterative scheme relying on the uncoupled Green operators for the elastic and electrostatics problems is proposed. This computational framework, which allows to consider composite materials with an infinite contrast on the local properties, is assessed in the case of porous and fiber-reinforced piezoelectric materials.     

Modal analysis of piezoelectric bodies with voids. II. Finite element simulation

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Modal analysis of piezoelectric bodies with voids. I. Mathematical approaches

The paper is concerned with the eigenvalue problems for piezoelectric bodies with voids in contact with massive rigid plane punches and coved by the system of open-circuited and short-circuited electrodes. The linear theory of piezoelectric materials with voids for porosity change properties according to Cowin–Nunziato model is used. The generalized statements for eigenvalue problem are obtained in the extended and reduced forms. A variational principle is constructed which has the properties of minimality, similar to the well-known variational principle for problems with pure elastic media. The discreteness of the spectrum and completeness of the eigenfunctions are proved. The orthogonality relations for eigenvectors are obtained in different forms. As a consequence of variational principles, the properties of an increase or a decrease in the natural frequencies, when the mechanical, electric and “porous” boundary conditions and the moduli of piezoelectric solid with voids change, are established.     

Experimental investigation and approximation of the temperature-dependent stiffness of short-fiber reinforced polymers

In order to predict the effective material properties of a short-fiber reinforced polymer (SFRP), homogenization of elastic properties with the self-consistent (SC) scheme and the interaction direct derivative (IDD) method is performed by means of µCT data describing the microstructure of the composite material. Using dynamic mechanical analysis (DMA), the material properties of both, polypropylene and fiber reinforced polypropylene are investigated by tensile tests under thermal load. The measured storage modulus of the matrix material is used as input parameter for the homogenization scheme. The effective properties of SFRP are compared to experimental results from DMA. (© 2015 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effect of parameters of the distribution of the length and orientation of short fibers on characteristics of the dynamic viscoelasticity of a polymer composite

Conclusion An algorithm for calculating the dynamic viscoelastic characteristics of a composite reinforced with short fibers was developed and realized in the form of a computer program. An analysis was made of the dependence of the characteristics of the composite on the volume content and length of its fibers, as well as on statistical distributions of fiber length and orientation in the material. It was shown that a change in the parameters of the statistical distributions has a significant effect on both the elastic and the dissi-pative properties of the composite. It was found that ignoring the statistical fiber-length distribution might lead to overestimation of the real component of the complex modulus and underestimation of the mechanical loss tangent.Translated from Mekhanika Kompozitnykh Materialov, No. 1, pp. 13–17, January–February, 1990.     

Application of the method of conditional moments to investigating the deformation properties of orthotropic composites with fiber microdamages

A model of deformation of stochastic composites subjected to microdamage is developed for the case of orthotropic materials with microdamages accumulating in the fibers. The composite is treated as a matrix strengthened with elliptic fibers with orthotropic elastic properties. The fractured microvolumes are modeled by a system of randomly distributed quasi-spherical pores. The porosity balance equation and relations for determining the effective elastic moduli for the case of a fibrous composite with orthotropic components are used as the fundamental relations. The fracture criterion is given as a limit value of the intensity of average shear stresses occurring in the undamaged part of the material, which is assumed to be a random function of coordinates and is described by the Weibull distribution. Based on an analytical and numerical approach, the algorithm for determining the nonlinear deformation properties of such a material is constructed. The nonlinearity of composite deformations is caused by the accumulation of microdamages in the fibers. By using a numerical solution, the nonlinear stress–strain diagrams for an orthotropic composite in uniaxial tension are obtained. Translated from Mekhanika Kompozitnykh Materialov, Vol. 45, No. 1, pp. 17–30, January–February, 2009.     

Composites based on cellulose fiber nonwovens and a water soluble polymer 1. Structure and strength-deformation characteristics of cellulose fiber nonwovens and structural characteristics of the composites

The results of a study on the strength-deformation characteristics (tensile elastic modulus, ultimate strength, elongation at break, and punching and tearing strengths) of two kinds of cellulose fiber nonwovens (CFNs) with dissimilar void content and different geometrical parameters of cellulose fibers are discussed. The structural characteristics of composites prepared by impregnation with poly(vinyl alcohol) water solutions are analyzed, too. Composites with volume fractions of polymer up to 0.4% and volume fractions of voids up to 0.3% were prepared. Filling of voids by the polymer occurred without significant changes in the structure of CFNs. The fraction of closed voids increased with polymer content.     

Computational approach for composite materials with coupled constitutive laws

A numerical scheme based on fast Fourier transform is presented to compute the effective response and the local fields within a heterogeneous material which exhibits a coupled constitutive law. It consists in the iterative resolution of periodic coupled Lippmann–Schwinger equations. This approach is illustrated in the case of electroelastic composite materials. By using an augmented Lagrangian formulation, a simple iterative scheme relying on the uncoupled Green operators for the elastic and electrostatics problems is proposed. This computational framework, which allows to consider composite materials with an infinite contrast on the local properties, is assessed in the case of porous and fiber-reinforced piezoelectric materials.     

Nonlinear flexural response of laminated composite plates under hygro-thermo-mechanical loading

The paper deals with Chebyshev series based analytical solution for the nonlinear flexural response of the elastically supported moderately thick laminated composite rectangular plates subjected to hygro-thermo-mechanical loading. The mathematical formulation is based on higher order shear deformation theory (HSDT) and von-Karman nonlinear kinematics. The elastic foundation is modeled as shear deformable with cubic nonlinearity. The elastic and hygrothermal properties of the fiber reinforced composite material are considered to be dependent on temperature and moisture concentration and have been evaluated utilizing micromechanics model. The quadratic extrapolation technique is used for linearization and fast converging finite double Chebyshev series is used for spatial discretization of the governing nonlinear equations of equilibrium. The effects of Winkler and Pasternak foundation parameters, temperature and moisture concentration on nonlinear flexural response of the laminated composite rectangular plate with different lamination scheme and boundary conditions are presented.     

Numerical Estimation of the Elastic Properties of Thin-Walled Structures Manufactured from Short-Fiber-Reinforced Thermoplastics

A model which allows us to estimate the elastic properties of thin-walled structures manufactured by injection molding is presented. The starting step is the numerical prediction of the microstructure of a short-fiber-reinforced composite developed during the filling stage of the manufacturing process. For this purpose, the Moldflow Plastic Insight^® commercial program is used. As a result of simulating the filling process, a second-rank orientation tensor characterizing the microstructure of the material is obtained. The elastic properties of the prepared material locally depend on the orientational distribution of fibers. The constitutive equation is formulated by means of orientational averaging for a given orientation tensor. The tensor of elastic material properties is computed and translated into the format for a stress-strain analysis based on the ANSYSÒ finite-element code. The numerical procedure and the convergence of results are discussed for a thin strip, a rectangular plate, and a shell of revolution. The influence of manufacturing conditions on the stress-strain state of statically loaded thin-walled elements is illustrated.     

Comparative micromechanical analysis of nonelastic behavior of unidirectional plastics with tetragonal and hexagonal structures

A mathematical model for determining the effective elastic properties and describing the processes of inelastic deformation and damage accumulation of unidirectional fiber-reinforced composites with tetragonal and hexagonal structures is developed. A comparative analysis of the effective elastic moduli of glass, boron, organic, and carbon unidirectional plastics shows that, if the fiber volume fraction does not exceed 0.5, the effective elastic properties calculated by the models presented give closely related results. The calculation results for nonlinear fields of deformation and failure are presented and the limiting strength surfaces of fibrous glass plastics with hexagonal and tetragonal structures are obtained for different transverse loading paths. It is found that the structure of a composite affects significantly its strength properties.Submitted to the 11th International Conference on Mechanics of Composite Materials (Riga, June 11–15, 2000).Perm' State Technical University, Perm', Russia. Translated from Mekhanika Kompozitnykh Materialov, Vol. 36, No. 4, pp. 455–464, July–August, 2000.     

Finite element based nonlinear dynamic response of elastically supported piezoelectric functionally graded beam subjected to moving load in thermal environment with random system properties

The second order statistics in terms of mean and standard deviation (SD) of normalized nonlinear transverse dynamic central deflection (NTDCD) response of un-damped elastically supported functionally graded materials (FGMs) beam with surface-bonded piezoelectric layers under the action of moving load are investigated in this paper. The random system properties such as Young's modulus, Poisson's ratio, density, thermal expansion coefficients, piezoelectric materials, volume fraction exponent and external loading are modeled as uncorrelated random variables. The basic formulation is based on higher order shear deformation theory (HSDT) with von-Karman nonlinear strain kinematics combined with Newton–Raphson technique through Newmark's time integrating scheme using finite element method (FEM). The non-uniform temperature distribution with temperature dependent material properties is taken into consideration for consideration of thermal loading. The one parameter Pasternak elastic foundation with Winkler cubic nonlinearity is considered as an elastic foundation. The stochastic based second order perturbation technique (SOPT) and direct Monte Carlo simulation (MCS) are adopted for the solution of nonlinear dynamic governing equation. The influences of volume fraction exponents, temperature increments, moving loads and velocity, nonlinearity, slenderness ratios, foundation parameters and external loadings with random system properties on the NTDCD are examined. The capability of present stochastic model in predicting the NTDCD statistics are compared by studying their convergence with the existing results those available in the literature.     

Mean and full field homogenization of artificial long fiber reinforced thermoset polymers

Based on two artificial microstructures representing a long fiber reinforced thermoset material, the effective linear elastic material properties are calculated by both a mean and a full field homogenization method. Concerning the mean field method, the effective elastic material properties are approximated using the homogenization scheme by Mori and Tanaka, formulated explicitly in terms of orientation averages. This allows to use orienation tensors of 2nd and 4th order describing the orientation information on the micro level. The full field method is based on the fast Fourier transformation (FFT), for which the effective material properties are determined by volume averaging. The comparison between both methods show good agreements, the deviations are in the range between 2% and 12%. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)