Variability response functions for apparent material properties

The computation of apparent material properties for a random heterogeneous material requires the assumption of a solution field on a finite domain over which the apparent properties are to be computed. In this paper the assumed solution field is taken to be that defined by the shape functions that underpin the finite element method and it is shown that the variance of the apparent properties calculated using the shape functions to define the solution field can be expressed in terms of a variability response function (VRF) that is independent of the marginal distribution and spectral density function of the underlying random heterogeneous material property field. The variance of apparent material properties can be an important consideration in problems where the domain over which the apparent properties are computed is smaller than the representative volume element and the approach introduced here provides an efficient means of calculating that variance and performing sensitivity studies with respect to the characteristics of the material property field. The approach is illustrated using examples involving heat transfer problems and finite elements with linear and nonlinear shape functions and in one and two dimensions. Features of the VRF are described, including dependency on shape and scale of the finite element and the order of the shape functions.     

Non-linear response statistics of composite laminates with random material properties under random loading

Laminated composite plates find extensive use in many engineering applications. Some of these incorporate large deflections that may not be in the linear range. The external loading may be random in nature. The laminate material properties show an inherent dispersion around a mean value. In this paper the static response of laminated composite flat plates to transverse random loading has been studied. The material properties have been taken as random variables for accurate prediction of the system behaviour. The basic formulation of the problem has been developed based on the classical laminate theory and the Von-Karman non-linear strain–displacement relationship. A first order perturbation technique has been used to obtain the second order response statistics. Typical results have been presented for a plate with all edges simply supported. A comparison has been drawn with Monte Carlo simulation results for validation of the proposed approach. The effects of side-to-thickness ratio, aspect ratio and change in standard deviation of input random variables have been investigated for cross-ply symmetric and anti-symmetric laminates.     

Modified linear estimation method for generating multi-dimensional multi-variate Gaussian field in modelling material properties

Although a number of methods have been developed to generate random fields, it remains a challenge to efficiently generate a large, multi-dimensional, multi-variate property field. For such problems, the widely used spectral representation method tends to require relatively longer computing time. In this paper, a modified linear estimation method is proposed, which involves mapping the linearly estimated field through a series of randomized translations and rotations from one realization to the next. These randomized translations and rotations enable the simulated property field to be stationary. The autocorrelation function of the simulated fields can be approximately described by a squared exponential function. The algorithms of the proposed method in both the rectangular and cylindrical polar coordinate systems are demonstrated and the results validated by Monte-Carlo simulations. Comparisons between the proposed method and spectral representation method show that the results from both methods are in good agreement, as long as the cut-off wave numbers of the spectral representation method are sufficiently large. However, the proposed method requires much less computational time than the spectral representation method. This makes it potentially useful for generating large multi-dimensional fields in random finite element analysis. Applications of the proposed method are exemplified in both rectangular and cylindrical polar coordinate systems.     

Estimates of effective characteristics of random cell structures in terms of effective characteristics of periodic structures

The bounds of effective characteristics of random cell structures are obtained in terms of effective characteristics of periodic structures.     

Non-Gaussian simulation of local material properties based on a moving-window technique

In this paper, a moving-window micromechanics technique, Monte Carlo simulation, and finite element analysis are used to assess the effects of microstructural randomness on the local stress response of composite materials. The randomly varying elastic properties are characterized in terms of a field of local effective elastic constitutive matrices using a moving-window technique based on a finite element model of a given digitized composite material microstructure. Once the fields are generated, estimates of the random properties are obtained for use as input to a simulation algorithm that was developed to retain spectral, correlation, and non-Gaussian probabilistic characteristics. Rapidly generated Monte Carlo simulations of the constitutive matrix fields are used in a finite element analysis to create a series of local stress fields associated with the random material sample under uniaxial tension. This series allows estimation of the statistical variability in the local stress response for the random composite. The identification of localized extreme stress deviations from those of the aggregate or effective properties approach highlight the importance of modeling the stochastic variability of the microstructure.     

Probabilistic bounds on effective elastic moduli for the superconducting coils

The main purpose of the paper presented is probabilistic characterization of the effective elastic characteristics of a superconducting four-component composite. The probabilistic moments of these characteristics up to the fourth-order have been estimated on the basis of experimentally measured expected values and standard deviations of component materials elastic properties (Young moduli and Poisson coefficients). The Monte-Carlo simulation technique has been used to carry out all computational experiments. The results computed have been discussed in details in order to verify the sensitivity of respective probabilistic moments to input random data and to total number of random trials used to estimate these values.     

Sparse polynomial chaos expansions of frequency response functions using stochastic frequency transformation

Frequency response functions (FRFs) are important for assessing the behavior of stochastic linear dynamic systems. For large systems, their evaluations are time-consuming even for a single simulation. In such cases, uncertainty quantification by crude Monte-Carlo simulation is not feasible. In this paper, we propose the use of sparse adaptive polynomial chaos expansions (PCE) as a surrogate of the full model. To overcome known limitations of PCE when applied to FRF simulation, we propose a frequency transformation strategy that maximizes the similarity between FRFs prior to the calculation of the PCE surrogate. This strategy results in lower-order PCEs for each frequency. Principal component analysis is then employed to reduce the number of random outputs. The proposed approach is applied to two case studies: a simple 2-DOF system and a 6-DOF system with 16 random inputs. The accuracy assessment of the results indicates that the proposed approach can predict single FRFs accurately. Besides, it is shown that the first two moments of the FRFs obtained by the PCE converge to the reference results faster than with the Monte-Carlo (MC) methods.     

Efficient uncertainty quantification of dynamical systems with local nonlinearities and uncertainties

Idealized modeling of most engineering structures yields linear mathematical models, i.e., linear ordinary or partial differential equations. However, features like nonlinear dampers and/or springs can render nonlinear an otherwise linear model. Often, the connectivity of these nonlinear elements is confined to only a few degrees-of-freedom (DOFs) of the structure. In such cases, treating the entire structure as nonlinear results in very computationally expensive solutions. Moreover, if system parameters are uncertain, their stochastic nature can render the analysis even more computationally costly. This paper presents an approach for computing the response of such systems in a very efficient manner. The proposed solution procedure first segregates the DOFs appearing in the nonlinear and/or stochastic terms from those DOFs that involve only linear deterministic operations. Second, the responses of nonlinear/stochastic terms are determined using a non-standard form of a nonlinear Volterra integral equation (NVIE). Finally, the responses of the remaining DOFs are computed through a convolution approach using the fast Fourier transform to further increase the computational efficiency. Three examples are presented to demonstrate the efficacy and accuracy of the proposed method. It is shown that, even for moderately sized systems (∼1000 DOFs), the proposed method is about three orders of magnitude faster than a conventional Monte Carlo sampling method (i.e., solving the system of ODEs repeatedly).     

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.     

On numerical evaluation of effective material properties for composite structures with rhombic fiber arrangements

This paper deals with unidirectional fiber reinforced composites with rhombic fiber arrangements. It is assumed, that there is a periodic structure on micro level, which can be taken by homogenization as a representative volume element (RVE) for the composite, where the composite phases have isotropic or transversely isotropic material characterizations. A special procedure is developed to handle the primary non-rectangular periodicity with common numerical homogenization techniques based on FE-models. Due to appropriate boundary conditions applied to the RVE elastic effective macroscopic coefficients are derived. Results are listed and compared with other publications and good agreements are shown. Furthermore new results are presented, which exhibit the special orthotropic behavior of such composites caused by the rhombic fiber arrangement.     

Variability response function approach for foundation reliability

This paper explores the applicability of variability response functions to nonlinear soil–structure interaction problems, focusing on the impacts of spatially variable soil properties on foundation reliability regarding the settlement response. An estimation scheme is proposed to obtain the response functions, which involves a periodic function to approximate the relationship between foundation response and phase angles representing the soil variability patterns. Using a single set of realizations, the response function approach enables the evaluation of foundation reliability under various spatial variability patterns, including different autocovariance distances in three dimensions and the rotated anisotropy features of soil variations. This leads to significant reductions in computational demands compared to previous attempts of random field modelling, which often involved individual Monte Carlo simulations for each combination of spatial variability parameters. The proposed approach is applied to both shallow foundation and piled foundation cases, illustrating its range of applicability. For linear-elastic systems, the approach is shown to be effective for various coefficients of variation in soil variability. For elastic–plastic pile group analyses, the approach leads to efficient evaluation of the statistics of average and differential settlements of the pile group, both of which compare favourably with conventional random field simulation techniques. Since it does not require multiple random field realizations, the approach is particularly efficient in identifying the worst-case scenario of autocovariance distances that corresponds to the largest uncertainty in foundation response. This can become a useful tool for conservative reliability assessments under a project setting, since site-specific estimates of random field parameters are often imprecise due to limited site data.     

Effective properties of linear random materials: application to Al/SiC and resin/glass composites

Here we present a statistical study on the effective linear properties of random materials, i.e., microstructures which are random lattices described by a stochastic process. The local numerical procedure associated with the homogenization methods used here was based on a wavelet-element method. The resulting numerical results are compared with those obtained using classical theories. A new approach was developed in order to determine the effective properties in cases where the characteristics of the microstructure are not known.     

Variability response functions for beams with nonlinear constitutive laws

The ability to determine probabilistic information of response quantities in structural mechanics (e.g. displacements, stresses) is restricted due to lack of information on the probabilistic characteristics of uncertain system parameters. The concept of the Variability Response Function (VRF) has been proposed as a means to systematically capture the effect of the stochastic spectral characteristics of uncertain system parameters modeled by homogeneous stochastic fields on the uncertain structural response. The key property of the VRF in its classical sense is its independence from the marginal probability distribution function (PDF) and the spectral density function (SDF) of the uncertain system parameters (it depends only on the deterministic structural configuration and boundary conditions). In this paper, the existence, the uniqueness, and the SDF- and PDF-independence of a variability response function is formally proven for the first time for statically determinate beam structures following a specific class of nonlinear constitutive laws (power laws). For statically indeterminate nonlinear structures, the generalized variability response function (GVRF) methodology is shown to produce GVRFs for statically indeterminate nonlinear beams with a square-root constitutive law that are almost SDF-independent and only mildly dependent on the marginal PDF. This PDF-dependence is not significant and all GVRFs computed in this study have very similar shapes. This is important as it implies that conclusions related to the effect of correlation length scales on the response uncertainty can be inferred in general. However, the GVRF methodology for nonlinear statically indeterminate structures is only suitable when a closed-form expression is known to exist for the VRF of statically determinate structures having the same constitutive law.     

Variability response functions for stochastic systems under dynamic excitations

The concept of variability response functions (VRFs) is extended in this work to linear stochastic systems under dynamic excitations. An integral form for the variance of the dynamic response of stochastic systems is considered, involving a Dynamic VRF (DVRF) and the spectral density function of the stochastic field modeling the uncertain system properties. As in the case of linear stochastic systems under static loads, the independence of the DVRF to the spectral density and the marginal probability density function of the stochastic field modeling the uncertain parameters is assumed. This assumption is here validated with brute-force Monte Carlo simulations. The uncertain system property considered is the inverse of the elastic modulus (flexibility). The same integral expression can be used to calculate the mean response of a dynamic system using a Dynamic Mean Response Function (DMRF) which is a function similar to the DVRF. These integral forms can be used to efficiently compute the mean and variance of the transient system response together with time dependent spectral-distribution-free upper bounds. They also provide an insight into the mechanisms controlling the dynamic mean and variability system response.     

Homogenization of transient heat transfer problems for some composite materials

The article presented is devoted to the homogenization of transient heat transfer problems in some composite materials. The mathematical model used in the FEM computation is based on the effective modules method introduced for periodic composites. The effective heat conductivity is calculated in the closed form; effective heat capacity and mass density for the composite are obtained by simple spatial averaging. Such a homogenization scheme makes it possible to significantly simplify the numerical analysis of transient heat phenomena in various types of composites. Computational experiments performed using symbolic mathematics show the variability of effective heat conductivity for 2D and 3D composites as a function of the reinforcement volume ratio, of composite components conductivity coefficients as well as of the probabilistic moments of material properties versus volume ratio. Finally, using the Finite Element Method program, the comparison of transient heat transfer problem for the real and homogenized composites models is carried out.     

On the shape of effective inclusion in the Maxwell homogenization scheme for anisotropic elastic composites

The paper focuses on the main uncertainty involved in classical Maxwell’s (1873) homogenization method for elastic composites. Maxwell’s scheme that equates the far fields produced by a set of inhomogeneities and by a fictitious domain with unknown effective properties (“effective inclusion”) is re-written in terms of the compliance and stiffness contribution tensors. It is shown that the shape of the effective inclusion substantially affects the overall elastic properties. The choice of this shape in the case of anisotropic composite is a non-trivial problem that has never been discussed in literature. In this paper, we show that the problem appears due to incompleteness of the Maxwell scheme and show that the problem can be realized when the effective inclusion is of ellipsoidal shape. We discuss how the aspect ratios of the ellipsoid have to be chosen and illustrates the approach by two examples – material with cracks having orientation scatter and a three-phase transversely-isotropic composite. It is also shown that tensor of the effective elastic constants calculated in the framework of Maxwell’s scheme is always symmetric with respect to couples of indices.     

Identification of material properties of composite plates using Fourier-generated frequency response functions

The present research adopts the use of Fourier-based plate model to synthesize FRFs for its proven prominent accuracy and incorporates with a hybrid optimization algorithm. The effectiveness of FRF error function in material identification consists in the trade-off range between those of natural frequency error function and mode shape error function with about 7% reduction in absolute relative error of the evaluated elastic moduli and shear modulus with respect to those of mode shape error function as well as approximately 25% reduction in absolute relative error of the identified Poisson's ratio with respect to that of natural frequency error function.     

Integrated shape and material selection for single and multi-performance criteria

A shape and material selection method, based on the concept of shape transformers, has been recently introduced to characterize the mass efficiency of lightweight beams under bending and shear. This paper extends this method to deal with the case of torsional stiffness design, and generalize it to single and multi-crieria selection of lightweight shafts subjected to a combination of bending, shear, and torsional load. The novel feature of the paper is the useful integration of shape and material to model and visualize multi-objective selection problems. The scheme is centered on concept selection in structural design, and hinges on measures that govern the shape properties of a cross-section regardless of its size. These measures, referred as shape transformers, can classify shapes in a way similar to material classification. The procedure is exemplified by considering torsional stiffness as a constraint. The performance charts are developed for single and multi-criteria to visualize in a glance the whole range of cross-sectional shapes for each material. Each design chart is explained with a brief example.     

Combined effects of relative density and material distribution on the mechanical properties of metallic honeycombs

This paper presents the combined effects of relative density and material distribution on the elastic constants and the yield strengths of metallic honeycombs. Periodic regular hexagonal cell is employed as the structural model. Cell wall bending, transverse shear and axial stretching/compression are taken as the deformation mechanisms in the analysis. Closed-form solutions for the yield strengths and all the five independent elastic constants are obtained for honeycombs with cell walls of uniform thickness. For honeycombs with cell walls of non-uniform thickness, the closed-form solutions would be too lengthy to use in practical applications. We instead provide numerical results to show the combined effects of relative density and material distribution on the initial and full yield strengths and all the five independent elastic constants of metallic honeycombs. The results can serve as a guide for the optimal design of metallic honeycombs.     

Free vibration of composite cylindrical panels with random material properties

Virtually in all structural systems, and in particular composites, there are uncertainties in the system parameters because of practical bounds on the quality control. In the present work the effect of variations in the mechanical properties of laminated composite cylindrical panels on its natural frequency has been obtained by modeling these as random variables. The transverse shear and rotatory inertia effects have been included in the governing equations. A perturbation approach is presented to obtain the mean and variance of the random natural frequencies. The effects of thickness ratios, edge support conditions and standard deviation of material properties on response of shallow square panels have been investigated. Results have been obtained by employing the finite element method. The approach has been validated by comparison of results with other approaches.