Derivatives of Green’s functions in piezoelectric media and their application in dislocation dynamics

Three-dimensional extended Green’s functions and their high-order derivatives in general anisotropic piezoelectric materials are derived and expressed in integral forms. They can be evaluated directly by the Gaussian numerical integration method. The extended Green’s functions and their derivatives by the present method have accuracy and computational efficiency. Using the extended Green’s functions, the stress field induced by an arbitrary dislocation in an anisotropic piezoelectric medium, is obtained and expressed as a line integral around the dislocation.     

The wave propagation in a beam on a random elastic foundation

This paper presents a study of wave propagation in an infinite beam on a random Winkler foundation. The spatial variation of the foundation spring constant is modelled as a random field and the influence of the correlation length is studied. As it is impossible to determine the general stochastic Green’s function, the configurational average of the Green’s function and its correlation function are considered. These functions are found through the transformation of the stochastic equation of motion into the Dyson equation for the mean or coherent field and the Bethe–Salpeter equation for the field correlation, as used in the study of wave propagation in random media. The approximate solutions of the Dyson and the Bethe–Salpeter equations are validated by means of a Monte Carlo simulation and compared with the results of a classical Neumann expansion method. Although both methods only involve the second order statistics of the random field, the approximation of the Dyson and the Bethe–Salpeter equations gives better results than the Neumann expansion, at the expense of a larger computational effort. Furthermore, the results show that a small spatial variation of the spring constant has an influence on the response if the correlation length and the wavelength have a similar order of magnitude, while the waves in the beam cannot resolve the spatial variation in the case where the correlation length is much smaller than the wavelength.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random variable case

This paper is the first of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. In this first paper, the concept of the variability response function (VRF) is discussed in some detail with respect to its strengths and its limitations. It is the first time that various limitations of the classical VRF are discussed. The concept of associated fields is then introduced as a potential tool for overcoming the limitations of the classical VRF. As a first step, the special case of material property variations modeled by a single random variable is examined. Specifically, beam structures with the elastic modulus being the only stochastic property are studied. Results yield a hierarchy of upper bounds on the mean, variance and exceedance values of the response displacement, obtained from zero-mean U-shaped beta-distributed random variables with prescribed standard deviation and lower limit. In the second paper that follows, the concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in this paper to more general problems involving stochastic fields.     

A hierarchy of upper bounds on the response of stochastic systems with large variation of their properties: random field case

This paper is the second of a two-part series that constitutes an effort to establish spectral- and probability-distribution-free upper bounds on various probabilistic indicators of the response of stochastic systems. The concept of the generalized variability response function is introduced and used with the aid of associated fields to extend the upper bounds established in the first paper for the special case of material property variations modeled by random variables to more general problems involving random fields. Specifically, a hierarchy of spectral- and probability-distribution-free upper bounds on the mean, variance, and exceedance values of the response of stochastic systems is established when only the coefficient of variation and lower limit of the stochastic material properties are known. Furthermore, a hierarchy of probability-distribution-free upper bounds on these quantities is established when the spectral density function describing the stochastic material properties is known in addition to the coefficient of variation and the lower limit.     

Cell size effect analysis of the effective Young’s modulus of sandwich core

In this paper, different methods of predicting the effective Young’s modulus of periodic cellular materials are taken into account and their abilities on revealing the size effect of the unit cell are analyzed. The bending energy method (BEM) is proposed to compare with the homogenization method (HM) and the G-A meso-mechanics method (G-A MMM). A variety of tests are carried out for different cell configurations and discussions are fully made to show its advantage. It is proved that the HM and G-A MMM results only compared to the situation of the scale factor n → ∞ and cannot reflect the size effect of the unit cell. The BEM results can perfectly reveal the underlying relationship between the effective Young’s moduli and the cell size and they are consistent with the HM and G-A MMM results for n → ∞.     

Experimental and numerical study of the Young’s modulus vs temperature for heterogeneous model materials with polygonal inclusions

This work is related to the thermomechanical behaviour of industrial refractory materials. The microstructural complexity of such materials and the strong influence of the elastic properties on the resistance to thermomechanical sollicitations, lead us to study first of all the Young’s modulus of heterogeneous model materials with a simplified microstructure. The studied materials are composed of a glass matrix surrounding polygonal alumina inclusions. These two materials exhibit a dilatometric dissension sufficiently large to induce, during a thermal cycle, thermal stresses able to damage the matrix/inclusions interfaces. The present study deals with the Young’s modulus variations of the studied model materials according to the temperature. The numerical simulation of the matrix/inclusions interfaces behaviour was carried out using the Abaqus FEM code whose contact tool “Debond” allows to account for the interface matrix damage during a thermal cycle. The results show the ability of this tool to well describe the evolution of damage in function of the temperature. The Young’s modulus of the model materials was also measured using ultrasonic technique. A good agreement of the numerical and experimental results is obtained.     

Stochastic finite element analysis of a cable-stayed bridge system with varying material properties

Stochastic seismic finite element analysis of a cable-stayed bridge whose material properties are described by random fields is presented in this paper. The stochastic perturbation technique and Monte Carlo simulation (MCS) method are used in the analyses. A summary of MCS and perturbation based stochastic finite element dynamic analysis formulation of structural system is given. The Jindo Bridge, constructed in South Korea, is chosen as a numerical example. The Kocaeli earthquake in 1999 is considered as a ground motion. During the stochastic analysis, displacements and internal forces of the considered bridge are obtained from perturbation based stochastic finite element method (SFEM) and MCS method by changing elastic modulus and mass density as random variable. The efficiency and accuracy of the proposed SFEM algorithm are evaluated by comparison with results of MCS method. The results imply that perturbation based SFEM method gives close results to MCS method and it can be used instead of MCS method, especially, if computational cost is taken into consideration.     

Reliability predictions of laminated composite plates with random system parameters

Reliability predictions of laminated composite plates with random system parameters subjected to transverse loads are performed using different methods. System parameters such as material properties, layer thicknesses, and lamina strengths of a laminated composite plate are treated as base-line random variables and an appropriate failure criterion is used to construct the limit state equation of the plate in the reliability analysis. Based on the statistics of the base-line random variables obtained from experiments, different methods, namely, Monte Carlo method, β method, and first-order second moment method, are used to calculate the reliability of the laminated composite plates. In the first-order second moment method, the stochastic finite element method is used to derive for the statistics of the first-ply failure load of the laminated composite plates from those of the base-line random variables. The reliability of the laminated plate is then computed using the theoretically determined statistics together with an assumed probability distribution function of the first-ply failure load. The feasibility and accuracy of the different methods are studied by means of the experimental data of centrally loaded laminated composite plates with different lay-ups. The suitability of several commonly used failure criteria for reliability analysis of laminated composite plates is also investigated by means of several examples.     

Fatigue damage assessment for a spectral model of non-Gaussian random loads

In this paper, a new model for random loads–the Laplace driven moving average–is presented. The model is second order, non-Gaussian, and strictly stationary. It shares with its Gaussian counterpart the ability to model any spectrum but has additional flexibility to model the skewness and kurtosis of the marginal distribution. Unlike most other non-Gaussian models proposed in the literature, such as the transformed Gaussian or Volterra series models, the new model is no longer derivable from Gaussian processes. In the paper, a summary of the properties of the new model is given and its upcrossing intensities are evaluated. Then it is used to estimate fatigue damage both from simulations and in terms of an upper bound that is of particular use for narrowband spectra.     

Non‐linear random response of laminated composite shallow shells using finite element modal method

This paper investigates the large‐amplitude multi‐mode random response of thin shallow shells with rectangular planform at elevated temperatures using a finite element non‐linear modal formulation. A thin laminated composite shallow shell element and the system equations of motion are developed. The system equations in structural node degrees‐of‐freedom (DOF) are transformed into modal co‐ordinates, and the non‐linear stiffness matrices are transformed into non‐linear modal stiffness matrices. The number of modal equations is much smaller than the number of equations in structural node DOF. A numerical integration is employed to determine the random response. Thermal buckling deflections are obtained to explain the intermittent snap‐through phenomenon. The natural frequencies of the infinitesimal vibration about the thermally buckled equilibrium positions (BEPs) are studied, and it is found that there is great difference between the frequencies about the primary (positive) and the secondary (negative) BEPs. All three types of motion: (i) linear random vibration about the primary BEP, (ii) intermittent snap‐through between the two BEPs, and (iii) non‐linear large‐amplitude random vibration over the two BEPs, can be predicted. Copyright © 2006 John Wiley & Sons, Ltd.     

Constructing models of random processes with specified properties by the Wiener method for identifying dynamic systems

It is shown that the use of the proposed models of random processes with specified statistical properties solves fundamental problems of identifying dynamic systems by Wiener’s method using testing actions of special form. Examples of the construction of these models in the time domain are presented. __________ Translated from Izmeritel’naya Tekhnika, No. 8, pp. 43–46, August, 2006.     

Elastic wave propagation in a 3‐D unbounded random heterogeneous medium coupled with a bounded medium. Application to seismic soil–structure interaction (SSSI)

We present a sub‐structuring method for the coupling between a large elastic structure, and a stratified soil half‐space exhibiting random heterogeneities over a bounded domain and impinged by incident waves. Both media are also weakly dissipative. The concept of interfaces classically used in sub‐structuring methods is extended to ‘volume interfaces’ in the proposed approach. The random dimension of the stochastic fields modelling the heterogeneities in the soil is reduced by introducing a Karhunen–Loéve expansion of these stochastic fields. The coupled overall problem is solved by Monte‐Carlo simulation techniques. A realistic example of a large industrial structure interacting with an uncertain stratified soil medium under earthquake is finally presented. This case study and others validate the presented methodology and its ability to handle complex mechanical systems. Copyright © 2002 John Wiley & Sons, Ltd.     

Modeling of viscoelastic structures with random material properties using time-separated stochastic mechanics

Modeling and simulation of materials with stochastic properties is an emerging field in both mathematics and mechanics. The most important goal is to compute the stochastic characteristics of the random stress, such as the expectation value and the standard deviation. An accurate approach are Monte Carlo simulations; however, they consume drastic computational power due to the large number of stochastic realizations that have to be simulated before convergence is achieved. In this paper, we show that a recently published approach for accurate modeling of viscoelastic materials with stochastic material properties at the material point level in the work of Junker and Nagel is also valid for macroscopic bodies. The method is based on a separation of random but time-invariant variables and time-dependent but deterministic variables for the strain response at the material point (time-separated stochastic mechanics [TSM]). We recall the governing equations, derive a simplified form, and discuss the numerical implementation into a finite element routine. To validate our approach, we compare the TSM simulations with Monte Carlo simulations, which provide the “true” answer but at unaffordable computational costs. In contrast, the numerical effort of our approach is in the same range as for deterministic viscoelastic simulations.     

Interval spectral stochastic finite element analysis of structures with aggregation of random field and bounded parameters

This paper presents the study on non‐deterministic problems of structures with a mixture of random field and interval material properties under uncertain‐but‐bounded forces. Probabilistic framework is extended to handle the mixed uncertainties from structural parameters and loads by incorporating interval algorithms into spectral stochastic finite element method. Random interval formulations are developed based on K–L expansion and polynomial chaos accommodating the random field Young's modulus, interval Poisson's ratios and bounded applied forces. Numerical characteristics including mean value and standard deviation of the interval random structural responses are consequently obtained as intervals rather than deterministic values. The randomised low‐discrepancy sequences initialized particles and high‐order nonlinear inertia weight with multi‐dimensional parameters are employed to determine the change ranges of statistical moments of the random interval structural responses. The bounded probability density and cumulative distribution of the interval random response are then visualised. The feasibility, efficiency and usefulness of the proposed interval spectral stochastic finite element method are illustrated by three numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Optimal representation of multi-dimensional random fields with a moderate number of samples: Application to stochastic mechanics

A significant amount of problems and applications in stochastic mechanics and engineering involve multi-dimensional random functions. The probabilistic analysis of these problems is usually computationally very expensive if a brute-force Monte Carlo method is used. Thus, a technique for the optimal selection of a moderate number of samples effectively representing the entire space of sample realizations is of paramount importance. Functional Quantization is a novel technique that has been proven to provide optimal approximations of random functions using a predetermined number of representative samples. The methodology is very easy to implement and it has been shown to work effectively for stationary and non-stationary one-dimensional random functions. This paper discusses the application of the Functional Quantization approach to the domain of multi-dimensional random functions and the applicability is demonstrated for the case of a 2D non-Gaussian field and a two-dimensional panel with uncertain Young modulus under plane stress.     

A comparative study of Young’s modulus of single-walled carbon nanotube by CPMD, MD and first principle simulations

Carbon nanotubes (CNTs), due to their exceptional magnetic, electrical and mechanical properties, are promising candidates for several technical applications ranging from nanoelectronic devices to composites. Young’s modulus holds the special status in material properties and micro/nano-electromechanical systems (MEMS/NEMS) design. The excellently regular structures of CNTs facilitate accurate simulation of CNTs’ behavior by applying a variety of theoretical methods. Here, three representative numerical methods, i.e., Car–Parrinello molecular dynamics (CPMD), density functional theory (DFT) and molecular dynamics (MD), were applied to calculate Young’s modulus of single-walled carbon nanotube (SWCNT) with chirality (3,3). The comparative studies showed that the most accurate result is offered by time consuming DFT simulation. MD simulation produced a less accurate result due to neglecting electronic motions. Compared to the two preceding methods the best performance, with a balance between efficiency and precision, was deduced by CPMD.     

Influence of continuum damage mechanics on the Bree’s diagram of a closed end tube

This paper extends the Bree’s cylinder behaviors, which is subjected to the constant internal pressure and cyclic temperature gradient loadings, with considering continuum damage mechanics coupled with nonlinear kinematic hardening model. The Bree’s biaxial stress model is modified using the unified damage and the Armstrong–Frederick nonlinear kinematic hardening models. With the help of the return mapping algorithm, the incremental plastic strain in axial and tangential directions is obtained. Continuum damage mechanics approach can be used to extend the Bree’s diagram to the damaging structures and reduce the plastic shakedown domain. Kinematic hardening behavior was considered in the material model which shifts the ratcheting zone. The role of the material constants in the Bree’s diagram is also discussed.     

Non‐isothermal ‘2‐D flow/3‐D thermal’ developments encompassing process modelling of composites: flow/thermal/cure formulations and validations

In the manufacturing process of large geometrically complex components comprising of fibre‐reinforced composite materials by resin transfer molding (RTM), the process involves injection of resin into a mold cavity filled with porous fibre preforms. The overall success of the RTM manufacturing process depends on the complete impregnation of the fibre mat by the polymer resin, prevention of polymer gelation during filling, and subsequent avoidance of dry spots. Since a cold resin is injected into a hot mold, the associated physics encompasses a moving boundary value problem in conjunction with the multi‐disciplinary study of flow/thermal and cure kinetics inside the mold cavity. Although experimental validations are indispensable, routine manufacture of large complex structural geometries can only be enhanced via computational simulations, thus eliminating costly trial runs and helping the designer in the set‐up of the manufacturing process. This study describes the computational developments towards formulating an effective simulation‐based design methodology using the finite element method. The specific application is for thin shell‐like geometries with the thickness being much smaller than the other dimensions of the part. Due to the highly advective nature of the non‐isothermal conditions involving thermal and polymerization reactions, special computational considerations and stabilization techniques are also proposed. Validations and comparisons with experimental results are presented whenever available. Copyright © 2001 John Wiley & Sons, Ltd.     

On the dual iterative stochastic perturbation‐based finite element method in solid mechanics with Gaussian uncertainties

The main idea is a dual mathematical formulation and computational implementation of the iterative stochastic perturbation‐based finite element method for both linear and nonlinear problems in solid mechanics. A general‐order Taylor expansion with random coefficients serves here for the iterative determination of the basic probabilistic characteristics, where linearization procedure widely applicable in stochastic perturbation technique is replaced with the iterative one. The expected values and, in turn, the variances are derived first, and then, they are substituted into the equations for higher central probabilistic moments and additional probabilistic characteristics. The additional formulas for up to the fourth‐order probabilistic characteristics are derived thanks to the 10th‐order Taylor expansion. Computational implementation of this idea in the stochastic finite element method is provided by using the direct differentiation method and, independently, the response function method with polynomial basis. Numerical experiments include the simple tension of the elastic bar, nonlinear elasto‐plastic analysis of the aluminum 2D truss, and solution to the homogenization problem of periodic fiber‐reinforced composite with random elastic properties. The expected values, coefficients of variation, skewness, and kurtosis of the structural response determined via this iterative scheme are contrasted with these estimated by the Monte Carlo simulation as well as with the results of the semi‐analytical probabilistic technique following the response function method itself. Although the entire methodology is illustrated here by using the Gaussian variables where all odd‐order terms simply vanish, it can be extended towards non‐Gaussian processes as well and completed with all the perturbation orders. Copyright © 2015 John Wiley & Sons, Ltd.