Analysis on the variances of material and structural properties based on random field theory

Based on the random field theory (RFT) and the stochastic finite element method (SFEM), the variances of the mechanical properties of materials and structures are studied. Manufacturing processes can easily lead to the spatial variations of the load and the material properties such as moduli and density. Characterizing the elastic moduli, load and density with one-dimensional random fields, the analytical solutions for the coefficient of variations (COVs) of effective material moduli, displacement and natural frequencies of beams are obtained. Then, with the fiber and matrix properties, volume fraction modeled by two-dimensional random fields and the fiber angle as a single random variable, a Monte Carlo simulation (MCS) is performed to generate the variances of effective modulus of fiber-reinforced composite laminar plate. Compared with the previous numerical conclusions, the present results reveal that the variances of effective material properties and structural displacement are greatly dependent on both the random fields and the sizes of structures in theory.     

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.     

Simulation of local material properties based on moving-window GMC

When analyzing the behavior of composite materials under various loading conditions, the assumption is generally made that the behavior due to randomness in the material can be represented by a homogenized, or effective, set of material properties. This assumption may be valid when considering displacement, average strain, or even average stress of structures much larger than the inclusion size. The approach is less valid, however, when considering either behavior of structures of size at the scale of the inclusions or local stress of structures in general. In this paper, Monte Carlo simulation is used to assess the effects of microstructural randomness on the local stress response of composite materials. In order to achieve these stochastic simulations, the mean, variance and spectral density functions describing the randomly varying elastic properties are required as input. These are obtained here by using a technique known as moving-window generalized method of cells (moving-window GMC). This method characterizes a digitized composite material microstructure by developing fields of local effective material properties. Once these fields are generated, it is straightforward to obtain estimates of the associated probabilistic parameters required for simulation. Based on the simulated property fields, a series of local stress fields, associated with the random material sample under uniaxial tension, is calculated using finite element analysis. An estimation of the variability in the local stress response for the given random composite is obtained from consideration of these simulations.     

Microstress estimate of stochastically heterogeneous structures by the functional perturbation method: A one dimensional example

A new functional perturbation method (FPM) for calculating the probabilistic response of stochastically heterogeneous, linear elastic structures is developed. The method is based on treating the governing differential operator as well as the unknown displacement function as a functional of material modulus field. By executing a functional perturbation around the homogeneous case, a set of successive differential equations is obtained and solved, from which the average and variance of any local parameter (displacements, stresses, strains) can be found. For a linear problem, the equations to be solved in each approximation order differ from the one for the homogeneous case by a pseudo external loading (right hand side) part only. Thus, only the Green function for the homogeneous case is needed for an analytical solution of the corresponding heterogeneous problem. A one dimensional stochastically heterogeneous rod embedded in a uniform shear resistant elastic medium is solved as an example. The statistical variance of displacements and stresses are found analytically, including the edge regions. Morphological (grain size) and material (modulus) effects on the stochastic response are demonstrated. The above results are essential for estimating the stochastic features of local stress concentrations, which are the source for many strength-related macro properties of materials. Extensive usage of generalized functions (Dirac operator and its derivatives) is needed for the analysis.     

Variability response functions for effective material properties

The variability response function (VRF) is a well-established concept for efficient evaluation of the variance and sensitivity of the response of stochastic systems where properties are modeled by random fields that circumvents the need for computationally expensive Monte Carlo (MC) simulations. Homogenization of material properties is an important procedure in the analysis of structural mechanics problems in which the material properties fluctuate randomly, yet no method other than MC simulation exists for evaluating the variability of the effective material properties. The concept of a VRF for effective material properties is introduced in this paper based on the equivalence of elastic strain energy in the heterogeneous and equivalent homogeneous bodies. It is shown that such a VRF exists for the effective material properties of statically determinate structures. The VRF for effective material properties can be calculated exactly or by Fast MC simulation and depends on extending the classical displacement VRF to consider the covariance of the response displacement at two points in a statically determinate beam with randomly fluctuating material properties modeled using random fields. Two numerical examples are presented that demonstrate the character of the VRF for effective material properties, the method of calculation, and results that can be obtained from it.     

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.     

Uncertainty analysis based on sensitivity applied to angle-ply composite structures

This article describes a finite element-based formulation for the statistical analysis of the response of stochastic structural composite systems whose material properties are described by random fields. A first-order technique is used to obtain the second-order statistics for the structural response considering means and variances of the displacement and stress fields of plate or shell composite structures. Propagation of uncertainties depends on sensitivities taken as measurement of variation effects. The adjoint variable method is used to obtain the sensitivity matrix. This method is appropriated for composite structures due to the large number of random input parameters. Dominant effects on the stochastic characteristics are studied analyzing the influence of different random parameters. In particular, a study of the anisotropy influence on uncertainties propagation of angle-ply composites is carried out based on the proposed approach.     

Stochastic homogenization of nano-thickness thin films including patterned holes using structural perturbation method

The aim of this paper is to investigate the statistical response of the homogenized mechanical behavior of nano-thickness thin films with circular holes. For this purpose, a stochastic multiscale framework is proposed. The proposed framework involves molecular dynamics simulation, surface modeling, asymptotic homogenization, moving-mesh technique, Monte-Carlo simulation, and a reduced computational scheme. The surface effect of thin-film material is predicted by the molecular dynamics (MD) approach. The volume fraction and location of each circular hole are considered as geometric uncertainties of a model. In order to investigate the statistical response of the homogenized mechanical behavior, Monte-Carlo simulation is performed to show the probability density distribution of the homogenized elastic modulus against geometric uncertainties. The reduced computational schematic based on the static reduction method and the structural perturbation method is proposed in order to overcome the issues of a cumbersome remeshing procedure and computational inefficiency of Monte-Carlo simulations involving a high number of repetitive trials. A guideline to minimize the coefficient of variation (CV) of the mechanical properties is suggested based on the parametric study.     

Dispersion of elastic waves in periodically inhomogeneous media

Propagation of time-harmonic elastic waves through periodically inhomogeneous media is considered. The material inhomogeneity exists in a single direction along which the elastic waves propagate. Within the period of the linear elastic and isotropic medium, the density and elastic modulus vary either in a continuous or a discontinuous manner. The continuous variations are approximated by staircase functions so that the generic problem at hand is the propagation of elastic waves in a medium whose finite period consists of an arbitrary number of different homogeneous layers. A dynamic elasticity formulation is followed and the exact phase velocity is derived explicitly as a solution in closed form in terms of frequency and layer properties. Numerical examples are then presented for several inhomogeneous structures.     

结构在水平与竖向随机地震同时作用下的相关函数和谱密度

对单自由度结构在水平与竖向地震同时作用下的随机稳定性、响应及其相关函数和谱密度函数进行系统研究。首先利用Stratonovich和It随机微分方程与响应矩微分方程的互相关转化关系,建立了结构响应矩方程;然后根据Hurwitz随机稳定准则,获得了结构一阶和二阶响应矩渐近稳定的解析判别式;继而,利用复模态法获得了结构响应二阶矩的解析瞬态解和平稳解;最后利用It随机微分方程解具有的非可料函数性质,获得了结构位移、速度响应的自相关函数、互相关函数以及谱密度函数、互谱密度函数的解析解,给出了算例,并综合分析了各种参数对结构响应、稳定性以及相关函数和谱密度函数的影响。     

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.     

Exact and numerical elastodynamic solutions for spherically symmetric problems of functionally graded thick-walled spheres subjected to pressure shocks

In the present paper, analytical and numerical elastodynamic solutions are developed for spherically symmetric problems of functionally graded thick-walled spheres subjected to arbitrary dynamic and shock loads. Both transient dynamic response and elastic wave propagation characteristics are studied in the mentioned nonhomogeneous structures. Variations of the material properties across the thickness are described according to both polynomial and power law functions. The numerical consistent transfinite element formulation is presented for both functions whereas the exact solution is presented for the power law function. The functionally graded material sphere is not divided into isotropic sub-spheres. An approach associated with dividing the dynamic radial displacement expression into quasi-static and dynamic parts and expansion of the transient wave functions in terms of a series of eigenfunctions is employed to propose the exact solution. Results are obtained for various exponents of the functions of the material properties distributions, various radius ratios, and variety of dynamic and shock loads.     

Stochastic shape functions and stochastic strain–displacement matrix for a stochastic finite element stiffness matrix

Summary For conventional finite element problems, element geometry is adequate to determine shape functions. However, to account for secondary effects due to material randomness, conventional shape functions need to be modified according to the spatial fluctuation of constitutive variables in each Monte Carlo sample. This paper develops a method to compute stochastic shape functions based on local equilibrium criteria when each simulated sample complies with the same order of accuracy as designated for the associated deterministic problem. The resulting stochastic stiffness matrix is then calculated via the stochastic strain–displacement matrix based on those stochastic shape functions. In order to attain high accuracy, which is the characteristic of the boundary element method, rational polynomial shape functions are used in this paper. The proposed formulation is indispensable when secondary effects (due to nano size and time scale in modern technology, fiber randomness in composites, thermodynamic interactions in biological tissues, to name a few) demand a high accuracy finite element formulation. The elasto-plastic deformation that introduces concavity motivated the numerical example elaborated here. An example of a concave quadrilateral element with spatial randomness for the modulus of elasticity is illustrated. Since isoparametric shape functions for concave quadrilaterals do not exist, the Wachspress rational polynomial shape functions with irrational terms are used. The computer algebra environment Mathematica is employed here. Dedicated to Professor Franz Ziegler on the occasion of his 70th birthday     

A solid-shell layerwise finite element for non-linear geometric and material analysis

The application of layerwise theories to correctly model the displacement field of sandwich structures or laminates with high modulus ratios usually employs plate or facet-shell finite element formulations to compute the element stiffness and mass matrices for each layer. In this work an alternative approach is proposed, using a high performance hexahedral finite element to represent the individual layer mass and stiffness. This eight-node hexahedral finite element is formulated based on the application of the enhanced assumed strain method (EAS) to solve several locking pathologies coming from the high aspect ratio of the finite element and the usual incompressibility condition of the core materials. The solid-shell finite element formulation is introduced in the layerwise theory through the definition of a projection operator, based on the finite element variables transformation matrix. The non-linear geometric and material capabilities are introduced into the finite element formulation, allowing for the representation of large displacements, large deformation and material non-linear behaviors. The developed formulation is numerically tested and benchmarked, being validated by using published experimental results obtained from sandwich specimens.     

Stochastic behavior of Mindlin plate with uncertain geometric and material parameters

The Mindlin assumption in the plate bending allows the shear deformation together with the bending one. The constitutive relation for shear behavior, however, has different order of contribution from the plate thickness t than that in bending. Therefore, when we have concern on the stochastic behavior of Mindlin plate taking into account the uncertain plate thickness, the bending and shear parts have to be handled in different ways. Furthermore, if the concern is put on the multiple uncertain parameters such as between elastic modulus and plate thickness, the influence of correlation between these parameters also has to be resolved. In this study, a formulation for stochastic analysis is given in the context of the weighted integral method for Mindlin plate with uncertainties in elastic modulus and plate thickness. The effects of correlation between these parameters on the structural behavior are examined. In addition, the stability of response variability in terms of the aspect ratio of a plate is investigated. Several square plates with various boundary conditions subjected to uniformly distributed or concentrated loads are taken as examples, and the results are compared with those of precedent works as well as those of classical Monte Carlo simulation.     

A first-principles study on the structural, elastic, electronic, and optical properties of CdRh2O4

The structural, elastic, electronic, and optical properties of CdRh₂O₄ with cubic $ (Fd\overline{ 3} m) $ and orthorhombic (Pnma) structures have been investigated using a pseudopotential plane wave (PP-PW) method within the local density approximation (LDA). The calculated lattice parameters agree reasonably with the experimental values. The single-crystal elastic stiffness constants C _ij s of the cubic and orthorhombic phases are investigated using the stress–strain method. In addition, the polycrystalline elastic properties including bulk modulus, shear modulus, Young’s modulus, bulk modulus–shear modulus ratio, Poisson’s ratio, and elastic anisotropy ratio are determined based on Voigt–Reuss–Hill approach. The use of the hybrid functional sX-LDA leads to considerably improved electronic properties compared to standard LDA approach. On the other hand, the dielectric function, refraction index, reflectivity, conductivity function, and energy-loss spectra were obtained and analyzed on the basis of electronic band structures and density of states.     

Modulus–density negative correlation for CNT-reinforced polymer nanocomposites: Modeling and experiments

Mechanical and weight properties of polymer nanocomposites (PNCs) are measured and modeled at the interlaminar region, predicting the density and elastic modulus of individual carbon nanotubes (CNTs). A simple model of the CNTs density and elastic modulus within the PNC, accounting for fundamental material properties, geometry, and interactions, is developed, capable of predicting CNT contributions in the PNCs. Furthermore, the model is validated with experimental results that demonstrate enhancement of the elastic modulus, while reducing density in the presence of aligned CNTs. By establishing an inverse relation of density and elastic modulus (negative correlation), it is demonstrated the potential of increasing mechanical properties while reducing weight. Therefore, by introducing controlled nanoporosity through suitable CNT distributions within the interlayer of multi-lamina structures, it is possible to simultaneously control effective weight reduction and enhanced modulus, toward bio-inspired carbon fiber reinforced polymer composites.     

Structural and elastic properties of TiN and AlN compounds: first-principles study

First-principles calculations of the lattice constants, bulk modulus, pressure derivatives of the bulk modulus and elastic constants of AlN and TiN compounds in rock-salt (B1) and wurtzite (B4) structures are presented. We have used the fullpotential linearized augmented plane wave (FP-LAPW) method within the density functional theory (DFT) in the generalized gradient approximation (GGA) for the exchange-correlation functional. Moreover, the elastic properties of cubic TiN and hexagonal AlN, including elastic constants, bulk and shear moduli are determined and compared with previous experimental and theoretical data. Our results show that the structural transition at 0 K from wurtzite to rock-salt phase occurs at 10 GPa and ?26 GPa for AlN and TiN, respectively. These results are consistent with those of other studies found in the literature.     

The structural and mechanical properties of CdN compound: A first principles study

First principles calculations are performed to investigate the structural, elastic, and mechanical properties of CdN for various structures: NaCI, CsCl, ZnS, wurtzite, WC, CdTe, NiAs, and CuS. The local density and generalized gradient approximations are used for modeling exchange–correlation effects. Our calculations indicate that CuS (B18) structure is energetically the most stable among the considered structures. The some basic physical properties such as lattice parameters, bulk modulus, and second-order elastic constants are calculated. We have also predicted the shear modulus, Young’s modulus, Poison’s ratio, Debye temperature, and sound velocities. Our structural and some other results are consistent with the available theoretical data.     

A formulation for the 4-node quadrilateral element

A formulation for the plane 4-node quadrilateral finite element is developed based on the principle of virtual displacements for a deformable body. Incompatible modes are added to the standard displacement field. Then expressions for gradient operators are obtained from an expansion of the basis functions into a second-order Taylor series in the physical co-ordinates. The internal degrees of freedom of the incompatible modes are eliminated on the element level. A modified change of variables is used to integrate the element matrices. For a linear elastic material, the element stiffness matrix can be separated into two parts. These are equivalent to a stiffness matrix obtained from underintegration and a stabilization matrix. The formulation includes the cases of plane stress and plane strain as well as the analysis of incompressible materials. Further, the approach is suitable for non-linear analysis. There, an application is given for the calculation of inelastic problems in physically non-linear elasticity. The element is efficient to implement and it is frame invariant. Locking effects and zero-energy modes are avoided as well as singularities of the stiffness matrix due to geometric distortion. A high accuracy is obtained for numerical solutions in displacements and stresses.