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.     

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.     

Response and eigenvalue analysis of stochastic finite element systems with multiple correlated material and geometric properties

The variability of the random response displacements and eigenvalues of structures with multiple uncertain material and geometric properties are studied in this paper using variability response functions. The material and geometric properties are assumed to be described by cross-correlated stochastic fields. Specifically, two types of problems are considered: the response displacement variability of plane stress/plane strain structures with stochastic elastic modulus, Poisson's ratio, and thickness, and the eigenvalue variability of beam and plate structures with stochastic elastic modulus and mass density. The variance of the displacement/eigenvalue is expressed as the sum of integrals that involve the auto-spectral density functions characterizing the structural properties, the cross-spectral density functions between the structural properties, and the deterministic variability response functions. This formulation yields separate terms for the contributions to the response displacement/eigenvalue variability from the auto-correlation of each of the material/geometric properties, and from the cross-correlation between these properties. The variability response functions are used to compute engineering-wise very important spectral-distribution-free realizable upper bounds of the displacement/eigenvalue variability. Using this formulation, it is also possible to compute the displacement/eigenvalue variability for prescribed auto- and cross-spectral density functions.     

Three-dimensional asymptotic mode I/II stress fields at the front of interfacial crack/anticrack discontinuities in trimaterial bonded plates

An eigenfuntion expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the front of a crack/anticrack discontinuity weakening/reinforcing an infinite pie-shaped trimaterial plate, of finite thickness, formed as a result of bimaterial (matrix/ARC plus reaction product/scatterer) deposit over a substrate (fiber/semiconductor). The wedge is subjected to mode I/II far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 or the substrate is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I/II eigenvalues (or stress singularities) with Young’s moduli ratio, as well as with the wedge aperture angle of the material 1 (reaction product/scatterer) are presented. Hitherto generally unavailable results, pertaining to the through-thickness variations of stress intensity factors or stress singularity coefficients for symmetric exponentially growing distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.     

The stochastic finite element method in structural reliability

First-order reliability and finite element methods are used to develop a methodology for reliability analysis of structures with stochastically varying properties and subjected to random loads. Two methods for discretization of random fields are examined and the influence of the correlation length of random property or load fields on the reliability of example structures are investigated. It is found that the correlation length of load fields has significant influence on the reliability against displacement or stress limit states. The correlation length of property fields is significant for displacement limit states, but may not be significant for stress limit states. Examples studied include a fixed ended beam with stochastic rigidity and a plate with stochastic elasticity.     

不同拉压模量及软化特性材料的球形孔扩张问题的统一解

对于具有不同的拉压模量及软化特性的岩土类材料,提出了不同拉压模量及软化特性的控制参数,采用双剪统一强度理论推导了球形孔扩张问题的应力及位移的统一解。分析了模量、模型和软化等控制参数对球形孔扩张时的扩张压力、塑性区开展规律及应力场的影响。结果表明:圆孔极限扩张压力,塑性区的发展规律,应力场,位移场等均随着模量控制参数、模型参数及软化参数的变化而变化,因此若采用经典的弹性理论、单一的模型参数及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

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.     

Stochastic post buckling analysis of laminated composite cylindrical shell panel subjected to hygrothermomechanical loading

In this paper, the effect of random system properties on the post buckling load of geometrically nonlinear laminated composite cylindrical shell panel subjected to hygrothermomechanical loading is investigated. System parameters are assumed as independent random variables. The higher order shear deformation theory and von-Karman nonlinear kinematics are used for basic formulation. The elastic and hygrothermal properties of the composite material are considered to be dependent on temperature and moisture concentration using micromechanical approach. A direct iterative based C⁰ nonlinear finite element method in conjunction with first-order perturbation technique proposed by present author for the plate is extended for shell panel subjected to hygrothermomechanical loading to compute the second-order statistics (mean and variances) of laminated composite cylindrical shell panel. The effect of random system properties, plate geometry, stacking sequences, support conditions, fiber volume fractions and temperature and moisture distributions on hygrothermomechanical post-buckling load of the laminated cylindrical shell panel are presented. The performance of outlined stochastic approach has been validated by comparing the present results with those available in the literature and independent Monte Carlo simulation.     

Stiffness tensor random fields through upscaling of planar random materials

Unique effective material properties are not possible for random heterogeneous materials at intermediate length scales, which is to say at some mesoscale above the microscale yet prior to the attainment of the representative volume element (RVE). Focusing on elastic moduli in particular, a micromechanical analysis based on the Hill–Mandel condition leads to the conclusion that two fields, stiffness and compliance, are required to bound the response of the material. In particular, we analyze means and correlation coefficients of a random planar material with a two-phase microstructure of random checkerboard type. We employ micromechanics, which can be viewed as an upscaling, smoothing procedure using the concept of a mesoscale “window”, and random field theory to compute the correlation structure of 4th-rank tensor fields of stiffness and compliance for a given mesoscale. Results are presented for various correlation distances, volume fractions, and contrasts in stiffness between phases. The main contribution of this research is to provide the data for developing analytical correlation functions, which can then be used at any mesoscale to generate micromechanically based inputs into analytical and computational mechanics models.     

考虑材料的拉压模量不同及应变软化特性的柱形孔扩张问题

建立了拉压模量不同及应变软化特性材料的柱形孔扩张理论。对于岩土类材料,提出用a及b分别作为拉压模量不同和软化特征的控制参数,运用不同模量弹性理论及应力跌落软化模型推导了Tresca和Mohr-Coulomb材料柱形圆孔扩张问题的应力及位移解。分析了不同模量及软化特性材料对柱形孔扩张的影响,结果表明:圆孔极限扩张压力,塑性区的发展规律,应力场,位移场等均随着模量参数a和软化系数b的变化而变化,因此若采用经典的弹性理论及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

A micromechanics based multiscale model for nonlinear composites

A micromechanics model for fiber-reinforced composites that can be used at the subscale in a multiscale computational framework is established to predict the effective nonlinear composite response. Using a fiber–matrix concentric cylinder model as the basic repeat unit to represent the composite, micromechanics is used to relate the applied composite strains to the fiber and matrix strains by a six by six transformation matrix. The resolved spatial variations of the matrix fields are found to be in good agreement with corresponding finite element analysis results. The evolution of the composite nonlinear response is assumed to be governed by two scalar, strain-based variables that are related to the extreme value of an appropriately defined matrix equivalent strain, and the matrix secant moduli are used to compute the composite secant moduli for nonlinear analysis. The results from the micromechanics model are compared well with a full finite element analysis. The predictive capability of the proposed model is illustrated by two distinct fiber-reinforced material systems, carbon and glass, for the fiber volume fraction varying from 50 to 70 %. Since fully analytical solutions are utilized for the micromechanical analysis, the proposed method offers a distinct computational advantage in a multiscale analysis and is therefore suitable for large-scale progressive damage and failure analyses of composite material structures.     

Three-dimensional asymptotic stress fields at the front of a trimaterial junction

A recently developed eigenfunction expansion method is employed for obtaining three-dimensional asymptotic displacement and stress fields in the vicinity of the junction corner front of an infinite pie-shaped trimaterial wedge, of finite thickness, formed as a result of bimaterial (matrix plus reaction product or contaminant) deposit over a substrate or reinforcement. The wedge is subjected to extension/bending (mode I), inplane shear/twisting (mode II) and antiplane shear (mode III) far field loading. Each material is isotropic and elastic, but with different material properties. The material 2 (substrate) is always taken to be a half-space, while the wedge aperture angle of the material 1 is varied to represent varying composition of the bimaterial deposit. Numerical results pertaining to the variation of the mode I, II, III eigenvalues (or stress singularities) with various moduli ratios as well as the wedge aperture angle of the material 1 (reaction product/contaminant), are also presented. Hitherto unavailable results, pertaining to the through-thickness variations of stress intensity factors for symmetric exponentially decaying distributed load and its skew-symmetric counterpart that also satisfy the boundary conditions on the top and bottom surfaces of the trimaterial plates under investigation, bridge a longstanding gap in the stress singularity/interfacial fracture mechanics literature.     

Ultrasonic wave propagation in two-phase media: Spherical inclusions

The scattering theory, recently developed via the extended method of equivalent inclusion, is used to study the propagation of time-harmonic waves in two-phase media of elastic matrix with randomly distributed elastic spherical inclusion materials. The elastic moduli and mass density of the composite medium are determined as functions of frequencies when given properties and concentration of the spheres and the matrix. Velocities and attenuation of ultrasonic waves in two-component media are determined. An averaging theorem that requires the equivalence of the strain energy and the kinetic energy between the effective medium and the original matrix with inhomogeneities is employed to derive the effective moduli and mass density. The functional dependency of these quantities upon frequencies and concentration provides a method of data analysis in ultrasonic evaluation of material properties. Numerical results for effective moduli, velocity and/or attenuation as functions of concentration of spherical inclusion material, or porosity, are graphically displayed.     

The use of boundary strip method (BSM) for evaluation of the transverse mechanical properties of fibrous composites

The boundary strip method (BSM) is applied for evaluation of the transverse mechanical properties of fibrous composites with random and periodical fiber distributions. This special semi numerical method helps find the link between the microscopic behavior of the composite material and its macroscopic response in a rather detailed manner, enabling definition of stress and strain magnitudes at each point of the cross section. Here, specifically statistical model based on the boundary strip method, is used for assessment of the transverse effective moduli of fibrous composites. Random fiber distributions are compared with periodic fiber distributions having square or hexagonal array arrangements. Those are the common models used nowadays and modeled by the finite element or the boundary element. A comparison with the bounds of the polarization extremum principles is conducted too. The influence of the randomly distributed fibers on the transverse effective moduli is investigated and a good correlation is found between the results of the present model and the lower bound of the polarization extremum principles.     

考虑界面影响的混凝土弹性模量的数值预测

提出了一种考虑界面过渡层影响的混凝土弹性模量的数值预测方法。将球形骨料与包裹它的界面过渡层作为二相复合球结构的等效颗粒,由广义自洽方法计算不同粒径骨料与界面过渡层组成复合球的有效模量。然后由等效颗粒生成的随机骨料模型建立体积表征单元,施加均匀位移边界条件,通过数值方法计算该体积表征单元中的应力和应变场,由细观力学数值均匀化方法预测体积表征单元的有效弹性模量。计算结果表明:对于不同骨料含量的混凝土,有效弹性模量的预测值与试验值非常接近,界面过渡层的厚度对混凝土的整体弹性性质有较大影响。     

Ply cracking and stiffness degradation in cross-ply laminates under biaxial extension, bending and thermal loading

Transverse ply cracking often leads to the loss of stiffness and reduction in thermal expansion coefficients. This paper presents the thermoelastic degradation of general cross-ply laminates, containing transverse ply cracks, subjected to biaxial extension, bending and thermal loading. The stress and displacement fields are calculated by using the state space equation method [Zhang D, Ye JQ, Sheng HY. Free-edge and ply cracking effect in cross-ply laminated composites under uniform extension and thermal loading. Compos Struct [in press].]. By this approach, a laminated plate may be composed of an arbitrary number of orthotropic layers, each of which may have different material properties and thickness. The method takes into account all independent material constants and guarantees continuous fields of all interlaminar stresses across interfaces between material layers. After introducing the concept of the effective thermoelastic properties of a laminate, the degradations of axial elastic moduli, Poisson’s ratios, thermal expansion coefficients and flexural moduli are predicted and compared with numerical results from other methods or available test results. It is found that the theory provides good predictions of the stiffness degradation in both symmetric and antisymmetric cross-ply laminates. The predictions of stiffness reduction in nonsymmetric cross-ply laminates can be used as benchmark test for other methods.     

Elastic Constants of Particle and Fiber Reinforced Metal Matrix Composites

Abstract

A model has been developed to predict the elastic moduli in composites reinforced with both particles and fibers. In the model the matrix material and the particles, which are assumed to be homogeneously distributed, form an effective matrix. The characteristics of this effective matrix is calculated using a theory formulated by Ledbetter and Datta. The effective matrix is then considered to be reinforced with fibers lying in one plane but randomly oriented in that plane. The effect of the 2-dimensionally random orientation of the fibers on the elastic moduli of the composites is determined in two steps. First the composite cylinders model by Hashin and Rosen for an aligned fiber system is employed, and then a geometric averaging procedure suggested by Christensen and Waals is performed. Using this model, the Young's and shear moduli were calculated for three samples with different aluminum matrices and volume fractions of particles (9, 13, and 17%) but the same fiber content (6%). The same elastic moduli were also determined using ultrasonic velocity measurements. The agreement between calculated and measured elastic moduli is found to be very good. Also, the elastic anisotropics between directions of the fiber rich plane and that normal to the plane could be predicted by the model.     

双模量复合材料层板的大挠度分析

本文用动力松弛法(DRM)分析了双模量复合材料层板的大挠度弯曲问题。文中介绍了求解该问题的主要公式和步骤;对轻度双模量材料及高度双模量材料的两层正交铺层矩形板在正弦载荷和均匀载荷作用下的挠度和内力进行了数值计算,并把其结果与小挠度结果及单模量结果做了比较,讨论了大挠度分析的必要性及模量性质对大挠度分析的影响。      

不同拉压模量及软化特性材料的柱形孔扩张问题的统一解

对于具有不同的拉压模量及软化特性的岩土类材料,提出了不同拉压模量及软化特性的控制参数,采用双剪统一强度理论推导了柱形孔扩张问题的应力及位移的统一解。分析了模量、模型和软化等控制参数对柱形孔扩张时的扩张压力、塑性区开展规律及应力场的影响。结果表明:圆孔极限扩张压力、塑性区的发展规律、应力场、位移场等均随着模量控制参数、模型参数及软化参数的变化而变化,因此若采用经典的弹性理论、单一的模型参数及传统的不考虑应变软化来对岩土类的工程材料进行设计计算,必会带来较大的误差。     

复合材料层合板临界屈曲载荷分散性

基于随机场理论, 将纤维和基体性能以及纤维体积分数作为随机场变量, 利用局部平均法对随机场进行离散。结合MATLAB与ANSYS的PDS模块对复合材料层合板临界屈曲载荷进行Monte-Carlo模拟, 分析各类随机场变量、随机场的相关长度、对称性和边界条件对临界屈曲载荷分散性的影响。结果表明: 不同随机场变量对层合板屈曲载荷分散系数影响的程度不同, 纤维体积分数的影响最大, 其次为纤维性能与基体性能; 屈曲载荷的分散系数存在尺寸效应, 随着板尺寸的增加, 屈曲载荷分散系数逐渐减小; 减小相关长度可有效地减小屈曲载荷的分散系数; 纤维正对称铺设所引起的屈曲载荷分散系数稍大于反对称铺设情况, 而两对边固支板的屈曲载荷分散系数一般大于四边简支板的结果。