Stochastic multiscale fracture analysis of three-dimensional functionally graded composites

A new moment-modified polynomial dimensional decomposition (PDD) method is presented for stochastic multiscale fracture analysis of three-dimensional, particle-matrix, functionally graded materials (FGMs) subject to arbitrary boundary conditions. The method involves Fourier-polynomial expansions of component functions by orthonormal polynomial bases, an additive control variate in conjunction with Monte Carlo simulation for calculating the expansion coefficients, and a moment-modified random output to account for the effects of particle locations and geometry. A numerical verification conducted on a two-dimensional FGM reveals that the new method, notably the univariate PDD method, produces the same crude Monte Carlo results with a five-fold reduction in the computational effort. The numerical results from a three-dimensional, edge-cracked, FGM specimen under a mixed-mode deformation demonstrate that the statistical moments or probability distributions of crack-driving forces and the conditional probability of fracture initiation can be efficiently generated by the univariate PDD method. There exist significant variations in the probabilistic characteristics of the stress-intensity factors and fracture-initiation probability along the crack front. Furthermore, the results are insensitive to the subdomain size from concurrent multiscale analysis, which, if selected judiciously, leads to computationally efficient estimates of the probabilistic solutions.     

A random field model for generating synthetic microstructures of functionally graded materials

This article presents a new level‐cut, inhomogeneous, filtered Poisson random field model for representing two‐phase microstructures of statistically inhomogeneous, functionally graded materials with fully penetrable embedded particles. The model involves an inhomogeneous, filtered Poisson random field comprising a sum of deterministic kernel functions that are scaled by random variables and a cut of the filtered Poisson field above a specified level. The resulting level‐cut field depends on the Poisson intensity, level, kernel functions, random scaling variables, and random rotation matrices. A reconstruction algorithm including model calibration and Monte Carlo simulation is presented for generating samples of two‐phase microstructures of statistically inhomogeneous media. Numerical examples demonstrate that the model developed is capable of producing a wide variety of two‐ and three‐dimensional microstructures of functionally graded composites containing particles of various sizes, shapes, densities, gradations, and orientations. An example involving finite element analyses of random microstructures, leading to statistics of effective properties of functionally graded composites, illustrates the usefulness of the proposed model. Copyright © 2008 John Wiley & Sons, Ltd.     

Antiplane crack analysis of a functionally graded material by a BIEM

Elastostatic analysis of an antiplane crack in a functionally graded material (FGM) is performed by using a hypersingular boundary integral equation method (BIEM). An exponential law is applied to describe the spatial variation of the shear modulus of the FGM. A Galerkin method is applied for the numerical solution of the hypersingular traction BIE. Both unidirectional and bidirectional material gradations are investigated. Stress intensity factors for an infinite and linear elastic FGM containing a finite crack subjected to an antiplane crack-face loading are presented and discussed. The influences of the material gradients and the crack orientation on the stress intensity factors are analyzed.     

Scattering of the harmonic anti-plane shear waves by a crack in functionally graded piezoelectric materials

In the present paper, the dynamic behavior of a Griffth crack in the functionally graded piezoelectric material (FGPM) is investigated. It is assumed that the elastic stiffness, piezoelectric constant, dielectric permittivity and mass density of the FGPM vary continuously as an exponential function, and that FGPM is under the anti-plane mechanical loading and in-plane electrical loading. By using the Fourier transform and defining the jumps of displacement and electric potential components across the crack surface as the unknown functions, two pairs of dual integral equations are derived. To solve the dual integral equations, the jumps of the displacement and electric potential components across the crack surface are expanded in a series of Jacobi polynomial. Numerical examples are provided to show the effects of material properties on the stress and the electric displacement intensity factors.     

The fracture analysis of a graded coating/substrate system of finite thickness with arbitrary spatial variations of coating properties: Plane deformation

A new multi-layered model for functionally graded materials (FGMs) with continuously varying elastic properties is developed. The model divides the FGM into multiple layers. In each layer the material properties vary linearly and are continuous on the sub-interfaces. With this new multi-layered model, we solve the crack problems of an FGM coated strip under the in-plane deformation. The method employs the Fourier integral transform technique and singular integral equation theory. The stress intensity factors are calculated. Comparisons between the present model and other existing models show some advantages of the new model: (i) it involves no discontinuities of the material properties at the sub-interfaces; and (ii) it can be used to analyze the crack problems of FGMs with properties of arbitrary variations.     

Analysis of an interface crack for a functionally graded strip sandwiched between two homogeneous layers of finite thickness

The interface crack problem for a composite layer that consists of a homogeneous substrate, coating and a nonhomogeneous functionally graded interphase was formulated for singular integral equations with Cauchy kernels, which were integrated using the Lobatto–Chebyshev collocation technique. Mixed-Mode Stress Intensity Factors (SIFs) and Strain Energy Release Rates were calculated. The SIFs were compared for accuracy with relevant results previously published. The parametric studies were conducted for the various thickness of each layer and for various nonhomogeneity ratios. Particular application to the Zirconia thermal barrier on steel substrate is demonstrated.     

Alternative test method for interlaminar fracture toughness of composites

Two methods of determining the mode I interlaminar fracture toughness for fiber-reinforced polymer matrix (FRPM) composites using a double cantilever beam (DCB) test are compared. The standard method of determining G _IC is based in linear-elastic fracture mechanics theory and requires a visual measurement of the crack length, presenting data acquisition and analysis difficulties. The proposed method makes use of elastic–plastic fracture mechanics theory and an analytical closed form solution to the J-integral to relate the fracture toughness J _IC , load, and angular displacement at the load application points. This method has the advantage of replacing visually acquired data with data easily obtained using inexpensive transducers as well as being applicable to a broader class of materials.     

A continuum shape sensitivity method for fracture analysis of isotropic functionally graded materials

This paper presents a new continuum shape sensitivity method for calculating mixed-mode stress-intensity factors for a stationary crack in two-dimensional, linear- elastic, isotropic FGMs with arbitrary geometry. The method involves the material derivative concept taken from continuum mechanics, the mutual potential energy release rate, and direct differentiation. Since the governing variational equation is differentiated prior to discretization, resulting sensitivity equations are independent of approximate numerical techniques, such as the finite element method, boundary element method, mesh-free method, or others. The discrete form of the mutual potential energy release rate is simple and easy to calculate, as it only requires multiplication of displacement vectors and stiffness sensitivity matrices. By judiciously selecting the velocity field, the method only requires displacement response in a subdomain close to the crack tip, thus making the method computationally efficient. Seven finite-element based numerical examples, which comprise mode-I and mixed-mode deformations and/or single or multiple interacting cracks, are presented to evaluate the accuracy of the fracture parameters calculated by the proposed method. Comparisons have been made between stress-intensity factors predicted by the proposed method and available reference solutions in the literature, generated either analytically or numerically using various other fracture integrals or analyses. Excellent agreement is obtained between the results of the proposed method and previously obtained solutions. Therefore, shape sensitivity analysis provides an attractive alternative to fracture analysis of cracks in homogeneous and non-homogeneous materials.     

Theoretical simulations of a propagating crack subjected to in-plane stress wave loading by caustic method

The optical method of caustics for measuring the dynamic stress intensity factor in a transient process is investigated in this study. The transient full-field solutions of a propagating crack contained in an infinite medium subjected to step-stress wave and ramp-stress wave loadings are used to establish the exact equations of the initial and caustic curves. The results of the stress intensity factor obtained from the caustic method are compared with theoretical predictions and some experiments. The results demonstrate that a significant deviation can occur in the determination of the dynamic stress intensity factor from shadow spot measurements. The factors, such as screen distance, magnitude of loading, crack speed and rising time which can influence the accuracy of the experimental measurements are discussed in detail. In addition, the valid region of the dynamic stress singular field for the propagating crack is discussed in detail and it gives a better understanding of the appropriate region of measurements for investigators. This revised version was published online in July 2006 with corrections to the Cover Date.     

Numerical study on fatigue crack growth in railway wheels under the influence of residual stresses

Accurate prediction of fatigue crack growth on railway wheels and the influence of residual stresses by finite element method (FEM) modeling can affect the maintenance planning. Therefore, investigation of rolling contact fatigue and its effect on rolling members life seem necessary. The objective of this paper is to provide a prediction of rolling contact fatigue crack growth in the rail wheel under the influence of stress field from mechanical loads and heat treatment process of a railway wheel. A 3D nonlinear stress analysis model has been applied to estimate stress fields of the railway mono-block wheel in heat treatment process. Finite element analysis model is presented applying the elastic–plastic finite element analysis for the rail wheel under variable thermal loads. The stress history is then used to calculate stress intensity factors (SIFs) and fatigue life of railway wheel. The effect of several parameters, vertical loads, initial crack length and friction coefficient between the wheel and rail, on the fatigue life in railway wheels is investigated using the suggested 3-D finite element model. Three-dimensional finite element analysis results obtained show good agreement with those achieved in field measurements.     

Automated simulation of fatigue crack propagation for two-dimensional linear elastic fracture mechanics problems by boundary element method

In this paper, automated simulation of multiple crack fatigue propagation for two-dimensional (2D) linear elastic fracture mechanics (LEFM) problems is developed by using boundary element method (BEM). The boundary element method is the displacement discontinuity method with crack-tip elements proposed by the author. Because of an intrinsic feature of the boundary element method, a general growth problem of multiple cracks can be solved in a single-region formulation. In the numerical simulation, for each increment of crack extension, remeshing of existing boundaries is not necessary. Local discretization on the incremental crack extension is performed easily. Further the new adding elements and the existing elements on the existing boundaries are employed to construct easily the total structural mesh representation. Here, the mixed-mode stress intensity factors are calculated by using the formulas based on the displacement fields around crack tip. The maximum circumferential stress theory is used to predict crack stability and direction of propagation at each step. The well-known Paris’ equation is extended to multiple crack case under mixed-mode loadings. Also, the user does not need to provide a desired crack length increment at the beginning of each simulation. The numerical examples are included to illustrate the validation of the numerical approach for fatigue growth simulation of multiple cracks for 2D LEFM problems.     

层板复合材料动态断裂韧性测试的SHPB技术研究

为测试层板复合材料的断裂韧性，对传统的Hopkinson压杆测试系统进行了改进，建立了应力波载荷作用下材料动态断裂韧性的测试方法。该方法采用三点弯曲试样进行动态断裂试验，应用动态断裂韧性测试系统ANSYSED5．4确定动态应力强度因子的响应曲线，进而测试材料动态断裂韧性。对层板复合材料试验结果的分析表明，设计的测试装置有效，建立的测试方法是对层板复合材料断裂韧性测试的有益尝试，有较大的参考价值。     

A comparative study of the effect of different rigid fillers on the fracture and failure behavior of polypropylene based composites

Polypropylene (PP) composites with 5 wt% of different rigid particles (Al₂O₃ nanoparticles, SiO₂ nanoparticles, Clay (Cloisite 20A) nanoparticles or CaCO₃ microparticles) were obtained by melt mixing. Composites with different CaCO₃ content were also prepared. The effect of fillers, filler content and addition of maleic anhydride grafted PP (MAPP) on the composites fracture and failure behavior was investigated. For PP/CaCO₃ composites, an increasing trend of stiffness with filler loading was found while a decreasing trend of strength, ductility and fracture toughness was observed. The addition of MAPP was beneficial and detrimental to strength and ductility, respectively mainly as a result of improved interfacial adhesion. For the composites with 5 wt% of CaCO₃ or Al₂O₃, no significant changes in tensile properties were found due to the presence of agglomerated particles. However, the PP/CaCO₃ composite exhibited the best tensile behavior: the highest ductility while keeping the strength and stiffness of neat PP. In general, the composites with SiO₂ or Clay, on the other hand, displayed worse tensile strength and ductility. These behaviors could be probably related to the filler ability as nucleating agent. In addition, although the incorporation of MAPP led to improved filler dispersion, it was damaging to the material fracture behavior for the composites with CaCO₃, Al₂O₃ or Clay, as a result of a higher interfacial adhesion, the retardant effect of MAPP on PP nucleation and the lower molecular weight of the PP/MAPP blend. The PP/MAPP/SiO₂ composite, on the other hand, showed slightly increased toughness respect to the composite without MAPP due to the beneficial concomitant effects of the presence of some amount of the β crystalline phase of PP and the better filler dispersion promoted by the coupling agent which favor multiple crazing. From modeling of strength, the effect of MAPP on filler dispersion and interfacial adhesion in the PP/CaCO₃ composites was confirmed.     

A two-step method for analysis of linear systems with uncertain parameters driven by Gaussian noise

A two-step method is proposed to find state properties for linear dynamic systems driven by Gaussian noise with uncertain parameters modeled as a random vector with known probability distribution. First, equations of linear random vibration are used to find the probability law of the state of a system with uncertain parameters conditional on this vector. Second, stochastic reduced order models (SROMs) are employed to calculate properties of the unconditional system state. Bayesian methods are applied to extend the proposed approach to the case when the probability law of the random vector is not available. Various examples are provided to demonstrate the usefulness of the method, including the random vibration response of a spacecraft with uncertain damping model.     

Theoretical study on fabrication of functionally graded material with density gradient by a centrifugal solid-particle method

Computer simulation was carried out for an estimation of the effect of centrifugal force on the formation of graded distribution of solid spherical particles within molten metal. The difference of migration rate between two kinds of spherical particles with different diameter and density was calculated taking account of the application of a centrifugal solid-particle method. Based on the systematic analysis, the capability of fabrication of the functionally graded materials ring with a unique density gradients which comprise the graded distributions of two kinds of solid particles is discussed. The unique gradient was essentially achieved due to the preferential movement of large particles with low density in comparison to small particles with high density.     

A virtual crack closure-integral method (VCCM) to compute the energy release rates and stress intensity factors based on quadratic tetrahedral finite elements

This paper proposes a new virtual crack closure-integral method (VCCM) for quadratic tetrahedral finite element to compute the energy release rates/stress intensity factors. The formulations, numerical implementations and some numerical results of proposed VCCM are presented in this paper. Proposed VCCM enables us to adopt the tetrahedral finite element in 3D crack problems and us to use automatic mesh generation programs. Therefore process time to perform 3D crack analysis drastically reduces compared with the case of hexahedral elements.