Mode-3 spontaneous crack propagation along functionally graded bimaterial interfaces

The effects of combining functionally graded materials (FGMs) of different inhomogeneous property gradients on the mode-3 propagation characteristics of an interfacial crack are numerically investigated. Spontaneous interfacial crack propagation simulations were performed using the newly developed spectral scheme. The numerical scheme derived and implemented in the present work can efficiently simulate planar crack propagation along functionally graded bimaterial interfaces. The material property inhomogeneity was assumed to be in the direction normal to the interface. Various bimaterial combinations were simulated by varying the material property inhomogeneity length scale. Our parametric study showed that the inclusion of a softening type FGM in the bimaterial system leads to a reduction in the fracture resistance indicated by the increase in crack propagation velocity and power absorbed. An opposite trend of increased fracture resistance was predicted when a hardening material was included in the bimaterial system. The cohesive tractions and crack opening displacements were altered due to the material property inhomogeneity, but the stresses ahead of the cohesive zone remained unaffected.     

In-plane elastic wave propagation and band-gaps in layered functionally graded phononic crystals

In-plane wave propagation in layered phononic crystals composed of functionally graded interlayers arisen from the solid diffusion of homogeneous isotropic materials of the crystal is considered. Wave transmission and band-gaps due to the material gradation and incident wave-field are investigated. A classification of band-gaps in layered phononic crystals is proposed. The classification relies on the analysis of the eigenvalues of the transfer matrix for a unit-cell and the asymptotics derived for the transmission coefficient. Two kinds of band-gaps, where the transmission coefficient decays exponentially with the number of unit-cells are specified. The so-called low transmission pass-bands are introduced in order to identify frequency ranges, in which the transmission is sufficiently low for engineering applications, but it does not tend to zero exponentially as the number of unit-cells tends to infinity. A polyvalent analysis of the geometrical and physical parameters on band-gaps is presented.     

A finite element formulation for analysis of functionally graded plates and shells

Summary A finite element formulation is derived for the thermoelastic analysis of functionally graded (FG) plates and shells. The power-law distribution model is assumed for the composition of the constitutent materials in the thickness direction. The procedure adopted to derive the finite element formulation contains the analytical through-the-thickness integration inherently. Such formulation accounts for the large gradient of the material properties of FG plates and shells through the thickness without using the Gauss points in the thickness direction. The explicit through-the-thickness integration becomes possible due to the proper decomposition of the material properties into the product of a scalar variable and a constant matrix through the thickness. The nonlinear heat-transfer equation is solved for thermal distribution through the thickness by the Rayleigh-Ritz method. According to the results, the formulation accounts for the nonlinear variation in the stress components through the thickness especially for regions with a variation in martial propperties near the free surfaces.     

The axisymmetric torsional contact problem of a functionally graded piezoelectric coated half-space

In this article, we study the axisymmetric tor-sional contact problem of a half-space coated with func-tionally graded piezoelectric material (FGPM) and subjected to a rigid circular punch. It is found that, along the thick-ness direction, the electromechanical properties of FGPMs change exponentially. We apply the Hankel integral trans-form technique and reduce the problem to a singular integral equation, and then numerically determine the unknown con-tact stress and electric displacement at the contact surface. The results show that the surface contact stress, surface azimuthal displacement, surface electric displacement, and inner electromechanical field are obviously dependent on the gradient index of the FGPM coating. It is found that we can adjust the gradient index of the FGPM coating to modify the distributions of the electric displacement and contact stress.     

粘结单元在模拟FRP层合板低速冲击响应中的应用

纤维增强树脂基复合材料层合板(fibre reinforced plastic composites,FRP)在航空、航天、交通、造船等诸多工程中得到了日益广泛的应用,而其在冲击载荷下的响应和破坏特别是分层一直为学术界所关注。本文中对FRP层合板在冲击载荷下的响应和破坏进行数值模拟,并通过引入粘结层重点研究其分层破坏。首先,介绍一种基于改进的粘结区域方法的粘结层损伤模型;其次,详细介绍了有限元模型建模过程和建模细节;最后,对有限元模型进行验证,并分析分层损伤发生的原因。模拟结果表明,该模型不仅能准确预测FRP层合板在低速冲击载荷下的载荷-时间曲线和载荷-位移曲线,还能成功地预测其分层破坏。     

Free vibration characteristics of sectioned unidirectional/bidirectional functionally graded material cantilever beams based on finite element analysis

Advancements in manufacturing technology, including the rapid development of additive manufacturing (AM), allow the fabrication of complex functionally graded material (FGM) sectioned beams. Portions of these beams may be made from different materials with possibly different gradients of material properties. The present work proposes models to investigate the free vibration of FGM sectioned beams based on onedimensional (1D) finite element analysis. For this purpose, a sample beam is divided into discrete elements, and the total energy stored in each element during vibration is computed by considering either the Timoshenko or Euler-Bernoulli beam theory. Then, Hamilton's principle is used to derive the equations of motion for the beam. The effects of material properties and dimensions of FGM sections on the beam's natural frequencies and their corresponding mode shapes are then investigated based on a dynamic Timoshenko model (TM). The presented model is validated by comparison with three-dimensional (3D) finite element simulations of the first three mode shapes of the beam.     

Cohesive modeling of transverse cracking in laminates under in-plane loading with a single layer of elements per ply

This study aims to bridge the gap between classical understanding of transverse cracking in cross-ply laminates and recent computational methods for the modeling of progressive laminate failure. Specifically, the study investigates under what conditions a finite element model with cohesive X-FEM cracks can reproduce the in situ effect for the ply strength. It is shown that it is possible to do so with a single element across the thickness of the ply, provided that the interface stiffness is properly selected. The optimal value for this interface stiffness is derived with an analytical shear lag model. It is also shown that, when the appropriate statistical variation of properties has been applied, models with a single element through the thickness of a ply can predict the density of transverse matrix cracks.     

Anti-plane problem of periodic cracks in a functionally graded coating–substrate structure

The static and dynamic anti-plane problem for a functionally graded coating–substrate structure containing a periodic array of parallel cracks, which are perpendicular to the boundary, is considered. Integral-transform techniques are employed to reduce the problem to the solution of an integral equation with hypersingular kernels. Numerical results are presented to show the influence of geometry, material properties and material gradient parameter on the fracture behavior.     

The behavior of a crack in functionally graded piezoelectric/piezomagnetic materials under anti-plane shear loading

Summary In this paper, the behavior of a crack in functionally graded piezoelectric/piezomagnetic materials subjected to an anti-plane shear loading is investigated. To make the analysis tractable, it is assumed that the material properties vary exponentially with the coordinate parallel to the crack. By using a Fourier transform, the problem can be solved with the help of a pair of dual integral equations in which the unknown variable is the jump of the displacements across the crack surfaces. These equations are solved using the Schmidt method. The relations among the electric displacement, the magnetic flux and the stress field near the crack tips are obtained. Numerical examples are provided to show the effect of the functionally graded parameter on the stress intensity factors of the crack.The authors are grateful for financial support from the Natural Science Foundation of Hei Long Jiang Province (A0301), the National Natural Science Foundation of China (50232030, 10172030), the Natural Science Foundation with Excellent Young Investigators of Hei Long Jiang Province(JC04-08) and the National Science Foundation with Excellent Young Investigators (10325208).     

Mechanical modeling and transient anti-plane fracture analysis for the graded inter-diffusion regions in a bonded structure

An inter-diffusion interface model (IDIM) is put forward for a bonded structure. Laplace and Fourier integral transforms are applied to reduce the transient anti-plane fracture problem of the structure as a Cauchy singular integral equation. Lobatto-Chebyshev collocation method and Laplace numerical inversion transform are employed to evaluate transient stress intensity factors (TSIFs). The effects of geometrical and physical parameters on TSIFs are studied. Specially discussed are the influences of the weak/micro-discontinuity of the interfaces. Comparison between IDIM and the graded interlayer model indicates that if the inter-diffusion between the two original materials is prominent, the former should be applied instead of the latter in fracture analyses of bonded structures.     

Free vibration analysis of functionally graded material beams based on Levinson beam theory

Free vibration response of functionally graded material (FGM) beams is studied based on the Levinson beam theory (LBT). Equations of motion of an FGM beam are derived by directly integrating the stress-form equations of elasticity along the beam depth with the inertial resultant forces related to the included coupling and higherorder shear strain. Assuming harmonic response, governing equations of the free vibration of the FGM beam are reduced to a standard system of second-order ordinary differential equations associated with boundary conditions in terms of shape functions related to axial and transverse displacements and the rotational angle. By a shooting method to solve the two-point boundary value problem of the three coupled ordinary differential equations, free vibration response of thick FGM beams is obtained numerically. Particularly, for a beam with simply supported edges, the natural frequency of an FGM Levinson beam is analytically derived in terms of the natural frequency of a corresponding homogenous Euler-Bernoulli beam. As the material properties are assumed to vary through the depth according to the power-law functions, the numerical results of frequencies are presented to examine the effects of the material gradient parameter, the length-to-depth ratio, and the boundary conditions on the vibration response.     

Elastodynamic analysis of a crack at an arbitrary angle to the graded interfacial zone in bonded half-planes under antiplane shear impact

A solution is provided for the elastodynamic problem of a crack at an arbitrary angle to the graded interfacial zone in bonded media under the action of antiplane shear impact. The interfacial zone is modeled by a nonhomogeneous interlayer with the spatially varying shear modulus and mass density in terms of power functions between the two dissimilar, homogeneous half-planes. Based on the use of Laplace and Fourier integral transforms and the coordinate transformations of basic field variables, formulation of the transient crack problem is reduced to solving a Cauchy-type singular integral equation in the Laplace transform domain. The crack-tip response in the physical domain is recovered via the inverse Laplace transform and the values of dynamic mode III stress intensity factors are obtained as a function of time. A comprehensive parametric study is then presented of the effects of crack obliquity on the overshoot behavior of the transient crack-tip response, by plotting the peak values of the dynamic stress intensity factors versus the crack orientation angle for various material and geometric combinations of the bonded system.     

Evaluation of strongly singular domain integrals for internal stresses in functionally graded materials analyses using RIBEM

An accurate evaluation of strongly singular domain integral appearing in the stress representation formula is a crucial problem in the stress analysis of functionally graded materials using boundary element method.To solve this problem,a singularity separation technique is presented in the paper to split the singular integral into regular and singular parts by subtracting and adding a singular term.The singular domain integral is transformed into a boundary integral using the radial integration method.Analytical expressions of the radial integrals are obtained for two commonly used shear moduli varying with spatial coordinates.The regular domain integral,after expressing the displacements in terms of the radial basis functions,is also transformed to the boundary using the radial integration method.Finally,a boundary element method without internal cells is established for computing the stresses at internal nodes of the functionally graded materials with varying shear modulus.     

Modeling the evolution of stress due to differential shrinkage in powder-processed functionally graded metal–ceramic composites during pressureless sintering

Pressureless sintering of powder-processed functionally graded materials is being pursued to economically produce metal–ceramic composites for a variety of high-temperature (e.g., thermal protection) and energy-absorbing (e.g., armor) applications. During sintering, differential shrinkage induces stresses that can compromise the integrity of the components. Because the strength evolves as the component is sintered, it is important to model how the evolution of the differential shrinkage governs the stress distribution in the component in order to determine when the strength will be exceeded and cracking initiated. In this investigation, a model is proposed that describes the processing/microstructure/property/performance relationship in pressurelessly sintered functionally graded plates and rods. This model can be used to determine appropriate shrinkage rates and gradient architectures for a given component geometry that will prevent the component from cracking during pressureless sintering by balancing the evolution of strength, which is assumed to be a power law function of the porosity, with the evolution of stress. To develop this model, the powder mixture is considered as a three-phase material consisting of voids, metal particles, and ceramic particles. A micromechanical thermal elastic–viscoplastic constitutive model is then proposed to describe the thermomechanical behavior of the composite microstructure. The subsequent evolution of the thermomechanical properties of the matrix material during sintering is assumed to obey a power law relationship with the level of porosity, which is directly related to the shrinkage strain, and was refined to account for the evolving interparticle cohesion of the matrix phase due to sintering. These thermomechanical properties are incorporated into a 2-D thermomechanical finite element analysis to predict the stress distributions and distortions that arise from the evolution of differential shrinkage during the pressureless sintering process. Differential shrinkage results were verified quantitatively through comparison with the shape profile for a pressurelessly sintered functionally graded nickel–alumina composite plate with a cylindrical geometry, and the stress distribution results verified from qualitative observations of the absence or presence of cracking as well as the location in specimens with different gradient architectures. The cracking was mitigated using a reverse gradient at one end of the specimen, and the resulting distortions associated with the shape profile were determined to be no more than 15% reduced from the predictions. The effects of geometry were also studied out-of-plane by transforming the plate into a rod through an increase in thickness, while in-plane effects were studied by comparing the results from the cylindrical specimen with a specimen that has a square cross-sectional geometry. By transforming from a plate to a rod geometry, the stress no longer exceeds critical levels and cracks do not form. The results from the in-plane geometric study indicated that critical stresses were reached in the square geometry at temperatures 100 °C less than in the cylindrical geometry. Additionally, the location of primary cracking was shifted towards the metal-rich end of the specimen, while the stress distribution associated with this shift and the lower temperature for the critical stress resulted in secondary cracking.     

Elliptical crack normal to functionally graded interface of bonded solids

This paper presents an analysis of an elliptical crack that is perpendicular to a functionally graded interfacial zone between two fully bonded solids. The functionally graded interfacial zone is treated as a non-homogeneous solid layer with its elastic modulus varying in the thickness direction. A generalized Kelvin solution based boundary element method is employed for the calculation of the stress intensity factors associated with the three-dimensional crack problem. The elliptical crack surface is subject to either uniform normal traction or uniform shear traction. The stress intensity factors are examined by taking into account the effects of the non-homogeneity parameter and thickness of the functionally graded interfacial zone, as well as the crack distance to the zone. The SIF values are further incorporated into the S-criterion for prediction of crack growth. The paper presents the most possible direction and location of the elliptical crack growth under an inclined tensile (or compressive) load. The paper further presents results of the critical external loads that would cause the elliptical crack to grow at the most possible location and along the most possible direction. The paper also examines the effects of external load direction and material and geometrical parameters on the critical loads.     

Dynamic behavior of two collinear anti-plane shear cracks in a functionally graded layer bonded to dissimilar half planes

In recent years, the functionally graded materials (FGMs) have been widely applied in extremely high temperate environment. In this paper, the dynamic behavior of two collinear cracks in FGM layer bonded to dissimilar half planes under anti-plane shear waves is studied by the Schmidt method. By using the Fourier transform technique, the present problem can be solved with a dual integral equation. These equations are solved using the Schmidt method. The present method is used to illustrate the fundamental behavior of the interacting cracks in FGMs under dynamic loading. Furthermore, the effects of the geometry of the interacting cracks, the shear stress wave velocity of the materials and the frequency of the incident wave on the Dynamic Stress Intensity Factor are investigated.     

Unified two-phase nonlocal formulation for vibration of functionally graded beams resting on nonlocal viscoelastic Winkler-Pasternak foundation

A nonlocal study of the vibration responses of functionally graded (FG) beams supported by a viscoelastic Winkler-Pasternak foundation is presented. The damping responses of both the Winkler and Pasternak layers of the foundation are considered in the formulation, which were not considered in most literature on this subject, and the bending deformation of the beams and the elastic and damping responses of the foundation as nonlocal by uniting the equivalently differential formulation of well-posed strain-driven (ε-D) and stress-driven (σ-D) two-phase local/nonlocal integral models with constitutive constraints are comprehensively considered, which can address both the stiffness softening and toughing effects due to scale reduction. The generalized differential quadrature method (GDQM) is used to solve the complex eigenvalue problem. After verifying the solution procedure, a series of benchmark results for the vibration frequency of different bounded FG beams supported by the foundation are obtained. Subsequently, the effects of the nonlocality of the foundation on the undamped/damping vibration frequency of the beams are examined.     

Stochastic analysis of a thermoelastic problem in functionally graded plates with uncertain material properties

The statistics (i.e., mean and variance) of temperature and thermal stress are analytically obtained in functionally graded material (FGM) plates with uncertainties in the thermal conductivity and coefficient of linear thermal expansion. These FGM plates are assumed to have arbitrary nonhomogeneous thermal and mechanical properties through the entire thickness of plate and are subjected to deterministic convective heating. The stochastic temperature and thermal stress fields are analysed by assuming the FGM plate is multilayered with distinct, random thermal conductivity and coefficient of linear thermal expansion in each layer. Vodicka’s method, which is a type of integral transform method, and a perturbation method are employed to obtain the analytical solutions for the statistics. The autocorrelation coefficients of each random property and cross-correlation coefficients between different random properties are expressed in exponential function forms as a non-homogeneous Markov random field of discrete space. Numerical calculations are performed for FGM plates composed of partially stabilised zirconia (PSZ) and austenitic stainless steel (SUS304), which have the largest dispersion of the random properties at the place where the volume fractions of the two constituent materials are both 0.5. The effects of the spatial change in material composition, thermal boundary condition and correlation coefficients on the standard deviations of the temperature and thermal stress are discussed.     

Cohesive zone laws for fatigue crack growth: Numerical field projection of the micromechanical damage process in an elasto-plastic medium

Cohesive zone failure models are widely used to simulate fatigue crack propagation under cyclic loading, but the model parameters are phenomenological and are not closely tied to the underlying micromechanics of the problem. In this paper, we will inversely extract the cohesive zone laws for fatigue crack growth in an elasto-plastic ductile solid using a field projection method (FPM), which projects the equivalent tractions and separations at the cohesive crack-tip from field information outside the process zone. In our small-scale yielding model, a single row of discrete voids is deployed directly ahead of a crack in an elasto-plastic medium subjected to cyclic mode I K-field loading. Damage accumulation under cyclic loading is captured by the growth of voids within the micro-voiding zone ahead of the crack, while the evolution of the cohesive zone law representing the micro-voiding zone is inversely extracted via the FPM. We show that the field-projected cohesive zone law captures the essential micromechanisms of fatigue crack growth in the ductile medium: from loading and unloading hysteresis caused by void growth and plastic hardening, to the softening damage locus associated with crack propagation via a void by void growth mechanism. The results demonstrate the effectiveness of the FPM in obtaining a micromechanics-based cohesive zone law in-place of phenomenological models, which opens the way for a unified treatment of fatigue crack problems.