For Functionally Graded Materials Research Articles

Thermomechanical problem of functionally graded spherical shells based on homogenization schemes: Data-driven volume fraction optimization with material uncertainties

Mechanical analysis and optimal design for functionally graded materials (FGMs) structures are significant. The present paper analyzes the parameter uncertainties problem for the thermomechanical response of the FGMs spherical shells with volume fraction power distribution. Here, the interval random uncertainty model is recommended to describe the uncertainties of each component of material parameters in FGMs, and a data-driven artificial neural network (ANN) fast calculation for FGMs spherical shells thermomechanical problems with uncertainties analysis is established and trained. The thermomechanical coupling response analysis and volume fraction optimization analysis of the FGMs spherical shell with material uncertainties are realized for zirconia-titanium FGMs, with the maximum normalized Von Mises stress as the structural safety index and the ANN as the fast calculation method. In addition, the Von Mises stress of the FGMs spherical shell and a ZrO2-Ti alloy double-layer spherical shell under different loads is compared. The results show that material parameter uncertainties, especially those in Young's modulus, thermal expansion coefficient, and uniaxial yield limit significantly affect structural safety. The structural safety index λ calculated based on the volume fraction optimization design of FGMs spherical shell is significantly lower than that of the ZrO2-Ti alloy double-layer. The ANN method greatly improves the computational efficiency for uncertain problems and maintains high accuracy, compared with the statistical analysis after many calculations by randomly selecting material parameters. This study provides a path for the safety design of FGMs structures with uncertain parameters.

Probabilistic damage identification of FGM structures using model updating procedure based on expansion of incomplete FRF data

The spatial incompleteness and uncertainties of measured data are unavoidable factors that could significantly affect the reliability of damage detection results. It is therefore essential to utilize a method that can effectively deal with the spatial incomplete and contaminated data, and presents probabilistic damage identification results instead of deterministic results. In this regard, a probabilistic damage identification approach for functionally graded materials (FGM) structures using model updating procedure based on expansion of incomplete frequency response function (FRF) data with measurement noise is presented. The model updating procedure is formulated as an optimization scheme which is accomplished by minimizing a cost function based on the changes in expanded FRF data. To expand the incompletely measured FRFs, an iterative order reduction method is performed, which makes the identification resistant to the adverse effects of measurement noise. For the minimization process, we adopt a novel meta-heuristic algorithm called bald eagle search algorithm (BES), which has not yet been tested in the field of model updating. Based on the statistical distributions of the identified stiffness parameters in the damaged and undamaged states, the probability of damage existence (PDE) is established to describe the damage probability for each element. The performance of the proposed model updating procedure is verified using three FGM structures: a simple beam, a two-span beam and a cantilever plate. The statistical results indicate that under a high level of noise (15%), the proposed procedure can provide the prediction of damage localization with a high level of confidence and yield damage estimation results with acceptable errors.

Machine Learning-Based Fatigue Life Prediction of Functionally Graded Materials Using Material Extrusion Technology

In this study, the research investigates the prediction of fatigue life for Functionally Graded Materials (FGM) specimens comprising Polylactic acid (PLA) and Thermoplastic Polyurethane (TPU). For this, Machine learning (ML) techniques, including Random Forest (RF), Support Vector Machine (SVM), and Artificial Neural Network (ANN) are utilized. A predictive in-house code is developed for each technique, thereby facilitating the fatigue performance of layered deposited specimens subjected to varying cyclic loadings. In order to verify the effectiveness of the ML technique, a comparative analysis among all is reported based on empirically determined fatigue life obtained values. RF is proven to be the most suitable technique with minimal error percentage in obtained results with optimally synchronized data sets in a minimum time frame. Subsequently, the application of ML in those predictions is reported for future aspects in augmenting the operational efficiency associated with fatigue life prediction.

Meshless variational method applied to fracture mechanics with functionally graded materials

With a variation in crack length, the stress intensity factors are obtained by a path independent contour integral in this work. The Chebyshev polynomials are employed using the finite block method for Functionally Graded Materials (FGMs) with two-dimensional fracture problems. By the mapping technique, a block of quadratic type is transformed from Cartesian coordinate to a normalised coordinate with 8 nodes. The new governing equations in terms of displacements are deduced in the mapping domain. All coefficients of Chebyshev polynomials are determined by considering governing equations, boundary conditions and connecting conditions for two blocks. The accuracy and convergence of the FBM with Chebyshev polynomials is demonstrated with four examples and comparison has been implemented with analytical solutions and different numerical approaches.

Fracture analysis of functionally graded materials by the method of fundamental solutions

In this paper the Method of Fundamental Solutions (MFS) incorporating Erdogan’s solutions for Functionally Graded Materials (FGM) is presented for analysis of 2D fracture problems subjected to static and dynamics loads. Erdogan derived analytical solutions for a pair of static concentrated force in an infinite isotropic plate with a straight cut. Based on homogenous isotropic analysis, the contribution from non-homogeneity in equilibrium equations is treated as body forces and domain integrals based on the Erdogan’s fundamental solutions are required. In the domain integrals, all singularities are cancelled by introduction of a polar coordinate system at collocation points and crack tips. In the dynamic cases, the Laplace transformation with the Durbin inversion technique is adopted to determine the time-dependent variables such as the stress intensity factors at crack tips. The domain integrals are obtained numerically with the sub-region technique. The accuracy of the MFS is demonstrated with four numerical examples and comparisons are implemented with different numerical approaches.

Vibration response of shear deformable functionally graded material plate with parabolic variable thickness

This work presents the semi-analytical solution of vibration response for functionally graded materials (FGM) plates with bi-linear and parabolic variable thickness. Continuous thickness variation may affect the design of plate structure which permits to modify the stiffness properties in the utmost stressed area within the domain, considering the overall weight parameter of the structure. As a result, a better dynamic response may be revealed. The first order shear deformation theory (FSDT) has been applied for a kinematic model of the variable thick plate. The material properties were calculated by the simple power law. Accuracy of this model is confirmed by various comparisons with outcomes available in the preceding literature. Frequency factor of the plates with boundary conditions are found by changing the various parameter like material volume exponent, aspect ratio, tapered ratio and thickness ratio.

New refined higher-order shear deformation theories for functionally graded plates conforming to graded variations of material properties

This paper proposes new refined higher-order shear deformation theories (RHSDTs) for functionally graded (FG) plates conforming to graded variations of material properties. The theoretical formulation starts with initial displacement descriptions based on general higher-order theories, determines in-plane stress fields using geometrical and constitutive relations, and further achieves transverse shear stress fields though equilibrium equations. By imposing tangential stress-free conditions and defining new variables, the transverse shear stress expressions without differential variables are obtained. Then, the uniform and quadratic distributions of the transverse displacement field are considered, respectively, and the final displacement expressions of the present 2D and quasi-3D RHSDTs are further formulated by integrating the associated shear strain expressions. This novel formulation approach first introduces gradient parameters of FG materials into the shear strain shape functions. Compared with traditional HSDTs, the present refined theories fully incorporate the influences of the graded variations of material properties and can better predict mechanical responses for FG plates. Moreover, the governing equations and associated analytical solutions are presented. Numerical examples demonstrate high accuracy of the present RHSDTs and reflect their obvious accuracy advantages over traditional HSDTs especially when predicting the transverse shear stresses for FG plates with arbitrary gradients. Furthermore, it is also validated that the present RHSDTs with more shear expansion terms have higher accuracy for the prediction of transverse shear stress fields.

A non-homogeneous multi-material topology optimization approach for functionally graded structures with cracks

This article presents an effective non–homogeneous multi-material topology optimization paradigm for functionally graded (FG) structures, considering both cracked and non-cracked cases for the first time. For that aim, an enrichment finite element concept known as the extended finite element method (X-FEM) is employed to analyze strong discontinuity states’ critical mechanical behavior. The preconditioned block-conjugate gradient (pb-CG) is considered to deal with the X-FEM’s natural block system form for saving computational efforts. In addition, a block Gauss–Seidel-based alternating active-phase algorithm is utilized to convert a multiphase topology optimization problem subjected to multiple constraints to many binary phase topology optimization sub-problems with only one constraint. Consequently, the current topology optimization methodology can dramatically reduce the number of design variables regardless of the number of material phases. Optimality Criteria (OC) optimizer is then utilized to update optimized design variables for such sub-problems. The study formulates in great detail mathematical expressions for topology optimization of cracked structures with multiple FG materials. Several numerical examples are tested to verify the efficiency and reliability of the current paradigm.

Stress Distributions for T-Section Beams Made of Functionally Graded Material (Dept.M)

This study introduces a theoretical analysis for functionally graded materials (FGMS) OF t-section beams. Analytical methods are set in the form of equations using the effective principal axes, in order to provide a method for predicting the normal stress distribution of the FGMS beam under both axial load and bending moments. Considering the elastic modulus to be an exponential function, the effect of the non-homogeneity parameter on the distribution of the normal stress, as well as on the position of the neutral axis along the beam height, is discussed for several different loading cases. The results obtained show that the non-homogeneity parameters have great effects on the normal stress distribution and the position of the neutral axis. This indicates that, with the less hard region increases, and the neutral axis transfers toward the harder region.

Phase field modeling of fracture in Functionally Graded Materials: [formula omitted]-convergence and mechanical insight on the effect of grading

A phase field (PF) approximation of fracture for functionally graded materials (FGM) using a diffusive crack approach incorporating the characteristic length scale as a material parameter is herein proposed. A rule of mixture is employed to estimate the material properties, according to the volume fractions of the constituent materials, which have been varied according to given grading profiles. In addition to the previous aspects, the current formulation includes the internal length scale of the phase field approach variable from point to point, to model a spatial variation of the material strength. Based on the ideas stemming from the study of size-scale effects, Γ-convergence for the proposed model is proved when the internal length scale is either constant or a bounded function. In a comprehensive sensitivity analysis, the effects of various model parameters for different grading profiles are analyzed. We first prove that the fracture energy and the elastic energy of FGM is bounded by their homogeneous constituents. Constitutive examples of boundary value problems solved using the BFGS solver are provided to bolster this claim. Finally, crack propagation events in conjunction with the differences with respect to their homogeneous surrogates are discussed through several representative applications, providing equivalence relationships for size-scale effects and demonstrating the applicability of the current model for structural analysis of FGMs.

Experimental validation of Scheil–Gulliver simulations for gradient path planning in additively manufactured functionally graded materials

Additive manufacturing (AM), through directed energy deposition, supports planned composition changes between locations within a single component, allowing for functionally graded materials (FGMs) to be developed and fabricated. The formation of deleterious phases along a particular composition path can cause significant cracking during the AM build process that makes the composition path unviable to produce structurally stable FGMs, but it is challenging to predict which phases will be present in as-built additively manufactured parts by analyzing only equilibrium phase relations. Solute segregation during solidification can lead to the formation of non-equilibrium phases that are stable at compositions far from the nominal composition of the melt, leading to crack formation. In this work, we developed and validated a Scheil–Gulliver simulation tool based on pycalphad. We used this tool to compare the non-equilibrium phases predicted to form during the AM build process using the Scheil–Gulliver model with experimentally measured phases at several locations with different composition in a Ti-6Al-4V to Invar-36 FGM and a commercially pure Ti to Invar-36 FGM. We showed that the phases predicted to form by the Scheil–Gulliver model agree better with the experimental results than the predictions made by assuming equilibrium solidification, proving that the Scheil–Gulliver model can be applied to FGMs. Further, we demonstrated the use of our Scheil–Gulliver simulation tool as a method of designing FGMs through screening potential FGM pathways by calculating the solidification phase fractions along the experimental gradient path in composition space.

Peridynamic analysis of dynamic fracture behaviors in FGMs with different gradient directions

In this article, a developed bond-based peridynamic model for functionally graded materials (FGMs) is proposed to simulate the dynamic fracture behaviors in FGMs. In the developed bond-based peridynamic model for FGMs, bonds are categorized into three different types, including transverse directionally peridynamic bond, gradient directionally peridynamic bond and arbitrary directionally peridynamic bond, according to the geometrical relationship between directions of peridynamic bonds and gradient bonds in FGMs. The peridynamic micromodulus in the gradient directionally and arbitrary directionally peridynamic bonds can be determined using the weighted projection method. Firstly, the standard bond-based peridynamic simulations of crack propagation and branching in the homogeneous PMMA plate are performed for validations, and the results are in good agreement with the previous experimental observations and the previous phase-field numerical results. Then, the numerical study of crack initiation, propagation and branching in FGMs are conducted using the developed bond-based peridynamic model, and the influence of gradient direction on the dynamic fracture behaviors, such as crack patterns and crack tip propagation speed, in FGMs is systematically studied. Finally, numerical results reveal that crack branching in FGMs under dynamic loading conditions is easier to occur as the gradient angle decreases, which is measured by the gradient direction and direction of the initial crack.

Free vibration response of P-FGM and S-FGM sandwich shell panels: A comparison

This research work presents the comparison of free vibration response for functionally graded materials (FGM) sandwich shell panels having bottom and top layers made of pure metal and ceramic and the core of the same is made of FGM. The material properties of the FGM core are varied according to the power-law model (P-FGM) and the sigmoid model (S-FGM). A finite element model (FEM) based on the displacement field according to first-order shear deformation theory for each layer has been used to develop the governing differential equation for free vibration analysis of P-FGM sandwich and S-FGM sandwich shell panels. The accuracy of the present finite element model has been demonstrated by comparing results with benchmark examples from with those published in literature. Next, the free vibration responses obtained from two different material modeling Viz., power-law variation and Sigmoid variation for FGM core of FGM sandwich shell have been compared by varying parameters like volume fraction index, curvature, and geometrical boundary conditions.

Functionally graded materials classifications and development trends from industrial point of view

Over the last few years, many classifications have been proposed for functionally graded materials (FGMs). In this Paper, critical review of different available classifications for FGM based on their physical, structural and manufacturing characteristics are presented. Advantages and limitations of each fabrication method for use in a given application is correspondingly considered. In addition, new classifications based on gradation control and accuracy, residual stresses, specific energy consumption, environmental impact evaluated throughout the complete life cycle and manufacturing costs are proposed. These classifications mainly reflect the needs of both FGM designers and industrial manufacturers. Based upon the presented classifications and the recent advances in analysis and production techniques, new major directions for FGMs research are proposed.

Numerical simulation of dynamic fracture in functionally graded materials using peridynamic modeling with composite weighted bonds

In this paper, the peridynamic (PD) method with the composite weighted bond for functionally graded materials (FGMs) was proposed. The cracks propagation in FGMs under dynamic loading was analyzed. Both uniform refinement grid (m-convergence) and decreasing radius of PD horizon (δ-convergence) of FGMs with double pre-crack were given. The effects of the micromodulus functions and material gradient form on the crack propagation were discussed, and it was found that the micromodulus function's effect on the crack propagation pattern was limited, while the effect of the gradient form of FGMs is existed. The numerical results show that there is almost no difference between the proposed composite weighted bond model and the previous model for FGMs with single pre-crack. By comparing the numerical results of the proposed model with the results from the experiment, the present PD model for FGMs is valid and effective.

Incompatible Graded Finite Elements for Analysis of Nonhomogeneous Materials

A six-node incompatible graded finite element is developed and studied. Such element is recommended for use since it is more accurate than four-node compatible element and more efficient than eight-node compatible element in two-dimensional plane elasticity. This paper presents comparison between six-node incompatible (QM6) and four-node compatible (Q4) graded elements. Numerical solution is obtained from abaqus using UMAT capability of the software and exact solution is provided as reference for comparison. A graded plate with exponential and linear gradation subjected to traction and bending load is considered. Additionally, three-node triangular (T3) and six-node triangular (T6) graded elements are compared to QM6 element. Incompatible graded element is shown to give better performance in terms of accuracy and computation time over other element formulations for functionally graded materials (FGMs).

A higher order theory for functionally graded shells

New theory of higher order for functionally graded (FG) shells, which is based on the expansion of the three-dimensional (3D) equations of elasticity for functionally graded materials (FGMs) into Legendre’s polynomials series is developed here. The stress and strain tensors, the displacement, traction and body force vectors of the 3D equations of elasticity, are expanded into Legendre’s polynomials series in terms of in the thickness coordinate. The mechanical parameters that describe the functionally graded material properties are also represented in the form of Legendre’s polynomials series expansion. As result the equations of the 3D elasticity are turned into the infinite number of two-dimensional (2D) equations for the Legendre’s polynomials series expansion coefficients. Considering finite number of the Legendre’s polynomials series coefficients and substituting kinematic relations into generalized Hooke’s law and the obtained result into the equations of motion the differential equations of motion the equations of motion in form of displacements have been obtained. The first order equations for the FG axisymmetric cylindrical plate and spherical shell are considered in more details. Corresponding boundary-value problems are solved using the finite element method (FEM) implemented in the MATEMATICA software. The numerical results are presented and discussed.

Exploration of the high order theory for functionally graded beams based on Legendre's polynomial expansion

A high order theory for functionally graded (FG) beams based on expansion of the two dimensional (2-D) equations of elasticity for functionally graded materials (FGMs) into Legendre's polynomials series has been developed. The 2-D equations of elasticity have been expanded into Legendre's polynomials series in terms of a thickness coordinate. In the same way functions that describe functionally graded relations also has been expanded. Thereby all equations of elasticity including Hook's law have been transformed to corresponding equations for coefficients of Legendre's polynomials expansion. Then system of differential equations in term of displacements and boundary conditions for the coefficients of Legendre's polynomials expansion coefficients has been obtained. Cases of the first and second approximations have been considered in more detail. For the obtained boundary-value problems solution, a finite element method (FEM) has been used and numerical calculations have been done with MATHEMATICA, MATLAB and COMSOL Multiphysics software. Numerical results are presented and discussed.

Elastoplastic nonlinear FEM analysis of FGM shells of Cosserat type

The paper is a continuation of [1] where the formulation of the elastic constitutive law for functionally graded materials (FGM) on the grounds of nonlinear 6-parameter shell theory with the 6th parameter (the drilling degree of freedom) was presented. Here the formulation is extended to the elasto-plastic range. The material law is based on J2 Cosserat plasticity and employs the well-known Tamura-Tomota-Ozawa (TTO) [2] mixture model with additional formulae for Cosserat material parameters. Formulation is verified by solving a set of demanding analyses of plates, curved and multi-branched shells, including geometry, thickness and material distribution variation parameter analyses.

Two-Step Sintering of Alumina/Zirconia Functionally Graded Materials

Abstract The interest for functionally graded materials (FGM) has grown up during the last 2 decades owing to the possibility of developing pieces and devices with gradual variation of properties along one or more dimensions. This allows for an optimum performance as a function of the intended use. On the other hand, two stage sintering is a relative new sintering technique, where the shaped pieces are heated up to a relative density of 75–92 % of theoretical density, followed by a fast cooling step to an inferior temperature, where the piece in maintained up to the end of the densification process. This results in the development of finer microstructures, responsible for better mechanical properties. In this work, we characterized alumina-ytria-stabilized zirconia composites, with different amounts of 3Y-ZrO2 by dilatometry. After fitting thermal behavior of the different composites during sintering and determining temperature of maximum densification rate, co-pressed FGMs were produced. Then the samples were sintered by pressureless traditional technique and two stage sintering, with a peak temperature of 1450 °C and holding temperature of 1350 °C. Microstructure was studied by SEM and mechanical properties (microhardness and indentation toughness) were determined in order to compare pressureless sintered FGMs at 1500 °C and two-stage sintered FGMs.