Functionally Graded Materials Structures Research Articles

Fracture parameter investigations of functionally graded materials by using ordinary state based peridynamics

Static and dynamic fracture parameter analyses of functionally graded materials (FGMs) are conducted by using the ordinary state based peridynamic theory (OSPD). As a meshfree method, OSPD applies an integral equation to describe the motion of objects, avoiding the geometrical singularity in conventional fracture analysis measures. Domain integral method by introducing material gradient terms is employed in evaluating the static and dynamic stress intensity factors (SIFs). Meanwhile, peridynamic differential operator (PDDO) is also applied for the calculation of physical components derivatives. Different FGM modeling schemes are also examined in the OSPD framework. Cracked FGM structures with mode-I and mixed-mode crack scenarios are under investigation and results are validated by reference solutions. Accuracy and reliability of the proposed method will be examined and discussed.

Investigation of the fracture problem of functionally graded materials with an inclined crack under strong transient thermal loading

In this paper, a special method for solving transient thermal stress intensity factors (TSIFs) in functionally graded materials (FGMs) with inclined cracks under strong, transient thermal shock loading is established based on the hyperbolic heat conduction equation. First, the Newmark method is employed to obtain the strong transient temperature field of FGMs containing inclined cracks. Then, the transient interaction energy integral method (IEIM) is used to solve the TSIFs for inclined cracks. In addition, the influences of the distribution patterns of the thermal relaxation parameter, the crack length and the crack angle on the TSIFs, and the crack growth angle are investigated. Moreover, we discuss the difference between using the hyperbolic heat conduction theory and the classical, Fourier heat conduction theory in calculating the TSIFs of FGMs with inclined cracks. The results show that the present, combined IEIM-Newmark method can be used to solve strong transient thermal shock crack problems with high efficiency. This work can be used for the thermal design, evaluation, and engineering applications of FGM structures.

Analytical Solution for Forced Vibration Characteristics of Rotating Functionally Graded Blades under Rub-Impact and Base Excitation.

This paper presents an analytical investigation on the forced vibration characteristics of a rotating functionally graded material (FGM) blade subjected to rub-impact and base excitation. Based on the Kirchhoff plate theory, the rotating blade is modelled theoretically. The material properties of the FGM blade are considered to vary continuously and smoothly along the thickness direction according to a volume fraction power-law distribution. By employing Hamilton’s principle, the equations of motion are derived. Then, the Galerkin method and the small parameter perturbation method are utilized to obtain the analytical solution for the composite blade under a combined action of radial force, tangential force and displacement load. Finally, special attention is given to the effects of power-law index, rub-impact location, friction coefficient, base excitation amplitude and blade aspect ratio on the vibration characteristics of the FGM structure. The obtained results can play a role in the design of rotating FGM blades to achieve significantly improved structural performance.

Three-dimensional thermal analysis of multidirectional (perfect/porous) functionally graded plate under in-plane heat flux

The present work is based on transient three-dimensional heat transfer analysis of multidirectional functionally graded material (FGM). Here, three dimensional (3D) FGM structure graded in multiple directions is considered. The extended Voigt’s micromechanical scheme along with multivariable power law is used to evaluate effective thermal properties. The imperfection within the material is incorporated by considering symmetric even porosity. Here, in-plane heat flux is applied at the top and right edges of the plate. The transient heat transfer response of a multidirectional functionally graded plate is computed using commercially available ANSYS APDL software. The elemental space is discretised by coupled field hexahedral element SOLID 226. The mesh refinement and verification tests are conducted to demonstrate the accuracy and robustness of the model. Finally, an inclusive report is presented to exemplify the effect of heterogeneity and imperfection of multidirectional functionally graded (metal/ceramic) panels through numerous numerical illustrations.

Numerical and experimental study of molten pool behaviour and defect formation in multi-material and functionally graded materials laser powder bed fusion

Laser powder bed fusion (LPBF) of multi-material and functionally graded materials (FGM) has attracted significant research interest due to its ability to fabricate components with superior performance compared with those manufactured with single powder material. However, the forming mechanisms of various defects remain unknown. In this paper, a DEM-CFD model was first established to obtain an in-depth understanding of this process. It was discovered that the defects including partially melted and un-melted Invar36 powder were embedded in the lower level of the powder layer; this was attributed to the low laser absorptivity, low melting point and high thermal conductivity of the Cu10Sn powder. Inter-layer defects were more likely to occur with an increased powder layer thickness. In addition, the scanned track width was found related to an equilibrium achieved among the thermal properties of the powder mixture. Process parameters were optimised to obtain FGM structures without defects in both horizontal and vertical directions. Invar36/Cu10Sn samples were fabricated with a multi-material LPBF system using different mixed powder contents and laser volumetric energy densities (VEDs). By increasing the VED, fewer defects were observed between the interface of two processed powder layers, which had a good agreement with the modelling results.

Effects of pores on nonlinear vibration and post buckling behavior of functionally graded material pipes conveying fluid

Functionally graded material (FGM) has an important application prospect in aircraft engineering, especially in smart aircraft. The dynamic behavior of FGM has been widely investigated so far but more work is needed for the porous FGM pipes conveying fluid. In this paper, a sensible pore distribution function related with the volume fraction of metal and ceramic is proposed for the dynamic modeling of porous FGM pipes conveying fluid. The maximum porosity and its corresponding position are taken into account in the present mechanical model. The material properties of the porous pipes are temperature dependent and can be affected by pore distribution. The governing equation of the porous FGM pipe is derived and then the exact solution of post buckling is obtained. The nonlinear primary resonance is determined by the multiple scale method. It is shown that the effect of the pore distribution is very significant on the post buckling behavior and nonlinear primary resonance of the porous FGM pipes. The current work is very helpful in understanding the influence of pore distribution on static and dynamic behavior of pores FGM structures in engineering practice.

Thermal free vibration analysis of functionally graded plates and panels with an improved finite shell element

This article deals with thermal free vibration analysis of functionally graded material (FGM) plates and panels using an improved first-order shear deformable (I-FSDT) shell model. The Effective material properties of the FGMs are estimated according to a power law distribution with temperature dependency. The FGM structure can be subjected to uniform or nonlinear temperature rise. The results obtained from the proposed formulation are validated with those available in the literature. Natural frequencies of FGM plates and cylindrical panels are computed for different power law indexes, thermal loadings, boundary conditions and aspect ratios. The present (I-FSDT) finite shell element model is much like the standard (FSDT) model but does not need shear correction factors and is of high accuracy. This work is the first application of the formulation to free vibration analysis of FG plate and shell structures including thermal effects.

Superior energy absorption of continuously graded microlattices by electron beam additive manufacturing

ABSTRACT Electron beam melted (EBM) Ti-6Al-4V functionally graded materials (FGM) with continuously graded densities are investigated for dimensional accuracy, compressive properties, fractography and build direction effect in comparison to uniform density counterparts of the same volume. It is found that FGMs exhibit progressive layer-by-layer deformation mode regardless of unit cell designs and build direction. This deformation behavior is highly favourable for uni-directional impact absorption applications compared to uniform density counterparts with random or diagonal failure. Overall, the EBM-built FGM exhibits superior energy absorption than counterparts of uniform density. Significant improvement in the quasi-elastic gradient and energy absorption is obtained by changing the build direction for specific designs. Compared with other FGM or uniform density lattice structures from the literature, the energy absorption of lattice structures with lower relative density could outperform those with higher relative density by changing the unit cell design or density profile.

Thermal-induced snap-through buckling of simply-supported functionally graded beams

The instability of functionally graded material (FGM) structures is one of major threats to their service safety in widely engineering applications. This paper aims to clarify a long-standing controversy on the type of thermal instability of simply-supported FGM beams. Firstbased on the Euler-Bernoulli beam theory and von Karman geometric nonlinearitya nonlinear governing equation of simply-supported FGM beams under uniform thermal loads by Zhang’s two-variable method is formulated. Secondan approximate analytic solution to the nonlinear integro-differential boundary value problem with a thermal-induced inhomogeneous force boundary condition is obtained by using a semi-inverse method when the coordinate axis is relocated to the bending axis (physical neutral plane), and then the analytical predictions are verified by the differential quadrature method (DQM). Finallybased on the free energy theoremit is revealed that the symmetry breaking caused by the material inhomogeneity can make the simply-supported FGM beam under uniform thermal loads occur snap-through postbuckling only in odd modes; furthermorethe nonlinear critical load of thermal buckling varies non-monotonically with the functional gradient index due to the stretching-bending coupling effect. These results are expected to provide new ideas and references for the design and regulation of FGM structures.

Mechanical characterization of functionally graded materials produced by the fused filament fabrication process

Abstract In this research study, the design and fabrication workflow of functionally graded materials (FGMs) were introduced using fused filament fabrication (FFF) process. Rigid polymers were mechanically characterized in three printing directions. Tensile strength and modulus of elasticity values for YZ and XY printing orientation specimens indicated similar results. Printing in the XZ direction showed poor strength values in comparison with other printing directions. Interface analysis with different transition patterns was accomplished and the advantage of gradient transition was achieved. Micrography of specimens was analyzed to investigate the internal structure and was used to validate the test results. Statistical methods were applied to investigate the relationship between microstructural descriptors (volume fractions) and process-related parameters (printing temperatures). The results of the analysis of variance (ANOVA) have confirmed that printing temperature and concentration have a significant impact on tensile test results. The data-driven approach was employed to construct a linear regression models in order to formulate input data for finite element analysis (FEA). As a result of FEA simulation, effective Young’s modulus was calculated using a MATLAB code. FEA-predicted Young’s moduli were validated through experimental test results. The comparison of the numerical method and experimental values showed less than 5% error for FGM specimens printed in 250, 260 and 270 °C temperatures. FFF process with FGM could have potential applications in medical, structural, and automotive industries by locally varying material properties. This study presents a unique method of fabricating FGM structures with low-cost manufacturing process.

The Homogenized Transformation Method for the Calculation of Stress Intensity Factor in Cracked FGM Structure

A convenient calculation method is proposed for the stress intensity factor (SIF) in cracked functionally graded material (FGM) structures. In this method, the complex computational problem for SIFs in cracked FGM plate and cylinder can be simplified as the calculation problem of empirical formulas of SIFs in cracked homogenous plate and cylinder with same loading conditions and the calculation problem of related transition parameters. The results show that the SIF in cracked FGM structure can be obtained accurately without using matrix and integral. The validity and usefulness of the present method are proved by comparing with the results of the conventional method.

Geometry Independent Direct and Converse Flexoelectric Effects in Functionally Graded Dielectrics: An Isogeometric Analysis

In the present study, a functionally graded material (FGM) based dielectric composite is investigated for direct and converse flexoelectric effects induced due to material inhomogeneity. An Isogeometric analysis based formulation is employed to effectively capture the gradient terms appearing in the electro-elastic coupling constitutive laws of flexoelectricity. Non-uniform stresses and strains generated due to non-uniform elastic properties in the domain induce flexoelectricity inherently without any special geometric and/or boundary conditions. A two-dimensional (2D) FGM structure made by the combination of polyvinylidene fluoride (PVDF) and barium titanate (BaTiO3) is studied under mechanical and thermal loading conditions for its flexoelectric response. The induced polarization and voltage along with its dependence on grading index (n) is analyzed. Further, to extend the study to converse flexoelectricity, the mechanical response of the FG flexoelectric material under an applied electric potential is studied. Finally, the actuation of the structure by an applied electric field is studied. As high as 10% actuation of original displacement is observed for higher values of n. Effectiveness of the grading concept to induce flexoelectric response is established through numerical studies. Longitudinal and transverse modes of flexoelectricity are scrutinized and the concept of grading is proved to inherently induce the flexoelectricity in dielectrics without special geometric modifications.

Characterization of interfacial transition zone of functionally graded materials with graded composition from a single material in electron beam powder bed fusion

Abstract Electron beam selective melting (EBSM), is powder bed fusion additive manufacturing (AM) technique by adding materials layer upon layer. The capacity of EBSM to fabricate complex and multi-functional parts has made it a realm of great interest for researchers in the domain of science. However, during the fabrication of functionally graded materials (FGMs) with graded composition by EBSM using at least two different materials, it is difficult to separate and reuse these materials after the fabrication. Additionally, it is difficult to vary the powder composition within a layer to fabricate a horizontal graded structure. In this research, an element evaporation concept based on electron beam powder bed fusion, is presented for fabricating one-dimensional (1D) FGMs of (α2 + γ) TiAl alloy/(α + β) titanium alloy from a single material powder. By irradiating different sub-regions of the powder bed with electron beam energy input of different magnitude to greatly reduce the relative atomic ratio of aluminum from 47% to less than 10%, the 1D FGMs structures with (α2 + α + β) phases in the interfacial transition zone (ITZ) were successfully produced. Since the microstructure of the ITZ has a significant effect on mechanical properties for the 1D FGMs, three different overlap scan distances (0.5, 1.0 and 2.0 mm), which result in different ITZs, were utilized for evaluating the microstructure of the ITZ and resulting mechanical properties. The results showed that no defects were found in the specimens of any overlap distance. Moreover, the width of the ITZ increases as the overlap scan distance, and the tensile strength decreases with the overlap scan distance.

Rationally designed functionally graded porous Ti6Al4V scaffolds with high strength and toughness built via selective laser melting for load-bearing orthopedic applications

Functionally graded materials (FGMs) with porosity variation strategy mimicking natural bone are potential high-performance biomaterials for orthopedic implants. The architecture of FGM scaffold is critical to gain the favorable combination of mechanical and biological properties for osseointegration. In this study, four types of FGM scaffolds with different structures were prepared by selective laser melting (SLM) with Ti6Al4V as building material. All the scaffolds were hollow cylinders with different three-dimensional architectures and had gradient porosity resembling the graded-porous structure of human bone. Two unit cells (diamond and honeycomb-like unit cells) were used to construct the cellular structures. Solid support structures were embedded into the cellular structures to improve their mechanical performances. The physical characteristics, mechanical properties, and deformation behaviors of the scaffolds were compared systematically. All the as-built samples with porosities of ~52–67% exhibited a radial decreasing porosity from the inner layer to the outer layer, and their pore sizes ranged from ~420 to ~630 μm. The compression tests showed the Young's moduli of all the as-fabricated samples (~3.79–~10.99 GPa) were similar to that of cortical bone. The FGM structures built by honeycomb-like unit cells with supporting structure in outer layer exhibited highest yield strength, toughness and stable mechanical properties which is more appropriate to build orthopedic scaffolds for load-bearing application.

An inclined FGM beam under a moving mass considering Coriolis and centrifugal accelerations

In this paper, the dynamic behaviour of an inclined functionally graded material (FGM) beam with different boundary conditions under a moving mass is investigated based on the first-order shear deformation theory (FSDT). The material properties vary continuously along the beam thickness based on the power-law distribution. The system of motion equations is derived by using Hamilton\'s principle. The finite element method (FEM) is adopted to develop a general solution procedure. The moving mass is considered on the top surface of the beam instead of supposing it on the mid-plane. In order to consider the Coriolis, centrifugal accelerations and the friction force, the contact force method is used. Moreover, the effects of boundary conditions, the moving mass velocity and various material distributions are studied. For verification of the present results, a comparative fundamental frequency analysis of an FGM beam is conducted and the dynamic transverse displacements of the homogeneous and FGM beams traversed by a moving mass are compared with those in the existing literature. There is a good accord in all compared cases. In this study for the first time in dynamic analysis of the inclined FGM beams, the Coriolis and centrifugal accelerations of the moving mass are taken into account, and it is observed that these accelerations can be ignored for the low-speeds of the moving mass. The new provided results for dynamics of the inclined FGM beams traversed by a moving mass can be significant for the scientific and engineering community in the area of FGM structures.

Study on Strength and Stiffness of WC-Co-NiCr Graded Samples.

This work deals with the investigation of the strength and the stiffness of samples built of step-variable functionally graded material (FGM). The considered FGM samples consist of two main components: WC and NiCr with some content of Co as a conjunction of structure. The samples were fabricated on the basis of the detonation gun layer deposition method which is regarded as a novelty in the case of FGM production. The analyzed samples possess a finite number of layers with different varying fractions of each constituent across the wall. The basic tests of bending were conducted to assess the influence of used components in adequate proportions on the stiffness and the total strength in bending. In addition, to validate the numerical approach, simulations of samples under similar loads with truly reflected material distributions were carried out. The material properties of components were determined due to micro-nano-hardness by instrumental indentation techniques. The numerical calculations were performed with the use of the material characteristics for each material and with a consideration of large deflections. Furthermore, by means of an electron microscope, the composition of materials and distribution of chemical elements across the thickness of samples were examined. This paper reveals the experimental results of FGM samples manufactured by detonation gun layer deposition which allows the creation of layer by layer moderately thin-walled structures. It was shown that the indentation method of a determination of Young’s modulus gave higher values in comparison to values attained in the bending test. Moreover, it was stated that a modelling of FGM still requires the study of each layer separately to clearly predict the strength of the whole FGM structure.

Thermal cracking simulation of functionally graded materials using the combined finite–discrete element method

The functionally graded materials (FGMs), characterized by spatially varying material properties, have been used in a wide range of engineering applications (i.e., aerospace, nuclear reactor and microelectronics) in high temperature and high-temperature gradient environments. The prediction of crack behavior under severe temperature conditions is essential for the safety and long-term service life of such critical components. In this paper, a novel thermal–mechanical coupling model for FGMs is proposed, which consists of a thermal part for the temperature field computation and the combined finite–discrete element method part for the crack evolution modeling. The spatially dependent material characteristics of the FGMs are captured in this model, together with typical property variation functions (quadratic, exponential and trigonometric). The accuracy and robustness of the proposed coupled TM model are validated by numerical tests. Then, this model is applied to investigate the thermal cracking process in FGMs under different kinds of thermal loads. The influence of the crack interaction on crack growth pattern is also discussed. The results show that the proposed method is useful to the fracture mechanics analysis and design of the FGMs structures.

Design and additive manufacture of functionally graded structures based on digital materials

Voxel-based multimaterial jetting additive manufacturing allows fabrication of digital materials (DMs) at the meso-scale (∼1 mm) by controlling the deposition patterns of soft elastomeric and rigid glassy polymers at the voxel-scale (∼90 μm). The digital materials can then be used to create heterogeneous functionally graded material (FGM) structures at the macro-scale (∼10 mm) programmed to behave in a predefined manner. This offers huge potential for design and fabrication of novel and complex bespoke mechanical structures.This paper presents a complete design and manufacturing workflow that simultaneously integrates material design, structural design, and product fabrication of FGM structures based on digital materials. This is enabled by a regression analysis of the experimental data on mechanical performance of the DMs i.e., Young’s modulus, tensile strength and elongation at break. This allows us to express the material behavior simply as a function of the microstructural descriptors (in this case, just volume fraction) without having to understand the underlying microstructural mechanics while simultaneously connecting it to the process parameters.Our proposed design and manufacturing approach is then demonstrated and validated in two series of design exercises to devise complex FGM structures. First, we design, computationally predict and experimentally validate the behavior of prescribed designs of FGM tensile structures with different material gradients. Second, we present a design automation approach for optimal FGM structures. The comparison between the simulations and the experiments with the FGM structures shows that the presented design and fabrication workflow based on our modeling approach for DMs at meso-scale can be effectively used to design and predict the performance of FGMs at macro-scale.

Radial Basis Function-Based Stochastic Natural Frequencies Analysis of Functionally Graded Plates

This paper deals with portraying the stochastic natural frequencies of cantilever plates made up of functionally graded materials (FGMs) by employing the radial basis function (RBF)-based finite element (FE) approach. The material modeling of FGM plates is carried out by employing three different distribution laws, namely power law, sigmoid law, and exponential law. A generalized algorithm is developed for uncertainty quantification of natural frequencies of the FGM structures due to stochastic variation in the material properties and temperature. The deterministic FE code is validated with the previous literature, whereas convergence study is carried out in between stochastic results obtained from full scale direct Monte Carlo Simulation (MCS) and MCS results obtained from RBF surrogate model of different sample sizes. The percentage of error present in the RBF model is also determined. The influence of crucial parameters such as distribution law, degree of stochasticity, power law index and temperature are determined for natural frequencies analysis of FGMs plates. The results illustrate the input parameters considered in the present study have significant effects on the first three stochastic natural frequencies of cantilever FGM plates.

3D shell model for the thermo-mechanical analysis of FGM structures via imposed and calculated temperature profiles

The exact three-dimensional (3D) shell model proposed in the present paper is able to perform the thermal stress analysis of simply-supported Functionally Graded Material (FGM) spherical and cylindrical shells, cylinders and plates. The model is based on the 3D equilibrium equations for spherical shells developed using an orthogonal mixed curvilinear coordinate system. The use of this reference system allows the investigation of cylindrical shells, cylinders and plates as particular cases of spherical shells by means of simple considerations on the radii of curvature. The 3D shell model uses a layer-wise approach and the exponential matrix method to calculate the general and the particular solutions through the thickness direction z. The system of second order differential equations in z is not homogeneous because of the thermal terms which are externally defined. The system is reduced to a group of first order differential equations in z simply redoubling the number of variables. The solution is in closed form in the in-plane directions α and β because of the hypotheses of simply-supported boundary conditions, harmonic forms for displacement and temperature fields, and isotropic behavior in the in-plane directions for functionally graded materials. In order to define the equivalent thermal load, the temperature profile through the thickness is separately defined by means of three possible ways. Using the hypothesis of temperature amplitudes imposed at the top and bottom external surfaces in steady-state conditions, the temperature profile can be: imposed as linear through the entire thickness direction, calculated by solving the 1D version of the Fourier heat conduction equation, or calculated by solving the 3D version of the Fourier heat conduction equation. The effects of different temperature profiles on the displacement and stress analyses of FGM plates and shells are here remarked. The first order differential equation system in z has not constant coefficients because of the presence of radii of curvature for shells and through-the-thickness variable elastic and thermal coefficients for the FGM layers. An appropriate number of mathematical layers is introduced to calculate the curvature influence for shells and the elastic and thermal material coefficients for FGM layers. Therefore, the system can be considered as differential equations with constant coefficients. The proposed results allow the evaluation of thickness ratio, geometry, lamination scheme, thickness material law and temperature profile effects in the related thermal stress analysis of single-layered and sandwich FGM plates, cylinders, spherical and cylindrical shells.