Compositional Gradient Exponent Research Articles

Ballistic performance of functionally graded B4C/Al composites without abrupt interfaces: experiments and simulations

Functionally graded materials (FGMs) are promising protective structures for lightweight armor plates because of their features and designability. However, traditional discretely layered FGMs with macroscopic interfaces can generate unfavorable interlaminar stresses under ballistic attacks, eventually leading to delamination or interlayer tearing. Herein, we constructed new B4C/2024Al FGMs without sharp interfaces between layers. The FGM armor systems with varying through-the-thickness distributions were tested against 7.62 mm armor-piercing incendiary projectiles. The effects of compositional gradient exponent on the damage characteristics, stress distribution and energy evolution were numerically and experimentally investigated. The results indicated that the smooth transition within through-the-thickness microstructures effectively avoided the occurrence of interlayer delamination or tearing. The compositional gradient exponent played an essential role in the impact response of the FGM targets. The FGM having a linear composition gradient outperformed other nonlinear types with respect to energy absorption and ballistic properties due to the reduced tensile wave strength and prolonged projectile-target interaction time. Furthermore, the structural optimization problem with the objective of maximizing ballistic efficiency was solved using the conjugate gradient method, and the optimal design of the FGM targets was achieved. The current research provides a promising design guideline for developing and optimizing new advanced lightweight composite armor.

Evaluation of the Performance of ANN Algorithms with the Bidirectional Functionally Graded Circular Plate Problem

Functionally graded materials (FGMs) are materials composed of metals and ceramics in which the distribution of material components varies according to a particular volumetric function. FGMs are often used in high-temperature applications. In our study, models were created in the Artificial Neural Network depending on the equivalent stress levels in the compositional gradient exponent, which is the most important parameter in determining the thermo-mechanical behavior of circular plates functionally staggered in two directions, and the performances of these models were evaluated. These models were obtained with four different training algorithms: Levenberg-Marquardt, Backpropagation Algorithm, Resilient Propagation Algorithm, Conjugate Gradient Backpropagation with Powell-Beale Restarts To train the ANN, equivalent stress levels were obtained by performing numerical analyzes at different compositional gradient upper values. The data sets were created by considering the largest value of the equivalent stress levels, the smallest value of the largest value, the largest value of the smallest value, and the smallest value of the smallest value. In this study, training stages and performance values were examined and interpreted with 4 training algorithms in detail.

Determination of three-dimensional thermo-mechanical behavior of bidirectional functionally graded rectangular plates with the theory of elasticity

In this study, the thermo-mechanical behavior of rectangular plates functionally graded (FG) in two directions along the plane is investigated with three-dimensional (3D) elasticity equations. In numerical analysis, the finite difference method (FDM) is used and the directional variation of the derivatives of the material properties is taken into account. The thermo-mechanical behavior of the plates is investigated for different compositional gradient exponents and it is emphasized that the variation of the compositional gradient exponent in both directions significantly changed the thermal stress distribution and the results are confirmed.

Stress Analysis of 2D-FG Rectangular Plates with Multi-Gene Genetic Programming

Functionally Graded Materials (FGMs) are designed for use in high-temperature applications. Since the mass production of FGM has not yet been made, the determination of its thermo-mechanical limits depends on the compositional gradient exponent value. In this study, an efficient working model is created for the thermal stress problem of the 2D-FG plate using Multi-gene Genetic Programming (MGGP). In our MGGP model in this study, data sets obtained from the numerical analysis results of the thermal stress problem are used, and formulas that give equivalent stress levels as output data, with the input data being the compositional gradient exponent, are obtained. For the current problem, efficient models that reduce CPU processing time are obtained by using the MGGP method.

Numerical investigation of the dynamic behavior of a Ti/TiB functionally graded material using Split Hopkinson Pressure Bar test

This work focuses on the mechanical behavior, at high strain rates, of metal-ceramic functionally graded materials (FGMs) using a Split Hopkinson Pressure Bar (SHPB) technique. The validity and the accuracy of the SHPB test results for these composite materials have not been meticulously studied due to their complex mechanical response under dynamic loadings. In this paper, numerical simulations of SHPB tests were performed to determine the dynamic response of specific FGMs at various strain rates. The effects of the compositional gradient exponent and the striker velocity, as well as increasing/decreasing stiffness through the thickness of the functionally graded specimen, were investigated. The considered material is composed of Titanium mono-boride ceramic (TiB) and Titanium metal (Ti) phases; the volume fraction (Vc) of the ceramic constituent varies through-thickness following a power-law distribution. The locally effective material properties were evaluated using a homogenization method based on the self-consistent method (SCM). The elastoplastic behavior of the FGMs under dynamic loading is described using dynamic Tamura-Tomota-Ozawa model (DTTO). The numerical simulations indicated that the compositional gradient exponent and striker velocity have significant effects on the dynamic response of the FGMs. On the other hand, the increasing/decreasing stiffness has a minor effect due to the low thickness of the experimental functionally graded specimen. Finally, the effect of the high strain rates the multiple interface reflection/transmission of the stress wave is very considerable for the design and the optimization of FGMs behavior under dynamic loading.

A peridynamic approach for modeling of two dimensional functionally graded plates

This study investigates the crack initiation and its progression in two-dimensional functionally graded (FG) plates under dynamic and quasi-static loading conditions by using an ordinary state-based PeriDynamic (OSB PD) theory. Functionally gradient materials (FGMs) are a new class of advanced materials that provide a smooth transition among the layers of materials to meet the desired requirements in engineering structures. The effective material properties in the FG plate were functionally tailored in two directions by employing a rule of mixture. The present OSB PD theory is very suitable for the failure analysis of materials without the use of surface and volume correction factors. The robustness of the present approach on monitoring the mixed-mode fractures is established with the experimental results under various loading conditions. The PD predictions successfully captured the critical load levels and damage evolutions. Subsequently, numerical analyses were carried out to assess the influence of the one- and two-dimensional material variations, boundary, and loading conditions on the fracture behavior of the FG plates. The compositional gradient exponent played a major role on the reaction force levels and crack trajectories. It was observed that two-dimensional material variations paved the way for increasing the plate strength and fracture resistance.

Evaluation of bending and post-buckling behavior of thin-walled FG beams in geometrical nonlinear regime with CUF

In this paper, the nonlinear behavior of beams with one-dimensional (1D) Functionally Graded Materials (FGM) is investigated by Carrera Unified Formulation (CUF). In the study, kinematic variants of CUF are extended with Lagrange Extension (LE), thus adopting Layer-Wise (LW) approach. In the CUF formulation combined with geometric nonlinear equations, the Lagrangian approximation and Newton–Raphson linearization scheme are used with the method based on arc length constraint. It is assumed that the variation of material properties in the thickness direction follows an exponential grading. The displacement and stress distributions of the Functionally Graded (FG) cantilever beam under transverse/axial loading are investigated for different compositional gradient exponents. The three-dimensional (3D) stress distributions of FG beams are investigated with the 1D CUF model and the results are confirmed by the literature. The effect of the compositional gradient exponent on the mechanical behavior is expressed and it is emphasized that the combined formulation is time-efficient and highly sensitive in determining the nonlinear behavior.

Improved Mathematical Models of Thermal Residual Stresses in Functionally Graded Adhesively Bonded Joints: A Critical Review

Functionally graded material (FGM) concept has been applied successfully in order to improve/design heat transfer, electric and electronic conductivity, static and dynamic strengths of adhesive joints by reliving stress distributions in both adhesive and adherend materials. This new approach relies on tailoring material composition of adhesive and adherends along one or more coordinate directions. Thermal residual stresses in adhesive joints are a vital issue in terms of the joint strength. FGM concept also allows to relieve/control thermal residual stresses encountered in adhesive joints due to mismatches between coefficients of thermal expansion of adhesive and adherend materials. Mathematical models and solutions on the thermal residual stress analysis have been continuously improved. This paper reviews the current status of mathematical models, and offers an improved mathematical model and numerical solution method by considering two-dimensional thermal stress and deformation states of adhesively bonded bi-directional functionally graded clamped plates subjected to an in-plane heat flux along one of the ceramic edges. This mathematical model assumes the material properties of the functionally graded plates to vary with a power law along two in-plane directions and not through the plate thickness direction, in particular, considers the spatial derivatives of thermal and mechanical properties of the material, and enables the investigation of the effects of the bi-directional composition variations and spatial derivative terms on the displacement, strain and stress distributions. The heat conduction and Navier equations describing the twodimensional thermo-elastic problem are discretized using finite-difference method, and the set of linear equations are solved using the pseudo singular value method. The functionally graded plates relieve both stress and strain distributions and levels in the adhesive layer and in the plates even though the adhesive layer is still ungraded. The spatial derivatives of mechanical and thermal properties of the local material become more effective on the strain and stress distributions of the plates and adhesive layer. The model, disregarding these derivative terms, exhibits sensitivity to small changes in the compositional gradients (n, m) by adjusting the variations of ceramic volume fraction along the x - and y-directions, respectively, and instability in the calculation of stress and strain distributions and levels. However, the improved model with material derivatives, which considers the effects of these derivative terms, predicts stress and strain distributions and levels complying with changes in the compositional gradient exponents.

A strain rate-dependent enhanced continuum model for elastic-plastic impact response of metal-ceramic functionally graded composites

An enhanced continuum model considering the strengthening of strain rate is developed to explore the particle size-dependent responses of the metal-ceramic functionally graded composite plates subjected to a drop-weight impact. The continuum model explicitly accounts for the effects of geometrically necessary dislocations and strain rate by combining Taylor dislocation model and Johnson–Cook model. The plastic constitutive relation of metal-ceramic FGM is derived from the Tamura–Tomota–Ozowa model subjected to different strain rates. The elastic-plastic impact responses of SiC/Al functionally graded plates with the micron-scale particle sizes and compositional gradient exponents are studied under varied impact velocities. The comparison at the numerical results for the particle size effects in the SiC/Al functionally graded plates and the experimental data reported in references demonstrates validity of the proposed continuum model.

Stress wave propagation in adhesively bonded functionally graded cylinders: an improved model

This study presents an improved mathematical model to analyse the stress wave propagation in adhesively bonded functionally graded (FG) circular cylinders (butt joint) under an axial impulsive load. The volume fractions of the material constituents in the upper and lower cylinders were functionally tailored through the thickness of each cylinder using a power-law. The effective material properties of both cylinders, which are made of aluminum (Al) and silicon carbide (SiC), at any point were predicted by using the Mori–Tanaka homogenization scheme. In this improved model, the governing equations of the wave propagation include the spatial derivatives of local mechanical properties and were discretized by means of the finite difference method. The influence of these spatial derivatives and the compositional gradient exponent on the displacement and stress distributions of the joint was investigated. The material composition variations of both cylinders affected the displacement and stress fields whereas the compositional gradient exponent had a minor effect. The stress concentrations were alleviated in time, the displacement and stress distributions/variations around/along the upper and lower cylinder-adhesive interfaces were significantly affected by the adhesive layer. The spatial derivatives also affected the temporal histories of the displacement and stress components evaluated at the selected critical points of the upper cylinder, adhesive layer and lower cylinder. The consideration of the spatial local material derivatives provided a more accurate mathematical model of wave propagations through the graded layered structures.

Investigation of the Thermo-elastic Response of Adhesively Bonded Two-Dimensional Functionally Graded Circular Plates Based on Theory of Elasticity

Compositional gradient exponents and grading direction in functionally graded materials designed for high-temperature applications play an important role in thermo-mechanical behavior. Herein, the effect of the material composition of both radial and angular directions on the thermo-mechanical behavior of the functionally graded circular plates adhesively bonded (FGCP-AB) and adhesive is investigated. The material composition varies along a plane direction, not in the plate thickness direction. Heat transfer and elasticity equations in polar coordinates are solved for plane stress and plane strain using the finite difference method. Herein, the coordinate-dependent change in material properties is taken into account in the partial differential equations. In the results obtained, it has been emphasized that the variation of the material composition is an important design criterion which must be taken into consideration both in the adhesive–adhesive interface and in the arrangement of the thermal residual stress regions occurring in the FGCP-AB. This study is an important source in determining the thermal stress behavior according to the material composition and grading direction of functionally graded circular plates adhesively bonded that have not yet been mass-produced.

Stress wave propagation in a functionally graded adhesive layer between two identical cylinders

ABSTRACTThis study investigates the stress wave propagation in circular aluminum cylinders bonded with a functionally graded adhesive layer subjected to an axial impulsive load. The adhesive joint consists of two identical (aluminum) cylinders and a functionally graded adhesive layer. The volume fractions of the two constituents: aluminum and epoxy in the adhesive layer were functionally tailored through the adhesive thickness by obeying a power-law. Therefore, the effective material properties at any point in the adhesive layer were predicted by the Mori-Tanaka homogenization scheme. The governing equations of the wave propagation in the joint were discretized by means of the finite difference method. The influence of the compositional gradient exponent on the displacement and stress distributions of the joint was examined. It was observed that changing the material composition of the adhesive layer had an evident effect on the displacement and stress levels, especially in the lower cylinder. On the contrary, the influence of the compositional gradient exponent was found to be minor on the displacement and stress distributions. The displacement and stress distributions were also investigated along the upper and lower cylinder-adhesive interfaces. Accordingly, with increasing the ductility of the adhesive layer the waves transmitted to the lower cylinder caused lower displacement levels. The normal stresses become peak at the bottom corners of the upper and lower cylinder-adhesive interfaces whereas the shear stresses concentrate in the middle region of the interfaces. In addition, the temporal variations of the displacement and stress components were evaluated at some critical points of the adhesive and lower cylinder. The compositional gradient exponent played an important role on the displacement and stress levels as well as the wave speeds in the adhesive and lower cylinder rather than in the upper cylinder. The stresses in the joints were observed to be alleviated by employing a functionally graded adhesive layer.

Numerical Solution of Thermal Residual Stress Analysis with Finite Difference Method of Functionally Graded Circular Plates

In this study, thermal residual stress analysis of functionally graded circular plates (FGCP) carried out. Finite difference equations are used in solving Navier's equations of elasticity and Fourier's heat conduction equation. The grading along the plate was made along the surface of plate and it was assumed that the material properties changed according to the Mori-Tanaka approach. Grading along the plate was made in both radial and tangential directions. In this study, the effect of the coordinate derivatives of material properties was taken into consideration in both Fourier's heat conduction equation and Navier's equations of elasticity, unlike the other studies. As a result, when the materials compositions of FGCP were changed from ceramic-rich to metal-rich compositions, the stress levels were not affected considerably. The strain levels increased significantly when the metal compound in the material composition of FGCP was increased. FGCP are emphasized that the change of material properties due to two-dimensional significantly affect distributions of thermal strain and stress. In this study, it was emphasized that changing the radial and tangential direction of the compositional gradient exponents of FGCP subjected to heat flux along the outer edge significantly influences the strain and stress distributions.

Thermal stress analysis of functionally graded plates with temperature-dependent material properties using theory of elasticity

Abstract Based on the three-dimensional elasticity theory, the transient thermal residual stress analyses of one-dimensional functionally graded rectangular plates (FGPs) have been performed under in-plane constant heat flux for different compositional gradient exponents. The thermo-mechanical properties of FGPs were assumed to be vary with a power law along an in-plane direction, not through the plate thickness direction and temperature-dependent/independent. The Heat Transfer and Navier's Equations in cartesian coordinates which represent the two-dimensional thermoelastic problem were resolved by means of the Finite-Difference Method (FDM), and the set of linear equations were solved using the pseudo singular value method. The effect of the coordinate derivatives of material properties were considered in both Heat Transfer and Navier's Equations. The current study aims at determining the effect of temperature dependence and temperature independence of material properties and the compositional gradient exponents in FGPs on the levels of temperature, strain and stress. The FGPs with temperature-dependent material properties showed higher levels of temperature, strain and stress than those with temperature-independent material properties. In order to verify the results of this study, two-dimensional thermo-elastic problem was resolved using the Finite Element Method (FEM) and the results are compared.

Thermal stress analysis of in-plane two-directional functionally graded plates subjected to in-plane edge heat fluxes

This study investigates the thermal stress and deformation states of bi-directional functionally graded clamped plates subjected to constant in-plane heat fluxes along two ceramic edges. The material properties of the functionally graded plates were assumed to vary with a power law along two in-plane directions not through the plate thickness direction. The spatial derivatives of thermal and mechanical properties of the material composition were considered, and the effects of the bi-directional composition variations and spatial derivative terms on the displacement, strain and stress distributions were also investigated. The heat conduction and Navier equations describing the two-dimensional thermo-elastic problem were discretized using finite-difference method, and the set of linear equations were solved using the pseudo singular value method. The compositional gradient exponents and the spatial derivatives of thermal and mechanical properties of the material composition were observed to play an important role especially on the heat transfer durations, the displacement and strain distributions, but had a minor effect on the temperature and stress distributions.

Experimental and numerical investigations of low velocity impact on functionally graded circular plates

This paper addresses low-velocity impact behaviour of functionally graded clamped circular plates. An experimental work was carried out to investigate the impact behaviour of FG circular plates which is composed of ceramic (SiC) and metal (Al) phases varying through the plate thickness by using a drop-weight impact test system. The influence of the compositional gradient exponent and impactor velocity on the contact forces and absorbed energies was concentrated on the tests. The explicit finite element method, in which a volume fraction based elastic–plastic model (the TTO model) was implemented for the functionally graded materials, was used to simulate their drop-weight impact tests. Effective material properties at any point inside FGM plates were determined using Mori–Tanaka scheme. The experimental and numerical results indicated that the compositional gradient exponent and impactor velocity more effective on the elasto-plastic response of the FG circular plates to a low-velocity impact loading. The comparison at the theoretical and experimental results showed that the use of the TTO model in modelling the elasto-plastic behaviour of FG circular plates results in increasing deviations between the numerical and experimental contact forces for ceramic-rich compositions whereas it becomes more successful for metal-rich compositions.

Thermal residual stresses in adhesively bonded in-plane functionally graded clamped circular hollow plates

This study investigates the effects of in-plane compositional gradient exponent and direction on the thermal residual stress and deformations in adhesively bonded functionally graded clamped circular plates. The material composition was assumed to vary with a power law along an in-plane direction not through the plate thickness direction. The transient heat conduction and Navier equations in polar coordinates describing the two-dimensional thermo-elastic problem were discretized using finite-difference method, and the set of linear equations were solved using the pseudo-singular-value method. The material composition direction is designed as Ceramic-Metal (CM)–CM, CM–Metal-Ceramic (MC), MC–CM, and MC–MC for the inner and outer plates. The temperature decreased radially along the plates, but exhibited a sharp decrease across the adhesive layer. The compositional gradient exponent and direction affected evidently temperature levels and heat transfer period. The compressive radial and shear strains are more effective on the deformation in the adhesive layer and the plate regions near the plate–adhesive interfaces. The adhesive layer is subjected to considerable shear deformations. The equivalent strain and stresses are very low in a large region of the plates but exhibit sharp peaks on the plate regions near the plate–adhesive interfaces, and decrease towards the adhesive interfaces. These stress and strain peaks in the plates and adhesive layer are affected by the compositional gradient and direction. For an outer edge flux, the largest equivalent strain and stresses are observed in the CM–MC joint but the lowest levels occur in the MC–CM or secondly CM–CM joint. In addition, an inner edge flux results in the lowest and highest peak strains and stresses in the MC–CM and CM–MC joints, respectively. The MC–MC and CM–CM joints result in lower temperature, stress and strain levels around the adhesive layer and along the adhesive interfaces for outer and inner edge fluxes, respectively.

Free Vibration Analysis of an Adhesively Bonded Functionally Graded Tubular Single Lap Joint

In this study, the three-dimensional free vibration analysis of an adhesively bonded functionally graded tubular single lap joint was carried out using the finite element method. The functionally graded tubes of the adhesive joint are composed of ceramic (Al2O3) and metal (Ni) phases varying through the tube thickness. The adhesive material properties, such as modulus of elasticity, Poisson's ratio, and density were found to have negligible effect on the first ten natural frequencies and mode shapes of the adhesive joint. The optimal design parameters of the adhesive joint, such as overlap length, inner radius of the inner tube, outer and inner tube thicknesses, and the through-the-thickness material composition variation were searched using both the artificial neural networks (ANNs) and the genetic algorithms (GAs). For this purpose, the natural frequencies and modal strain energy values were calculated for an adhesive joint with random geometrical properties and material compositions through the tube thicknesses, and were used for training the proposed artificial neural network models. The outer tube thickness, the inner tube-inner radius, and the compositional gradient exponent had considerable effect on the natural frequencies, mode shapes, and modal strain energies of the functionally graded tubular single lap joint, whereas the overlap length and the inner tube thickness had a minor effect. The GAs integrated with ANNs was employed to determine optimal design parameters satisfying both maximum natural frequency and minimum modal strain energy conditions for each natural mode of the tubular adhesive joint.

Indentation behavior of functionally graded Al–SiC metal matrix composites with random particle dispersion

This study addresses the effects of number of layers, compositional gradient exponent (material composition) and random particle dispersion on the indentation behavior of a functionally graded metal matrix composite. The ceramic particles in a certain volume fraction were randomly distributed within each layer depending on the compositional gradient exponent in order to have a similar structure to an actual microstructure as possible. Increasing the number of layers caused an evident increase in indentation depths for all compositional gradient exponents. The compositional gradient exponent had an important effect on the indentation depth. The random particle dispersion also played an important role on the central indentation depth ( n < 3) as well as on the deformed surface profiles for both high layer numbers ( N = 75 and 150 layers) and the material compositions approaching to a metal-rich structure resulted in some fluctuations in the deformed surface profiles. Increasing the number of layers at a certain compositional gradient exponent resulted in larger mean residual stresses and plastic deformations. Increasing the compositional gradient exponent at a certain layer number decreased mean residual stresses and strains since the ceramic volume fraction is increased. The random particle dispersion also resulted in non-uniform residual stress and strain levels and distributions.

The elasto-plastic impact analysis of functionally graded circular plates under low-velocities

Abstract In this study, three-dimensional elasto-plastic analysis of functionally graded (FG) circular plates under low-velocity impact loads is presented. The FG circular plate is composed of ceramic (SiC) and metal (Al) phases, varying in a predetermined fashion through the plate thickness. The elasto-plastic behaviour of the FG circular plate is described by the TTO model. In the analyses, the ceramic phase is taken to be an isotropic elastic material whereas the metal phase is taken to be elasto-plastic material in accordance with the TTO model. The locally effective material properties were evaluated using homogenization method which is based on the Mori–Tanaka scheme. The effects of compositional gradient exponent and impactor velocity on the elasto-plastic impact response of the FG circular plates are investigated, and results are presented graphical form. The compositional gradient exponent and impactor velocity have significant effects on the elasto-plastic impact response of the FG circular plates. The elasto-plastic impact response of FG circular plates is similar to those of homogeneous plates. Therefore, the TTO model can be used to describe the mechanical behaviour of FG plates beyond the elastic range, assuming that the material response is essentially governed by the spreading of plasticity in the metal phase.