Graded Material Properties Research Articles

Adaptive PF-CZM for multiphysics fracture analysis in functionally graded materials

Functionally graded materials (FGM) hold significant relevance in engineering due to their tailored material property gradation, designed for specific engineering applications. The challenge in fracture analysis of FGM stems from the spatial variation of material properties, which complicates the prediction of crack topology. Local and global refinement strategies are impractical for fracture analysis in FGM due to the unpredictable nature of crack topology, which renders local refinement infeasible. Additionally, global refinement is not advisable as it leads to a significant increase in degrees of freedom, adversely affecting computational efficiency.The novelty of this research lies in the incorporation of spatial variation in both the length scale and material properties, enhancing the realism of FGM domain modeling. To preserve the required length scale, it is necessary to adopt a minimum mesh size, which consequently results in a substantial increase in the degrees of freedom and, thereby, escalates the computational cost. To address these challenges, the study employs an adaptive mesh refinement (AMR) algorithm integrated with a phase-field cohesive zone model (PF-CZM), providing a robust solution for accurate fracture analysis in FGM. Based on the ideas stemming from the need for efficient and realistic modeling, the AMR-PF-CZM framework refines the mesh efficiently in regions of crack growth based on crack-driving energy and phase field variable, thereby eliminating the need for pre-refinement. The findings demonstrate a 76%–85% increase in computational efficiency and accuracy of the AMR-PF-CZM approach compared to the non-adaptive PF-CZM. Furthermore, the developed algorithm’s applicability to dynamic fracture and multi-physics problems, specifically addressing mechanical and thermal fracture in FGM, underscores the importance of this approach in capturing complex fracture phenomena.

Free Vibration of the Magneto-Electro-Elastic Plates Resting on Elastic Foundation Using the Refined Plate Theory with Two Variables

The objective of this article is to investigate the free vibration analyses of the functionally graded magneto-electro-elastic (FG MEE) plates supported by an elastic foundation using the refined plate theory (RPT) with two variables. The elastic foundation is modeled utilizing the Winkler-Pasternak theory. The power-law model is employed to characterize the graded material properties of the FG MEE plates. According to the RPT and Hamilton's principle, the governing equations of the FG MEE plate are derived. The displacement fields and electric and magnetic potentials are approximated using the Non-Uniform Rational B-Splines (NURBS) basic functions of the isogeometric approach (IGA). The proposed model’s advantages and accuracy are demonstrated by comparing the obtained results with those reported in the existing literature. The study comprehensively examines and discusses the impact of several parameters, including the power index, initial external magnetic potential and electric voltage, and the geometry, on the vibration frequency of the FG MEE plates. The numerical findings indicate that an increase in the power index leads to a decrease in the frequency of the FG MEE plates. Besides, the stiffness of FG MEE plates decreases with an increase in the initial external electric voltage, whereas it increases with an increase in the initial external magnetic potential. This article presents valuable perspectives on examining vibration analysis of the FG MEE plates, which can inform the design of innovative materials and structures with customized properties.

Flutter in functionally graded conical shell under follower force

Truncated conical shells are essential components of rocket booster nozzles and thrust vector control (TVC) systems for the propulsion of multi-stage launch vehicles. The TVCs assist the ascent of launch vehicles by directing the resultant thrust from the boosters, acting as a follower force that might trigger instability. Previous studies on the instability of aerospace structures have mostly focused on beams, plates, and cylindrical shells. This study analyses a truncated conical shell under a follower force of constant magnitude, considering thickness-wise gradation of material properties employed for thermal management. The governing equations are derived following Hamilton's principle, considering first-order shear deformation theory, and solved using the finite element method. Clamped and free boundaries are assumed at the small and large ends. The influence of mass and stiffness proportional damping is considered. Although strong flutter appears as the dominant instability mode, instances of erratic weak instabilities are also observed for undamped and lightly damped cases. Damping enhances stability, but its effect becomes saturated. The flutter loads are presented for varying non-dimensional parameters, characterizing the shell geometry and material properties.

Effect of bi-directional material gradation on thermo-mechanical bending response of metal-ceramic FGM sandwich plates using inverse trigonometric shear deformation theory

PurposeThe purpose of this study is to investigate the bending analysis of metal (Ti-6Al-4V)-ceramic (ZrO2) functionally graded material (FGM) sandwich plate with material property gradation along length and thickness direction under thermo-mechanical loading using inverse trigonometric shear deformation theory (ITSDT). FGM sandwich plate with a ceramic core and continuous variation of material properties has been modelled using Voigt’s micro-mechanical model following the power law distribution method. The impact of bi-directional gradation of material properties over the bending response of FGM plate under thermo-mechanical loading has been investigated in this work.Design/methodology/approachIn this study, gradation of material properties for FGM plates is considered along length and thickness directions using Voigt’s micromechanical model following the power law distribution method. This type of FGM is called bi-directional FGMs (BDFGM). Mechanical and thermal properties of BDFGM sandwich plates are considered temperature-dependent in the present study. ITSDT is a non-polynomial shear deformation theory which requires a smaller number of field variables for modelling of displacement function in comparison to poly-nominal shear deformation theories which lead to a reduction in the complexity of the problem. In the present study, ITSDT has been utilized to obtain the governing equations for thermo-mechanical bending of simply supported uni-directional FGM (UDFGM) and BDFGM sandwich plates. Analytical solution for bending analysis of rectangular UDFGM and BDFGM sandwich plates has been carried out using Hamilton’s principle.FindingsThe bending response of the BDFGM sandwich plate under thermo-mechanical loading has been analysed and discussed. The present study shows that centre deflection, normal stress and shear stress are significantly influenced by temperature-dependent material properties, bi-directional gradation exponents along length and thickness directions, geometrical parameters, sandwich plate layer thickness, etc. The present investigation also reveals that bi-directional FGM sandwich plates can be designed to obtain thermo-mechanical bending response with an appropriate selection of gradation exponents along length and thickness direction. Non-dimensional centre deflection of BDFGM sandwich plates decreases with increasing gradation exponents in length and thickness directions. However, the non-dimensional centre deflection of BDFGM sandwich plates increases with increasing temperature differences.Originality/valueFor the first time, the FGM sandwich plate with the bi-directional gradation of material properties has been considered to investigate the bending response under thermo-mechanical loading. In the literature, various polynomial shear deformation theories like first-order shear deformation theory (FSDT), third-order shear deformation theory (TSDT) and higher-order shear deformation theory (HSDT) have been utilized to obtain the governing equation for bending response under thermo-mechanical loading; however, non-polynomial shear deformation theory like ITSDT has been used for the first time to obtain the governing equation to investigate the bending response of BDFGM. The impact of bi-directional gradation and temperature-dependent material properties over centre deflection, normal stress and shear stress has been analysed and discussed.

Thermoelastic bending analysis of thick functionally graded sandwich plates with arbitrary graded material properties using a novel quasi-3D HSDT

Thermoelastic bending analysis of thick functionally graded sandwich plates with arbitrary graded material properties using a novel quasi-3D HSDT

The Casimir, Van der Waals, and electrostatic forces’ effects on the response of magneto-electro-elastic nanosensor/switch beams under thermal environment

This study investigates the impact of Casimir, Van der Waals, and electrostatic forces on nanomechanical switches’ thermomechanical and free vibration behavior. The analysis is conducted using a novel higher-order beam theory and the nonlocal strain gradient elasticity. The motion equations of the nanosensor/switch beam are derived using Hamilton’s principle and solved using Navier’s method for general boundary conditions. The nanoswitch is composed of electroelastic barium-titanate (BaTiO3) and magnetostrictive cobalt-ferrite (CoFe2O4) materials, which are modeled using a power-law approach to account for functionally graded material property variations across the beam’s thickness. The impact of different parameters, such as Casimir, Vander Waals, electrostatic forces, and variations in material composition, size parameters, and gap distance, on a nanoswitch system’s bucking and free vibration is comprehensively examined. With the intermolecular and electrostatic forces, the temperature dependency of barium-titanate and cobalt-ferrite nanoswitch materials, which have not been extensively studied in any previous research, is considered in the modeling of free vibration, and the buckling behavior of a nanoswitch for the first time. This research represents the first comprehensive analysis of these factors. Considering the investigated parameters, the study’s findings can provide helpful insights into developing micro/nano-electromechanical systems, including switches, sensors, and actuators.

An open-source moose implementation of phase-field modeling of fracture in functionally graded materials

Fracture analysis of functionally graded plates is a challenging problem in materials engineering due to the complex material behavior and the presence of cracks. In this study, we propose an efficient open-source MOOSE implementation of phase field fracture analysis of functionally graded. Automatically oriented exponential finite elements are used to discretize the model, which allows for efficient and accurate computation of the fracture behavior. The functionally graded material properties are modeled using a power law function, which captures the variation of material properties along the thickness direction of the plate. The phase field model is coupled with a finite element solver to simulate the mechanical response of the plate under loading. The proposed implementation is validated using several benchmark problems available in literature. The results show that the proposed MOOSE implementation provides accurate predictions of the crack propagation and fracture behavior of functionally graded plates.

Surface and subsurface material modifications and graded material properties in sheet steel 1.0338 (DC04) and their influence on forming processes

For bulk material it is very well known, that the material properties differ between the material near the surface and the inner material (i.e. segregations). However, for sheet material a gradient of material properties in depth direction and its influence on forming processes was not yet investigated. Within experiments on sheet steel 1.0338 (DC04) a gradient in material properties was proven by hardness measurements for both unprocessed material as well as after conducting forming and milling processes. The characterisation shows a gradient in material hardness over the entire sheet thickness. The experimental results regarding forming processes, such as rolling and bending, additionally show an increase of the materials hardness due to work hardening. The problem from the inhomogeneous material properties leads to an inaccurate prediction of the forming behaviour of the workpiece in further processing. Machining experiments were conducted for transferring the knowledge of modified surface and subsurface layers onto forming processes and their simulation. The bending experiments and the corresponding process simulation of the spring back behaviour show, that the implementation of an inhomogeneous material property increases the prediction accuracy. Notably the prediction of the spring back behaviour of thinner sheet metals is highly improved. Therefore, an analysis of sheet metal properties in thickness direction and their consideration in forming process simulations is inevitable for the implementation of accurate material models.

On the nonlinear dynamics of a multi-scale flexoelectric cylindrical shell

The flexoelectric property shows the relationship between the strain gradient and the electric polarization and has a significant effect on static and dynamic responses of structures at small scales such as micro and nano. Nowadays, experimental studies for intelligent/composite ultra-small mechanical systems are complex, challenging, and in most cases still impossible, especially for nonlinear dynamic behavior analysis. For the first time, this study presents an advanced and complex generalized model for the nonlinear vibrations of a piezo-flexoelectric mechanical system based on a cylindrical shell. It should be noted, that the formulated boundary value problem is generalized, well-posed, and can be applied to the system at micro and nano scales. Hamilton's variational principle as well as assumptions of the first-order shear deformation theory (FSDT) and reformulated flexoelectric theory were used to derive equations of motion of the multi-scale intelligent shell system. To introduce nonlinear effects, von-Kármán strains are applied. Appropriate classical and non-classical mechanical/electric boundary conditions as well as higher-order forces and moments are determined to fully close the formulated problem. The Galerkin method combined with the GDQM, supported by the displacement control strategy and the weighted residual method, have been used to discretize and solve the nonlinear governing equations of the shell with different boundary conditions. Good convergence between the current results and the results from previous studies for simpler cases of the shell was proved. Furthermore, the results showed that geometric ratios of the structure, gradation of material properties, thickness of the flexoelectric layers, and the size effect parameter significantly influence the nonlinear frequency of the structure under the direct flexoelectric effect.

A novel solution to thermo-elasto-dynamic response for inhomogeneous orthotropic hollow cylinders

This paper proposes a novel solution for the radially FGM cylinder subjected to thermo-mechanical shock. By discretizing the state space equation in the spatial domain and applying the time marching technique based on central difference, a semi-analytical solution to the thermoelastic dynamic behavior of cylinders is developed. The proposed solution technique has two advantages, one is suitable for the arbitrariness of material properties varying in the radial direction, and the other is applicable to different boundary conditions. Numerical examples for the cylinders with different graded material properties and two different boundary conditions are examined. The obtained results not only verify the correctness and effectiveness of the present solution, but also demonstrate the flexibility of the present method. The influences of gradient functions and boundary conditions on dynamic behavior are clearly displayed. The solution method is convenient and practical, which may help to freely select the gradient function for the design of FGM cylinders under various conditions.

Influence of Composite Structure on Temperature Distribution-An Analysis Using the Finite Difference Method.

Among composites, we can distinguish periodic structures, biperiodic structures, and structures with a functional gradation of material properties made of two or more materials. The selection of the composite's constituent materials and the way they are distributed affects the weight of the composite, its strength, and other properties, as well as the way it conducts heat. This work is about studying the temperature distribution in composites, depending on the type of component material and its location. For this purpose, the Tolerance Averaging Technique and the Finite Difference Method were used. Differential equations describing heat conduction phenomena were obtained using the Tolerance Averaging Technique, while the Finite Difference Method was used to solve them. In terms of results, temperature distribution plots were produced showing the effect of the structure of the composite on the heat transfer properties.

Thermodynamically consistent volumetric–deviatoric decomposition-based phase-field model for thermo-electro-mechanical fracture

Using a volumetric–deviatoric decomposition of strain, we propose a novel phase-field model for thermo-electro-mechanical fracture and provide an open-source finite element implementation of the proposed model. Considering displacement, electric potential, temperature field, and phase-field variable, we have obtained the governing partial differential equations (PDEs), by utilizing the virtual power principle. We have obtained the constitutive relations for the necessary thermodynamic fluxes on satisfying the first and second laws of thermodynamics. The proposed model can accommodate any dissipative effect, such as viscous damping, whenever necessary, in a thermodynamically consistent manner. We have developed open-source finite element codes for the proposed model using the Gridap package in Julia. Here, we have used the proposed model for fracture simulation in transversely isotropic material and functionally graded material for compact tension and three-point bending tests. We have considered variations in thermal and electrical loading conditions and investigated the corresponding changes in the structural responses. For the functionally graded specimens, we have also considered the variation in the material property gradation directions, along with the thermal and electrical boundary conditions to evaluate their effect on the structural responses of specimens.

Mixed-mode fracture analysis of FGMs using [formula omitted]-integral: Formulation and FE implementation

This paper introduces a computational method based on the Jk-integral for mixed-mode fracture analysis of isotropic functionally graded materials (FGMs), subjected to mixed-mode loading. The present development is implemented by means of the finite element method (FEM), by couplingofAnsys and Matlabsoftware’s. The continuous variations in the material parameters of the functionally graded medium are treated by specifying the material properties at the centroid of each finite element. The Jk-integral implementation is validated numerically by considering two different problems, the edge-cracked FGM plate and the four-point bending specimen with crack parallel to material gradation. The presented results illustrate the influences of material property gradation and crack geometry on the mixed-mode SIFs. The comparisons reported by available reference solutions indicate also that the numerical results obtained by present method have a high precision and that the expanded form of the Jk-integral is independent of the domain.

Buckling analysis of stiffened FGM plates

The addition of stiffeners is a common way to achieve higher strength for plates. The linear buckling response of stiffened Functionally Graded Material (FGM) plates is investigated in this work. FGM plates having a centrally attached stiffener plate are subjected to in-plane uniform uniaxial loads. The gradation of material properties in FGM plates along the plate thickness is based on power-law, sigmoid and exponential functions. The critical buckling loads are obtained using finite element method based on Classical Plate Theory in ABAQUS software. The influence of material, plate aspect ratio, stiffener depth and width is studied in detail.

Improvement of process control in sheet metal forming by considering the gradual properties of the initial sheet metal

In scientific studies, sheet metal is usually considered as a two-dimensional body. Thus, it is accepted that material properties are in most cases regarded homogeneous in thickness direction. However, a gradation of certain properties becomes apparent when going beyond the standard characterization methods for sheet metals, which can for example, influence the springback behavior and the thinning of the sheet after forming. Thus, the aim of this work is to further improve the prediction accuracy of springback after forming in simulations, by implementing several inhomogeneous properties over the sheet thickness in an existing material model. For this purpose, the entire procedure from the identification of the inhomogeneous properties for describing the gradation to the implementation in a numerical model and its validation by comparing experimental and simulated bending operations is carried out on a DC04 cold-forming steel in order to prove its influence on the springback behavior. It is shown that including graded material properties in simulations does indeed have an impact on the prediction quality of springback and that the information about inhomogeneous properties can be provided by existing characterization methods with a high local resolution like electron backscatter diffraction or X-ray stress analysis. In a further step, it was possible to validate the improvement in numerical accuracy by comparing the prediction of the springback angle from both the existing and the extended model with experimental bending results. Both the initial model as well as the model supplemented with the 3D properties provide a good prediction accuracy in the solution heat treated material state. For the predeformed material, however, the initial numerical model predicts a springback angle of about 13°, which deviates remarkably from the experimentally obtained mean value of about 17°. The extended model delivers a significantly improved accuracy in springback prediction in relation to the initial prediction (deviation of 4°) with a minor deviation of only about 0.8°, which proves the importance of considering the gradation of material properties in thickness direction for an overall higher dimensional accuracy of sheet metal products.

Global buckling of axially functionally graded columns with variable boundary conditions

The problem of a stability loss in compressed Euler columns with variable gradation of material properties along their length is addressed. Axial gradation causes that the cross-section has a transverse symmetry, i.e., the coupling matrix of cross-sectional forces is equal to zero. Prismatic columns with I-sections, differing in length, minimal moment of inertia and boundary conditions, were analysed in detail. A continuous change in material mechanical properties was discretized by assuming different numbers of column segments of the given length, with a linear alternation in them. The eigenproblem was solved with an analytical method, and the obtained results were compared to the solutions attained with the FEM Abaqus package. In the analytical solution, the Euler-Bernoulli beam theory for conservative systems was adopted. A particularly good correspondence of the results from those methods was obtained, the error in eigenvalues did not exceed 3%. The adoption of 20 segments on the FG column length can be treated as a continuous distribution of the material gradation along the column length. A further increase in their number is pointless. Finally, a procedure for determining minimal global eigenloads (i.e., fundamental buckling eigenloads) in the form of a modified Euler formula was put forward.

Analysis of Smart Functionally Graded Beams combined with piezoelectric material using Finite Element Method

The present work devotes the static and vibration behaviour of Smart Functionally Graded (SFG) Beams combined with piezoelectric material. In this work, Lead Zirconate Titanate (PZT-4) is taken as the piezoelectric material which is commercially available. The efficiency of piezoelectric material integrated onto the functionally graded (FG) beams is evaluated. A finite element model is made for the FG beam combined with piezoelectric material considering first order theory (FSDT). The material properties of the FG beam is graded along the direction of its thickness following power law distribution method. Simulation models are also made employing finite element software (ANSYS) for the functionally graded beams. Different boundary conditions as well as and power law indexes are taken for analysis. Validation of results is presented with the results available in earlier research publications. Effects of gradation of material properties on static and dynamic behaviour of smart functionally graded beam are studied. Efforts are also made to examine the performance of piezoelectric patches towards control of deflections and vibrations.

A graded interphase enhanced phase-field approach for modeling fracture in polymer composites

The superiority of nanosized filler particles in improving the elastic and fracture behavior of polymers, in comparison to microsized inclusions, has been substantiated by several experimental observations. Accurate modeling of crack propagation in such heterogeneous materials, however, involves resolution of complex crack topologies while taking into account the coalescence and branching of multiple cracks. Such complexity renders the traditional sharp crack modeling approaches, such as those based on the idea of partition of unity, to be of limited suitability, especially in 3D, since these involve explicit tracking of the evolving crack surfaces. Consequently, phase-field fracture approaches have come up as an attractive alternative to the traditional sharp crack models, especially when complex crack topologies need to be handled. Furthermore, standard two-phase continuum models, owing to their lack of the necessary length scale, are unable to capture the previously mentioned smaller is stronger size effect. In order to remedy this shortcoming, in context of finite strain hyperelasticity, a graded interphase based enhancement of the standard first-order continuum model was introduced in our previous work. The present contribution extends the idea of graded interphases to the phase-field fracture approach. Herein, an interphase region around filler particles, as observed experimentally, having continuously varying or graded material properties is considered. Fracture behavior of the composite material can be controlled by means of the degree of grading employed in determining the elastic and fracture properties within the interphase region. An optimal combination of the graded interphase parameters yields a tougher macroscopic response, as observed in case of tougher interphases, in comparison to the one obtained with a standard phase-field fracture model, The appropriateness of the introduced technique for modeling a wide spectrum of experimentally observed fracture behaviors, depending upon the degree of adhesion between the filler and the matrix phases, is depicted by means of extensive numerical experimentation.

Adaptive approximation-based multi-objective hybrid optimization method for dual-gradient top-hat structures

In this article, an adaptive approximation-based multi-objective hybrid optimization method, integrating a multi-objective decision-making method, adaptive approximation method and hybrid optimization method, is developed. In particular, a multi-objective decision-making method developed by combining a weight strategy inspired by grey relational analysis and entropy analysis is applied to normalize multiple objectives; the trust region is introduced into the support vector regression model to solve the problem of the sequential sampling for the approximation model; a hybrid optimization algorithm combining the genetic algorithm pattern search algorithm is developed to further assist the surrogate model-based optimization. Finally, a novel dual-gradient structure with graded thickness and graded material property along the axial direction is applied to the crashworthiness optimization design. The results demonstrate that the developed hybrid method has faster optimization efficiency than the Pointer algorithm built into Isight commercial software, and the final optimized structure is superior to the original one.

Computational elastodynamics of functionally graded thick-walled cylinders and annular coatings subjected to pressure shocks

A computational technique based on domain-boundary element method (D-BEM) is developed for elastodynamic analysis of functionally graded thick-walled cylinders and annular coatings subjected to pressure shock type of loadings. The formulation is built on the wave equation, which is derived in accordance with plane elastodynamics. Weighted residual statement for the wave equation is expressed by using the static fundamental solution as the weight function. Applying integration by parts and incorporating the boundary conditions, the problem is reduced to an integral equation. Problem domain is discretized by quadratic cells to transform the integral equation into a system of ordinary differential equations in time. Equation system is solved numerically applying Houbolt's method. Developed procedure is verified through comparisons to the analytical results available in the literature. Parametric analyses are carried out considering short-time ramp and exponential variation types of pressure shocks. Presented numerical results illustrate the influence of material property gradation on time histories and spatial distributions of displacement and stress components in FGM thick-walled cylinders and annular coatings.