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

Beam element with spatial variation of material properties for multiphysics analysis of functionally graded materials

The present study focuses on thermal buckling and post-buckling analysis of functionally graded material (FGM) rods for various uniform temperature rises under different sets of boundary condition. Only chemical gradient FGMs are analyzed and power law functions are considered for material gradient along the thickness of the rod. The governing equations are developed relative to the neutral plane, and displacement fields are considered to be of first order to coincide with the Timoshenko beam theory. The set of highly nonlinear equations are solved using a commercial software package, ANSYS based on finite element method (FEM). To tackle the problem of spatial variation of material properties along the thickness of the FGM rod, the entire geometry is divided into finite number of layers, each layer being homogeneous and material properties are assigned to the layers from the power function making the spatial variation of any material property a piecewise function across the thickness of the whole geometry. Variation of critical buckling temperature and post-buckled configurations for different material gradient under different thermal loading and rod end conditions are observed and presented.

Considering that the statistic numerical characteristics are often required in the probability-based damage identification and safety assessment of functionally graded material (FGM) structures, an stochastic model updating-based inverse computational method to identify the second-order statistics (means and variances) of material properties as well as distribution of constituents for damaged FGM structures with material uncertainties is presented by using measurable modal parameters of structures. The region truncation-based optimization method is employed to simplify the computational process in stochastic model updating. In order to implement the forward propagation of uncertainties required in the stochastic model updating and avoid large error resulting in the nonconvergence of the iteration process, an algorithm is proposed to compute the covariance between the modal parameters and the identified parameters for damaged FGM structures. The proposed method is illustrated by a numerically simulated damaged FGM beam with continuous spatial variation of material properties and verified by comparing with the Monte-Carlo simulation (MCS) method. The influences of the levels and sources of measured data uncertainties as well as the boundary conditions on the identification results are investigated. The numerical simulation results show the efficiency and effectiveness of the presented method for the identification of material parameter variability by using the measurable modal parameters of damaged FGM structures.

Two-dimensional contact mechanics of functionally graded materials with arbitrary spatial variations of material properties

The high temperature occurs due to the combustion gas from engine in unmanned aerial vehicles (UAV). The high temperature may cause serious damages in UAV structure. The Functionally Graded Material (FGM) is chosen as a candidate material of the engine duct structure. A functionally graded material (FGM) is a two- component mixture composed by compositional gradient materials from one material to the other. In contrast, traditional composite materials are homogeneous mixtures, and involve compositions between the desirable properties of the component materials. Since significant proportions of an FGM contain the pure form of each material, the need for compromise is eliminated. The properties of both components can be fully utilized. Thermal stress analysis of FGM layers (20, 40, 60, 80 and 100) is performed in this paper. In addition, the creep behavior of FGM applied in duct structure of an engine is analyzed for better understanding of FGM characteristics.

Material engineering has become one of the corner stones in various engineering applications. The scientific alteration of base materials into various inorganic and organic compounds has resulted in the development of Functionally Graded Material (FGM). FGM can be described as a cutting edge material in which the particular material's properties could be made varied as the dimension changes. The aims of this study was to review on FGM's applications and its manufacturing processes. In addition, the mathematical idealization of the FGM was also reviewed. Based on the review, since FGMs are highly heterogeneous, it can be idealized as their mechanical properties changes smoothly with respect to their spatial coordinates. Nonetheless, in order to analyze FGMS in an efficient manner, the homogenization schemes are needed to be simplified from their complicated heterogeneous microstructures. Closed-form solutions of some fundamental solid mechanics' problems can be obtained by this idealization. Conclusively, FGM is unique in relation to any of the individual materials that structures it. Suggestively, this provide a limitless possibility for future engineering applications.

Natural frequency analysis of anisotropic functionally graded material (FGM) annular sector plates on Winkler elastic foundations based on three-dimensional theory of elasticity was investigated. The three-dimensional graded finite element formulation was derived based on the principle of minimum potential energy and the Rayleigh-Ritz method. For an orthotropic FGM, the material properties were assumed to have in-plane polar orthotropy and transverse heterogeneity according to an exponential law, whereas the mass density was assumed to be constant. For an isotropic FGM, material properties varied continuously through the thickness direction according to a power-law distribution, whereas Poisson’s ratio was set to be constant. The effects of material gradient exponents, different sector angles, different thickness ratio, Winkler parameter and two different boundary conditions on the natural frequencies and mode shapes of FGM annular sector plates have been investigated. Numerical solution was compared with the result of an FGM annular circular plate, which showed good agreement.

We describe the development and implementation of a block-based adaptive mesh refinement (AMR) algorithm for solving the discrete ordinates neutral particle transport equation. AMR algorithms allow mesh refinement in areas of interest without requiring the extension of this refinement throughout the entire problem geometry, minimizing the number of computational cells required for calculations. The block-based AMR algorithm described here is a hybrid between traditional cell or patch-based approaches and is designed to allow an efficient parallel solution of the transport equation while still reducing the cell count.This paper discusses the data structure implementation and CPU/memory efficiency for our Block AMR method, the equations and procedures used in mapping edge fluxes between blocks of different refinement levels for both diamond and linear discontinuous spatial discretizations, effects of AMR on mesh convergence, and our approach to parallelization. Comparisons between our Block AMR method and a traditional single-level mesh are presented for a sample brachytherapy problem. The Block AMR results are shown to be significantly faster for this problem (on at least a few processors), while still returning an accurate solution.

A parallel implicit adaptive mesh refinement (AMR) algorithm is described for the system of partial-differential equations governing steady two-dimensional compressible gaseous flows. The AMR algorithm uses an upwind finite-volume spatial discretization procedure in conjunction with limited linear solution reconstruction and Riemann-solver based flux functions to solve the governing equations on multi-block mesh composed of structured curvilinear blocks with quadrilateral computational cells. A flexible block-based hierarchical data structure is used to facilitate automatic solution-directed mesh adaptation according to physics-based refinement criteria. A Jacobian-free inexact Newton method is used to solve the system of nonlinear equations arising from this finite-volume spatial discretization procedure and a preconditioned generalized minimal residual (GMRES) method is used to solve the resulting non-symmetric system of linear equations at each step of the Newton algorithm. An additive Schwarz global preconditioner with variable overlap is used in conjunction with block-fill incomplete lower-upper (BFILU) type preconditioners based on the Jacobian of the first-order upwind scheme for each sub-domain. The Schwarz preconditioning and block-based data structure readily allow efficient and scalable parallel implementations of the implicit AMR approach on distributed-memory multi-processor architectures. Numerical results will be described for several flow cases, demonstrating both the effectiveness of the mesh adaptation and algorithm parallel performance. The proposed parallel implicit AMR method allows for anisotropic mesh refinement and is well suited for predicting complex flows with disparate spatial and temporal scales.

A new adaptive local mesh refinement algorithm and its application on fourth order thin film flow problem

The mechanics of equal channel angular extrusion (ECAE) in a rectangular die was studied by upper bound method and rigid block model of deformation with two variables (two degrees of freedom). The appearance of a dead zone in the die external corner was assumed and verified by physical simulation using plasticine as a modeling material. The obtained dependences of ECAE pressure, total ECAE shear and geometrical parameters of dead zone upon friction factor were compared with corresponding results of upper bound rigid block analysis with one variable (one degree of freedom) and with the results of slip line analysis published by the authors elsewhere. It was demonstrated that an increase in degrees of freedom in an ECAE rigid block model from one to two results in much better approximation of the reference data of slip line analysis. The authors believe that an increase in the numbers of degrees of freedom and flexibility of a rigid block model generally leads to a better accuracy in modeling of other metal forming processes as well. As an example, the ECAE pressure of the aluminum 6061 alloy was calculated and compared with the experimental data from literature. A good agreement between them was found. The importance of consideration of friction in the inlet and outlet die channels was emphasized.

This work presents a finite element formulation for the vibration analysis of functionally graded material (FGM) plates containing microstructural defects operating in thermal environments. The material properties are assumed to be temperature-dependent and are modeled as higher-order functions of temperature. The volume fractions of the constituent materials are varied through the plate thickness according to a power-law distribution. The effective material properties of the FGM are evaluated using both the Mori–Tanaka homogenization scheme and the rule of mixtures. The kinematics of the FGM plates are described using an efficient layerwise theory. The governing equations of motion, constitutive relations, and boundary conditions are derived using on Hamilton’s principle. The stiffness coefficients of the plates are computed using a sub-laminate mechanics approach, wherein the plate thickness is discretized into several thin sublayers, each assumed to possess uniform material properties. This approach is particularly effective for micromechanical models where exact analytical integration of stiffness coefficients is difficult due to the complex polynomial variation of properties across the thickness. A four-noded quadrilateral finite element, previously developed for laminated and FGM shells, is extended in this study to analyze the dynamic behavior of FGM plates with microstructural defects and temperature dependent material properties. The natural frequencies are obtained using a subspace iteration technique, while the transient responses under step and impulse excitations are computed using the state-space form of the governing dynamic equations. The proposed finite element model is validated against three-dimensional elasticity and Ritz-based semi-analytical solutions, demonstrating excellent accuracy and reliability across varying thermal environments. A detailed parametric study is also carried out to investigate the effects of temperature, inhomogeneity index, porosity concentration, boundary conditions, and geometric parameters on the natural frequencies and transient responses of porous FGM plates.

An electromechanically coupled zigzag theory is presented for static and free vibration responses in functionally graded material (FGM) plates bonded with piezoelectric layers. It has the capability to consider continuity as well as discontinuity in material properties at all interfaces. The displacement field approximation, which is initially layer based, has been made layer independent by using continuity conditions on displacements and transverse shear stresses as well as zero traction conditions on top and bottom surfaces. Material properties in FGM layers vary continuously in the thickness direction. Piezoelectric layers have constant property across their thickness. Extended Hamilton’s principle for piezoelectric continuum has been used in deriving governing equations of motion and boundary conditions. An analytical solution is obtained for elastic as well as piezoelectric layer bonded FGM plates. Static and free vibration responses have been obtained in a number of FGM plates, with each plate considered to contain a number of FGM layers with discontinuity or continuity in material properties at the interfaces. Wherever available, present responses have been compared with three-dimensional exact solutions and other shear deformation theories. The effects of the aspect ratio, length-to-width ratio, and different schemes for effective material property calculation have been illustrated.

Quantitative 3D phase field modelling of solidification using next-generation adaptive mesh refinement

A micromechanical model for predicting thermal properties and thermo-viscoelastic responses of functionally graded materials