Thermomechanically coupled sensitivity analysis and design optimization of functionally graded materials

This paper presents a systematic numerical technique for performing sensitivity analysis of coupled thermomechanical problem of functionally graded materials (FGMs). General formulations are presented based on finite element model by using the direct method and the adjoint method. In the modeling of spatial variances of material properties, the graded finite element method is employed to conduct the heat transfer analysis and structural analysis and their sensitivity analysis. The design variables are the volume fractions of FGMs constituents and structural shape parameters. The design optimization model is then constructed and solved by the sequential linear programming (SLP). Numerical examples are presented to demonstrate the accuracy and the applicability of the present method.     

Large deflection analysis of functionally graded magneto-electro-elastic porous flat panels

In this article, an attempt has been made on evaluating the large/nonlinear deflection of functionally graded magneto-electro-elastic porous (FG-MEEP) flat panels taking geometric skewness into consideration. Further, the flat panel is subjected to combined loads which include mechanical, electrical and magnetic loads. The mathematical formulation is derived through higher order shear deformation theory and von-Karman's geometric nonlinearity under the framework of finite element method (FEM). The effective material properties of FG-MEEP material are determined using modified power law. Two forms of material gradation such as ‘B’ rich bottom and ‘F’ rich bottom are modelled and implemented in the analysis. The numerical assessment is carried out to investigate the effect of prominent parameters such as skew angle, porosity distribution, gradient index, porosity volume, functionally graded pattern, electromagnetic loads on the nonlinear deflection of FG-MEEP flat panels. In addition, this study also makes an attempt to evaluate the degree of coupling associated with these parameters.

     

Flow-based fabrication: An integrated computational workflow for design and digital additive manufacturing of multifunctional heterogeneously structured objects

Structural hierarchy and material organization in design are traditionally achieved by combining discrete homogeneous parts into functional assemblies where the shape or surface is the determining factor in achieving function. In contrast, biological structures express higher levels of functionality on a finer scale through volumetric cellular constructs that are heterogeneous and complex. Despite recent advancements in additive manufacturing of functionally graded materials, the limitations associated with computational design and digital fabrication of heterogeneous materials and structures frame and limit further progress. Conventional computer-aided design tools typically contain geometric and topologic data of virtual constructs, but lack robust means to integrate material composition properties within virtual models. We present a seamless computational workflow for the design and direct digital fabrication of multi-material and multi-scale structured objects. The workflow encodes for and integrates domain-specific meta-data relating to local, regional and global feature resolution of heterogeneous material organizations. We focus on water-based materials and demonstrate our approach by additively manufacturing diverse constructs associating shape-informing variable flow rates and material properties to mesh-free geometric primitives. The proposed workflow enables virtual-to-physical control of constructs where structural, mechanical and optical gradients are achieved through a seamless design-to-fabrication tool with localized control. An enabling technology combining a robotic arm and a multi-syringe multi nozzle deposition system is presented. Proposed methodology is implemented and full-scale demonstrations are included.     

Non-classical plate model for FGMs

Engineering with Computers - In this work, we developed a non-classical plate model for the analysis of the behavior of microplates prepared from functionally graded materials (FGMs). It was...     

Multi-dimensional optimization of functionally graded material composition using polynomial expansion of the volume fraction

The work of this paper proposes a method for multi-dimensional optimization of functionally graded materials (FGMs) composition. The method is based on using polynomial expansion of the volume fraction of the constituent materials. In this approach, the design variables are the coefficients of the polynomial expansion which to be determined through the optimization process. This method provides much more flexibility in the design compared to the methods based on the power-law, or the exponential-law which will in turn lead to more optimal designs. Also it requires much less number of design variables compared to the grid based approaches which is also utilized for two-dimensional optimization of FGMs structures. As an application of the proposed method, the optimization of a simply supported Aluminum plate reinforced with Silicon Carbide nano-particles is considered. Cost plays a very important role for this type of structures, since the cost of the reinforcements such as Silicon Carbide nano-particles, or carbon nano-tubes is too high. So the aim of the optimization process is to minimize the amount of the reinforcement required to satisfy certain performance criteria. Both static, and dynamic cases are considered in this work; a plate under a transverse pressure distribution is considered as the static case, and the panel flutter problem as the dynamic case.     

Size-dependent vibration analysis of multi-span functionally graded material micropipes conveying fluid using a hybrid method

Microscale fluid-conveying pipes and functionally graded materials (FGMs) have many potential applications in engineering fields. In this paper, the free vibration and stability of multi-span FGM micropipes conveying fluid are investigated. The material properties of FGM micropipes are assumed to change continuously through thickness direction according to a power law. Based on modified couple stress theory, the governing equation and boundary conditions are derived by applying Hamilton’s principle. Subsequently, a hybrid method which combines reverberation-ray matrix method and wave propagation method is developed to determine the natural frequencies, and the results determined by present method are compared with those in the existing literature. Then, the effects of material length scale parameter, volume fraction exponent, location and number of supports on dynamic characteristics of multi-span FGM micropipes conveying fluid are discussed. The results show that the size effect is significant when the diameter of micropipe is comparable to the length scale parameter, and the natural frequencies determined by modified couple stress theory are larger than those obtained by classical beam theory. It is also found that natural frequencies and critical velocities increase rapidly with the increase of volume fraction exponent when it is less than 10, and the intermediate supports could improve the stability of pipes conveying fluid significantly.     

功能梯度薄壁圆柱壳的自由振动

研究了由功能梯度材料制成的薄壁圆柱壳的自由振动.采用幂律分布规律描述功能梯度材料沿厚度的梯度性质,根据Donnell壳体理论,导出了功能梯度材料薄壁圆柱壳线性振动的简化控制方程.基于此理论分析了功能梯度圆柱壳的自由振动特性,给出了两端简支功能梯度材料薄壁圆柱壳小挠度固有振动的频率公式.以简支圆柱壳作为算例,与前人结果及有限元法对比验证了该简化功能梯度薄壁圆柱壳理论的正确性,同时讨论了周向波数及梯度指数对其频率的影响.     

Free vibration analysis of functionally graded conical, cylindrical shell and annular plate structures with a four-parameter power-law distribution

Based on the First-order Shear Deformation Theory (FSDT) this paper focuses on the dynamic behavior of moderately thick functionally graded conical, cylindrical shells and annular plates. The last two structures are obtained as special cases of the conical shell formulation. The treatment is developed within the theory of linear elasticity, when materials are assumed to be isotropic and inhomogeneous through the thickness direction. The two-constituent functionally graded shell consists of ceramic and metal. These constituents are graded through the thickness, from one surface of the shell to the other. A generalization of the power-law distribution presented in literature is proposed. Two different four-parameter power-law distributions are considered for the ceramic volume fraction. Some material profiles through the functionally graded shell thickness are illustrated by varying the four parameters of power-law distributions. For the first power-law distribution, the bottom surface of the structure is ceramic rich, whereas the top surface can be metal rich, ceramic rich or made of a mixture of the two constituents and on the contrary for the second one. Symmetric and asymmetric volume fraction profiles are presented in this paper. The homogeneous isotropic material can be inferred as a special case of functionally graded materials (FGM). The governing equations of motion are expressed as functions of five kinematic parameters, by using the constitutive and kinematic relationships. The solution is given in terms of generalized displacement components of the points lying on the middle surface of the shell. The discretization of the system equations by means of the Generalized Differential Quadrature (GDQ) method leads to a standard linear eigenvalue problem, where two independent variables are involved without using the Fourier modal expansion methodology. Numerical results concerning six types of shell structures illustrate the influence of the power-law exponent, of the power-law distribution and of the choice of the four parameters on the mechanical behaviour of shell structures considered.     

Optimal design of periodic functionally graded composites with prescribed properties

The computational design of a composite where the properties of its constituents change gradually within a unit cell can be successfully achieved by means of a material design method that combines topology optimization with homogenization. This is an iterative numerical method, which leads to changes in the composite material unit cell until desired properties (or performance) are obtained. Such method has been applied to several types of materials in the last few years. In this work, the objective is to extend the material design method to obtain functionally graded material architectures, i.e. materials that are graded at the local level (e.g. microstructural level). Consistent with this goal, a continuum distribution of the design variable inside the finite element domain is considered to represent a fully continuous material variation during the design process. Thus the topology optimization naturally leads to a smoothly graded material system. To illustrate the theoretical and numerical approaches, numerical examples are provided. The homogenization method is verified by considering one-dimensional material gradation profiles for which analytical solutions for the effective elastic properties are available. The verification of the homogenization method is extended to two dimensions considering a trigonometric material gradation, and a material variation with discontinuous derivatives. These are also used as benchmark examples to verify the optimization method for functionally graded material cell design. Finally the influence of material gradation on extreme materials is investigated, which includes materials with near-zero shear modulus, and materials with negative Poisson’s ratio.     

Functionally graded materials for prescribed field evolution

The present communication is concerned with a numerical procedure to determine an optimal material layout of a functionally graded material (FGM) within the context of a transient phenomenon. In particular, the physical problem considered here is transient heat conduction. The idea is to determine the effective material properties (via volume fraction of an isotropic composite) with the goal of controlling the evolution of the corresponding field quantity. The difference between the actual temperature field and a prescribed target field is minimized in a mean squares sense in space and time. Results show that the proposed methodology can be used for development of FGMs in a variety of practical applications.     

An effective multilayered sandwich beam-link finite element for solution of the electro-thermo-structural problems

The first part of this contribution deals with deriving the effective matrices of the multilayered sandwich 2D-link/beam finite element made of FGM (functionally graded material) with variation of material properties and rectangular cross-section for solution of weak-coupled electric–thermal–structural problems. The variation of effective material properties is caused by both continuous longitudinal and layer-wise symmetric transversal variation of the constituent’s volume fractions and constituent’s material properties. The second part of this contribution is completed by numerical validation, which documents the high accuracy and effectiveness of our proposed electro-thermo-structural composite (FGMs) link/beam finite element.     

Optimal tailoring of 2D volume-fraction distributions for heat-resisting functionally graded materials using FDM

Thermoelastic behavior of heat resisting functionally graded materials (FGMs), under given thermal loading and boundary conditions, is definitely characterized by the spatial distribution of volume fractions of constituent particles. Hence, the determination of volume-fraction distribution becomes a crucial step in tailoring an optimal heat-resisting FGM. In this paper, we address a two-dimensional volume-fraction optimization procedure for relaxing the effective thermal stress distribution. In order for the optimization efficiency and the volume-fraction continuity, we approximate the volume-fraction field with bilinear elements of n-times larger size than for the thermoelastic analysis. As well, the refined material-property estimate is employed to assure the design quality. Numerical experiments illustrating our theoretical work are also presented.     

T-stress,mixed-mode stress intensity factors,and crack initiation angles in functionally graded materials: a unified approach using the interaction integral method

For linear elastic functionally graded materials (FGMs), the fracture parameters describing the crack tip fields include not only stress intensity factors (SIFs) but also T-stress (nonsingular stress). These two fracture parameters are important for determining the crack initiation angle under mixed-mode loading conditions in brittle FGMs (e.g. ceramic/ceramic such as TiC/SiC). In this paper, the mixed-mode SIFs and T-stress are evaluated by means of the interaction integral, in the form of an equivalent domain integral, in combination with the finite element method. In order to predict the crack initiation angle in brittle FGMs, this paper makes use of a fracture criterion which incorporates the T-stress effect. This type of criterion involves the mixed-mode SIFs, the T-stress, and a physical length scale r_c (representative of the fracture process zone size). Various types of material gradations are considered such as continuum models (e.g. exponentially graded material) and micromechanics models (e.g. self-consistent model). Several examples are given to show the accuracy and efficiency of the interaction integral scheme for evaluating mixed-mode SIFs, T-stress, and crack initiation angle. The techniques developed provide a basic framework for quasi-static crack propagation in FGMs.     

基于自然元法和精细积分法的功能梯度材料瞬态热传导问题求解

为促进无网格法分析技术在热传导分析中的应用,提出空间离散采用自然单元法、时间离散采用精细积分法求解功能梯度材料瞬态热传导问题的数值计算方法.在计算过程中,取高斯点的材料参数模拟功能梯度材料特性的变化.温度场采用自然邻接点插值形函数进行离散插值.数值算例验证该数值算法的正确性和有效性.     

Demonstration of optimizing piezoelectric polarization in the design of a flextensional actuator

Much effort has gone into amplifying the displacements of actuators built around piezoelectric materials (PZTs). Some researchers have used topology optimization to design compliant mechanisms that best magnify either the geometric or mechanical advantage of piezoelectric wafers or “stack” actuators. PZTs are generally poled through the “thin” direction, so actuation by an electric field in that direction only induces eigenstrains normal to the free edges. Some researchers have shown advantages of “shear mode” actuation, and material scientists have demonstrated the ability to pole a PZT in an arbitrary direction. This work attempts to justify the inclusion of the PZT polarization vector as a design variable in the design of a flextensional actuator. We present two examples based on the “cymbal” actuator: one using a simplified model to justify off-angle polarization and another using the polarization vector as a design variable to optimize the topology of a compliant mechanism.     

Fluid–structure–soil interaction effects on the free vibrations of functionally graded sandwich plates

This study investigates the effects of fluid–structure and soil–structure interaction on the free vibration response of functionally graded sandwich plates. To this aim, an exemplary problem is analyzed, whereas a metal/ceramic sandwich plate is placed at the bottom of a tank filled in with fluid. Two cases are considered: (i) soft core, i.e., a sandwich plate with metal core and ceramic skins, and (ii) hard core, i.e., a sandwich plate with ceramic core and metal skins. In both cases, the skins are modelled as suitable functionally graded materials (FGMs). The soil is modelled as a Pasternak foundation. The free vibration analysis is carried out according to the extended higher order sandwich plate theory (EHSAPT). The fluid is assumed to be inviscid, incompressible, and irrotational. Hamilton’s principle is exploited to deduce the governing equations and the corresponding boundary conditions. The Rayleigh–Ritz method with two-variable orthogonal polynomials is used to compute the natural frequencies of the sandwich plate. The adopted approach is first validated through comparison with results published in the literature. Then, the effects are studied of several parameters on the dynamic response of the system.

     

A geometrically nonlinear size-dependent hypothesis for porous functionally graded micro-plate

The static bending behavior of porous functionally graded (PFG) micro-plate under the geometrically nonlinear analysis is studied in this article. A small-scale nonlinear solution is established using the Von-Kármán hypothesis and the modified couple stress theory (MCST). To obtain the deflection of the plate, the Reddy higher-order plate theory coupled with isogeometric analysis (IGA) is utilized. The distribution of porosities is assumed to be even and uneven across the plate’s thickness and the effective material properties of porous functionally graded micro-plate are calculated using the refined rule-of-mixture hypothesis. The influence of power index, porosity parameter and material length scale parameter on the nonlinear behaviors of static bending of porous FGM micro-plates are also investigated using several numerical examples.

     

A design model to predict optimal two-material composite structures

A method is presented for the prediction of optimal configurations for two-material composite continuum structures. In the model for this method, both local properties and topology for the stiffer of the two materials are to be predicted. The properties of the second, less stiff material are specified and remain fixed. At the start of the procedure for computational solution, material composition of the structure is represented as a pure mixture of the two materials. This design becomes modified in subsequent steps into a form comprised of a skeleton of concentrated stiffer material, together with a nonoverlapping distribution of the second material to fill the original domain. Computational solutions are presented for two example design problems. A comparison among solutions for different ratios of stiffness between the two materials gives an indication of how the distribution of concentrated stiffer material varies with this factor. An example is presented as well to show how the method can be used to predict an efficient layout for rib-reinforcement of a stamped sheet metal panel.     

Vibration analysis of porous magneto-electro-elastically actuated carbon nanotube-reinforced composite sandwich plate based on a refined plate theory

In this article, the free vibration response of sandwich plates with porous electro-magneto-elastic functionally graded (MEE-FG) materials as face sheets and functionally graded carbon nanotube-reinforced composites (FG-CNTRC) as core is investigated. To this end, four-variable shear deformation refined plate theory is exploited. The properties of functionally graded material plate are assumed to vary along the thickness direction of face sheets according to modified power-law expression. Furthermore, properties of FG-CNTRC layer are proposed via a mixture rule. Hamilton’s principle with a four-variable tangential–exponential refined theory is used to obtain the governing equations and boundary conditions of plate. An analytical solution approach is utilized to get the natural frequencies of embedded porous FG plate with FG-CNTRC core subjected to magneto-electrical field. A parametric study is led to fulfill the effects of porosity parameter, external magnetic potential, external electric voltage, types of FG-CNTRC, and different boundary conditions on dimensionless frequencies of porous MEE-FG sandwich plate. It is noteworthy that the numerical consequences can serve as benchmarks for future investigations for this type of structures with porous mediums.

     

Nonlinear thermo-mechanical response of temperature-dependent FG sandwich nanobeams with geometric imperfection

In this paper, the nonlinear dynamic response of functionally graded (FG) sandwich nanobeam associated with temperature-dependent material properties by considering the initial geometric imperfection is investigated. The size-dependent behavior of the FG sandwich nanobeam is simulated based on the nonlocal strain gradient theory, and Von Karman nonlinear hypothesis is used to model the geometrical nonlinearity. Moreover, the geometric imperfection is considered as a slight curvature satisfying the first mode shape, and four different FG sandwich patterns including two asymmetric configurations and two symmetric configurations are taken into account. The governing equation of the FG sandwich nanobeam subjected to thermal and harmonic external excitation loadings is derived on the basis of Hamilton’s principle. The numerical results are obtained by employing the multiple-scale method, which are also validated by comparison with two previous studies. Furthermore, comprehensive investigations into the influences of size-dependent parameters, external temperature variation, geometric imperfection amplitude, gradient index and sandwich configuration on the nonlinear characteristics of imperfect FG sandwich nanobeams are conducted through numerical results.