Bending and free vibration response of layered functionally graded beams: A theoretical model and its experimental validation
Introduction
The concept of functionally graded material (FGM) emerged from the need to fabricate a new composite for high temperature structural applications by using a heat resistant ceramic on the high temperature side and a metal on the low temperature side to provide mechanical toughness [1]. The gradient compositional variation of the constituents from one surface to the other provides an elegant solution to the problem of high transverse shear stresses that are induced when two dissimilar materials with large difference in material properties are bonded. Since its inception, the concept has found many potential applications like thermal and corrosion barriers, dental and orthopedic implants, plasma facing bio materials, sensors, lightweight armour material with high ballistic efficiency etc. [2], [3], [4], [5]. The gradation of the volume fractions of the constituents along the thickness direction can have a continuous or a step-wise variation [6].
Several studies have been performed to analyze the static and dynamic behavior of functionally graded beams, plates and shells. The various models available for the analysis of FGM structures can be categorized based on two criteria (i) the type of displacement field approximations across the thickness (kinematic modelling) and (ii) the direct use of assumed variations of the material properties across the thickness or use of a material model for computing the effective properties at a point with given volume fractions of the constituents. The accuracy of prediction of response of FGM structures will depend on both the kinematic modelling and the correctness of estimated effective material properties of the two-phase system. Exact analytical solutions based on three dimensional (3D) elasticity have been presented for simply supported functionally graded infinite panel [7] and rectangular plate [8] under mechanical load, and for flat panel under transient thermal load [9], considering an assumed exponential variation of Young’s modulus and thermal expansion coefficient, and constant Poisson’s ratio over the thickness. Vel and Batra [10] have presented a 3D exact solution for free and forced vibrations of simply supported rectangular functionally graded plates, considering the Mori–Tanaka [11] and the self consistent [12] methods for computing the effective Young’s modulus and Poisson’s ratio with power law variation of the volume fractions of the constituents across the thickness. Such 3D analytical solutions are available only for specific geometry and boundary conditions. For analysis of structures or arbitrary shape and boundary conditions, several 2D models for plates and shells and 1D models for beams have been developed. The classical plate theory (CPT) has been employed for static bending [13] and transient [14] nonlinear response of FGM plates, using a power law variation for the volume fractions and employing linear rule of mixtures [15] for computing all effective material properties. The classical theory neglects shear deformation. Reddy and his coworkers have employed first order shear deformation theory (FSDT) (the simplest one to incorporate shear deformation effect) for thermoelastic finite element analysis of FGM plates [16] and for axisymmetric bending of circular FGM plates [17] using the linear rule of mixtures (ROM) for computing effective material properties with a power law variation of the volume fractions. Reddy [18], and Yang and Shen [19] have employed the refined third order theory (TOT) with an assumed power law variation for Young’s modulus, density and thermal expansion coefficient and a constant Poisson’s ratio for static and dynamic response of FGM plates under thermal loading. The same material model has been used by Chakraborty et al. [20], who presented a FSDT based shear locking free element for FGM beams. To the best of the authors’ knowledge, no experimental validation of the theoretical predictions of static and dynamic response of FGM beams and plates has been reported in open literature.
In a recent study, the authors [21] showed that the experimentally obtained values for the modulus of elasticity for Al/SiC composite system with different volume fractions of the constituents are considerably lower than those predicted using the ROM. It was also shown that the modified rule of mixtures (MROM) proposed by Tomota et al. [22] for two-phase systems, which involves a parameter for the stress to strain transfer between the two phases, can be used to accurately predict the Young’s modulus of this system. This parameter was found to be consistent across different volume fractions of Al and SiC.
This paper presents a finite element model for the dynamic analysis of layered FGM beams using an efficient third order zigzag theory [23] for the layerwise mechanics and the MROM for estimating the effective modulus of elasticity, and its experimental validation for the static deflection and natural frequencies of two different FGM systems under various boundary conditions. The two FGM systems considered are Al/SiC and Ni/Al2O3, both of which have generated considerable interest as heat resistent materials [24], [25]. Powder processes are ideally suited for fabricating gradient materials because of the excellent microstructural control and the versatility in these techniques. Several studies have been reported on the fabrication of ceramic–metal gradient materials by different powder metallurgy methods such as consolidation of powder stack [26], [27], consolidation of green sheet lamination [28], [29], powder slurry spraying [30], and combustion synthesis [31]. Reviews of these methods have been recently presented by Kieback et al. [6] and Watanabe et al. [32]. In the present study, Al/SiC FGM beam samples with three and five layers were fabricated using the powder metallurgy route following consolidation of powder stack with cold isostatic pressing (CIP) followed by sintering. The study on the beams with two different numbers of layers illustrates the effect of the gradation step size on the applicability of the theoretical model. Five-layered Ni/Al2O3 beam samples were prepared using combustion powder thermal spray process [33]. The two systems fabricated with different methods ascertain the consistency of the theoretical model.
Section snippets
Third order zigzag beam theory
The zigzag one dimensional (1D) beam theory presented by Kapuria et al. [23] is used to predict the response of the layered FGM beam. Consistent with the fabrication process, the FGM is modelled in this theory as a laminate of multiple perfectly bonded layers of isotropic material of layerwise constant composition. Let the beam be made of L such layers of discretely varying compositions. In general, the volume fractions Vc and Vm of the ceramic and the metal are assumed to vary along the
Effective material properties
The effective Young’s modulus of the two-constituent composite systems is computed using the modified rule of mixtures (MROM), which was originally proposed by Tomota et al. [22] for cemented carbides, and subsequently used for FGMs by many researchers (e.g. [25], [35]). According to this approach, the two-phase material in each layer in the FGM is treated as an isotropic composite for which the uniaxial stress σ and strain ε are related to the average uniaxial stresses σm, σc and strains εm, εc
Specimen geometry and design
FGM beam samples with isotropic homogeneous layers of metal and ceramic with their volume fractions varying from layer to layer were prepared for the experimental investigation. Al/SiC FGM samples were prepared by powder metallurgy route and Ni/Al2O3 FGM samples were made by thermal spray technique. For the Al/SiC system, 99.8% pure aluminum powders of grade 1180 and 95% pure silicon carbide powder with 400 mesh size were used as the raw materials. To study the effect of number of layers on the
Results and discussions
The material properties of the basic constituents Al, SiC, Ni and Al2O3 of the FGM systems, considered for computing the effective properties of the layers, are listed in Table 2. The overall densities of the samples computed using the ROM and their measured densities are listed in Table 3. It is observed that the measured densities differ from the theoretical densities by a maximum of 3.2% for Al/SiC samples and 5.2% for the Ni/Al2O3 samples. This difference is due to porosity in the samples.
Conclusions
The static and free vibration response of layered FGM beams of Al/SiC and Ni/Al2O3 prepared with powder metallurgy and combustion powder thermal spray processes, respectively, have been studied experimentally as well as theoretically. For the theoretical prediction of response, a finite element based on an efficient zigzag theory is used in conjunction with a modified rule of mixtures for predicting the effective elastic modulii of the layers, with the experimentally determined value (91.6 GPa)
Acknowledgement
The authors are grateful to International Advanced Research Center for Powder Metallurgy and New Materials, Hyderabad, India, and Metallizing Equipment Co. Pvt. Ltd., Jodhpur, India, for their generous help in fabrication by providing their facilities for sample preparation.
References (39)
- et al.
Viscoplastic deformations and compressive damage in an A359/SiCP metal–matrix composite
Acta Mater
(2000) - et al.
Functionally graded material: a parameter study on thermal stress characteristics using the Crank–Nicolson–Galerkin scheme
Comput Methods Appl Mech Eng
(2000) - et al.
Functionally graded ZrO2–NiCrAlY coatings prepared by plasma spraying using pre-mixed, spheroidized powders
Surf Coat Technol
(1997) - et al.
Preparation and characterization of Al2O3–Ti3SiC2 composites and its functionally graded materials
Mater Res Bull
(2003) - et al.
Curvature changes during thermal cycling of a compositionally graded Ni–Al2O3 multilayered material
Mater Sci Eng A
(1996) - et al.
Assessment of coupled 1D models for hybrid piezoelectric layered functionally graded beams
Compos Struct
(2006) - et al.
Tensile deformation of two-ductile-phase alloys: flow curves of α–γ Fe–Cr–Ni alloys
Mater Sci Eng
(1976) - et al.
A new beam finite element for the analysis of functionally graded materials
Int J Mech Sci
(2003) - et al.
Nonlinear bending analysis of shear deformable functionally graded plates subjected to thermo-mechanical loads under various boundary conditions
Composites Part B
(2003) - et al.
Axisymmetric bending of functionally graded circular annular plates
Eur J Mech Solids
(1999)
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