A spectrally formulated finite element for wave propagation analysis in functionally graded beams

Abstract

In this paper, spectral finite element method is employed to analyse the wave propagation behavior in a functionally graded (FG) beam subjected to high frequency impulse loading, which can be either thermal or mechanical. A new spectrally formulated element that has three degrees of freedom per node (based upon the first order shear deformation theory) is developed, which has an exact dynamic stiffness matrix, obtained by exactly solving the homogeneous part of the governing equations in the frequency domain. The element takes into account the variation of thermal and mechanical properties along its depth, which can be modeled either by explicit distribution law like the power law and the exponential law or by rule of mixture as used in composite. Ability of the element in capturing the essential wave propagation behavior other than predicting the propagating shear mode (which appears only at high frequency and is present only in higher order beam theories), is demonstrated. Propagation of stress wave and smoothing of depthwise stress distribution with time is presented. Dependence of cut-off frequency and maximum stress gradient on material properties and FG material (FGM) content is studied. The results are compared with the 2D plane stress FE and 1D Beam FE formulation. The versatility of the method is further demonstrated through the response of FG beam due to short duration highly transient temperature loading.

Introduction

An ideal material combines the best properties of metals and ceramics––the toughness, electrical conductivity, and machinability of metals, and the low density, high strength, high stiffness, and temperature resistance of ceramics. In recent years, these type of advanced materials are no longer dreams but properly conceived and developed. These materials are known as metal matrix composites (MMCs) or ceramic matrix composites (CMCs) and they have incredible promise in many engineering applications. Demand for such materials comes from the automotive industry (lightweight and strong materials would increase fuel efficiency and last longer), electronics, telecommunications, and the aerospace and defense industries. The gradation in properties of the material reduces the thermal stresses, the residual stresses, and the stress concentration factors. If two dissimilar materials are bonded together, there is a very high chance that debonding will occur at some extreme loading conditions, be it static, dynamic, or thermal load. Cracks are likely to initiate at interfaces and grow into the weaker material section. Another problem associated with such structures is the presence of residual stresses due to the difference in coefficients of thermal expansion between the materials of adjacent layers. By gradually varying the volume fraction of the constituents rather than abruptly changing them over an interface can resolve these problems. The gradual variation results in a very efficient material tailored to suit the needs.

Advanced materials can be “functionally graded” to provide the ideal combination of characteristics desired. FGMs are materials in which the material properties vary with location in such a way as to optimize some function of the overall FGM. The matrix alloy (the metal), the reinforcement material (the ceramic), the volume, shape, and location of the reinforcement, and the fabrication method can all be tailored to achieve particular desired properties. In MMCs, for example, ceramic reinforcements in the form of either fibers, whiskers, or particulates are introduced into the metal; the structure is controlled at scales varying from 100 nm to several millimeters.

FGM has gained widespread applicability as thermal-barrier structures, wear and corrosion resistant coatings, and they are also used for joining dissimilar materials (Suresh and Mortensen, 1998). FGMs consisting of metallic and ceramic components are well known to improve the properties of thermal-barrier systems, caused because of cracking or delamination. These are often observed in the conventional two-layer systems, which are avoided by the smooth transition between the properties of the components in FGMs. In defense applications, for example, faster transportation of armor is necessary. The fundamental problem that limits its applicability is the weight of the armor. That is, the weight of materials inhibit fast movement and incur heavy consumption of fuel. MMCs have received considerable attention to alleviate this problem due to their lightness and ability to work harden under dynamic loading. However, damage to the armor from shockwaves can limit the work hardening. An FGM can be used here to tailor the microstructure of the MMC. A typical modern composite armor consists a hard outer surface, typically an Al₂O₃ tile, backed by a ductile material such as aluminium. These type of combinations, in impact environment, are often subjected to tensile wave, generated because of the reflection at the interface between the hard and ductile material due to acoustic impedance mismatch. This is of great concern because ceramic materials typically have low tensile strengths. FGM can be used here to diffuse the reflection by smoothly varying the properties from ceramic to metal.

In another application of FGM, thin walled members like plates and shells, which are used in reactor vessels, turbines and other machine parts, are susceptible to failure from buckling, large amplitude deflections, or excessive stresses induced by thermal or combined thermo-mechanical loading. Functionally gradient coatings on these structural elements may help reduce the failures.

Analysis of FGM involves consideration of temperature change, which imparts thermal loading of significant amount due to mismatch in thermal coefficients between metallic and ceramic materials. Reddy and Chin (1998) have already dealt with this problems for static and transient loading. El-Abbasi and Meguid (2000) analysed the thermoelastic behavior of functionally graded plates and shells. In this work, explicit coupling of thermal and mechanical field is not considered. Only the external thermal effect in the form of strain is considered. Power law and exponential variation of the material properties through the depth of the beam are considered in this study.

One of the fundamental characteristics of the wave propagation problem is that the incident pulse duration is very small (of the order of micro seconds) and hence the frequency content of pulse is very high (of the order of kHz). When such a pulse is applied to the structure, it will force all the higher order modes to participate in the response. At higher frequencies, the wave lengths are small. Hence, in order to capture all the higher order modes, the conventional finite element method requires very fine mesh to match the wavelengths. This makes the system size enormously large.

The spectral element approach (SEA) could be the nice alternative for such problems (Doyle, 1999). In SEA, first the governing equation is transformed in frequency domain using discrete Fourier transform (DFT). In doing so, for 1D waveguides, the governing partial differential equation (PDE) is reduced to a set of ordinary differential equation’s (ODE) with constant coefficients, with frequency as a parameter. The resulting ODEs are much easier to solve than the original PDE. The SEA begins with the use of exact solution to governing ODEs in the frequency domain as interpolating function. The use of exact solution results in exact mass distribution and hence the resulting dynamic stiffness matrix is exact. Hence, in the absence of any discontinuity, one single element is sufficient to handle a beam of any length. This substantially reduces the system size and they are many order smaller than the sizes involved in the conventional FEM. First, the exact dynamic stiffness is used to determine the system transfer function (frequency response function). This is then convolved with load. Next, inverse fast Fourier transform (IFFT) is used to get the time history of the response. Spectral element for elementary rod (Doyle, 1988), elementary beam (Doyle and Farris, 1990a, Doyle and Farris, 1990b), Timoshenko beam (Gopalakrishnan et al., 1992), for higher order Mindlin Hermann rod (Martin et al., 1994), for elementary composite beam (Roy Mahapatra et al., 2000) and for 2D membrane element (Rizzi and Doyle, 1991) are reported in literature. Till date, to the best of authors’ knowledge no spectral finite element formulation is available in the literature for FGM beams.

In addition to the smaller system sizes, there are many advantages that are inherent to the SEA. These are summarized below: (1) Treatment of non-reflecting boundary conditions are simple and straight-forward. This is done by leaving out the terms associated with the reflected coefficients from the interpolating displacement functions. Such an element is called the “Throw-off” element and it amounts to adding maximum damping to the structure. (2) Since the system transfer function is direct by product of the approach, performing inverse problems, such as, system identification and force identification, is straight forward. (3) The method is easily amenable for control related problem. A novel active spectral element (Roy Mahapatra et al., 2001) was formulated for this purpose. This has great utility in smart structure research for noise and vibration suppression.

Wave propagation analysis of FGM beam poses tremendous challenge due to the presence of material anisotropy. Because of this, an additional shear wave gets created beyond a certain high frequency, called the cut-off frequency. Due to this, there will be a three way (axial shear bending) coupling of modes. Tracking these individual waves is a very difficult task specially for a dispersive system such as a FGM beam. In this work, we have devised an efficient methodology to capture such coupled waves.

The literature on the response of such advanced materials to dynamic and impact loadings (severe mechanical environments) are limited in numbers. No results existed for the case of through-thickness material property varying plates using shear deformation plate theories with the von Kármán non-linearity until the works of Reddy and Chin (1998), Praveen and Reddy (1998) and Reddy (2000) appeared. Non-linear transient thermoelastic analysis of FGM was carried out in Reddy and Chin (1998) by plate finite element for moderately large rotations (i.e., the von-Kármán non-linear theory of plates). Gong et al. (1999) studied the elastic response of FGM shell subjected to low velocity impact. The existing literature on the responses of FGM to high frequency impact loading are very limited in numbers. Among them, there are only methods proposed to analyse wave propagation problems in FGM plates. Liu et al., 1991a, Liu et al., 1991b and Liu and Tani, 1992, Liu and Tani, 1994, used strip element method for FGM plates. An analytical method was proposed by Ohyoshi et al. (1996), where the wave reflection and the transmission coefficients were obtained for FGM plate. Thermomechanical behavior of FGM plates and shells was investigated by Reddy and Chin (1998). Non-linear transient thermoelastic analysis of FGM was carried out by Praveen and Reddy (1998) using plate finite element for moderately large rotation in von-Kármán sense. Liu et al. (1999) proposed a method for analysing stress waves in FGM plates, where it is shown that the variation of material properties can be approximated by a piecewise linear function. Recently, Han et al. (2002a) presented an analytical–numerical method for analysing characteristics of waves in a cylinder composed of FGM. They have also proposed a numerical method for analysing transient waves in plates of FGM excited by impact loads (Han et al., 2002b). Here, the displacement response is determined by employing Fourier transformation and the modal analysis.

Till date, the only work involving wave propagation in FGM beam is that of Chakraborty et al. (2002), where an exact finite element is developed which takes thermal strain and depthwise property variation into consideration.

The organization of the paper is as follows. In the next two sections, the details of the formulation of the spectral element is given. Next, the numerical studies are performed. First, the predicting capability of the element is tested by observing the normal wave propagation behavior and the result is compared with the 2D finite element (FE) solution. Second, the stress wave propagation through FG beam and the variation of stress with depth and the smoothening effect of FGM is inspected closely. Effect of thermal loading is investigated next. Here, a thermal burst kind of loading is applied to the beam structure and its response is observed. Lastly, the presence of bending-axial-shear coupling in FG beam is demonstrated. Based on the results, a number of important conclusions are drawn.