Reduction of thermal stresses by developing two-dimensional functionally graded materials

https://doi.org/10.1016/j.ijsolstr.2003.08.017Get rights and content

Abstract

Modern aerospace shuttles and craft are subjected to super high temperatures, that have variation in two or three directions, which need to introduce new materials that can stand with such applications. Therefore, in the present work a two-dimensional functionally graded materials, 2D-FGM, are introduced to withstand super high temperatures and to give more reduction in thermal stresses. The suitable functions that can represent volume fractions of the introduced 2D-FGM are proposed. Then the rules of mixture of the 2D-FGM are derived based on the volume fractions of the 2D-FGM and the rules of mixture of the conventional FGM. The introduced volume fractions and rules of mixture for 2D-FGM were used to calculate the thermal stresses in 2D-FGM plate. Comparison between 2D-FGM and conventional FGM was carried out and showed that 2D-FGM has high capability to reduce thermal stresses than the conventional FGM.

Introduction

The high level of operating temperature involved in many industrial machine elements, such as space shuttles, combustion chambers, nuclear plants and ovens, requires effective high temperature resistant materials to improve the strength of such machine elements. In recent years, the concept of functionally graded material (FGM) has been introduced as a thermal barrier material. FGMs were adopted for the development of structural components subjected to severe thermal loading. Day by day FGMs prove their high capability as high temperature resistant material and quietly gain this position. Many authors, such as; Noda and Tsuji (1991), Arai et al. (1991), Tang et al. (1993), Erdogan and Wu (1996) and Jin and Batra (1996), analyzed FGM problems in different cases with and without crack under thermal and mechanical loads. Their results were very exciting and enhanced their aims in the development of new temperature resisting materials.

During the investigation of the FGM problems different functions for the continuous gradation of the material properties were considered. In analytical solution of FGM problems, under thermal and mechanical loads, exponential functions for continuous gradation of the material properties were considered, see for example; Erdogan et al. (1991), Erdogan and Wu (1997), Choi (1996), Jin and Noda (1994), Noda and Jin, 1993, Noda and Jin, 1995 and Wang et al. (2000), as: E=E0eβx, ν=ν0eβx, k=k0eδx and α=α0eγx, where E: Young’s modulus, ν: Poisson’s ratio, α: coefficient of linear thermal expansion, k: heat conductivity and E0, ν0, α0, k0, β, γ and δ are material constants. Exponential functions for representation of the material properties usually facilitate the analytical solution but don’t give real representation for material properties, expect for the upper and lower surfaces of FGM. The most realistic way for representation of the continuous gradation of the material properties of FGM is the volume fractions and rules of mixture. The use of the volume fractions and rules of mixture complicate the analytical solution of the FGM problems and may make it impossible. The use of finite element method in such problems is the most effective tools to overcome such difficulties. Many authors, such as Fuchiyama et al. (1993), Noda (1997), Sumi and Sugano (1997), Hassab-Allah and Nemat-Alla (2002) and Fujimoto and Noda (2001), analyzed FGM problems, in different cases with and without crack, using the volume fractions and rules of mixture.

Steinberg (1986) showed the variations of the temperature at various places on the outer surface, of the new aerospace craft when the plane is in sustained flight at a speed of Mach 8 and altitude of 29 km. The temperature on the outer surface of such a plane ranges from 1033 K along the top of the fuselage to 2066 K at the nose and from outer surface temperature to room temperature inside the plane. Such kind of aerospace craft added a new challenge to introduce and develop more high temperature resistant materials that can stand with high external temperatures that have variation in two or three directions. In the design requirement of such problem Callister (2001) suggested to design several different thermal protection materials that satisfy the required criteria for a specific region of the spacecraft surface. Callister design has the same drawbacks of the composite layers, cracking, separation through layers interface and low mechanical strength, that were overcome by functionally graded materials. Recently, Colombia space shuttle was loosed in a catastrophic break up. According to International Herald Tribune (2003) the speculation of such failure is that some kind of structural damage took place––perhaps caused by outer surface insulation that fell loose when the Columbia lifted off. Also, Hindustan Times (2003) reported that, damage to the space shuttle’s protective thermal tiles is emerging as a key focus of the probe into the Colombia tragedy.

Conventional functionally graded material may also not be so effective in such design problems since all outer surface of the body will have the same composition distribution. Since, temperature distribution in such advanced machine element changes in two or three directions. Therefore, if the FGM has two-dimensional dependent material properties, more effective high-temperature resistant material can be obtained. In other words it is necessary to add a material that has more strength to the thermal stresses in the places that have maximum values of the thermal stresses or the places where yielding may occur. Based on such fact a two-dimensional, 2D, FGM whose material properties are two-directional dependent is introduced. 2D-FGM takes a great interest as higher order theory for the thermo-elastic response of composite materials in order to achieve more relaxation of the thermal stresses and their intensity factors. The two-dimensional models of the material properties of FGM increase the number of thermal/mechanical non-homogeneous parameters. The wide variety of thermal/mechanical non-homogeneous parameters will increase the ability of reducing thermal stresses and their concentration factors.

Recently, few authors investigated 2D-FGM. Dhaliwal and Singh (1978) solved the equation of equilibrium for non-homogeneous isotropic elastic solid under shearing force in rectangular Cartesian coordinates as well as in cylindrical polar coordinates. The modulus of rigidity of their considered material was varied exponentially in lateral and vertical directions. They considered the problem of determining the state of stresses in an infinite non-homogenous elastic medium containing a Griffith crack under shearing force. Clements et al. (1997) introduced the solution of the equations of antiplane inhomogeneous elasticity when the shear modulus varied continuously with two Cartesian coordinates. A particular crack problem was considered for a certain multi-parameter form for the shear modulus. Aboudi et al., 1996a, Aboudi et al., 1996b, studied thermo-elastic/plastic theory for the response of materials functionally graded in two directions. Their studies circumvented the problematic use of the standard micro-mechanical approach, based on the concept of a representative volume element, commonly employed in the analysis of functionally graded composites by explicitly couple the local (micro-structural) and global (macro-structural) responses. The response of symmetrically laminated plates subjected to temperature change in one-dimension was investigated. It was possible to reduce the magnitude of thermal stress concentrations by a proper management of the microstructure of the composite. Nemat-Alla and Noda, 1996a, Nemat-Alla and Noda, 1996b, Nemat-Alla and Noda, 2000 and Nemat-Alla et al. (2001) analyzed the crack problem in semi-infinite and finite FGM plate with bi-directional coefficient of thermal expansion under one and two-dimensional steady thermal loads. They showed that the thermal stresses and the thermal stress intensity factors could be decreased by the proper selection of the mechanical and thermal non-homogeneous parameters. Cho and Ha (2002) optimize the volume fractions distributions of FGM for relaxing the effective thermal stresses. They obtained the optimal volume fractions distribution in two directions for the FGM. The obtained optimum volume fractions have not continues, direct or function, representation as conventional FGM. It looks like a random distribution, which is very difficult to represent or simulate FGM that have continues variations of the composition. Also, their obtained optimum volume fractions was investigated under steady thermal loading, cooling to 300 K uniform temperature from uniform initial temperature of 1000 K. Such kind of thermal loading can be resisted by conventional FGM.

From the forgoing review of literature, one can see that several works have been carried out to investigate the thermal stresses and the thermal stress intensity factors in 2D-FGM under thermal and mechanical loading. The material properties were considered to be exponential functions in two-directions. Unfortunately, the rules of mixture and the volume fractions relations that can represent the 2D-FGM are not available. The main objective of the current investigation is to introduce the rules of mixture and the volume fractions relations that can represent the 2D-FGM. This is very important in order to characterize the behavior of the 2D-FGM under 2D and 3D-thermal loading. The developed volume fractions and rules of mixture relations of the 2D-FGM will be used to calculate the thermal stresses in a 2D-FGM plate. Then a comparison of the thermal stresses in 2D-FGM plate with the conventional FGM plate will be carried out.

Section snippets

Volume fractions and rules of mixture of FGM

Consider a plate of FGM with porosity that functionally graded from ceramic and metal. The volume fractions of FGM with porosity can be represented as:Vm=(x/t)mVc=(1−Vm)where Vm, Vc, x and t are volume fraction of the metal, volume fraction of ceramic, Cartesian coordinate x and plate thickens respectively.

Also, m is a non-homogenous parameter that control the composition variations through the thickens. If m=1 the variations of the composition of ceramics and metal are linear. The composition

Volume fractions of 2D-FGM

2D-FGM is made of continuous gradation of three distinct material phases at least one of them is ceramics and the others are metal alloy phases. 2D-FGM is fabricated in such a way that the volume fractions of the constituents are varied continuously in a predetermined composition profile.

Now let us discuss the volume fractions and porosity of the 2D-FGM at any arbitrary point A on the 2D-FGM plate shown in Fig. 2. Firstly, the volume fractions of point A may be treated as one-dimensional FGM

Rules of mixture of 2D-FGM

The rules of mixture for the 2D-FGM with porosity can be obtained using the same way used to obtain the porosity of 2D-FGM with some mathematical manipulation. For any point on the 2D-FGM plate with volume fractions V1, V2 andV3 as shown in Fig. 1, using Eqs. , , , , , , , , , , , , , , , , , the rules of mixture for the different thermal and mechanical properties may be obtained as:
For Poisson’s ratio:ν=ν1V12V23V3For modulus of elasticityE=E0y(1−py)1+py(5+8ν)(37−8ν)/8(1+ν)(23+8ν)whereE0y=Ex

Transient thermal loading

According to Fuchiyama et al. (1993) and Kokini et al. (1997) the crack in FGM plate may not open or initiate during the heating stage while it may initiate and propagate during the cooling stage after heating. Therefore, in the current study it will be more realistic to investigate the considered 2D-FGM under cooling thermal loading conditions. According to Steinberg (1986) and Callister (2001) it will be more realistic if the plate has temperature variations along the upper surface.

Finite element model

A 2D-FGM plate having 15 mm thickness and 300 mm length was modeled using 2D eight-node thermal solid element. Due to the inhomogeneity of the material in x- and y-directions, every element on the finite element mesh was assigned by its own thermal and mechanical properties according to the volume fractions and the rules of mixture of 2D-FGM, Eqs. , , , , , , , , , , , , , , . The plate was initially kept at the temperature distributions generated from the following initial boundary conditions.

Results and discussions

Since the main advantage of the 2D-FGM over the conventional FGM is that it has variations of the compositions in two directions by adding a new material that add more strength and consequently can give more reduction to the thermal stresses or make a delay to yielding. Therefore, for proper comparison between the 2D-FGM and conventional FGM both of them are investigated under the same thermal loading. The conventional FGM constituent materials are SiC and Al1100 with linear variation in y

Conclusions

Through the current investigation for the introduced 2D-FGM the volume fractions are proposed and rules of mixture are obtained. The volume fractions and rules of mixture for the introduced 2D-FGM were used to calculate the thermal stresses in SiC/Al1100/Ti-6Al-4V 2D-FGM plate, with temperature independent material properties. The SiC/Al1100/Ti-6Al-4V 2D-FGM plate was subjected to steady thermal load that has temperature variations along the ceramic surface, SiC. Then the 2D-FGM plate was

Acknowledgements

The authors would like to express his deepest gratitude to professor Karam Emara, Mechanical Engineering Department, Assiut university, Egypt, for his guidance, suggestions and valuable discussions through the course of this work.

References (35)

  • F. Erdogan et al.

    The crack problem in bonded non-homogeneous material

    ASME Journal of Applied Mechanics

    (1991)
  • F. Erdogan et al.

    Crack problems in FGM layers under thermal stresses

    Journal of Thermal Stresses

    (1996)
  • F. Erdogan et al.

    The surface crack problem for a plate with functionally graded properties

    ASME Journal of Applied Mechanics

    (1997)
  • T. Fuchiyama et al.

    Analysis of thermal stress and stress intensity factor of functionally gradient materials

  • Fujimoto, T., Noda, N., 2001. Multiple crack growths in the functionally graded plate under thermal shock. In:...
  • H. Hassab-Allah et al.

    Transient thermal stress intensity factors for edge and internal cracks in FGM plate

    Journal of Sciences and Engineering

    (2002)
  • Hindustan Times, 2003. Available from...
  • Cited by (0)

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