Overall Elastic Stiffness of Advanced Anisotropic Materials

Mohamed S. Gaith and Imad Alhayek

Department of Mechanical Engineering, Al-Balqa Applied University, Amman, Department of Applied Sciences, Al-Balqa Applied University, Amman

Jordan

1. Introduction

Most of the elastic materials in engineering are, with acceptable accuracy, considered as anisotropic materials; metal crystals (due to the symmetries of the lattice), fiber-reinforced composites, polycrystalline textured materials, biological tissues, rock structure etc. can be considered as orthotropic materials. In recent years fiber reinforced composite materials have been paid considerable attentions due to the search for materials of light weight, great strength and stiffness. Consequently the determination of their mechanical properties, i. e. stiffness effect, becomes important.

Piezoelectric materials nowadays have been widely used to manufacture various sensors, conductors, actuators, and have been, extensively, applied in electronics, laser, ultrasonics, naval and space navigation as well as biologics, smart structures and many other high-tech areas.

The wide-gap II-VI semiconductors, well known anisotropic materials used in high technology, have received much attention in the past decades since they have important applications in short-wavelength light-emitting diodes (LEDs), laser diodes and optical detectors (Okuyama, 1992). Moreover, semiconductor materials constitute today basic components of emitters and receivers in cellular, satellite, fiberglass communication, solar cells, and photovoltaic systems. Their electronic and structural properties of such systems are subject of considerable interest in nanotechnology as well. For the semiconductor compounds ZnX and CdX (X=S, Se, and Te), the zinc-blend structure (ZB) has the lowest minimum total energy. With respect to classical II – VI semiconductors, these semiconductor compounds have attracted much attention in recent years for their great potential in technological applications (Reich, 2005). They have a high melting point, high thermal conductivity, and large bulk modulus. The hardness and large bulk modulus of these anisotropic materials make them ideal protective coating materials in photovoltaic applications and in machine tools (Reich, 2005). These materials can, therefore, be used for optoelectric devices in which the availability of light sources in the mid-infrared spectral region is crucial for many applications, i. e., molecular spectroscopy and gas-sensor systems for environmental monitoring or medical diagnostics (Leitsmann et al., 2006). Being stable to high temperatures and can be made of sufficiently insulating allows precise measurements

of piezoelectric, elastic, and dielectric constants. For such data eventually a fully quantum – mechanical description is essential in order to serve to verify a quantitative theory of piezoelectricity and elasticity in these structures. The window layer compounds CuInSe2, CuInS2, AgGaSe2, and AgGaS2 have a chalcopyrite structure that is closely related those of zinc blend and wurtzite structures. These compounds are found in many applications such as fiberglass communication, thin film solar cells, and photovoltaic systems. Thin film solar cells made from ternary chalcopyrite compounds, such as the aforementioned hetrojunction layers, are characterized by low-cost and clean energy sources. Their high absorption and resistance to deterioration make them good candidate materials for solar cell absorbers. Moreover, due to their flexible optical properties and good stabilities, they are promising compounds for fabricating polycrystalline thin film hetrojunction solar cells (Ramesh et al., 1999; Ramesh et al., 1998; Murthy et al.; 1991). Yet, the significant discrepancy in the efficiency of solar cells between the laboratory scale, over 19% (Murthy et al.; 1991), and the commercial one, around 13 %, is due to the lack of fundamental understanding of interface and junction properties in the film.

Historically, the study of anisotropic elastic materials has been synonymous with study of crystals. For a deep understanding of the physical properties of these anisotropic materials use of tensors is inevitable. Tensors are the most apt mathematical entities to describe direction-dependent-physical properties of solids, and the tensor components characterize physical properties which must be specified without reference to any coordinate system. When all the directions in the material can be considered as equivalent the physical property is isotropic, and consequently, the choice of the reference frame is of no real importance. More often the medium is anisotropic and tensor notation can not be avoided. Specifying the values of the tensor components which represent physical properties of crystals, as Nowack(Nowacki, 1962) points out, do not determine the material constants directly since their values vary with the direction of the coordinate axes. It is, therefore, natural to seek to characterize physical properties of crystals by constants whose values do not depend upon the choice of the coordinate system, i. e. constants which are invariant under all coordinate transformations. Some of such invariants have been obtained using different decomposition methods in the case of photoelastic coefficients ((Srinivasan & Nigam, 1968)), piezoelectric coefficients (Srinivasan, 1970) and elastic stiffness coefficients (Srinivasan et al. (Srinivasan & Nigam, 1968; Srinivasan, 1969; Srinivasan, 1985). A physical property is characterized by n rank tensor that has two kinds of symmetry properties. The first kind is due to an intrinsic symmetry derives from the nature of the physical property itself, and this can be established by the thermodynamical arguments or from the indispensability of some of the quantities involved. The second kind of symmetry is due to the geometric or crystallographic symmetry of the system described.

The purpose of this chapter, hence, is to develop the existing methods of decomposing Cartesian tensors into orthonormal basis using invariant-form to decompose some well – known tensors into orthonormal tensor basis. Next, as an outcome of these decompositions, to investigate the contributions to the formulation of the physical properties of elastic stress, strain, piezoelectric and elastic stiffness anisotropic materials. Finally, the concept of norm and norm ratios is introduced to measure the overall effect of material properties and to measure the degree of anisotropy. Numerical engineering applications are introduced for several engineering materials like semiconductor compounds, piezoelectric ceramics and fiber reinforced composites.