Steady State Motion and Surface Waves

The Stroh formalism for two-dimensional elastostatics can be extended to elastodynamics when the problem is a steady state motion. Most of the identities in Chapters 6 and 7 remain applicable. The Barnett-Lothe tensors S, H, L now depend on the speed υ of the steady state motion. However S(υ), H(υ), L(υ) are no longer tensors because they do not obey the laws of tensor transformation when υ≠0. Depending on the problems the speed υ may not be prescribed arbitrarily. This is particularly the case for surface waves in a half-space where υ is the surface wave speed. The problem of the existence and uniqueness of a surface wave speed in anisotropic materials is the crux of surface wave theory. It is a subject that has been extensively studied since the pioneer work of Stroh (1962). Excellent expositions on surface waves for anisotropic elastic materials have been given by Farnell (1970), Chadwick and Smith (1977), Barnett and Lothe (1985), and more recently, by Chadwick (1989d).

The Stroh Formalism

In this chapter we study Stroh's sextic formalism for two-dimensional deformations of an anisotropic elastic body. The Stroh formalism can be traced to the work of Eshelby, Read, and Shockley (1953). We therefore present the latter first. Not all results presented in this chapter are due to Stroh (1958, 1962). Nevertheless we name the sextic formalism after Stroh because he laid the foundations for researchers who followed him. The derivation of Stroh's formalism is rather simple and straightforward. The general solution resembles that obtained by the Lekhnitskii formalism. However, the resemblance between the two formalisms stops there. As we will see in the rest of the book, the Stroh formalism is indeed mathematically elegant and technically powerful in solving two-dimensional anisotropic elasticity problems. The possibility of extending the formalism to three-dimensional deformations is explored in Chapter 15.

The Lekhnitskii Formalism

A two-dimensional deformation means that the displacements ui, (i= 1,2,3) or the stresses σij depend on x1 and x2 only. Among several formalisms for two-dimensional deformations of anisotropic elastic materials the Lekhnitskii (1950, 1957) formalism is the oldest, and has been extensively employed by the engineering community. The Lekhnitskii formalism essentially generalizes the Muskhelishvili (1953) approach for solving two-dimensional deformations of isotropic elastic materials. The formalism begins with the stresses and assumes that they depend on x1 and x2 only. The Stroh formalism, to be introduced in the next chapter, starts with the displacements and assumes that they depend on x1 and x2 only. Therefore the Lekhnitskii formalism is in terms of the reduced elastic compliances while the Stroh formalism is in terms of the elastic stiffnesses. It should be noted that Green and Zerna (1960) also proposed a formalism for two-dimensional deformations of anisotropic elastic materials. Their approach however is limited to materials that possess a symmetry plane at x3=0. The derivations presented below do not follow exactly those of Lekhnitskii.

Linear Anisotropic Elastic Materials

The relations between stresses and strains in an anisotropic elastic material are presented in this chapter. A linear anisotropic elastic material can have as many as 21 elastic constants. This number is reduced when the material possesses a certain material symmetry. The number of elastic constants is also reduced, in most cases, when a two-dimensional deformation is considered. An important condition on elastic constants is that the strain energy must be positive. This condition implies that the 6×6 matrices of elastic constants presented herein must be positive definite. Referring to a fixed rectangular coordinate system x1, x2, x3, let σij and εks be the stress and strain, respectively, in an anisotropic elastic material. The stress-strain law can be written as . . . σij = Cijksεks . . . . . .(2.1-1). . . in which Cijks are the elastic stiffnesses which are components of a fourth rank tensor. They satisfy the full symmetry conditions . . . Cijks = Cjiks, Cijks = Cijsk, Cijks = Cksij. . . . . . .(2.1-2). . .

Matrix Algebra

We will present in this chapter some aspects of matrix algebra that are needed in this book. Most results presented here can be found in standard books on matrix algebra. Proofs are provided for those results that are either easily derived or not readily available elsewhere. For readers who have no knowledge of matrix algebra, this chapter is essential for the rest of the book. They may find the one-chapter treatment of matrix algebra in the book by Hildebrand (1954) helpful and informative. For readers who have some knowledge of matrix algebra this chapter can be skimmed or skipped altogether, depending on how familiar they are with the subject. If they want to find the proofs omitted in this chapter or want to devote more time on the subject, the books by Hohn (1965) and Pease (1965) are recommended. The notations employed in this chapter have no relations, in most cases, with the notations adopted in the rest of the book. This point should be kept in mind in referring back to this chapter.

Degenerate and Near Degenerate Materials

The Stroh formalism presented in Sections 5.3 and 5.5 assumes that the 6×6 fundamental elasticity matrix N is simple, i.e., the three pairs of eigenvalues pα are distinct. The eigenvectors ξα (α=l,2,3) are independent of each other, and the general solution (5.3-10) consists of three independent solutions. The formalism remains valid when N is semisimple. In this case there is a repeated eigenvalue, say p2=p1 ,but there exist two independent eigenvectors ξ2 and ξ1 associated with the repeated eigenvalue. The general solution (5.3-10) continues to consist of three independent solutions. Moreover one can always choose ξ2 and ξ1 such that the orthogonality relations (5.5-11) and the subsequent relations (5.5-13)-(5.5- 17) hold. When N is nonsemisimple with p2=p1, there exists only one independent eigenvector associated with the repeated eigenvalue. The general solution (5.3-10) now contains only two independent solutions. The orthogonality relations (5.5-11) do not hold for α,β=l,2 and 4,5, and the relations (5.5-13)-(5.5-17) are not valid. Anisotropic elastic materials with a nonsemisimple N are called degenerate materials. They are degenerate in the mathematical sense, not necessarily in the physical sense. Isotropic materials are a special group of degenerate materials that happen to be degenerate also in the physical sense. There are degenerate anisotropic materials that have no material symmetry planes (Ting, 1994). It should be mentioned that the breakdown of the formalism for degenerate materials is not limited to the Stroh formalism. Other formalisms have the same problem. We have seen in Chapters 8 through 12 that in many applications the arbitrary constant q that appears in the general solution (5.3-10) can be determined analytically using the relations (5.5-13)-(5.5- 17). These solutions are consequently not valid for degenerate materials. Alternate to the algebraic representation of S, H, L in (5.5-17), it is shown in Section 7.6 that one can use an integral representation to determine S, H, L without computing the eigenvalues pα and the eigenvectors ξα. If the final solution is expressed in terms of S, H, and L the solution is valid for degenerate materials.

Anisotropic Media with a Crack or a Rigid Line Inclusion,

A crack, or cracks, in a material is perhaps one of the most studied problems in solid mechanics. This is due to the fact that many structural failures are related to the presence of a crack in the material. The knowledge of stress distribution near a crack tip is indispensable in a fracture mechanics analysis (Rice, 1968; Sih and Liebowitz, 1968; Sih and Chen, 1981; Kanninen and Popelar, 1985; K. C. Wu, 1989a). A crack is represented by a slit cut whose surfaces are assumed traction-free. This is a mathematical idealization. For a composite material that consists of stiff short fibers or whiskers in the matrix, we have rigid line inclusions. A rigid line inclusion is the counterpart of a crack. It is sometimes called an anticrack. The displacement at a rigid line inclusion either vanishes or has a rigid body translation and rotation. One of the puzzling problems for a crack is the one when it is located on the x1-axis that is an interface between two dissimilar materials. The displacement of the crack surfaces near the crack tips may oscillate, creating a physically unacceptable phenomenon of interpenetration of two materials. The bimaterial tensor Š introduced in Section 8.8 plays a key role in the analysis. If Š vanishes identically, there is no oscillation. If Š is nonzero, we may decompose the tractions applied on the crack surfaces into two components, one along the right null vector of Š denoted by to and the other on the right eigenplane of Š denoted by tγ . The solution associated with to is not oscillatory. It has the property that the traction on the interface x2=0 is polarized along the right null vector of Š while the crack opening displacement is polarized along the left null vector of Š. The solution associated with tγ is oscillatory. It has the property that the traction on the interface x2=0 is polarized on the right eigenplane of Š while the crack opening displacement is polarized on the left eigenplane of Š.

The Structures and Identities of the Elasticity Matrices

The matrices Q, R, T, A, B, N1, N2, N3, S, H, L, and M introduced in the previous chapter are the elasticity matrices. They depend on elastic constants only, and appear frequently in the solutions to two-dimensional problems. The matrices A, B, and M are complex while the others are real. We present their structures and identities relating them in this chapter. In Chapter 7 we will show that A and B are tensors of rank one and S, H, L, and M are tensors of rank two when the transformation is a rotation about the x3-axis. Readers who are not interested in how the structures of these matrices and the identities relating them are derived may skip this chapter. They may return to this chapter when they read later chapters on applications where the results presented here are employed.

Three-Dimensional Deformations

There appears to be very little study, if any, on the extension of Stroh's formalism to three-dimensional deformations of anisotropic elastic materials. In most three-dimensional problems the analyses employ approaches that are remotely related to Stroh's two-dimensional formalism. This is not unexpected, since this has been the situation between two-dimensional and three-dimensional isotropic elasticity. However it needs not be the case for three-dimensional anisotropic elasticity. Much can be gained if a connection to the Stroh formalism can be established. Barnett and Lothe (1975a) appeared to be the only ones who made a connection between a three-dimensional solution and Stroh's two-dimensional formalism. Earlier, several investigators obtained the Green's function for the infinite anisotropic medium in term of a line integral on an oblique plane in the three-dimensional space. That line integral, as we will see here, is one of Barnett-Lothe tensors on an oblique plane. We propose in this chapter extensions and applications of Stroh's two-dimensional formalism to certain three-dimensional deformations of anisotropic elastic solids.

Antiplane Deformations

As a starter for anisotropic elastostatics we study special two-dimensional deformations of anisotropic elastic bodies, namely, antiplane deformations. Not all anisotropic elastic materials are capable of an antiplane deformation. When they are, the inplane displacement and the antiplane displacement are uncoupled. The deformations due to inplane displacement are plane strain deformations. Associated with plane strain deformations are plane stress deformations. After defining these special deformations in Sections 3.1 and 3.2 we present some basic solutions of antiplane deformations. They provide useful references for more general deformations we will study in Chapters 8, 10, and 11. The derivation and motivation in solving more general deformations in those chapters become more transparent if the reader reads this chapter first. The solutions obtained in those chapters reduce to the solutions presented here when the materials are restricted to special materials and the deformations are limited to antiplane deformations. In a fixed rectangular coordinate system xi (i=1, 2, 3), let ui, σij, and εij be the displacement, stress, and strain, respectively. The strain-displacement relations and the equations of equilibrium are . . .εij = 1/2 (ui,j + uj,i),. . . . . . (3.1 -1) . . . . . .σij,j =0,. . . . . . (3.1 - 2). . . in which repeated indices imply summation and a comma stands for differentiation. The stress-strain laws for an anisotropic elastic material can be written as σij = Cijks εks or εij = Sijksσks, . . .(3.1 - 3). . . where Cijks and Sijks are, respectively, the elastic stiffnesses and compliances.