General Anisotropic Elastic Material Research Articles

Crack propagation induced by a plane stress wave in a general anisotropic elastic material

Abstract The stress intensity factors of a semi-infinite crack propagating with constant speed in an anisotropic elastic solid under a uniform stress wave loading are considered. The crack is assumed to start propagating at some arbitrary time after an obliquely incident plane stress wave strikes the crack tip. It is shown that the stress intensity factor of the propagating crack has the form of the product of a universal matrix function of the crack speed and an equivalent stationary crack stress intensity factor of t*, which is the time that would have elapsed since the incident wavestruck the crack tip if the crack tip had been always at its instantaneous position. The present result is a generalization of that obtained by Freund for isotropic materials.

On a consistent rod theory for a linearized anisotropic elastic material: I. Asymptotic reduction method

An asymptotic reduction method is introduced to construct a rod theory for a linearized general anisotropic elastic material for space deformation. The starting point is Taylor expansions about the central line in rectangular coordinates, and the goal is to eliminate the two cross-section spatial variables in order to obtain a closed system for displacement coefficients. This is first achieved, in an ‘asymptotically inconsistent’ way, by deducing the relations between stress coefficients from a Fourier series for the lateral traction condition and the three-dimensional (3D) field equation in a pointwise manner. The closed system consists of 10 vector unknowns, and further refinements through elaborated calculations are performed to extract bending and torsion terms and to obtain recursive relations for the first- and second-order displacement coefficients. Eventually, a system of four asymptotically consistent rod equations for four unknowns (the three components of the central-line displacement and the twist angle) are obtained. Six physically meaningful boundary conditions at each edge are obtained from the edge term in the 3D virtual work principle, and a one-dimensional rod virtual work principle is also deduced from the weak forms of the rod equations.

A new secular equation for slip waves along the interface of two dissimilar anisotropic elastic half-spaces in sliding contact

We consider wave propagation along the interface of two dissimilar anisotropic elastic half-spaces that are in sliding contact. A new secular equation is obtained that covers all special cases in one equation. One special case is when a Rayleigh wave (called the R-wave) can propagate in both half-spaces with the same wave speed. Another special case is when a slip wave (called the S-wave) can propagate in each of the half-spaces with the same wave speed. If a Rayleigh wave and a slip wave can propagate in one of the half-spaces it is called the RS-wave. In this case an interfacial slip wave exists in which the other half-space is at rest unless an RS-wave can also propagate in the other half-space. The results for general anisotropic elastic materials are applied to orthotropic materials.

Surface waves in an exponentially graded, general anisotropic elastic material under the influence of gravity

In a recent paper Destrade [1] studied surface waves in an exponentially graded orthotropic elastic material. He showed that the quartic equation for the Stroh eigenvalue p is, after properly modified, a quadratic equation in p 2 with real coefficients. He also showed that the displacement and the stress decay at different rates with the depth x 2 of the half-space. Vinh and Seriani [2] considered the same problem and added the influence of gravity on surface waves. In this paper we generalize the problem to exponentially graded general anisotropic elastic materials. We prove that the coefficients of the sextic equation for p remain real and that the different decay rates for the displacement and the stress hold also for general anisotropic materials. A surface wave exists in the graded material under the influence of gravity if a surface wave can propagate in the homogeneous material without the influence of gravity in which the material parameters are taken at the surface of the graded half-space. As the wave number k → ∞, the surface wave speed approaches the surface wave speed for the homogeneous material. A new matrix differential equation for surface waves in an arbitrarily graded anisotropic elastic material under the influence of gravity is presented. Finally we discuss the existence of one-component surface waves in the exponentially graded anisotropic elastic material with or without the influence of gravity.

Uniform Stress Inside an Anisotropic Elliptic Inclusion with Imperfect Interface Bonding

In this paper we study the two-dimensional deformation of an anisotropic elliptic inclusion embedded in an infinite dissimilar anisotropic matrix subject to a uniform loading at infinity. The interface is assumed to be imperfectly bonded. The surface traction is continuous across the interface while the displacement is discontinuous. The interface function that relates the surface traction and the displacement discontinuity across the interface is a tensor function, not a scalar function as employed by most work in the literature. We choose the interface function such that the stress inside the elliptic inclusion is uniform. Explicit solution for the inclusion and the matrix is presented. The materials in the inclusion and in the matrix are general anisotropic elastic materials so that the antiplane and inplane displacements are coupled regardless of the applied loading at infinity.

Dispersion equations for steady waves in an anisotropic elastic plate or a layered plate

Steady waves propagating in a plate that consists of one or more layers of general anisotropic elastic material are studied. The surface of the plate can be a traction-free (F), rigid (R) or slippery surface (S). The interface between any two layers in the plate can be perfectly bonded (b) or in sliding contact (s). The thickness of the layers need not be the same. The purpose of this paper is to present dispersion equations for all possible combinations of the boundary and interface conditions. If the thickness h of one of the layers is very small, the dispersion equation allows us to expand the solution in an infinite series in the power of h from which an approximate solution can be obtained by keeping the terms up to O ( h n ) for any n . The special case of a sandwich plate that consists of a centre layer and two identical outside layers is studied. In the literature, the dispersion equations for a sandwich plate were studied for special elastic materials. The results presented here are for elastic materials of general anisotropy.

Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies

When a very thin elastic layer is bonded to an elastic body, it is desirable to have effective boundary conditions for the interface between the layer and the body that take into account the existence of the layer. In the literature, this has been done for special anisotropic elastic layers. We consider here the layer that is a general anisotropic elastic material. The mechanics of a thin layer is studied for elastostatics as well as steady state waves. It is shown that one-component surface waves cannot propagate in a semi-infinite thin layer. We then present Love waves in an anisotropic elastic half-space bonded to a thin anisotropic elastic layer. The dispersion equation so obtained is valid for long wavelength. Finally, effective boundary conditions are presented for two thin layers bonded to two surfaces of a plate and a thin layer bonded between two anisotropic elastic half-spaces.

Transverse waves in anisotropic elastic materials

With few exceptions, a transverse wave can propagate in an anisotropic elastic material along certain directions only. Equations that allow us to compute the direction n, the polarization vector a and the wave speed c are presented for a general anisotropic elastic material. In contrast to longitudinal waves, the directions along which a transverse wave can propagate are not finite isolated directions. There is a one-parameter family of n along which a transverse wave can propagate. On the surface of the unit sphere n 1 2 + n 2 2 + n 3 3 = 1 , the n along which a transverse wave can propagate traces out at least three curves. The intersection of two or more curves is the direction along which a longitudinal wave can propagate. We present solutions for n, a and c for orthotropic, tetragonal and trigonal materials. The corresponding solutions for hexagonal and cubic materials have been obtained earlier.

Explicit Expression of the Stationary Values of Young's Modulus and the Shear Modulus for Anisotropic Elastic Materials

AbstractExplicit expressions of the directionnand the stationary values (maximum, minimum and saddle point) of Young's modulusE(n) for orthotropic, tetragonal, trigonal, hexagonal and cubic materials are presented. For the shear modulusG(n, m), explicit expressions of the extrema (maximum and minimum) and the two mutually orthogonal unit vectorsn, mare given for cubic and hexagonal materials. We also present a general procedure for computing the extrema ofG(n, m) for more general anisotropic elastic materials. It is shown that Young's modulusE(n) can be made as large as we wish for certainnwithout assuming that the elastic compliance s11, s22or s33is very small. As to the shear modulusG(n,m), it can be made as large as we wish for certainnandmwithout assuming that any one of the elastic compliance sαβis very small. We also show that Young's modulusE(n) can be independent ofnfor orthotropic and hexagonal materials while the shear modulusG(n, m) can be independent ofnandmfor hexagonal materials.

The Stationary Values of Young's Modulus For Monoclinic and Triclinic Materials

AbstractThe stationary values (maximum, minimum, saddle point) of Young modulusE(n) for a general anisotropic elastic materials is studied. The general results are then spcialized for monoclinic materials. Equations that provide the directionnfor a stationary value are given. Some have an explicit solution. Other may require a numerical computation. The equations that required numerical solutions are two coupled polynomials of degree no more than four.

Negative Poisson’s Ratios in Anisotropic Linear Elastic Media

Poisson’s ratio for an anisotropic linear elastic material depends on two orthogonal directions n and m. Materials with negative Poisson’s ratios for all (n,m) pairs are called completely auxetic while those with positive Poisson’s ratios for all (n,m) pairs are called nonauxetic. Simple necessary and sufficient conditions on elastic compliances are derived to identify if any given material of cubic or hexagonal symmetry is completely auxetic or nonauxetic. When these conditions are not satisfied, the medium is auxetic for some (n,m) pairs. Several simple necessary conditions for completely auxetic or nonauxetic media are derived for a general anisotropic elastic material.

Diffraction of a plane stress wave by a semi-infinite crack in a general anisotropic elastic material

The problem of a semi-infinite crack subjected to an incident stress wave in a general anisotropic elastic solid is considered. The plane wave impinges the crack at a general oblique angle and is of any of the three types propagating in that direction. A related problem of a semi-infinite crack loaded by a pair of concentrated forces moving along the crack surfaces is also considered. In contrast to the conventional approach by Laplace transforms, a Stroh-like formalism is employed to construct the solution directly in the time domain. The solution is shown to depend on a Wiener–Hopf factorization of a symmetric matrix. Closed-form solution of the stress intensity factors is derived. A remarkably simple expression for the energy release rate is obtained for normal incidence.

The polarization vector and secular equation for surface waves in an anisotropic elastic half-space

The displacement at the free surface of an anisotropic elastic half-space x 2>0 generated by a surface wave propagating in the direction of the x 1-axis traces an elliptic path. It is represented by the polarization vector a R= e 1+ i e 2 , where e 1 , e 2 are the conjugate radii of the ellipse on the polarization plane. The displacement traces the ellipse in the direction from e 1 to e 2 . We present explicit expressions of e 1 , e 2 and the secular equation without computing the Stroh eigenvalues p and the associated eigenvectors a and b . After presenting the expressions for a general anisotropic elastic material, the special cases are studied separately. For monoclinic materials with the symmetry plane at x 3=0, the secular equation and the conjugate radii e 1 , e 2 are identical to that for orthotropic materials when s ′ 16=0 but s ′ 26 need not vanish. For monoclinic materials with the symmetry plane at x 1=0, e 1 is along the x 1-axis while e 2 is on the plane x 1=0. If the symmetry plane is at x 2=0, e 1 is on the plane x 2=0 while e 2 is along the negative x 2-axis. In both cases, e 1 , e 2 are the principal radii of the ellipse. We also present the derivative of a R with respect to x 2, the depth from the free surface, that provides information on (i) whether the conjugate radii of the ellipse increase as x 2 increases and (ii) whether the polarization plane rotates as x 2 increases. New secular equations are obtained for monoclinic materials with the symmetry plane at x 1=0 or x 2=0.

A unified formalism for elastostatics or steady state motion of compressible or incompressible anisotropic elastic materials

For a two-dimensional deformation of an anisotropic elastic material, the Lekhnitski formalism applies to elastostatics only. It does not apply to steady state motion such as surface waves or a moving line dislocation. The Stroh formalism applies to elastostatics as well as steady state motion. However, it does not apply to an incompressible material because some of the elastic stiffnesses become unbounded in the incompressible limit. Modifying a new formalism studied recently, a more explicit and remarkably simple expression of the matrices in the new formalism is given, and a unified formalism that is applicable to elastostatics as well as steady state motion of a compressible or incompressible anisotropic elastic material is obtained. As an application, the secular equation is presented for surface waves propagating in a monoclinic material with the symmetry plane at x 3=0 that has a more explicit expression than what is available in the literature. For a general anisotropic elastic material, explicit expressions are obtained for the sextic equation for the eigenvalues p and the eigenvectors b and a for any steady state wave speed. The explicit expressions recover results related to one-component surface waves. When the material is incompressible, it is shown that the eigenvalues p remain complex for elastostatics. We also show that, when the material is incompressible, a monoclinic material with the symmetry plane at x 3=0 behaves like that of an orthotropic material.

A state space formalism for anisotropic elasticity. Part I: Rectilinear anisotropy

A state space formalism for anisotropic elasticity including the thermal effect is developed. A salient feature of the formalism is that it bridges the compliance-based and stiffness-based formalisms in a natural way. The displacement and stress components and the thermoelastic constants of a general anisotropic elastic material appear explicitly in the formulation, yet it is simple and clear. This is achieved by using the matrix notation to express the basic equations and grouping the stress in such a way that it enables us to cast neatly the three-dimensional equations of anisotropic elasticity into a compact state equation and an output equation. The homogeneous solution to the state equation for the generalized plane problem leads naturally to the eigen relation and the sextic equation of Stroh. Extension, twisting, bending, temperature change and body forces are accounted for through the particular solution. Based on the formalism the general solution for generalized plane strain and generalized torsion of an anisotropic elastic body are determined in an elegant manner.

An Explicit Secular Equation for Surface Waves in an Elastic Material of General Anisotropy

An explicit secular equation for surface waves in an elastic half-space was obtained by Rayleigh in 1885 (Proc. London Math. Soc. 17) for isotropic materials and by Stoneley in 1963 (Geophys. J. Astron. Soc. 8) for orthotropic materials. Since then no explicit secular equation was found for a material more general than orthotropic materials. Recently, Destrade (J. Acoust. Soc. Amer. 109, 2001) employed the first integrals (originally proposed by Mozhaev) to obtain an explicit secular equation for monoclinic materials with the symmetry plane at x 3 = 0, which is the plane of motion. The first integral approach does not work for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. Ting (Proc. R. Soc. 32, 2002) proposed a new approach that not only recovers Destrade's secular equation easily, but also provides an explicit expression of the Stroh eigenvalues p and the eigenvectors a, b without solving the quartic equation for p. These eigenvalues and eigenvectors are needed in the surface wave solution. Employing this new approach, Ting also obtained explicit secular equations for monoclinic materials with the symmetry plane at x 1 = 0 or x 2 = 0. This new approach is extended here to derive an explicit secular equation for surface waves in a general anisotropic elastic material. The secular equation for a general anisotropic material is then specialized to monoclinic materials with the symmetry plane at x 1 = 0, x 2 = 0 or x 3 = 0. It does recover the secular equations of Destrade, Stoneley and Rayleigh.

Anisotropic Elastic Materials With a Parabolic or Hyperbolic Boundary: A Classical Problem Revisited

When an anisotropic elastic material is under a two-dimensional deformation that has a hole of given geometry Γ subjected to a prescribed boundary condition, the problem can be solved by mapping Γ to a circle of unit radius. It is important that (i) each point on Γ is mapped to the same point for the three Stroh eigenvalues p1,p2,p3 and (ii) the mapping is one-to-one for the region outside Γ. In an earlier paper it was shown that conditions (i) and (ii) are satisfied when Γ is an ellipse. The paper did not address to the case when Γ is an open boundary, such as a parabola or hyperbola that was studied by Lekhnitskii. We examine the mappings employed by Lekhnitskii for a parabola and hyperbola, and show that while the mapping for a parabola satisfies conditions (i) and (ii), the mapping for a hyperbola does not satisfy condition (i). Nevertheless, a valid solution can be obtained for the problem with a hyperbolic boundary, although the prescription of the boundary condition is restricted. We generalize Lekhnitskii’s solutions for general anisotropic elastic materials and for more general boundary conditions. Using known identities and new identities presented here, real form expressions are given for the displacement and hoop stress vector at the parabolic and hyperbolic boundary.

Recent developments in anisotropic elasticity

Anisotropic elasticity has been an active research subject for the last thirty years due to its applications to composite materials. There are essentially two formalisms for two-dimensional deformations of a general anisotropic elastic material. The Lekhnitskii formalism [Lekhnitskii, S.G., 1950. Theory of Elasticity of an Anisotropic Elastic Body. Gostekhizdat, Moscow (in Russian)] has been the favorite among the engineering community, while the newer Stroh formalism is well-known in the material sciences, applied mathematics and physics community. The Stroh formalism (Stroh, A.N., 1958. Dislocations and cracks in anisotropic elasticity. Phil. Mag. 3, 625–646.) is mathematically elegant and technically powerful. It began to be noticed by the engineering community in recent years, specially among the younger researchers. A comprehensive treatment of both formalisms and applications of the theory have been presented in a book by Ting. Since the appearance of the book in 1996, there have been several new developments in the theory and applications of anisotropic elasticity. We present here new results that have appeared since 1996. Only linear anisotropic elasticity is considered here; for nonlinear elasticity, the reader is referred to the book by Antman (Antman, S.S., 1995. Nonlinear Problems in Elasticity. Springer–Verlag, New York).

Anisotropic Elastic Materials that Uncouple All Three Displacement Components, and Existence of One-Displacement Green's Function

It was shown in an earlier paper that, under a two-dimensional deformation, there are anisotropic elastic materials for which the antiplane displacement u3 and the inplane displacements u1, u2 are uncoupled but the antiplane stresses σ31, σ32 and the inplane stresses σ11, σ12, σ22 remain coupled. The conditions for this to be possible were derived, but they have a complicated expression. In this paper new and simpler conditions are obtained, and a general anisotropic elastic material that satisfies the conditions is presented. For this material, and for certain monoclinic materials with the symmetry plane at x3 = 0, we show that the unnormalized Stroh eigenvectors ak for k = 1, 2, 3 are all real. The matrix A =[a1, a2, a3] is a unit matrix when the material has a symmetry plane at x2 = 0. Thus any one of the u1, u2, u3 can be the only nonzero displacement, and the solution is a one-displacement field. Application to the Green's function due to a line of concentrated force f and a line dislocation with Burgers vector v in the infinite space, the half-space with a rigid boundary, and the infinite space with an elliptic rigid inclusion shows that one can indeed have a one-displacement field u1, u2 or u3. One can also have a two-displacement field polarized on a plane other than the (x1, x2)-plane. The material that uncouples u1, u2, u3 is not as restrictive as one might have thought. It can be triclinic, monoclinic, orthotropic, tetragonal, transversely isotropic, or cubic. However, it cannot be isotropic.

Isotropy of Strain Energy Functions Which Depend Only on a Finite Number of Directional Strain Measures

Motivated by studies of low-density materials and fiber-dominated composites, we consider an elastic material whose strain energy function depends only on a finite number of directional strain measures, which correspond to the strains of material fibers in specific material directions. It is shown that an arbitrary set of six distinct direction vectors can be used to define six symmetric base tensors which span the space of all symmetric second-order tensors. Using these base tensors and their reciprocal tensors we develop a representation for the strain tensor in terms of six directional strain measures. The functional form of the strain energy of a general anisotropic nonlinear elastic material may then be expressed in terms of these directional strain measures. Next, we consider general nonlinear isotropic response by developing explicit functional forms for three independent strain invariants in terms of these directional strain measures. Finally, with reference to previous work, we discuss isotropy of specific functional forms of directional strain measures associated with up to 15 directions in space.