## EN224: Linear Elasticity

Division of Engineering

Brown University

##### 9. Solutions for anisotropic materials

9.1 General Principles: Constitutive law and field equations

We turn next to develop the theory of elasticity for anisotropic solids.  The theory is far less well developed for anisotropic materials than it is for isotropic solids.  Apart from a few special cases, almost all the known solutions are two-dimensional (plane strain or anti-plane shear).  Even simple 3D problems such as a point force in an infinite solid or on the surface of a half-space are generally only known in Fourier transform space.  The theory is still evolving, however, and a small but dedicated group of elasticians have made substantial progress in recent years.

Here, we will present briefly the various techniques and analytical solutions that are available today. We will summarize approaches to solving plane problems first, before turning to 3D boundary value problems.

Constitutive Equation

The constitutive equation for anisotropic materials has the form  where  are components of stress, and  are the components of the infinitesimal strain tensor.  The elasticity tensor  has symmetries .

For anisotropic materials the constitutive equations are best expressed in matrix form.  Thus, we write

where , etc.  The inverse has the form

Note that the vector definitions of strain  and stress  used in this section differ slightly from our earlier formulation (for consistency with Ting `Anisotropic Elasticity’)

Basis changes

The material constants  or  for a particular material are usually specified in a basis with coordinate axes aligned with particular symmetry planes (if any) in the material.  When solving problems involving anisotropic materials it is frequently necessary to transform these values to a coordinate system that is oriented in some convenient way relative to the boundaries of the solid.  Since  is a fourth rank tensor, the basis change formulas are highly tedious, unfortunately.  We will list them below for completeness.

To this end, let us suppose that the components of the stiffness tensor are given in a basis , and we wish to determine its components in a second basis, .  We define the transformation tensor  with components , or in matrix form

This is an orthogonal tensor satisfying . In practice, the tensor can be computed in terms of the angles between the basis vectors. It is straightforward to show that stress, strain and elasticity tensors transform as

The basis change formula for the elasticity tensor is more conveniently expressed in matrix form as

where the rotation matrix K is computed as

where the modulo function satisfies

Although these expressions look cumbersome they are quite convenient for computer implementation.

The basis change for the compliance tensor follows as

where

The proof of these expressions is merely tiresome algebra and will not be given here.  Ting’s book (1996) has a nice clear discussion.

For the particular case of rotation through an angle  in a counterclockwise sense about the  axes, respectively, the rotation matrix reduces to

where . The inverse matrix  can be obtained simply by changing the sign of the angle  in each rotation matrix.  Clearly, applying the three rotations successively can produce an arbitrary orientation change.

9.3 Governing Equations

Our mission is to find solutions to the Navier equation of elasticity

subject to the usual boundary conditions.  In subsequent discussions, we will largely need to abandon index notation and instead use matrix notation.  Thus, the equilibrium equations will be written as

Plane deformations

We now seek either anti-plane shear solutions of the form , or plane strain deformations of the form .

Note first that such solutions cannot exist in general anisotropic materials.  We will first establish the conditions necessary for the existence of plane deformations.

For anti-plane shear deformations the equilibrium equations reduce to

The three equations cannot all be satisfied by the same displacement; moreover elastic constants must satisfy .  Thus, anti-plane shear deformations can exist if and only if

The governing equation is thus

For Plane strain deformations the equilibrium equations reduce to

In this case,  can be chosen to satisfy two out of the three equations, but not all three.  The elastic constants must satisfy .  Consequently, the third equation can only be satisfied by setting

The governing equations for plane deformations are

Plane deformations therefore only exist in a material with elastic constants and orientation satisfying

The most common class of crystals – cubic materials – satisfies these conditions for appropriate orientations.