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Chapter 5
Analytical techniques and solutions for linear elastic solids
5.5 Solutions to generalized plane problems for anisotropic linear elastic solids
Materials such as wood, laminated composites, and single crystal metals are stiffer when loaded along some material directions than others.  Such materials are said to be anisotropic, and cannot be modeled using the procedures described in the preceding sections.  In this chapter, we describe briefly the most widely used method for calculating elastic deformation and stress in two dimensional anisotropic solids. As you might expect, these calculations are difficult, and while the solutions can be expressed in a surprisingly compact form, the resulting expressions can usually only be evaluated using a computer. In many practical situations it is simplest to calculate solutions for anisotropic materials using direct numerical computations (e.g. using the finite element method, discussed in Chapters 7 and 8).  Nevertheless, analytical solutions are useful: for example, the finite element method cannot easily be applied to problems involving cracks, dislocations, or point forces, because they contain singularities; in addition exact calculations can show how the solutions vary parametrically with elastic constants and material orientation.  Â
5.5.1 Governing Equations of elasticity for anisotropic solids
A typical plane elasticity problem is illustrated in the
picture. The solid is two dimensional:
in this case we are concerned with plane strain solutions, which means that the
solid is very long in the
To simplify calculations, we shall assume that (i) The solid is free of body forces; (ii) thermal strains can be neglected.  Under these conditions the general equations of elasticity listed in Section 5.1.2 reduce to
subject to the usual boundary conditions. In subsequent discussions, it will be convenient to write the  equilibrium equations in matrix form as
Conditions necessary for strict plane
strain deformation of anisotropic solids.Â
For Plane strain deformations the displacement field has the form
In this case,
Strict plane deformations therefore only exist in a material with elastic constants and orientation satisfying
The
most common class of crystals
Generalized plane strain deformations. A generalized plane strain displacement field can exist in any general anisotropic crystal. In this case the displacement field has the form
i.e. the displacement is independent of position along the length of the cylindrical solid, but points may move out of their original plane when the solid is loaded.
5.5.2 Stroh representation for fields in anisotropic solids
The Stroh solution is a compact, complex variable, representation for generalized plane strain solutions to elastically anisotropic solids. To write the solution, we need to define several new quantities: 1. We define three new 3x3 matrices of elastic constants, as follows
2. We introduce three complex valued eigenvalues
The eigenvalues can be computed by solving the equation
Since Q, R and T are 3x3 matrices,
this is a sextic equation for p, with 6 roots. It is possible to show that for a material
with physically admissible elastic constants p is always complex, so the 6 roots are pairs of complex
conjugates 3. To calculate the stresses, it is helpful to
introduce three further vectors
4. It is often convenient to collect the eigenvectors
Note also that, as always, while the eigenvalues
where I is the identity matrix.
General representation of displacements: The displacement
where
General representation of stresses: The stresses can be expressed in terms of a vector
valued stress function
The stresses can be
calculated from the three components of
Combined matrix representation for displacement and stresses:Â The solution for the displacement field and stress function can be expressed in the form
where
Simpler representation for stresses and displacements: The solutions given above are the most general form of the generalized plane strain solution to the governing equations of linear elasticity. However, not all the solutions of this form are of practical interest, since the displacements and stresses must be real valued.  In practice most solutions can be expressed in a much simpler the form as
where Re(z) denotes the real part of z,
and
5.5.3 Demonstration that the Stroh representation satisfies the governing equations
Our first objective is to
show that a displacement field of the form
To see this, 1. Note thatÂ
2. Substituting this result into the governing equation shows that
3. This can be re-written as
or in matrix form as
where
Our next objective is to show that stresses can be computed from the formulas given in Section 5.5.2. To see this, 1. Note that the stresses can be obtained from the
constitutive equation 2. Recall that for each of the six characteristic
solutions we may obtain displacements as
where Q, R and T are the matrices defined in the preceding section. 3. To simplify this result, define
and note that the governing equations require that
4. Combining the results of (2) and (3) shows that stresses can be computed from
5. Finally, recall that the stress function
5.5.4 Stroh eigenvalues and anisotropy matrices for cubic materials
Since
the eigenvalues p for a general
anisotropic material involve the solution to a sextic equation, an explicit
general solution cannot be found. Even
monoclinic materials (which have a single symmetry plane) give solutions that
are so cumbersome that many symbolic manipulation programs cannot handle them. The solution for cubic materials is
manageable, as long as one of the coordinate axes is parallel to the
whence
The characteristic equation therefore has the form
giving
whence
For
The matrices A and B can be expressed as
There
are some materials for which the general procedure outlined in the preceding
sections breaks down.  We can
illustrate this by attempting to apply it to an isotropic material. In this case we find that
The physical significance of this degeneracy is not known. Although isotropic materials are degenerate, isotropy does not appear to be a necessary condition for degeneracy, as fully anisotropic materials may exhibit the same degeneracy for appropriate values of their stiffnesses.
S. T. Choi, H. Shin and Y. Y. Earmme, Int J. Solids Structures 40, (6) 1411-1431 (2003) have found a way to re-write the complex variable formulation for isotropic materials into a form that is identical in structure to the Stroh formulation. This approach is very useful, because it enables us to solve problems involving interfaces between isotropic and anisotropic materials, but it does not provide any fundamental insight into the cause of degeneracy, nor does it provide a general fix for the problem.
In many practical situations the problems associated with degeneracy can be avoided by re-writing the solution in terms of special tensors (to be defined below) which can be computed directly from the elastic constants, without needing to determine A and B.
5.5.6 Fundamental Elasticity Matrix
The
vector
where the matrices
To see this, note that the expressions relating vectors a and b
can be expressed as
Since T is positive definite and symmetric its inverse can always be computed. Therefore we may write
and therefore
This is an eigenvalue equation, and multiplying out the matrices gives the required result.
The second identity may be proved in exactly the same way.  Note that
so
again, giving the required answer.
For non-degenerate materials N has six distinct eigenvectors. A matrix of this kind is called simple. For some materials N has repeated eigenvalues, but still has six distinct eigenvectors. A matrix of this kind is called semi-simple. For degenerate materials N does not have six distinct eigenvectors. A matrix of this kind is called non semi-simple.
5.5.7 Orthogonal properties of Stroh matrices A and B
The
observation that
In addition. the vectors can always be normalized so that
If this is done, we see that the matrices A and B must satisfy
Clearly the two matrices are inverses of each other, and therefore we also have that
These results give the following relations between A and B
5.5.8 Barnett-Lothe tensors and the Impedance Tensor.
In this section we define four important tensors that can be calculated from the Stroh matrices A and B. Specifically, we introduce: The
Barnett-Lothe
tensors The
Impedance
Tensor with properties The following relations between the Barnett-Lothe tensors and the impedance tensor are also useful
Many solutions can be expressed in terms of S, H and L directly, rather than in terms of A and B. In addition, Barnett and Lothe devised a procedure for computing S, H and L without needing to calculate A and B (See Sect. 5.5.11). Consequently, these tensors can be calculated even for degenerate materials.
As an example, for cubic materials, with coordinate axes aligned with coordinate directions, Â
5.5.9 Useful properties of matrices in anisotropic elasticity
We collect below various useful algebraic relations between the various matrices that were introduced in the preceding sections.
By
definition, a matrix
·
·
·
·
The matrices ·
· The Barnett-Lothe tensors are real (i.e. they have zero imaginary part). To see this, note that the orthogonality of A and B (see sect. 5.5.7) implies that
Therefore
· The impedance tensor can be expressed in terms of the Barnett Lothe tensors as
To see the first result, note that ·
·
The
second row of this equation is
5.5.10 Basis Change formulas for matrices used in anisotropic elasticity
The various tensors and matrices defined in the preceding sections are all functions of the elastic constants for the material. Since the elastic constants depend on the orientation of the material with respect to the coordinate axes, the matrices are functions of the direction of the coordinate system. Â
To this end: 1. Let 2. Let 3. Let 4. Let  5. Let 6. Similarly, let
In
addition, define rotation matrices
The following alternative
expressions for
The basis change formulas can then be expressed as
Derivation: These results can be derived as follow: 1. Note that the displacements transform as vectors, so
that
which
shows that 2. To find the expression for p, we note that
Therefore,
we may write
as required. 3. The basis change formulas for Q, R and T follow directly from the definitions of these matrices. 4. The basis change formula for B is a bit more cumbersome. By definition
Substituting
for
and
finally recalling that 5. The basis change formulas for the Barnett-Lothe tensors and impedance tensor follow trivially from their definitions. The basis change formulas justify our earlier assertion that these quantities are tensors.
5.5.11 Barnett-Lothe integrals
The basis change formulas in the preceding section lead to a remarkable direct procedure for computing the Barnett-Lothe tensors, without needing to calculate A and B. The significance of this result is that, while A and B break down for degenerate materials, S, H, and L are well-behaved. Consequently, if a solution can be expressed in terms of these tensors, it can be computed for any combination of material parameters.
Specifically, we shall show that S, H, and L can be computed by integrating the sub-matrices of the fundamental elasticity matrix over orientation space, as follows. Let
and define
Then
Derivation: To
see this, we show first that
where
and
which can be expressed as
as before, we can arrange this into an Eigenvalue problem by writing
whence
This shows that [a,b] are eigenvectors of the rotated elasticity matrix. Following standard procedure, we obtain the diagonalization stated.
Now, we examine
Integrating gives
(the sign of the integral
is determined by Im(p) because the branch cut for
5.5.12 Stroh representation for a state of uniform stress
A
uniform state of stress (with generalized plane strain deformation) provides
a very simple example of the Stroh representation. The solution can be expressed in several
different forms. Note that for a
uniform state of stress
In terms of these vectors the Stroh representation is given by
or, in matrix form
Derivation: To see this, recall that a and b form eigenvectors of the fundamental elasticity matrix N as
therefore we can write (for each pair of eigenvectors/values)
Hence
Recall that
so
and finally, defining
gives the required result.
5.5.13 Line load and Dislocation in an Infinite Anisotropic Solid
The figure illustrates the problem to be solved. We consider an infinite, anisotropic, linear elastic solid, whose elastic properties will be characterized using the Stroh matrices A and B.
The
solid contains a straight dislocation, with line direction
At
the same time, the solid is subjected to a line of force (with line direction
extending out of the plane of the figure). The force per unit length acting
on the solid will be denoted byÂ
The displacement and stress function can be expressed in terms of the Stroh matrices as
where
 The solution can also be expressed as
Derivation: Â We must show that the solution satisfies the following conditions: 1. The displacement field for a dislocation with
burgers vector b must satisfy 2. The resultant force exerted by the stresses acting on any contour surrounding the point force must balance the external force F. For example, taking a circular contour with radius r centered at the origin, we see that
3. We can create the required solution using properties of log(z). We try a solution of the form
where
4. Recalling the orthogonality properties of A and B
we can solve for q
giving
5.5.14 Line load and dislocation below the surface of an anisotropic half-space
The
figure shows an anisotropic, linear elastic half-space. The elastic properties of the solid are
characterized by the Stroh matrices A,
B and P defined in SectionÂ
5.5.2. The solid contains a dislocation with Burgers vector b and is also subjected to a linear
load with force per unit length F at
a point
The solution can be computed from the simplified Stroh representation
where
and
The first term in the expression for f will be recognized as the solution for a dislocation and point force in an infinite solid; the second term corrects this solution for the presence of the free surface.
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(c) A.F. Bower, 2008 |