Division of Engineering
Brown University
9.4 Solutions to selected boundary value problems for anisotropic solids
Uniform stress
The Stroh representation for a uniform state of stress (with generalized plane strain deformation) can be expressed in several different forms. Note that for a uniform state of stress and corresponding strain we may write
In terms of these vectors the Stroh representation is given by
or, in matrix form
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, setting
gives the required result.
Point Force and Dislocation in and Infinite Solid
We construct dislocation and point force solutions in the usual way. The displacements for a dislocation should be multiple valued – if we introduce a branch cut along any radial line L from the origin, the displacements should satisfy , where b is the Burger’s vector of the dislocation. The resultant force acting on any closed curve in the solid should vanish. In contrast, the displacements induced by a point force (really a line load extending out of the plane) should be single valued, and the resultant force acting on any closed curve encircling the force should be equal and opposite to the point force. These conditions can be expressed as
As usual we can create the required solution using properties of log(z). We try a solution of the form
where and q is a vector to be determined. Using the properties of log(z), we have that
Recalling the orthogonality properties of A and B
we can solve for q
giving
Some simple algebra can be used to express the displacement and stress fields in terms of the impedance tensor M. Since M can be expressed in terms of Barnett-Lothe tensors, it can be evaluated without computing A and B, and can therefore be evaluated for degenerate materials. To see this, note that the equations
can be solved to give
The displacement and stress function can then be computed as
In addition, we recall that M is given in terms of the (real valued) Barnett-Lothe tensors S, H, L as
Half-space subjected to prescribed tractions
Half-space problems can be solved using the same analytic continuation procedure that we developed for isotropic elasticity.
We assume that the solution is to be found in the upper half-plane , and that the solution will be expressed using the Stroh representation
where
and
is a vector of functions (analytic in ) to be determined. Following the usual procedure, we first devise an analytic continuation that automatically generates a solution such that the real axis is traction free. This requires
This can be expressed as
The function on the left is analytic in , while the function on the right is analytic in . We therefore conclude that
is analytic everywhere. We can therefore set
To set up a solution with prescribed tractions on the real axis we must find f’(z) satisfying
which can be expressed in terms of as
This is a Hilbert problem, with solution
The stresses and displacements can then be determined from as
where . The expression for can also be expressed in a form that does not involve the eigenvalues p using the expression
which leads to
Point force and dislocation near a free surface
The procedure outlined in the preceding section can be used to construct the solution to a point force and/or dislocation in an anisotropic half-space. Assume that the singularity lies in the upper half-plane.
We will construct the solution as the sum of the solution to a dislocation/point force in an infinite solid, together with a correction that is added to satisfy the traction free boundary condition.
We can denote the solution to a point force/dislocation in an infinite solid by a vector of complex functions , where
where and denotes the position of the dislocation/point force. The singularities induce tractions
The corrective solution will be derived from a complex function following the procedure outlined in the preceding section. The function must satisfy
Note that is analytic in and is analytic in . Therefore, setting
satisfies
and therefore satisfies the boundary conditions. The solution can be written in a unified form by recalling the continuation established in the preceding section
whence . The combined infinite space solution with the corrective solution can be expressed as
with tractions and displacements obtained from g using the usual expressions
.
Crack in an infinite anisotropic solid under prescribed crack-face tractions
This problem can be solved using exactly the same analytic continuation that we used to solve the corresponding problem for an isotropic material. The solution will be constructed using vectors of two analytic functions which are analytic in , respectively. The functions must generate a continuous displacement and traction across the real axis, which requires that
We obtain a suitable continuation by rearranging these conditions in terms of functions , which are analytic in
which shows that
are analytic in the whole plane. These equations can be solved for the vectors . In view of the boundary conditions, it is convenient to express the results as
We can simplify these conditions by defining a (real valued) matrix G in terms of the impedance tensors for the solid
in which case
We now return to the boundary conditions on the crack faces
Noting that we can use the continuation to set , and using a similar expression for we find that the boundary conditions reduce to
Subtracting the second equation from the first, and using the definition of G, we see that
whence
where Q(z) is an arbitrary vector of analytic functions. If the stresses vanish at infinity, then Q(z)=0. If, in addition, the tractions acting on the crack faces are equal and opposite, we see that .
For the particular case of equal and opposite tractions acting on the crack faces, the governing equation for the remaining complex potential reduces to
Since is related to through it is convenient to let in which case the governing equation reduces to
This is a Hilbert problem – the different components in h decouple – and can be solved in the usual way
where , P(z) is an arbitrary polynomial, and it is understood that denotes for the three components of h. If tractions vanish at infinity then P(z)=0.
For the particular case of a uniform traction, we find that the various components of h are
where denote the components of traction acting on the upper crack face. Tractions and displacements can then be computed as
The tractions on for are of particular interest. Clearly
We see the usual square-root singularity at the crack tip. Stress intensity factors can be defined in the usual way
giving results that are identical to those for an isotropic solid. Although the stresses along are identical in both isotropic and anisotropic solids, the full crack tip fields differ.
Interface crack between two anisotropic solids
The solution for an interface crack between two anisotropic solids can be constructed using the same method. We characterize the two materials using the Stroh matrices , with corresponding notation for all matrices derived from these. The solution will be constructed using vectors of two analytic functions which are analytic in , respectively.
As usual, our first order of business is to find a continuation that satisfies traction and displacement continuity across the interface. Traction and displacement continuity follow as
We obtain a suitable continuation by rearranging these conditions in terms of functions , which are analytic in
which shows that
are analytic in the whole plane. These equations can be solved for the vectors . As before, it is convenient to express the results as
Again, we can simplify these conditions by defining a bi-material matrix G in terms of the impedance tensors for the two solids
in which case
We now return to the boundary conditions on the crack faces
Noting that we can use the continuation to set , and using a similar expression for we find that the boundary conditions reduce to
Subtracting the second equation from the first, and using the definition of H, we see that
whence
where Q(z) is an arbitrary vector of analytic functions. If the stresses vanish at infinity, then Q(z)=0. If, in addition, the tractions acting on the crack faces are equal and opposite, we see that .
For the particular case of equal and opposite tractions acting on the crack faces, the governing equation for the remaining complex potential reduces to
Since is related to through it is convenient to let in which case the governing equation reduces to
This is a vector valued Hilbert problem, but the components decouple only if G is real. If not, some further manipulations are required before the problem can be solved. The equations may be decoupled by diagonalizing , as follows. We first calculate eigenvectors and eigenvalues satisfying
or equivalently
Since G is Hermitian, the eigenvalue-eigenvector pairs have a special structure: , where is real. In terms of and eigenvalues we define
We can therefore write the Hilbert problem as
Finally, define to get
The equations for the three components of g now decouple, and are thus standard Hilbert problems. The equations for the three components of g can be expressed as
The three equations are solved using the usual procedure. Recall that the Plemelj function satisfies , so dividing both sides of the first equation by and using this result gives
so choosing gives
The solutions for the remaining components of g can be calculated the same way.
Because of the structure of the three eigenvalues of , the powers appearing in the Plemelj functions also have a very special structure. We find that
These three values of determine the structure of the asymptotic crack tip fields. Because two of the values are complex, and give rise to oscillatory crack tip singularities that resemble those for a crack on the interface between two isotropic solids. The solution associated with is just the square-root singularity encountered in homogeneous materials.