EN222: Mechanics of Solids

  

 

 

 

 

Division of Engineering

    Brown University

 

 

4.4 Potential and complex variable representations for 2D static linear elasticity

 

Potential representations can be used to simplify the governing equations for 2D problems as well as 3D problems.  We summarize three commonly used techniques below.

 

Complex variable methods for static anti-plane shear problems

 

The governing equations for static anti-plane shear deformation are

where

These equations are so simple there’s not much need to simplify them further.  There are lots of ways to solve the Laplace equation, of course.  Here we will illustrate just one approach, which is to exploit properties of functions of a complex variable.  The motivation is to lead up to more sophisticated complex variable methods for general plane stress or plane strain problems.

 

Complex variable methods are among the most powerful methods for solving Laplace’s equation.  The approach is based on the observation that both the real and imaginary parts of an analytic function satisfy Laplace’s equation.  To see this, consider a complex function

If f is analytic, the derivative with respect to z is path independent, which requires

(these are known as the Cauchy-Riemann conditions) so that

since, for example

 

Thus, both the real and imaginary parts of an analytic function satisfy the Laplace equation.  To exploit this, we let  characterize the position of a point in the plane of the solid of interest.  Then let  denote any analytic function of position.  We can set

and  then automatically satisfies the equilibrium equation .  We can solve our elasticity problem by finding an analytic function that satisfies appropriate boundary conditions.  Of course, we can just as well use

instead.

 

 

Stresses determined from the complex potential

If we use the real or imaginary part of an analytic function as a solution to an anti-plane shear problem, we can determine the stresses from f(z) directly.  Note that

Suppose we choose   , then

 

Where   denotes the complex conjugate.

 

Similar expressions can be determined for .  In this case

 

 

Some examples of complex potentials for Anti-Plane shear

 

An immediate consequence of this observation is that the real and imaginary parts of any analytic function will solve some anti-plane shear problem.  We can generate solutions by superposing analytic functions.

 

 

The fundamental solution for an infinite solid

 

Find the displacement and stress fields induced in an infinite solid by a line load  acting at the origin of an infinite solid.  This is clearly an anti-plane shear problem.  Guided by the solution to the 3D Kelvin problem, we might try generating the solution from the function

where C must be determined.

This is analytic everywhere except the origin. Let us choose . Using the results outlined in the preceding subsection, we may readily compute the resultant force acting on a circular arc enclosing the origin:

 

and note furthermore that

so that, plugging back into the integral, we find that

Therefore, if we choose , then

will therefore generate the solution we need.  Of course we can simplify this as

where .

 

 

 

Screw Dislocation

 

What happens if we choose

Recall that , so that the displacement field is evidently multiple valued, with a jump of magnitude b across any radial branch cut.   Furthermore, applying the results of the preceding section, we see that the resultant force on any closed contour vanishes.  This is therefore the displacement for a screw dislocation at the origin.  The stresses follow as

Here we’ve just guessed complex functions and looked at what problems they solve.  But the real power of complex variable methods is that a number of clever techniques exist for finding analytic functions that satisfy prescribed boundary conditions.  These are

1.                              Expansion of analytic functions in Laurent series, and matching coefficients in the series to satisfy boundary conditions.  This procedure can be made more or less automatic using Fourier transform techniques

2.                              Use of the Cauchy integral formula

3.                              Conformal mapping techniques

4.                              Use of analytic continuation

among others.  These are discussed in detail in the linear elasticity notes.

 

 

Airy Stress Functions Solution to Plane Stress and Plane Strain Problems

 

 

The governing equations for static plane strain or plane stress problems are

 (Plane strain)

  (Plane stress)

subject to boundary conditions

 

 

One way to solve the equations is to generate the stresses from a scalar potential, chosen so as to satisfy the equilibrium equations automatically.  For simplicity, we will neglect body forces (the solution with body forces can be found in the linear elasticity notes).

 

To generate an equilibrium stress field, let   be a scalar potential on  , and let

 

Then you can verify by direct substitution that

thereby satisfying equilibrium.

 

Of course, the stress state in a solid must satisfy more than just equilibrium conditions.  One must also ensure that the strain field associated with it is compatible.  Recall the stress equations of compatibility, which reduce to

in 2D.  To satisfy this condition,  must satisfy

 

Completeness.  As for 3D problems, one must also show that a potential representation is complete, that is, all 2D elastostatic states may be derived from an Airy function.  It is not hard to do this, but the details are left to the linear elasticity course.

 

 

 

 

 

 

 

Airy Functions in Cylindrical-Polar Coordinates

 

 

Boundary value problems involving cylindrical regions are best solved using Cylindrical-polar coordinates.  It is worth recording the governing equations for this coordinate system.

 

 

 

The state of stress is related to the Airy function by

 

 

Example of Airy Functions

 

As an example, the stress fields due to a line load magnitude P per unit out-of-plane length acting on the surface of a homogeneous, isotropic half-space can be generated from the Airy function

A straightforward exercise shows that

 

A major disadvantage of Airy function solutions to plane elasticity problems is that it is a big pain to calculate displacement fields.  To do this, you need to determine the strains by substituting the expression for stress into the constitutive relation, and then integrating strains to calculate displacements.  A formal procedure to do this can be found in the linear elasticity notes, in the section discussing compatibility (there’s a closed form 3D expression that can be used to get displacements for any 3D strain distribution).

 

For the point force solution here, one can show that

to within an arbitrary rigid motion.  Note that the displacements vary as log(r) so they are unbounded both at the origin and at infinity.  Moreover, the displacements due to any distribution of traction that exerts a nonzero resultant force on the surface will also be unbounded.

 

 

Complex Variable Representations for Plane Elastostatics

 

Airy functions have been used to find many useful solutions to plane elastostatic boundary value problems.  The method does have some limitations, however.  The biharmonic equation is not the easiest field equation to solve, for one thing. Another limitation is that displacement components are difficult to determine from Airy functions, so that the method is not well suited to displacement boundary value problems.

 

We found the complex variable methods led to a systematic procedure for solving anti-plane shear problems.  It is natural to attempt to develop a similar approach for more general plane problems.  It turns out that this can indeed be done, although the results are rather more complex (in one sense of the word) than those for anti-plane shear.

 

 

 

 

 

The basic idea is very similar to the technique discussed for anti-plane shear deformation earlier.  Let

 be two analytic functions on the plane region . 

Define a complex displacement field

 

Then a displacement field generated from

can be shown (by substitution and some algebra) to satisfy the two dimensional Cauchy-Navier equation

Furthermore, the stress state can be computed easily from

 

As always we need to show that all solutions can be represented in this way.  This is not hard.  See the linear elasticity notes if you are curious.

 

 

Here are two examples of complex potential representations for two dimensional elastostatic states.

 

Line load in an infinite solid

We are looking for  the stress and displacement field induced by a line load acting at the origin of a plane region.  The solution must satisfy

Condition (a) ensures that the resultant force due to tractions acting on any closed arc enclosing the origin are in equilibrium with the applied force.  The second two boundary conditions ensure that stresses and displac

When we solved the anti-plane shear version of this problem, we found that f(z)=log(z) produced a displacement field of the correct form.  It is worth trying the same thing here.

Suppose that

where A and B are complex coefficients, to be determined. Both potentials are analytic everywhere except at the origin.

 

These potentials clearly satisfy conditions (b) and (c) above.  In addition, the displacement field must be single valued.  Recall that

Note that

so we must set

to ensure that the displacement field is single valued.

 

Finally, we need to satisfy condition (a).  As for the equivalent anti-plane shear problem, we do this by computing the resultant force acting on a circular curve enclosing the origin.  It is left as an exercise to show that

We must therefore select  A and B so that

 

 

 

Where F is the magnitude of the force acting at the origin.

 

 

Finally, solving for A and B shows that

 

 

 

 

 

Edge Dislocation in an Infinite Solid

 

 

Our potentials for a point force evidently have the potential (excuse me) to generate multiple valued displacement fields.  Indeed, they can be combined to generate a displacement field with a constant jump across any radial branch cut.  This is the displacement field associated with an edge dislocation in an infinite solid.

 

Thus, choose A and B so that

 

 

(the sign convention for the Burger’s vector b is arbitrary. The statement above defines what we mean by a positive Burger’s vector).

 

This requires

For the dislocation, we require the resultant force on any closed curve to vanish. From the point force problem, we see that this requires

 

 

Solving for A and B shows that

 

Here we’ve just guessed complex functions and looked at what solutions they solve.  But the real power of complex variable methods is that a number of clever techniques exist for finding analytic functions that satisfy prescribed boundary conditions.  These are

1.                              Expansion of analytic functions in Laurent series, and matching coefficients in the series to satisfy boundary conditions.  This procedure can be made more or less automatic using Fourier transform techniques

2.      Use of the Cauchy integral formula

3.      Conformal mapping techniques

4.      Use of analytic continuation

among others.  These are discussed in detail in the linear elasticity notes.