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Exact solutions for testing programs

There are many exact solutions of composite problems that are quite useful for testing programs like the ones in this package. Some of these are for special microstructures at general volume fractions, some are for general microstructures with special choices of the phase properties, while the others are for dilute microstructures, where a certain shaped inclusion with different properties is introduced into an initially uniform matrix. A general review of many of these dilute limits can be found in [11,12,13]. The relations most useful for testing programs are given in this manual. These kinds of programs can also be useful in exploring the dilute limit for inclusions with shapes that cannot be solved analytically [11].

The importance of having such exact solutions available for checking numerical computations should not be underestimated. It is usually easy to prepare numerical methods for uniform regions. Even including a simple boundary, such as exists in series or parallel problems, is not hard to do. But for random material problems, there are usually many boundaries, of general shape. A proper test of a numerical method will include such boundaries. However, random problems usually have no analytical solution, leaving the problem of how can a numerical result be assessed as to its accuracy? A numerical result for a random system can seem perfectly reasonable, and yet be 100% wrong. The problems discussed in this section give exact results for non-trivial boundaries and choices of phase moduli. These exact solutions can then be used to rigorously test if a numerical method will give the correct answer or not. These solutions can also be used to test things like the effect of resolution (number of pixels describing a microstructural feature) or finite size effects, which result when using periodic boundary conditions and a finite system size to simulate an infinite system.

• Definition of effective properties
• Series and parallel
• Small contrast of properties
• Keller and Mendelsen 2-D result for conductivity
• Field fluctuation result
• Equal shear modulus
• Intrinsic properties for spheres, circles, and cubes
• Vegard's law and Goodier result for thermal strains
• Hashin and Rosen thermal strain result
• Mackenzie result for pressurized pore space
• CLM Theorem

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