Exact relations for effective moduli of polycrystals

Composite materials play an increasingly important role in our everyday life and technology from skis and golf clubs to sensors and actuators in high tech components. By “composite” we mean a perfectly bonded homogeneous mixture of two or more materials on a length scale much smaller than human size and much larger than interatomic distances. The physical properties of a composite (conducting, elastic, etc.) are described by a tensor—the effective tensor of a composite. In order to create a composite with desired properties two basic problems become important: prediction of the effective tensors of composite materials and determination of the properties of a given composite by as few measurements as possible. The principal difficulty in prediction is the universally recognized fact that the effective tensors of composites in general depend on the microstructure (spatial arrangement of component materials). Therefore, the object of importance is the set of all possible effective properties of a composite made with given materials taken in prescribed volume fractions (a so called G-closure set). Unfortunately, aside from a few particular cases the G-closures are extremely difficult to compute analytically. Usually a G-closure has a non-empty interior in the space of all tensors of appropriate type and may be described by a set of inequalities. On rare occasions researchers have found that a G-closure has empty interior, i.e. becomes part of a hyper-surface. The equations describing such a hyper-surface are called exact relations for effective moduli of a composite. We will also use the same term for the hyper-surface itself. When an exact relation is present the variability of an effective tensor with the microstructure is affected drastically: at or near an exact relation the efforts of achieving certain properties by varying the microgeometry may be futile. At the same time the number of expensive measurements needed to determine material moduli may be significantly reduced. For such important materials as piezoelectrics the number of constants needed for its description is 45 in general but it is only 9 on a 9-dimensional exact relation. There is a large plethora of known exact relations in various contexts (too large to give a fair list of references). The two most famous ones are Keller-Dykhne-Mendelson relation for 2-D conductivity: detσ∗ = const, [4, 16, 18], and Hill’s result for elasticity that a mixture of isotropic materials with the fixed shear modulus is isotropic with