Inverse problem for the structure of composite materials
Elena Cherkaev
Abstract:
In heterogeneous media such as artificial composites and
porous geological or biological materials, the scale of microstructure
is much smaller than the wavelength of the applied electromagnetic
or acoustic signal. In this situation, the wave cannot resolve all the
details of the microgeometry, and the response of the medium is homogenized.
The talk discusses a problem of deriving information about the fine
scale structure of a two phase composite from homogenized or
effective data provided by low frequency measurements.
The approach is based on reconstruction of the spectral measure of
a self-adjoint operator that depends on the geometry of the composite.
This measure contains all information about the microgeometry.
I show that the problem of identification of the spectral function
from effective measurements known in an interval of frequency, has a unique
solution. In particular, the volume fractions of materials in the
composite and an inclusion separation parameter, as well as the spectral
gaps at the ends of the spectral interval, can be uniquely recovered.
I will discuss reconstruction of microstructural parameters
from electromagnetic and viscoelastic measurements, application to
coupling of different effective properties, and show an extension to
nonlinear composites.