The fourth day: solid mechanics
Periodic homogenisation in solid mechanics
Most solid materials exhibit a distinct microstructure – i.e. a
characteristic arrangement of different phases – at one or several
spatial scales. Examples are glass fibres embedded in an epoxy
matrix, the polycrystalline grain structure of metals, or the
network of fibres of which paper consists. However, predictive
analyses in most applications require the effective material
properties at a much larger spatial scale which we term
macroscopic. These macroscopic properties depend heavily on the
properties of the individual microstructural phases, as well as on
their geometrical arrangement.
Characterising the relationship between (micro-)structure and
(macro-)properties allows one to establish macroscopic
constitutive laws and the parameters in them without extensive
experimental characterisation. Furthermore, and perhaps more
importantly, such structure–property relations reveal how the
microstructure should be changed in order to obtain certain
desired properties and thus allow one to design the ideal material
for a given application.
For linear elastic materials with (more or less) periodic
microstructures, an elegant and rigorous method exists to
determine structure–property relations: periodic homogenisation.
Based on a known heterogeneous distribution of elastic stiffness
at the microscale, it allows one to determine the properties of an
equivalent, homogeneous macroscopic elastic continuum. To this
end, the ratio of the relevant microscopic and macroscopic scales
is identified as a small parameter. The solution of the detailed
problem is written as a series of terms of increasing order of
this small parameter. By requiring that the governing equations
hold at each order, the problem may be split into a set of
microstructural problems and a macroscopic problem. The former can
be solved once and for all for a given microstructure. The latter
must be solved again for each new application. The connection
between the two scales is provided by the effective elastic
properties of the material, which can be computed based on the
solutions of the microstructural problems.
In this lecture we first illustrate the role played by
microstructure in various engineering materials. We then consider
in detail the periodic homogenisation method as outlined above and
illustrate its use. Extensions to higher orders and to nonlinear
behaviour are discussed briefly. We finish by a brief review of
other methods of establishing structure–property relations, which
are partially based on ideas taken from periodic homogenisation.