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.