Homogenization theory 
for cellular and composite structures
at Narvik University College

Slides-presentasjon på norsk

Strongly non-homogeneous structures (see the examples given in the figures) have fascinated people for a very long time. Archaeological observations in Finland show that fibre- reinforced ceramics were made about 4000 years ago, and that people already at this time had ideas and theories for ‘‘intelligent’’ combinations of materials and structures. Analysis of the macroscopic properties of composites was initiated by the physicists Rayleigh, Maxwell and Einstein. Around 1970 one managed to formulate the physical problems of material structures and composites in such a way that this field became interesting from a purely mathematical point of view.  This formulation initiated a new mathematical discipline called homogenization theory. 

Optimal rank 3-laminate

Independently of this development scientists within the field of micro-mechanics have developed their own theory concerning the mechanics of composites and structures, often referred to as the theory of cellular solids. At Narvik Institute of Technology, HiN, scientists from both these fields work closely together. The local industri (Natech A/S) is also involved in this reseach activity (at the moment one Natech-employee is a doctorand in this field). 
Typical problems
By using these theories we can determine locale and global (effective) properties of inhomogeneous structures which are too complex to be treated by conventional computational methods. 
The theories make it possible to design material structures with optimal properties with respect to weight, strength, stiffness, heat conductivity, electric conductivity, magnetic permeability, viscosity etc.
Isotropic structure with negative Poisson’s ratio.
(Graeme Milton 1992) 

The discoveries so far even include isotropic materials with negative Poisson’s ratio close to -1 (i.e. when the material is expanded  in one direction it will expand in the orthogonal direction as well in almost the same ratio). 

Many of the structures illustrated on  this page are socalled iterated structures. Concerning the recent developement of the mathematical theory of such structures e.g. Braides-Lukkassen, 2000Lions, Lukkassen, Persson, Wall 2001 and Lukkassen-Milton, 2002

Properties of a chiral honeycomb with a Poisson's ratio -1 of Lakes. (for moreinformationseethehomepageofRod Lakes)

Homogenization theory in Narvik and Luleå

In Scandinavia the homogenization-groups at Luleå University of Technology and Narvik Institute of Technology, HiN, play an important role in the development of the homogenization theory and have since 1992 generated 5 PhD’s  within this area (for further information concerning Ph.D program click here). 

Optimal reiterated structure with respect to in-plane shear stiffness
(Dag Lukkassen 1998 pdf-file ).

Activities at HiN
Optimal reiterated structure with respect to out-of-plane shear stiffness.
(Dag Lukkassen  1997pdf-file).

The scientific staff at HiN doing research on homogenization and composites has good connections with foreign research institutions within homogenization theory, composites and cellular solids, such as Cambridge University, Moscow State University, SISSA in Italy, Collège de France, Royal Institute of Technology (KTH), Chalmers University, Lund University, New Dehli University. 

Moreover, they have been members of Norfa Network on structural optimization

Visiting Research Scientists at HiN

The research has been financially supported by a number of institutions and enterprises, e.g.  the oil company ELF, Natech A/S, The Norwegian Research Council, Nordland Fylke and the Ministry of Education and Research (KUF).

A link to related research groups (and individuals) can be found here.
Sandwich-construction with optimal stiffness properties
(Lukkassen 1998).


Further examples

Below we give some further illustrations of physical problems which can be analysed by the theory. 

A micro-structured I-beam optimized with respect to stiffness and strength 

(Zeuthen 1997). For similar structures see also a publication in <<Nature>> by R. Lakes 

Optimal reiterated cell-structure in 3D
(Dag Lukkassen  1996).

The ''Lavrentiev phenomenon'':

The conductivity r(r) takes values as specified in the figure on the right side, where r denotes the distance from the center of each circle and a(r)=r^(a), where 0<a<2 and k is a very small positive number.  It appears to be two natural ways of defining the effective conductivity of the structure. It might come as a surprise that one of these formulations implies a very high effective conductivity wheras the other implies a very small effective conductivity. 
The mathematical treatment of this problem can be found in 
Jikov-Lukkassen 2001


Low speed flow in porous media (Stokes flow) is an other example of physical problem which can be analysed by using the homogenization method. In the figure the structure is periodic. In the figure on the right side we show the stream-lines in one period.
Numerical computation by using ''Flow 3D'' (Bang-Lukkassen 1999)

Last Update:July 3, 2002 by Dag Lukkassen.