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Preprocessing and postprocessing for materials based on the homogenization method with adaptive finite element methods. (English) Zbl 0737.73008
The paper deals with composite materials. Their high degree of heterogeneity needs to find some kind of equivalent medium with the same average mechanical behavior. The homogenization method is a rigorous mathematical theory to compute the equivalent material properties. It is usually assumed that the composite is locally formed by the spatial repetition of small microstructures, the microscopic cells, when compared with the macroscopic dimensions of the structure of interest. This enables the computation of equivalent material properties by a limiting process when the microscopic cell size, $$\varepsilon$$, tends to zero.
In this paper the method is applied to a linear elastic material with a periodic distribution of holes. The paper gives at first a clear review of the homogenization theory. The elastic connected domain is subjected to body forces, to traction and prescribed displacement on its boundary, together with traction inside the holes. Homogenization leads to the solution of three distinct problems: two in the microscopic cell, which give the homogenized elastic coefficients $$D_{ijkl}$$ and the residual stress $$\tau_{ij}$$ (due to the traction inside the holes), and the other on the macroscopic level, to obtain the macroscopic displacement field $$\vec u^ 0$$ of the homogenized problem.
The finite element solution procedure to obtain the homogenized elastic coefficients $$D_{ijkl}$$, the residual stress $$\tau_{ij}$$ and the macroscopic displacement field $$\vec u^ 0$$ is presented. Then n the homogenization method is used to introduced the idea of a material preprocessor (PREMAT) for the computation of the homogenized properties and a material postprocessor (POSTMAT) for the computation of stress and strain distribution within the composite microstructure after a finite element procedure has been used to solve the general structure. Several examples are presented: fibers in elasticity, honeycomb structures, woven fiber reinforced composites. Finally, an adaptive finite element method is introduced in order to improve the accuracy of the numerical results.
Reviewer: Th.Lévy (Paris)

##### MSC:
 74E05 Inhomogeneity in solid mechanics 74S05 Finite element methods applied to problems in solid mechanics 74E30 Composite and mixture properties
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