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A homogenization method for heterogeneous materials using a multi‐scale technique
Author(s) -
Müller Wolfgang H.,
Brandmair Andreas
Publication year - 2011
Publication title -
pamm
Language(s) - English
Resource type - Journals
ISSN - 1617-7061
DOI - 10.1002/pamm.201110196
Subject(s) - homogenization (climate) , asymptotic homogenization , microstructure , microelectronics , materials science , hardening (computing) , material properties , composite material , mechanics , mathematical analysis , mathematics , composite number , physics , nanotechnology , biodiversity , ecology , layer (electronics) , biology
For modern microelectronics solders lifetime and stability predictions are important. To perform such an analysis material properties are required. As electronic devices and the corresponding amount of matter used become smaller, the influence of a changing microstructure on mechanical properties must be considered. First some analytical methods were conducted for upper and lower bounds ignoring the exact geometric distribution of the solder phases. Second, analytical equations derived for geometries such as laminate structures were applied to examine the influence of the geometry on homogenized properties. Third, a multi‐scale approach for periodic media was presented allowing for a more general analysis of structures. We assume that the solder material is composed of periodic cells, which represent the properties of the whole structure. Composite materials with periodic structures can be investigated by using at least two scales. A global scale is related to the whole piece of material whereas a local scale is related to the periodic cell only. The constitutive equations are stated and a homogenization technique for the elastic properties of arbitrary structures is derived. The resulting equations are solved numerically and results are presented. Again, for layered materials closed‐form formulas are derived and compared to the numerical results. The method is also used to obtain effective mechanical properties for materials with linear hardening. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)