84 3-D textile reinforcements in composite materials micro-cell (xyz) combi-cell(123) LEVEL 3 matrix layer impregnated yarn layer LEVEL 4 matrix+fibre WV IL:OE LEVEL 5 3.8 Schematic of a micro-cell or combi-cell (level 3),matrix and yarn layers(level 4)and a unidirectional lamina with fibre and matrix 210e phases (level 5). 公 impregnated yarn layer(Fig.3.8).This is level 4.Finally,each impregnated yarn is treated as a unidirectional lamina with matrix and fibre phases.This % is level 5. 3.3.4 The complementary energy elastic model Previously,by constructing a multilevel decomposition scheme,the com- posite unit cell was split automatically into matrix and yarn cells.Presently, by a multistep homogenization procedure,a link is established between the external loading and the internal stresses.The principal idea lies in the inter- pretation that the stress 'concentration factors'can be computed at each step by applying the complementary variational principle.This principle states that from all the admissible stress fields,the true field is that which minimizes the total complementary energy (hence the name:complemen- tary energy model,CEM).We achieve a straightforward analytical stress model for woven fabric composites. The four-step homogenization model CEM is now developed.The Venn diagrams in Fig.3.9 show the link between the fractional cell volumes k,the stress concentration factors A and the compliance matrices S.With thisimpregnated yarn layer (Fig. 3.8). This is level 4. Finally, each impregnated yarn is treated as a unidirectional lamina with matrix and fibre phases. This is level 5. 3.3.4 The complementary energy elastic model Previously, by constructing a multilevel decomposition scheme, the com￾posite unit cell was split automatically into matrix and yarn cells. Presently, by a multistep homogenization procedure, a link is established between the external loading and the internal stresses.The principal idea lies in the inter￾pretation that the stress ‘concentration factors’ can be computed at each step by applying the complementary variational principle. This principle states that from all the admissible stress fields, the true field is that which minimizes the total complementary energy (hence the name: complemen￾tary energy model, CEM). We achieve a straightforward analytical stress model for woven fabric composites. The four-step homogenization model CEM is now developed. The Venn diagrams in Fig. 3.9 show the link between the fractional cell volumes k, the stress concentration factors A and the compliance matrices S. With this 84 3-D textile reinforcements in composite materials 3.8 Schematic of a micro-cell or combi-cell (level 3), matrix and yarn layers (level 4) and a unidirectional lamina with fibre and matrix phases (level 5). RIC3 7/10/99 7:37 PM Page 84 Copyrighted Material downloaded from Woodhead Publishing Online Delivered by http://woodhead.metapress.com Hong Kong Polytechnic University (714-57-975) Saturday, January 22, 2011 12:30:11 AM IP Address: 158.132.122.9