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P.A. Belov
JSC «Moscow Experimental Machine building Plant – Composite Technologies»,
Moscow, Russia
THE THEORY OF MEDIA WITH CONSERVED DISLOCATIONS:
GENERAL AND APPLIED THEORIES OF INTERFACE LAYER
This work presents the formulation and the proof of the theorems, establishing equivalence of
gradient models and classical model of the non-uniform media, for models of media with fields of conserved dislocations. Tensor of moduli of the equivalent classical non-uniform media is presented in the
form of obvious function of components of tensor of dislocation damaging and components of tensors of
moduli, reflecting dislocation properties of media. It is shown, that the area, where the this tensor essentially depends on coordinates, is localized around of surfaces, lines and points of indignation. Such area
is treatment as the interphase layer.
Key words: damaged media mechanics, fields of conserved dislocations, interface layer, effective properties of composites, nonclassical moduli.
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(* /$)$&0()'< – 8'&/'-0-('< (0-=')'< Cijnm
, Cijnm
, Cijnm
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8 )$=-@& '>N08($& ((0-=')$& Cijnm
, Cijnm
, Cijnm
) * -0 &',A( >@(; '>N0-
3*-0-@. J0.B$=-@1 %+'1 3+5 8$.3'1 B$=@ 5<+50(%5 *='()'/-@& -08+$%%*C0%8*& '>N08('&, '>+$3$67*& -08+$%%*C0%8*&* &04$-*C0%8*&* %<'1%(<$&*, 8'(')@0 < '>70& %+AC$0 4$)$8(0)*=A6(%5 30<5(;6 /$)$&0()$&*, $ < /)*>+*.0--'& (/)*8+$3-'&) <$)*$-(0 – /5(;6.
5'6%'*0"+1'7#.4'8 .!'.*4
1. MA);0 I.O., P0+'< E.O. F0')*5 %)03 % %'4)$-567*&*%5 3*%+'8$9*5&*. Q$%(-@0 %+AC$*: %)03@ "'%%0)$ * OG)'-"A<2*-%8',', /')*%(@0 %)03@, %)03@ % «3<'1-*8'<$-*0&» / I'<)0&0--@0 /)'>+0&@ &04$-*8* ,0(0)',0--@4 %)03: %>. (). 8'-B.; K--( /)*8+. &04$-*8* #OR. –
J., 2005. – I. 235–268.
2. Lurie S.A., Belov P.A. Gradient theory of media with conserved
dislocations: application to microstructure materials. BOOK series «Advances in Mechanics and Mathematics». Generalized Continua. Springer. –
N. Y., 2010.
12
3. Lurie S., Belov P., Tuchkova N. The Application of the multiscale
models for description of the dispersed composites // Int. J. Comp Mater
Sci, A. – 2005. – Vol. 36(2). – P. 145–152.
4. Interphase layer theory and application in the mechanics of composite
materials / S.A. Lurie, P.A. Belov, D.B. Volkov-Bogorodsky, N.P. Tuchkova //
J. Mat. Sci. – 2006. – 41(20). – P. 6693–6707.
5. Eshelby’s inclusion problem in the gradient theory of elasticity.
Applications to composite materials / S. Lurie, D. Volkov-Bogorodsky, A. Leontiev, E. Aifantis // International Journal of Engineering Science. – 2011. –
Vol. 49. – P. 1517–1525.
6. P0+'< E.O., MA);0 I.O. "'-(*-A$+;-$5 (0')*5 $3,0=*'--@4
<=$*&'301%(<*1 /'<)0.30--@4 %)03 // J04$-*8$ 8'&/'=*9*'--@4 &$(0)*$+'< * 8'-%()A89*1. – 2009. – F. 15, S 4. – I. 610–629.
References
1. Lurie S.A., Belov P.A. The theory of continuums with conserved
dislocations. Particular cases: Cosserat and Aero-Kuvshinsky continuums,
continuums with porous and twinning [Teorija sred s sohranjajuwimisja dislokacijami. Chastnye sluchai: sredy Kossera i Ajero-Kuvshinskogo, poristye
sredy, sredy s "dvojnikovaniem"]. The collection of works of conference
«Modern problems of mechanics of heterogeneous environments», 2005, Institute of Applied Mechanics of Russian Academy of Sciences, pp. 235–268.
2. Lurie S.A., Belov P.A. Gradient theory of media with conserved
dislocations: application to microstruc-tured materials // One hundred years
after the Cosserats. Series: Advances in Mechanics and Mathematics. 2010.
Vol. 21. Maugin, G.A., Metrikine, A.V. (Eds.).1st Ed. Springer.pp. 223–
232.
3. Lurie S., Belov P., Tuchkova N. The Application of the multiscale
models for description of the dispersed composites // Int. J. Comp Mater
Sci, A. 2005. Vol. 36(2). 145–152.
4. Lurie S.A., Belov P.A., Volkov-Bogorodsky D.B., Tuchkova N.P.
Interphase layer theory and application in the mechanics of composite materials // J. Mat. Sci. 2006. 41(20). P. 6693–6707.
5. Lurie S., Volkov-Bogorodsky D., Leontiev A., Aifantis E.
Eshelby’s inclusion problem in the gradient theory of elasticity. Applications to composite materials // International Journal of Engineering Science.
2011. Vol. 49. #. 1517–1525.
13
6. Belov P.A., Lurie S.A. The theory of adhesive interactions of the
damaged continuums [Kontinual'naja teorija adgezionnyh vzaimodejstvij
povrezhdennyh sred] // Jechanics of composite materials and designs.
2009. Vol. 15. No. 4. #. 610–629.
6 !"#$ %
&'(#! )'"$ *+ "#(,'!-. ( !"#$%, &!""'() – #%)*'*%+ ,'-'#!.%+/.%+'0/"#'1 )%2#, )%0%34)'# !+*/3% 56!0)!"+' 787 « !"#!$"#'9
.%:')!"+6!'+/34);9 <#"5/6'./)+%34);9 -%$!* – #!.5!-'='!));/
+/1)!3!>''» >!". #!)=/6)% «&!"?/1)!3!>''» (>. !"#$%, %>'"+6%34);9 56!/-*, *. 9., e-mail: BelovPA@yandex.ru).
About the authors
Belov Petr Anatolevich (Moscow, Russia) – PhD, Head of Stress and
Resource Team JSC «Moscow Experimental Machine building Plant –
Composite Technologies» (123290, 9, 1-st Magistralniy passage, Moscow,
e-mail: BelovPA@yandex.ru).
@!320/)! 28.10.2011
14
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