Стр. 11 - Ю.И. Димитриенко, А.П. Соколов, С.В. Сборщиков - Моделирование микроразрушения тканевых композитов

ISSN 1812-3368. ȼɟɫɬɧɢɤ ɆȽɌɍ ɢɦ. ɇ.ɗ. Ȼɚɭɦɚɧɚ. ɋɟɪ. «ȿɫɬɟɫɬɜɟɧɧɵɟ ɧɚɭɤɢ». 2012 15
Ʉɪɢɬɟɪɢɣ ɩɪɨɱɧɨɫɬɢ ɤɨɦɩɨɡɢɬɚ. Ⱦɥɹ ɪɚɫɱɟɬɚ ɩɪɨɱɧɨɫɬɢ ɤɨɦɩɨ-
ɡɢɬɚ ɜ ɰɟɥɨɦ ɧɟɨɛɯɨɞɢɦɨ ɜɵɱɢɫɥɢɬɶ ɩɪɟɞɟɥɶɧɵɟ ɡɧɚɱɟɧɢɹ ɫɪɟɞɧɢɯ
ɧɚɩɪɹɠɟɧɢɣ ,
kl V ɩɪɢ ɤɨɬɨɪɵɯ ɩɪɨɢɫɯɨɞɢɬ ɩɟɪɜɢɱɧɨɟ ɦɢɤɪɨɪɚɡɪɭɲɟ-
ɧɢɟ ɜ ɨɞɧɨɦ ɢɡ ɤɨɦɩɨɧɟɧɬɨɜ (ɜ ɦɚɬɪɢɰɟ ɢɥɢ ɜ ɧɢɬɹɯ) ɜ ɤɚɤɨɣ-ɥɢɛɨ
ɨɞɧɨɣ ɬɨɱɤɟ *
k V  [D [ ɜ ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ t*, ɚ ɡɚɬɟɦ ɩɨɥɧɨɟ ɪɚɡɪɭɲɟ-
ɧɢɟ, ɩɪɢ ɤɨɬɨɪɨɦ ɩɪɨɢɡɨɣɞɟɬ ɧɚɪɭɲɟɧɢɟ ɰɟɥɨɫɬɧɨɫɬɢ ɜɫɟɣ əɉ. Ⱦɥɹ
ɜɵɱɢɫɥɟɧɢɹ ɩɪɟɞɟɥɶɧɵɯ ɡɧɚɱɟɧɢɣ ɧɚɩɪɹɠɟɧɢɣ, ɤɚɤ ɩɪɚɜɢɥɨ, ɪɟɚɥɢ-
ɡɭɸɬ ɩɪɨɰɟɫɫ ɥɢɧɟɣɧɨɝɨ ɧɚɝɪɭɠɟɧɢɹ (ɷɬɨ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɭɫɥɨɜɢɹɦ
ɩɪɨɜɟɞɟɧɢɹ ɷɤɫɩɟɪɢɦɟɧɬɨɜ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ ɩɪɨɱɧɨɫɬɢ ɦɚɬɟɪɢɚɥɨɜ),
ɩɪɢ ɤɨɬɨɪɨɦ ɫɪɟɞɧɢɟ ɧɚɩɪɹɠɟɧɢɹ ɢɡɦɟɧɹɸɬɫɹ ɜɨ ɜɪɟɦɟɧɢ ɩɪɨɩɨɪɰɢ-
ɨɧɚɥɶɧɨ: ( ) ,
kl kl t t V V ɝɞɟ kl V — ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɚ ɫɤɨɪɨɫɬɟɣ ɢɡ-
ɦɟɧɟɧɢɹ ɧɚɩɪɹɠɟɧɢɣ. ɉɨɞɫɬɚɜɥɹɹ ɫɨɨɬɧɨɲɟɧɢɹ (11) ɜ (14), ɚ ɡɚɬɟɦ
ɩɨɥɭɱɢɜɲɢɟɫɹ ɜɵɪɚɠɟɧɢɹ — ɜ ɤɪɢɬɟɪɢɢ ɩɪɨɱɧɨɫɬɢ ɦɚɬɪɢɰɵ (19) ɢɥɢ
ɧɢɬɟɣ (22), ɩɨɥɭɱɚɟɦ ɭɫɥɨɜɢɟ ɩɟɪɜɢɱɧɨɝɨ ɪɚɡɪɭɲɟɧɢɹ ɤɨɦɩɨɡɢɬɚ:
^ `
( ) ( ) * ( ) * ( ) * *
1 max ( ( ) ( ) ( ) ( ) ; l n
i r j r lnkm r km z Q Q B t D D D D
[ [ [ V
( ) ( ) * ( ) * ( ) * * (3) (3) * *
2 ( ( ) ( ) ( ) ( ); ( ( ) ( ))} 1 l n
i r j r lnkm r km lnkm r km z Q Q B t z B t D D D D
[ [ [ V [ V ,
ɝɞɟ *
r [ — ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɜ əɉ, ɜ ɤɨɬɨɪɨɣ ɜɵɩɨɥɧɹɸɬɫɹ ɭɫɥɨɜɢɹ
(23); *
t — ɦɨɦɟɧɬ ɜɪɟɦɟɧɢ, ɩɪɢ ɤɨɬɨɪɨɦ ɜɩɟɪɜɵɟ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨ-
ɜɢɟ (23); * ( )
km t V — ɩɪɟɞɟɥɶɧɵɟ ɧɚɩɪɹɠɟɧɢɹ, ɩɪɢ ɤɨɬɨɪɵɯ ɩɪɨɢɫɯɨ-
ɞɢɬ ɩɟɪɜɢɱɧɨɟ ɪɚɡɪɭɲɟɧɢɟ ɤɨɦɩɨɡɢɬɚ.
ɉɟɪɜɢɱɧɨɟ ɪɚɡɪɭɲɟɧɢɟ ɦɨɠɟɬ ɩɪɨɢɫɯɨɞɢɬɶ ɜ ɦɚɬɪɢɰɟ ɢɥɢ ɜ ɧɢɬɹɯ
(ɜ ɧɢɬɹɯ ɫɧɚɱɚɥɚ ɧɚɛɥɸɞɚɟɬɫɹ ɱɚɫɬɢɱɧɨɟ ɪɚɡɪɭɲɟɧɢɟ ɜɫɥɟɞɫɬɜɢɟ ɪɚɡ-
ɪɭɲɟɧɢɹ ɦɚɬɪɢɰɵ, ɨɤɪɭɠɚɸɳɟɣ ɦɨɧɨɜɨɥɨɤɧɚ). ɉɨɫɥɟ ɩɟɪɜɢɱɧɨɝɨ ɪɚɡ-
ɪɭɲɟɧɢɹ ɨɬɦɟɱɚɟɬɫɹ ɢɡɦɟɧɟɧɢɟ ɦɨɞɭɥɟɣ ɭɩɪɭɝɨɫɬɢ ɜ ɪɚɡɪɭɲɟɧɧɵɯ ɨɛ-
ɥɚɫɬɹɯ ɦɚɬɪɢɰɵ ɢɥɢ (ɢ) ɧɢɬɟɣ ɜ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɨɩɢɫɚɧɧɨɣ ɜɵɲɟ ɦɨɞɟ-
ɥɶɸ. ɉɨ ɦɟɪɟ ɞɚɥɶɧɟɣɲɟɝɨ ɭɜɟɥɢɱɟɧɢɹ ɫɪɟɞɧɢɯ ɧɚɩɪɹɠɟɧɢɣ ( )
km t V
ɭɫɥɨɜɢɟ ɪɚɡɪɭɲɟɧɢɹ (23) ɜɵɩɨɥɧɹɟɬɫɹ ɜ ɛɨɥɶɲɟɦ ɱɢɫɥɟ ɬɨɱɟɤ *
r [ əɉ,
ɬ. ɟ ɩɪɨɢɫɯɨɞɢɬ ɩɪɨɰɟɫɫ ɪɚɫɩɪɨɫɬɪɚɧɟɧɢɹ ɦɢɤɪɨɪɚɡɪɭɲɟɧɢɹ. ɉɪɢ ɬɟɯ
ɡɧɚɱɟɧɢɹɯ ** ( ),
km t V ɩɪɢ ɤɨɬɨɪɵɯ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ
( ) ( ) ** ( ) ** ( ) ** **
2 max{ ( ( ) ( ) ( ) ( ))} 1, l n
i r j r lnkm r km z Q Q B t D D D D [ [ [ V (24)
ɩɪɨɢɫɯɨɞɢɬ ɱɚɫɬɢɱɧɨɟ ɪɚɡɪɭɲɟɧɢɟ ɧɢɬɟɣ ɜ ɪɟɡɭɥɶɬɚɬɟ ɪɚɡɪɭɲɟɧɢɹ
ɧɚɯɨɞɹɳɟɣɫɹ ɜ ɧɢɯ ɱɚɫɬɢ ɦɚɬɪɢɰɵ. ɉɪɢ ɡɧɚɱɟɧɢɹɯ ***
( ),
km t V ɩɪɢ ɤɨ-
ɬɨɪɵɯ ɜɵɩɨɥɧɹɟɬɫɹ ɭɫɥɨɜɢɟ
( ) ( ) *** ( ) *** ( ) *** ***
1 max{ ( ( ) ( ) ( ) ( ))} 1, l n
i r j r lnkm r km z Q Q B t D D D D [ [ [ V (25)
ɧɚɛɥɸɞɚɟɬɫɹ ɩɨɥɧɨɟ ɪɚɡɪɭɲɟɧɢɟ ɤɨɦɩɨɡɢɬɚ ɜɫɥɟɞɫɬɜɢɟ ɪɚɡɪɵɜɚ
ɦɨɧɨɜɨɥɨɤɨɧ ɜ ɧɢɬɹɯ. Ⱦɚɥɶɧɟɣɲɟɟ ɭɜɟɥɢɱɟɧɢɟ ɧɚɩɪɹɠɟɧɢɣ
(23)