General solution for 3-D viscoelastic four –  element models that can be used at both numerical analysis of stressedly – deformed condition of road coatings and experimental investigations are offered

S. Bogomolov

V. Bogomolov

V. Zhdaniuk

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ǮȐȠȜȚȜȏȖșȪțȩȗ ȠȞȎțȟȝȜȞȠ, Ȑȩȝ. 29, 2011 43ɍȾɄ 624.048:625.7/.8  だゐべぎぎ づぎぷぎぞごぎ がずé ぶぎどぼづぎびÄずぎぜぎぞどぞぼび, ずごぞぎざぞぼび,  ゑéげとだばぢづばゎごび 3-D ぜだがぎずぎざ  ゑ.ん. ゐÜÇÜ½Ü¿Üç, äëÜâñïïÜë, Ñ.ö.Ö., ゑ.と. ぐÑíÖ0¡, äëÜâñïïÜë, Ñ.ö.Ö.,  で.ゑ. ゐÜÇÜ½Ü¿Üç, óÖ¢ñÖñë, びぞんがば  Ⱥɧɧɨɬɚɰɢɹ. ɉɪɟɞɥɨɠɟɧɵ ɨɛɳɢɟ ɪɟɲɟɧɢɹ ɞɥɹ 3-D ɜɹɡɤɨɭɩɪɭɝɢɯ ɱɟɬɵɪɟɯɷɥɟɦɟɧɬɧɵɯ ɦɨɞɟɥɟɣ, ɤɨɬɨɪɵɟ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨɜɚɧɵ ɤɚɤ ɩɪɢ ɱɢɫɥɟɧɧɨɦ ɚɧɚɥɢɡɟ ɧɚɩɪɹɠɟɧɧɨ-ɞɟɮɨɪɦɢɪɨɜɚɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɞɨɪɨɠɧɵɯ ɨɞɟɠɞ, ɬɚɤ ɢ ɩɪɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ.  Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɜɹɡɤɨɭɩɪɭɝɚɹ ɦɨɞɟɥɶ, ɬɟɧɡɨɪ, ɞɟɜɢɚɬɨɪ, ɲɚɪɨɜɨɣ ɬɟɧɡɨɪ, ɞɟɮɨɪɦɚɰɢɢ, ɧɚ-ɩɪɹɠɟɧɢɹ.  げんゎんずぽぞぎ づやぷぎぞぞé がずé ぶだどごづごぎずぎぜぎぞどぞごび,  ずやぞやざぞごび, ゑ’éげとだぢづばぐぞごび 3-D ぜだがぎずぎざ  ゑ.だ. ゐÜÇÜ½Ü¿Üç, äëÜâñïÜë, Ñ.ö.Ö., ゑ.と. ぐÑíÖ0¡, äëÜâñïÜë, Ñ.ö.Ö., で.ゑ. ゐÜÇÜ½Ü¿Üç, ：Ö¢ñÖñë, びぞんがば  Ⱥɧɨɬɚɰɿɹ. Ɂɚɩɪɨɩɨɧɨɜɚɧɨ ɡɚɝɚɥɶɧɿ ɪɿɲɟɧɧɹ ɞɥɹ 3-D ɜ’ɹɡɤɨɩɪɭɠɧɢɯ ɱɨɬɢɪɢɟɥɟɦɟɧɬɧɢɯ ɦɨɞɟ-ɥɟɣ, ɳɨ ɦɨɠɭɬɶ ɛɭɬɢ ɜɢɤɨɪɢɫɬɚɧɿ ɹɤ ɩɪɢ ɱɢɫɥɨɜɨɦɭ ɚɧɚɥɿɡɿ ɧɚɩɪɭɠɟɧɨ-ɞɟɮɨɪɦɨɜɚɧɨɝɨ ɫɬɚɧɭ ɞɨɪɨɠɧɶɨɝɨ ɨɞɹɝɭ, ɬɚɤ ɿ ɩɪɢ ɟɤɫɩɟɪɢɦɟɧɬɚɥɶɧɢɯ ɞɨɫɥɿɞɠɟɧɧɹɯ.  Ʉɥɸɱɨɜɿ ɫɥɨɜɚ: ɜ’ɹɡɤɨɩɪɭɠɧɚ ɦɨɞɟɥɶ, ɬɟɧɡɨɪ, ɞɟɜɿɚɬɨɪ, ɲɚɪɨɜɢɣ ɬɟɧɡɨɪ, ɞɟɮɨɪɦɚɰɿʀ, ɧɚɩɪɭɝɢ.  GENERAL SOLUTION FOR FOUR –ELEMENT LINEAR  VISCOELASTIC 3-D MODELS  V. Bogomolov, Professor, Doctor of Technical Science, V. Zhdaniuk, Professor,  Doctor of Technical Science, S. Bogomolov, engineer, KhNAHU  Abstract. General solution for 3-D viscoelastic four – element models that can be used at both numerical analysis of stressedly – deformed condition of road coatings and experimental investigations are offered.    Key words: viscoelastic model, tensor, deviator, spherical tensor, deformation, stress.   ゑçñÑñÖóñ  ɂɡɜɟɫɬɧɨ, ɱɬɨ ɱɟɬɵɪɟɯɷɥɟɦɟɧɬɧɵɟ ɪɟɨɥɨɝɢ-ɱɟɫɤɢɟ ɦɨɞɟɥɢ ɜ ɞɨɫɬɚɬɨɱɧɨ ɩɨɥɧɨɣ ɦɟɪɟ ɫɩɨɫɨɛɧɵ ɨɩɢɫɵɜɚɬɶ ɜɹɡɤɨɭɩɪɭɝɨɟ ɩɨɜɟɞɟɧɢɟ ɪɚɡɥɢɱɧɵɯ ɦɚɬɟɪɢɚɥɨɜ [1], ɢ, ɜ ɱɚɫɬɧɨɫɬɢ,  ɚɫɮɚɥɶɬɨɛɟɬɨɧɚ [2, 3, 4].  ɍɠɟ ɜ ɪɚɛɨɬɟ [1] ɨɬɦɟɱɚɟɬɫɹ, ɱɬɨ ɭɜɟɥɢɱɟɧɢɟ ɤɨɥɢɱɟɫɬɜɚ ɷɥɟɦɟɧɬɨɜ ɫɜɵɲɟ ɱɟɬɵɪɟɯ «… ɧɟ ɞɚɟɬ ɤɚɱɟɫɬɜɟɧɧɨɝɨ ɢɡɦɟɧɟɧɢɹ ɦɟɯɚɧɢɱɟɫɤɨ-ɝɨ ɩɨɜɟɞɟɧɢɹ ɦɨɞɟɥɢ».  ɉɨɷɬɨɦɭ ɜ ɞɚɥɶɧɟɣɲɢɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ ɰɟɥɟ-ɫɨɨɛɪɚɡɧɨ ɨɝɪɚɧɢɱɢɬɶɫɹ ɢɦɟɧɧɨ ɬɚɤɢɦɢ ɤɨɧ-ɫɬɪɭɤɰɢɹɦɢ. んÖí¿ó£ äÜß¿ó¡íîóú  ȼ ɪɚɛɨɬɟ [5] ɩɨɤɚɡɚɧɨ, ɱɬɨ ɥɢɧɟɣɧɭɸ ɜɹɡɤɨ ɭɩɪɭɝɭɸ ɦɨɞɟɥɶ ɥɸɛɨɣ ɫɥɨɠɧɨɫɬɢ ɦɨɠɧɨ ɩɪɢɜɟɫɬɢ  ɤ ɨɛɨɛɳɟɧɧɨɣ ɦɨɞɟɥɢ Ɇɚɤɫɜɟɥɥɚ. ɇɚ ɪɢɫ. 1 ɩɨɤɚɡɚɧɵ ɞɜɚ ɜɨɡɦɨɠɧɵɯ ɬɢɩɚ ɬɚ-ɤɢɯ ɱɟɬɵɪɟɯɷɥɟɦɟɧɬɧɵɯ ɦɨɞɟɥɟɣ. ȼ ɪɚɛɨɬɟ [6] ɞɚɧɨ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɥɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶ-ɧɵɯ ɭɪɚɜɧɟɧɢɣ, ɨɩɢɫɵɜɚɸɳɢɯ ɫɯɟɦɵ (ɪɢɫ. 1).  ぴñ¿á ó äÜïöíÖÜç¡í £íÑíôó  ȼɦɟɫɬɟ ɫ ɷɬɢɦ ɩɪɚɤɬɢɤɚ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɪɚɫ-ɫɦɚɬɪɢɜɚɟɦɵɯ ɦɨɞɟɥɟɣ ɩɨɤɚɡɵɜɚɟɬ, ɱɬɨ ɨɱɟɧɶ ɱɚɫɬɨ ɛɵɜɚɟɬ ɭɞɨɛɧɵɦ ɦɨɞɟɥɶ (ɪɢɫ. 1, ɚ) ǮȐȠȜȚȜȏȖșȪțȩȗ ȠȞȎțȟȝȜȞȠ, Ȑȩȝ. 29, 2011  44 ɩɪɟɞɫɬɚɜɥɹɬɶ ɜ ɜɢɞɟ ɫɯɟɦ ɪɢɫ. 2, ɚ, ɛ, ɜ [7, 8]; ɚ ɦɨɞɟɥɶ ɪɢɫ. 1, ɛ, ɩɨ ɫɯɟɦɟ  ɪɢɫ. 2, ɝ [7].           Ɋɢɫ. 1. Ⱦɜɚ ɬɢɩɚ ɱɟɬɵɪɟɯɷɥɟɦɟɧɬɧɵɯ ɥɢɧɟɣ-ɧɵɯ ɜɹɡɤɨɭɩɪɭɝɢɯ ɦɨɞɟɥɟɣ: ɚ – ɫ ɦɝɧɨ-ɜɟɧɧɨɣ ɭɩɪɭɝɨɫɬɶɸ ɢ ɨɫɬɚɬɨɱɧɨɣ ɞɟɮɨɪ-ɦɚɰɢɟɣ; ɛ – ɛɟɡ ɧɢɯ; 1 4...G G , 1 4...η η  – ɤɨɷɮɮɢɰɢɟɧɬɵ, ɯɚɪɚɤɬɟɪɢɡɭɸɳɢɟ ɠɟɫɬ-ɤɨɫɬɧɵɟ ɢ ɜɹɡɤɨɫɬɧɵɟ ɫɜɨɣɫɬɜɚ ɦɚɬɟ-ɪɢɚɥɚ                         Ɋɢɫ. 2. Ɋɚɡɧɨɜɢɞɧɨɫɬɢ ɱɟɬɵɪɟɯɷɥɟɦɟɧɬɧɵɯ ɦɨɞɟɥɟɣ: ɚ, ɛ, ɜ – ɫ ɦɝɧɨɜɟɧɧɨɣ ɭɩɪɭɝɨ-ɫɬɶɸ ɢ ɨɫɬɚɬɨɱɧɨɣ ɞɟɮɨɪɦɚɰɢɟɣ; ɝ – ɛɟɡ ɧɢɯ  ɇɚɩɪɢɦɟɪ, ɦɨɞɟɥɶ ɪɢɫ. 2, ɚ ɨɱɟɧɶ ɭɞɨɛɧɚ ɞɥɹ ɢɫɩɨɥɶɡɨɜɚɧɢɹ ɜ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɢɫɫɥɟ-ɞɨɜɚɧɢɹɯ, ɚ ɪɢɫ. 2, ɛ ɞɥɹ ɫɨɡɞɚɧɢɹ ɧɟɥɢɧɟɣ-ɧɨɣ ɜɹɡɤɨɭɩɪɭɝɨɣ ɦɨɞɟɥɢ  ɢ ɬ.ɞ.  Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɧɟɨɛɯɨɞɢɦɨ ɩɨɫɬɪɨɢɬɶ ɨɛ-ɳɟɟ ɪɟɲɟɧɢɟ ɞɥɹ ɫɯɟɦ ɪɢɫ. 2.   がñçóíöÜë▲ Öíäë　¢ñÖóú ó ÑñâÜë½íîóú  ȼ ɪɚɛɨɬɟ [5] ɩɨɥɭɱɟɧɵ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ, ɫɜɹɡɵɜɚɸɳɢɟ ɞɟɜɢɚɬɨɪɵ ɧɚ-ɩɪɹɠɟɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ. Ⱦɥɹ ɫɯɟɦɵ ɪɢɫ. 1, ɚ  ( )1 2 1 2 2 1 1 21 2 1 2 1 2 1 22 ( ) 2 ( ) .H H Hd dD G G D G G DG G D G G Dη η + η + η + == η η + + η + η$$ $$$ $  (1)  Ⱦɥɹ ɫɯɟɦɵ ɪɢɫ. 2, ɚ  ( )5 6 5 5 5 6 6 55 6 5 5 6 5 6 52 2 .H HH d dD G G G DG G D G D G G Dη η + η + η + η ++ = η η + η$$ $$$ $    (2)  Ⱦɥɹ ɫɯɟɦɵ ɪɢɫ. 2, ɛ  [ ]7 8 7 8 8 7 87 8 7 8 7 8 7 8 7( )2( ) 2 .H HH d dD G G DG G D G G D G G Dη η + η + η +η ++ = + η η + η$$ $$$ $(3)   ȼ ɫɨɨɬɜɟɬɫɬɜɢɢ ɫ ɦɟɬɨɞɢɤɨɣ, ɩɪɟɞɥɨɠɟɧɧɨɣ ɜ [5] ɞɥɹ ɪɢɫ. 2, ɜ ɩɨɥɭɱɚɟɦ  [ ]9 10 10 9 10 9 109 10 9 9 10 9 10 9 10( ) ) (4)2 2 ( ) .H HH d dD G G DG G D G D G G Dη η + η +η + η ++ = η η + η +η$$ $$$ $ Ʉɚɤ ɜɢɞɧɨ ɢɡ (1)–(4), ɩɨɥɭɱɟɧɧɵɟ ɞɢɮɮɟɪɟɧ-ɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɞɥɹ ɪɟɲɟɧɢɹ ɨɬɧɨɫɢ-ɬɟɥɶɧɨ HD  ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ  0 1 2 ( )H H Ha D a D a D f t+ + =$$ $ .             (5)  ɉɨɷɬɨɦɭ, ɪɟɲɟɧɢɟ ɬɚɤɨɝɨ ɭɪɚɜɧɟɧɢɹ ɦɨɠɧɨ ɩɨɫɬɪɨɢɬɶ ɩɨ ɨɛɳɟɢɡɜɟɫɬɧɨɣ ɫɯɟɦɟ [9]  0H H HchD D D= + ,                   (6)  ɝɞɟ 0HD  – ɪɟɲɟɧɢɟ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ; HchD  – ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɧɟɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ (5).  ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (5)  20 1 2 0a k a k a+ + = .                 (7)  Ɉɬɤɭɞɚ  21 1 0 21,2042a a a aka− ± −= ;            (8) a G2 Ș1 G1 Ș2 G4ɛ Ș4Ș3 G3 a G5 Ș5 G6 Ș6 Ș7 G7ɛ Ș8 G8 G9, K9 Ș9,ȘV9 ɜ Ș10,ȘV10 G10, K10 ɝ G12 Ș12G11 Ș11ǮȐȠȜȚȜȏȖșȪțȩȗ ȠȞȎțȟȝȜȞȠ, Ȑȩȝ. 29, 2011 45ɜ ɞɚɥɶɧɟɣɲɟɦ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɫɥɭɱɚɣ, ɤɨɝɞɚ 21 0 24 0a a a− > .  Ⱦɥɹ ɨɞɧɨɪɨɞɧɨɝɨ ɭɪɚɜɧɟɧɢɹ  1 20 1 2k t k tHD c e c e= + ,                  (9)  ɨɬɤɭɞɚ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ ɦɨɝɭɬ ɛɵɬɶ ɨɩ-ɪɟɞɟɥɟɧɵ 1c ɢ 2c .  Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɟɫɥɢ ɩɪɢ 0t = ; 0H HD D=  ɢ 0H HD D=$ $ , ɬɨ  0 0122 1H HD k Dck k−=−$; 01 2Hc D c= − .       (10)  ȿɫɥɢ ɱɚɫɬɧɨɟ ɪɟɲɟɧɢɟ ɢɫɤɚɬɶ ɜ ɜɢɞɟ  1 21 2( ) ( )k t k tHchD c t e c t e= + ,              (11)  ɬɨ, ɩɨɫɥɟ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɩɪɟɨɛɪɚɡɨɜɚɧɢɣ, ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ   2 1( ) ( )0 0 2 11( )( ) ( ),tk t k tHchD e ea k kf d−ξ −ξ⎡ ⎤= − ×⎣ ⎦−× ξ ξ∫  (12)  ɝɞɟ ( )f ξ – ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (5).  Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɨɤɨɧɱɚɬɟɥɶɧɨ ɞɥɹ ɞɟɜɢɚɬɨɪɚ ɧɚɩɪɹɠɟɧɢɣ ɦɨɠɧɨ ɡɚɩɢɫɚɬɶ  1 22 11 20 2 1 0( ) ( )1 ( )( ) .tk t k tHk t k tD c e c ek k ae e f d−ξ −ξ= + + ×−⎡ ⎤× − ξ ξ⎣ ⎦∫    (13)  Ⱦɥɹ ɪɟɲɟɧɢɹ ɨɬɧɨɫɢɬɟɥɶɧɨ ɞɟɜɢɚɬɨɪɚ ɞɟ-ɮɨɪɦɚɰɢɣ, (1)–(4) ɧɟɨɛɯɨɞɢɦɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ  0 1 ( )d da D a D f t+ =$$ $ .                  (14)  Ɍɨɝɞɚ ɢɡ ɪɟɲɟɧɢɹ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɝɨ ɭɪɚɜ-ɧɟɧɢɹ (8)  1 0k = ; 120aka= − ,                 (15)  ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ  2 2 ( )1 20 111 ( ) ,tk t k tdD c c e e f da−ξ⎡ ⎤= + + × − ξ ξ⎣ ⎦∫ (16)  ɝɞɟ ɩɨɫɬɨɹɧɧɵɟ 1 c ɢ 2c ɬɚɤ ɠɟ, ɤɚɤ ɢ ɜ ɩɪɟɞɵ-ɞɭɳɟɦ ɫɥɭɱɚɟ, ɨɩɪɟɞɟɥɹɸɬɫɹ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ. ɇɚɩɪɢɦɟɪ, ɟɫɥɢ ɩɪɢ 0t = ; 0d dD D=  ɢ 0d dD D=$ $ , ɬɨ, ɩɨ ɚɧɚɥɨɝɢɢ ɫ (10),   022dDck=$; 01 2dc D c= − .            (17)  Ⱦɥɹ ɫɯɟɦ ɪɢɫ. 1, ɛ, 2, ɝ.  Ɇɨɞɟɥɶ ɪɢɫ. 1, ɛ  ( )3 3 3 43 3 4 4 3 3 422 2 .H H dd dD G D DG G D G G Dη + = η η +⎡ ⎤+ η + η + η +⎣ ⎦$ $$$    (18)  Ɇɨɞɟɥɶ ɪɢɫ. 2, ɝ  11 12 11 1211 12 11 12 12 1111 12( ) ( )2 2( )2 .H Hd ddD G G DD G G DG G Dη +η + + == η η + η + η ++$$$ $      (19)  Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɩɪɢ ɪɟɲɟɧɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ HD  ɭɪɚɜɧɟɧɢɹ (18, 19) ɩɪɢɜɨɞɹɬɫɹ ɤ ɜɢɞɭ  0 1 ( )H Ha D a D f t+ =$ .                 (20)  ɏɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɞɥɹ (20)  1 1 0a k a+ = ,                     (21)  ɨɬɤɭɞɚ  100aka= − ,                       (22)  ɢ ɨɛɳɟɟ ɪɟɲɟɧɢɟ  0k tHD ce= + 0( )0 01 ( )tk te f da−ξ ξ ξ∫ ,     (23)  ɝɞɟ c  ɧɚɯɨɞɢɦ ɢɡ ɧɚɱɚɥɶɧɵɯ ɭɫɥɨɜɢɣ, ɧɚ-ɩɪɢɦɟɪ, ɟɫɥɢ ɩɪɢ 0t = , 0H HD D= , ɬɨ  0Hc D= .                            (24)  ɉɪɢ ɪɟɲɟɧɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ dD  ɢɦɟɟɦ  0 1 2 ( )d d da D a D a D f t+ + =$$ $ ,           (25) ǮȐȠȜȚȜȏȖșȪțȩȗ ȠȞȎțȟȝȜȞȠ, Ȑȩȝ. 29, 2011  46 ɢ ɬɨɝɞɚ ɯɚɪɚɤɬɟɪɢɫɬɢɱɟɫɤɨɟ ɭɪɚɜɧɟɧɢɟ ɢɦɟɟɬ ɜɢɞ (7), ɟɝɨ ɤɨɪɧɢ (8), ɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ  1 22 11 20 2 1 0( ) ( )1 ( )( ) .tk t k tdk t k tD c e c ek k ae e f d−ξ −ξ= + + ×−⎡ ⎤× − ξ ξ⎣ ⎦∫        (26)  ɉɪɢ ɷɬɨɦ 1c ɢ 2c , ɩɪɢ 0t = , 0d dD D=  ɢ 0d dD D=$ $ , ɛɭɞɭɬ ɢɦɟɬɶ ɜɢɞ  0 0122 1d dD k Dck k−=−$; 01 2dc D c= − .          (27)  ぷíëÜçÜú öñÖ£Üë Öíäë　¢ñÖóú  ó ÑñâÜë½íîóú  Ʉɚɤ ɢɡɜɟɫɬɧɨ [8, 10], ɞɥɹ ɬɨɝɨ, ɱɬɨɛɵ ɨɩɪɟɞɟ-ɥɢɬɶ ɲɚɪɨɜɨɣ ɬɟɧɡɨɪ, ɧɟɨɛɯɨɞɢɦɨ ɡɧɚɬɶ ɡɧɚ-ɱɟɧɢɟ ɫɪɟɞɧɢɯ ɩɚɪɚɦɟɬɪɨɜ  3x y zsrσ + σ + σσ = ; 3x y zsrε + ε + εε = ;  2 (1 )(1 2 )sr srG +µσ = ε− µ,  (28)  ɝɞɟ ...x zσ ε  – ɤɨɦɩɨɧɟɧɬɵ ɬɟɧɡɨɪɨɜ ɧɚɩɪɹɠɟ-ɧɢɣ ɢ ɞɟɮɨɪɦɚɰɢɣ ɩɨ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɦ ɤɨɨɪ-ɞɢɧɚɬɧɵɦ ɨɫɹɦ; µ  – ɤɨɷɮɮɢɰɢɟɧɬ ɉɭɚɫɫɨɧɚ.  ȼ ɪɚɛɨɬɟ [5] ɩɪɟɞɥɨɠɟɧɚ ɦɟɬɨɞɢɤɚ ɫɨɫɬɚɜɥɟ-ɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɭɪɚɜɧɟɧɢɣ ɞɥɹ ɪɚɫ-ɫɦɚɬɪɢɜɚɟɦɵɯ ɜɟɥɢɱɢɧ.  Ɍɚɤ, ɧɚɩɪɢɦɟɪ, ɞɥɹ ɫɯɟɦɵ ɪɢɫ. 2, ɜ ɩɨɥɭɱɚɟɦ  [ ]9 109 10 10 9 109 10 9 9 109 10 9 10( )33 ( ) ,V V srV V V srsr V V srV V srK KK K KK Kη η σ ++ η + η + η σ ++ σ = η η ε ++ η + η ε$$$$$$             (29)  ɝɞɟ 9 10,  V Vη η  – ɤɨɷɮɮɢɰɢɟɧɬɵ ɨɛɴɟɦɧɨɝɨ ɜɹɡɤɨɝɨ ɫɨɩɪɨɬɢɜɥɟɧɢɹ; 9 10,  K K  – ɨɛɴɟɦɧɵɟ ɦɨɞɭɥɢ ɭɩɪɭɝɨɫɬɢ (ɫɦ. ɪɢɫ. 2, ɜ).  Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɫɯɟɦ ɧɚ ɪɢɫ. 1, ɚ, 2, ɚ–ɜ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɟ ɭɪɚɜɧɟɧɢɹ ɢɦɟɸɬ ɜɢɞ (5), ɨɬɤɭɞɚ ɨɛɳɟɟ ɪɟɲɟɧɢɟ ɞɥɹ srσ  ɩɨɥɭɱɢɦ ɜ ɜɢɞɟ  1 22 11 20 2 1 0( ) ( )1( )( ) ,tk t k tsrk t k tc e c ek k ae e f d−ξ −ξσ = + + ×−⎡ ⎤× − ξ ξ⎣ ⎦∫         (30)  ɝɞɟ 1 2,  k k  ɧɚɯɨɞɢɦ ɩɨ ɚɧɚɥɨɝɢɢ ɫ (10);  0,0 122 1sr srkck kσ − σ=−$; 1 ,0 2src c= σ − ;      (31)  ( )f ξ  – ɩɪɚɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (29); ,0 ,srσ  ,0srσ$  – ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ.  ɉɪɢ ɪɟɲɟɧɢɢ ɨɬɧɨɫɢɬɟɥɶɧɨ srε  ɢɫɯɨɞɧɨɟ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɨɟ ɭɪɚɜɧɟɧɢɟ ɩɨɥɭɱɚɟɦ ɜ ɜɢɞɟ (14), ɢ ɪɟɲɟɧɢɟ  2 2 ( )1 20 111 ( ) ,tk t k tsr c c e e f da−ξ⎡ ⎤ε = + + − ξ ξ⎣ ⎦∫   (32)  ɝɞɟ 2k ɧɚɯɨɞɢɦ ɩɨ ɚɧɚɥɨɝɢɢ ɫ (15);  ,022srckε=$; 1 ,0 2src c= ε − ;             (33)  ( )f ξ  – ɥɟɜɚɹ ɱɚɫɬɶ ɭɪɚɜɧɟɧɢɹ (29); ,0srε , ,0srε$  – ɧɚɱɚɥɶɧɵɟ ɭɫɥɨɜɢɹ.  ȼ ɫɥɭɱɚɟ, ɟɫɥɢ 9 10V Vη = η = ∞ , ɬ.ɟ. ɫɪ 0ε =$  [10], ɭɪɚɜɧɟɧɢɟ (29) ɜɵɪɚɠɚɟɬɫɹ [5] ɜ: 93sr srKε = ε .  ゑ▲çÜÑ▲  ɉɨɥɭɱɟɧɧɵɟ ɪɟɲɟɧɢɹ ɦɨɝɭɬ ɛɵɬɶ ɢɫɩɨɥɶɡɨ-ɜɚɧɵ ɜ ɱɢɫɥɟɧɧɵɯ ɪɟɲɟɧɢɹɯ 3-D ɦɨɞɟɥɟɣ, ɚ ɬɚɤɠɟ ɜ ɩɪɨɰɟɫɫɟ ɚɧɚɥɢɡɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶ-ɧɵɯ ɞɚɧɧɵɯ.  ずóöñëíöÜëí  1. Ƚɨɥɶɛɟɪɝ ɂ.ɂ. Ɇɟɯɚɧɢɱɟɫɤɨɟ ɩɨɜɟɞɟɧɢɟ ɩɨɥɢɦɟɪɧɵɯ ɦɚɬɟɪɢɚɥɨɜ (ɦɚɬɟɦɚɬɢɱɟ-ɫɤɨɟ ɨɩɢɫɚɧɢɟ) / ɂ.ɂ. Ƚɨɥɶɛɟɪɝ. – Ɇ. : ɏɢɦɢɹ, 1970. – 192 ɫ. 2. Ɋɟɣɧɟɪ Ɇ. Ⱦɟɮɨɪɦɚɰɢɹ ɢ ɬɟɱɟɧɢɟ / Ɇ. Ɋɟɣɧɟɪ ; ɩɟɪ. ɫɨ ɜɬɨɪ. ɚɧɝɥ. ɢɡɞ. Ʌ.ȼ. ɇɢɤɢɬɢɧɚ, Ⱥ.Ɇ. Ʉɨɱɟɬɤɨɜɚ, ȼ.ɇ. Ʉɭ-ɤɭɞɠɚɧɨɜɚ. – Ɇ. : Ƚɨɫ. ɧɚɭɱɧ.-ɬɟɯɧ. ɢɡɞ-ɜɨ ɧɟɮɬɹɧ. ɢ ɝɨɪɧɨ-ɬɨɩɥɢɜɧɨɣ ɥɢɬ-ɪɵ, 1963. – 381 ɫ.  ǮȐȠȜȚȜȏȖșȪțȩȗ ȠȞȎțȟȝȜȞȠ, Ȑȩȝ. 29, 2011 473. Ɂɨɥɨɬɚɪɟɜ ȼ.Ⱥ. ɂɫɫɥɟɞɨɜɚɧɢɟ ɫɜɨɣɫɬɜ ɚɫ-ɮɚɥɶɬɨɛɟɬɨɧɨɜ ɪɚɡɥɢɱɧɨɣ ɦɚɤɪɨɫɬɪɭɤ-ɬɭɪɵ: ɞɢɫ… ɤɚɧɞ. ɬɟɯɧ. ɧɚɭɤ: 05.23.05 / ȼ.Ⱥ. Ɂɨɥɨɬɚɪɟɜ. – ɏɚɪɶɤɨɜ, 1967. – 207 ɫ. 4. Ȼɨɝɨɦɨɥɨɜ ȼ.Ɉ. Ɋɟɨɥɨɝiɱɧɚ ɦɨɞɟɥɶ ɪɨɛɨɬɢ ɚɫɮɚɥɶɬɨɛɟɬɨɧɭ ɩɪɢ ɫɬɢɫɤɚɧɧi / ȼ.Ɉ. Ȼɨ-ɝɨɦɨɥɨɜ, ȼ.Ʉ. ɀɞɚɧɸɤ, ȼ.Ɇ. Ɋɹɩɭɯiɧ, ɋ.ȼ. Ȼɨɝɨɦɨɥɨɜ // Ⱥɜɬɨɲɥɹɯɨɜɢɤ ɍɤɪɚʀɧɢ. –  2010. – ʋ 3. – ɋ. 34–37. 5. Ȼɨɝɨɦɨɥɨɜ ȼ.Ⱥ. Ɉɛɳɢɣ ɦɟɬɨɞ ɩɨɥɭɱɟɧɢɹ ɞɢɮɮɟɪɟɧɰɢɚɥɶɧɵɯ ɡɚɜɢɫɢɦɨɫɬɟɣ ɞɟ-ɮɨɪɦɚɰɢɣ ɨɬ ɧɚɩɪɹɠɟɧɢɣ ɞɥɹ ɥɢɧɟɣɧɵɯ ɪɟɨɥɨɝɢɱɟɫɤɢɯ 3-D ɦɨɞɟɥɟɣ / ȼ.Ⱥ. Ȼɨɝɨ-ɦɨɥɨɜ, ȼ.Ʉ. ɀɞɚɧɸɤ, ɋ.ȼ. Ȼɨɝɨɦɨɥɨɜ // ȼɟɫɬɧɢɤ ɏɇȺȾɍ: ɫɛ. ɧɚɭɱɧ. ɬɪ. – 2011. – ʋ 52. – ɋ. 54–59.  6. Ȼɨɝɨɦɨɥɨɜ ȼ.Ⱥ. Ɉɛɳɟɟ ɪɟɲɟɧɢɟ ɞɥɹ ɥɢ-ɧɟɣɧɨɣ, ɬɪɟɯɦɟɪɧɨɣ, ɜɹɡɤɨɭɩɪɭɝɨɣ ɦɨ-ɞɟɥɢ Ɇɚɤɫɜɟɥɥɚ // ȼɟɫɬɧɢɤ ɏɇȺȾɍ : ɫɛ. ɧɚɭɱɧ. ɬɪ. / ȼ.Ⱥ. Ȼɨɝɨɦɨɥɨɜ, ȼ.Ʉ. ɀɞɚɧɸɤ, ɋ.ȼ. Ȼɨɝɨɦɨɥɨɜ. – 2011. – ʋ 53. – ɋ. 70–72. 7. Ȼɨɝɨɦɨɥɨɜ ȼ.Ⱥ. ɍɧɢɜɟɪɫɚɥɶɧɵɣ ɦɟɬɨɞ ɫɨ-ɫɬɚɜɥɟɧɢɹ ɥɢɧɟɣɧɵɯ ɜɹɡɤɨɭɩɪɭɝɢɯ ɫɬɪɭɤɬɭɪɧɵɯ ɦɨɞɟɥɟɣ / ȼ.Ⱥ. Ȼɨɝɨɦɨɥɨɜ,  ȼ.Ʉ. ɀɞɚɧɸɤ, ɋ.ȼ. Ȼɨɝɨɦɨɥɨɜ // Ⱥɜɬɨ-ɦɨɛɢɥɶɧɵɣ ɬɪɚɧɫɩɨɪɬ : ɫɛ. ɧɚɭɱɧ. ɬɪ. –2011 – ʋ28. – ɋ. 125–131. 8. Ⱦɠ. Ɇɟɣɡ Ɍɟɨɪɢɹ ɢ ɡɚɞɚɱɢ ɦɟɯɚɧɢɤɢ ɫɩɥɨɲɧɵɯ ɫɪɟɞ / Ⱦɠ. Ɇɟɣɡ; ɩɟɪ. ɫ ɚɧɝɥ. ȿ.ɂ. ɋɜɟɲɧɢɤɨɜɨɣ. – Ɇ. : Ɇɢɪ, 1974. – 318 ɫ. 9. Ʉɨɪɧ Ƚ. ɋɩɪɚɜɨɱɧɢɤ ɩɨ ɦɚɬɟɦɚɬɢɤɟ ɞɥɹ ɧɚɭɱɧɵɯ ɪɚɛɨɬɧɢɤɨɜ ɢ ɢɧɠɟɧɟɪɨɜ. Ɉɩ-ɪɟɞɟɥɟɧɢɹ, ɬɟɨɪɟɦɵ, ɮɨɪɦɭɥɵ / Ƚ. Ʉɨɪɧ, Ɍ. Ʉɨɪɧ ; ɩɟɪ. ɫɨ 2-ɝɨ ɚɦɟɪɢɤɚɧɫɤ. ɩɟɪɟ-ɪɚɛ. ɢɡɞ. ɇ.Ƚ. Ⱥɪɚɦɚɧɨɜɢɱɚ, Ⱥ.Ɇ. Ȼɟ-ɪɟɡɦɚɧɚ ɢ ɞɪ. – Ɇ. : ɇɚɭɤɚ. Ƚɥɚɜɧ. ɪɟɞ. ɮɢɡ.-ɦɚɬ. ɥɢɬ-ɪɵ, 1984. – 831 ɫ. 10. Ȼɟɡɭɯɨɜ ɇ.ɂ. Ɉɫɧɨɜɵ ɬɟɨɪɢɢ ɭɩɪɭɝɨɫɬɢ ɩɥɚɫɬɢɱɧɨɫɬɢ ɢ ɩɨɥɡɭɱɟɫɬɢ / ɇ.ɂ. Ȼɟɡɭ-ɯɨɜ. – Ɇ. : ȼɵɫɲɚɹ ɲɤɨɥɚ, 1968. – 512 ɫ.  Ɋɟɰɟɧɡɟɧɬ: ȼ.ȼ. Ɏɢɥɢɩɩɨɜ, ɩɪɨɮɟɫɫɨɪ, ɞ.ɬ.ɧ. ɏɇȺȾɍ.  ɋɬɚɬɶɹ ɩɨɫɬɭɩɢɥɚ ɜ ɪɟɞɚɤɰɢɸ 1 ɫɟɧɬɹɛɪɹ 2011 ɝ.       

GENERAL SOLUTION FOR FOUR –ELEMENT LINEAR VISCOELASTIC 3-D MODELS

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GENERAL SOLUTION FOR FOUR –ELEMENT LINEAR VISCOELASTIC 3-D MODELS

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