О ПРЕДЕЛАХ УПРУГОСТИ И ПРОЧНОСТИ ГОРНЫХ ПОРОД

ȼȿɋɌɇɂɄ ɉɇɂɉɍ
Ɇɟɯɚɧɢɤɚ
2013
ʋ2
ɍȾɄ 539.215; 539.374
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
Ʉɵɪɝɵɡɫɤɨ-Ɋɨɫɫɢɣɫɤɢɣ ɋɥɚɜɹɧɫɤɢɣ ɭɧɢɜɟɪɫɢɬɟɬ, ɝ. Ȼɢɲɤɟɤ, Ʉɵɪɝɵɡɫɬɚɧ
Ɉ ɉɊȿȾȿɅȺɏ ɍɉɊɍȽɈɋɌɂ
ɂ ɉɊɈɑɇɈɋɌɂ ȽɈɊɇɕɏ ɉɈɊɈȾ
ȼ ɫɬɚɬɶɟ ɢɫɫɥɟɞɭɟɬɫɹ ɩɨɜɟɞɟɧɢɟ ɨɛɪɚɡɰɨɜ ɝɨɪɧɵɯ ɩɨɪɨɞ, ɢɫɩɵɬɚɧɧɵɯ ɩɪɢ ɧɟɪɚɜɧɨɦɟɪɧɨɦ
ɬɪɟɯɨɫɧɨɦ ɫɠɚɬɢɢ ɩɨ ɫɯɟɦɟ Ʉɚɪɦɚɧɚ. ɂɫɩɨɥɶɡɨɜɚɧɵ ɢɡɜɟɫɬɧɵɟ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɞɚɧɧɵɟ
Ⱥ.ɇ. ɋɬɚɜɪɨɝɢɧɚ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɝɨɪɧɵɯ ɩɨɪɨɞ. Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜ [1, 2], ɷɬɢ ɦɚɬɟɪɢɚɥɵ ɜ ɢɫɯɨɞɧɨɦ
ɫɨɫɬɨɹɧɢɢ ɨɛɥɚɞɚɥɢ ɭɩɪɭɝɨɣ ɨɪɬɨɬɪɨɩɧɨɣ ɫɢɦɦɟɬɪɢɟɣ. ɉɪɟɞɟɥɵ ɭɩɪɭɝɨɫɬɢ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɵɯ
ɧɚɩɪɹɠɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹɯ ɨɩɪɟɞɟɥɟɧɵ ɱɟɪɟɡ ɩɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ ɧɚ ɨɫɧɨɜɟ
ɫɢɧɬɟɡɚ ɤɪɢɬɟɪɢɹ Ʉɭɥɨɧɚ-Ɇɨɪɚ ɢ ɤɨɧɰɟɩɰɢɢ ɫɤɨɥɶɠɟɧɢɹ ɜ ɬɪɚɤɬɨɜɤɟ Ɇ.ə. Ʌɟɨɧɨɜɚ. ɍɱɬɟɧ ɬɚɤɠɟ
ɜɵɜɨɞ [3]: ɧɚɱɢɧɚɹ ɫ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɤɨɝɞɚ ɫɪɟɞɧɟɟ ɝɥɚɜɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɞɨɫɬɢɝɚɟɬ ɡɧɚɱɟɧɢɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ, ɢ ɩɪɢ ɛɨɥɟɟ ɜɵɫɨɤɢɯ ɞɚɜɥɟɧɢɹɯ, ɝɨɪɧɵɟ ɩɨɪɨɞɵ
ɜɟɞɭɬ ɫɟɛɹ ɤɚɤ ɩɥɚɫɬɢɱɧɵɟ ɦɚɬɟɪɢɚɥɵ. ɉɪɢ ɷɬɨɦ ɧɚɛɥɸɞɚɟɦɨɟ ɩɪɢ ɦɟɧɶɲɢɯ ɞɚɜɥɟɧɢɹɯ ɪɚɡɪɵɯɥɟɧɢɟ ɦɚɬɟɪɢɚɥɚ ɢɫɱɟɡɚɟɬ, ɚ ɤɪɢɬɟɪɢɣ Ʉɭɥɨɧɚ-Ɇɨɪɚ ɩɟɪɟɯɨɞɢɬ ɜ ɤɪɢɬɟɪɢɣ Ɍɪɟɫɤɚ. ɗɬɢ ɭɫɥɨɜɢɹ
ɨɬɨɛɪɚɠɟɧɵ ɩɪɢ ɨɩɪɟɞɟɥɟɧɢɢ ɩɪɟɞɟɥɨɜ ɩɪɨɱɧɨɫɬɢ ɢ ɬɟɨɪɟɬɢɱɟɫɤɨɦ ɭɫɬɚɧɨɜɥɟɧɢɢ ɭɝɥɚ ɫɪɟɡɚ ɩɪɢ
ɪɚɡɪɭɲɟɧɢɢ ɨɛɪɚɡɰɨɜ. ɉɨɫɬɪɨɟɧɵ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɨɝɢɛɚɸɳɢɟ ɤɪɭɝɨɜ Ɇɨɪɚ.
Ʉɥɸɱɟɜɵɟ ɫɥɨɜɚ: ɝɨɪɧɵɟ ɩɨɪɨɞɵ, ɩɥɚɫɬɢɱɧɨɫɬɶ, ɩɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ, ɩɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ,
ɢɡɨɬɪɨɩɧɵɣ ɦɚɬɟɪɢɚɥ, ɨɪɬɨɬɪɨɩɧɵɣ ɦɚɬɟɪɢɚɥ.
B.A. Rychkov, N.M. Komartsov, T.A. Luzhanskaya
Kyrgyz-Russian Slavic University, Bishkek, Kyrgyzstan
ABOUT ELASTIC AND STRENGTH LIMITS OF ROCKS
In this paper we research how rock samples respond to irregular triaxial compression according
to Karman’s scheme. We have used known A. N. Stavrogin’s experimental data for some type of rock.
As shown [1, 2], these materials originally possessed elastic orthotropic symmetry. Elastic limits in arbitrary stressed states was determined using elastic limit under uniaxial compression on the basis of synthesis Mohr-Kulon’s criterion and the ɫoncept of sliding in the treatment of M.J. Leonov. We used [3]
conclusion: rocks behave as ductile materials from the state of stress when the average principal stress
reaches the maximum shear stress to at higher pressures. In other words, in this situation ripping of
materials disappears and Mohr-Kulon’s criterion converses in Treska’s criterion. These conditions was
realized when establishing strength limits and teoretical determination of cutting angle at failure. Curves
of Mohr's circles were constructed.
Keywords: rock, plasticity, elastic limit, strength limit, isotropic material, orthotropic material.
Ɋɚɫɫɦɚɬɪɢɜɚɸɬɫɹ ɩɪɨɱɧɨɫɬɧɵɟ ɫɜɨɣɫɬɜɚ ɝɨɪɧɵɯ ɩɨɪɨɞ ɩɨ ɪɟɡɭɥɶɬɚɬɚɦ ɢɫɩɵɬɚɧɢɹ ɩɨ ɫɯɟɦɟ Ʉɚɪɦɚɧɚ ɰɢɥɢɧɞɪɢɱɟɫɤɢɯ ɨɛɪɚɡɰɨɜ, ɤɨɝɞɚ
V1 t V2 V3 ( Vi – ɝɥɚɜɧɵɟ ɧɚɩɪɹɠɟɧɢɹ, i 1,2,3 ). ȼɢɞ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨ110
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
ɫɬɨɹɧɢɹ ɯɚɪɚɤɬɟɪɢɡɭɟɬɫɹ ɨɬɧɨɲɟɧɢɟɦ ɦɢɧɢɦɚɥɶɧɨɝɨ ɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ
ɝɥɚɜɧɵɯ ɧɚɩɪɹɠɟɧɢɣ c V1 V3 . Ʉɚɤ ɷɬɨ ɱɚɫɬɨ ɩɪɢɧɢɦɚɟɬɫɹ ɜ ɬɟɨɪɟɬɢɱɟɫɤɢɯ ɪɚɫɱɟɬɚɯ, ɩɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ ɛɭɞɟɦ ɨɬɨɠɞɟɫɬɜɥɹɬɶ ɫ ɩɪɟɞɟɥɨɦ
ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɫɬɢ ɢ ɩɪɟɞɟɥɨɦ ɬɟɤɭɱɟɫɬɢ. ɉɪɟɞɟɥɵ ɩɪɨɱɧɨɫɬɢ ɨɩɪɟɞɟɥɢɦ, ɢɫɩɨɥɶɡɭɹ ɤɪɢɬɟɪɢɣ Ɇɨɪɚ.
ɋɥɟɞɭɹ Ɇɨɪɭ [4], ɛɭɞɟɦ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɟɪɜɨɧɚɱɚɥɶɧɵɟ ɥɨɤɚɥɶɧɵɟ
ɫɤɨɥɶɠɟɧɢɹ ɜɨɡɧɢɤɚɸɬ ɧɚ ɩɥɨɳɚɞɤɟ, ɧɚ ɤɨɬɨɪɨɣ ɤɚɫɚɬɟɥɶɧɨɟ WQ ɢ ɧɨɪɦɚɥɶɧɨɟ VQ ɧɚɩɪɹɠɟɧɢɹ ɫɜɹɡɚɧɵ ɫɥɟɞɭɸɳɢɦ ɫɨɨɬɧɨɲɟɧɢɟɦ:
WȞ S0 PVȞ ,
(1)
ɝɞɟ S0 ɢ P – ɩɚɪɚɦɟɬɪɵ ɦɚɬɟɪɢɚɥɚ; Q – ɧɨɪɦɚɥɶ ɤ ɪɚɫɫɦɚɬɪɢɜɚɟɦɨɣ
ɩɥɨɳɚɞɤɟ. ɉɪɚɜɚɹ ɱɚɫɬɶ ɜɵɪɚɠɟɧɢɹ ɩɪɟɞɫɬɚɜɥɹɟɬ ɫɨɛɨɣ ɫɨɩɪɨɬɢɜɥɟɧɢɟ
ɫɤɨɥɶɠɟɧɢɸ (ɫɞɜɢɝɭ).
ɇɨɪɦɚɥɶɧɨɟ ɢ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɹ ɜ ɩɥɨɫɤɨɫɬɢ, ɩɨɜɟɪɧɭɬɨɣ
ɨɬɧɨɫɢɬɟɥɶɧɨ ɩɥɨɳɚɞɤɢ ɞɟɣɫɬɜɢɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ ɧɚ ɭɝɨɥ E , ɨɩɪɟɞɟɥɹɸɬɫɹ ɩɨ ɮɨɪɦɭɥɚɦ
Vȕ V0 Ɍ sin 2E,
Wȕ Ɍ cos2E,
(2)
ɝɞɟ V0 – ɫɪɟɞɧɟɟ ɧɚɩɪɹɠɟɧɢɟ; T – ɦɚɤɫɢɦɚɥɶɧɨɟ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ, ɬ.ɟ.
V0 1 (V1 V3 ) , T
2
1 (V V ) .
1
3
2
(3)
ɋɨɝɥɚɫɧɨ ɨɩɪɟɞɟɥɟɧɢɸ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɫɞɜɢɝɭ ( Sȕ ) ɜɵɪɚɠɚɟɬɫɹ ɜ ɞɚɧɧɨɦ ɤɨɧɬɟɤɫɬɟ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Sȕ S0 P(V0 T sin 2E) .
ɉɪɢ ɩɟɪɜɨɦ ɫɤɨɥɶɠɟɧɢɢ ɤɚɫɚɬɟɥɶɧɵɟ ɤ ɤɪɢɜɵɦ
(4)
Sȕ Sȕ (E)
ɢ Wȕ Wȕ (E) ɫɨɜɩɚɞɭɬ, ɬ.ɟ. ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ Wȕ (E) ɞɨɫɬɢɝɚɟɬ ɡɧɚɱɟɧɢɹ ɫɨɩɪɨɬɢɜɥɟɧɢɹ ɫɞɜɢɝɭ ɜ ɧɚɩɪɚɜɥɟɧɢɢ ɷɬɨɝɨ ɫɤɨɥɶɠɟɧɢɹ:
dSȕ
d Wȕ
dE
dE
ɩɪɢ E E0 ,
(5)
111
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
ɝɞɟ E0 – ɭɝɨɥ, ɨɩɪɟɞɟɥɹɸɳɢɣ ɧɚɩɪɚɜɥɟɧɢɟ ɩɟɪɜɨɝɨ ɫɤɨɥɶɠɟɧɢɹ, ɨɬɫɱɢɬɵɜɚɟɦɵɣ ɨɬ ɧɚɩɪɚɜɥɟɧɢɹ ɞɟɣɫɬɜɢɹ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ T .
ɂɡ (5), ɭɱɢɬɵɜɚɹ (1), (2), (3) ɢ (4), ɜɵɬɟɤɚɟɬ, ɱɬɨ
P tg2E0 .
(6)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɜɟɥɢɱɢɧɚ ɭɝɥɚ E0 ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɚɪɚɦɟɬɪɨɦ P ,
ɜɟɥɢɱɢɧɭ ɤɨɬɨɪɨɝɨ ɛɭɞɟɦ ɫɱɢɬɚɬɶ ɡɚɜɢɫɹɳɟɣ ɨɬ ɜɢɞɚ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.
ȿɫɬɟɫɬɜɟɧɧɨ ɫɱɢɬɚɬɶ, ɱɬɨ ɩɪɢ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɨɦ ɧɚɝɪɭɠɟɧɢɢ
ɜ ɩɥɨɳɚɞɤɚɯ, ɨɬɦɟɱɚɟɦɵɯ ɭɝɥɨɦ E0 , ɢɧɬɟɧɫɢɜɧɨɫɬɶ ɫɤɨɥɶɠɟɧɢɣ ɫ ɪɨɫɬɨɦ ɭɪɨɜɧɹ ɧɚɩɪɹɠɟɧɢɣ ɧɚɢɛɨɥɶɲɚɹ, ɢ ɢɦɟɧɧɨ ɩɨ ɷɬɢɦ ɩɥɨɳɚɞɤɚɦ
ɩɪɨɢɫɯɨɞɢɬ ɫɪɟɡ ɩɪɢ ɪɚɡɪɭɲɟɧɢɢ ɨɛɪɚɡɰɚ.
ɋɨɝɥɚɫɧɨ ɨɩɵɬɧɵɦ ɞɚɧɧɵɦ [5], ɧɚɱɢɧɚɹ ɫ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ c 1 3 ɢ ɜɵɲɟ, ɭɝɨɥ E0 0 , ɬ.ɟ. ɩɪɢ ɞɚɧɧɨɦ ɧɚɩɪɹɠɟɧɧɨɦ ɫɨɫɬɨɹɧɢɢ ɫɪɟɡ ɩɪɨɢɫɯɨɞɢɬ ɩɨ ɩɥɨɫɤɨɫɬɢ ɦɚɤɫɢɦɚɥɶɧɨɝɨ ɤɚɫɚɬɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɢɹ (Ɍ), ɢ ɝɨɪɧɵɟ ɩɨɪɨɞɵ ɞɟɮɨɪɦɢɪɭɸɬɫɹ ɤɚɤ ɩɥɚɫɬɢɱɧɵɣ ɦɚɬɟɪɢɚɥ. Ɍɨɝɞɚ ɩɪɢ c 1 3 ɢɡ ɫɨɨɬɧɨɲɟɧɢɣ (2) ɢ (3) ɜɵɬɟɤɚɟɬ, ɱɬɨ
Vȕ0
Wȕ0
2.
(7)
ɉɪɢ ɫ, ɨɬɥɢɱɧɵɯ ɨɬ 1 3 , ɷɬɨ ɨɬɧɨɲɟɧɢɟ ɜɵɝɥɹɞɢɬ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ:
Vȕ0
Wȕ0
1 c 1 csin 2ȕ0
1 c cos2ȕ0
k c .
(8)
ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɨɬɧɨɲɟɧɢɟ (8) ɥɢɧɟɣɧɨ ɢɡɦɟɧɹɟɬɫɹ ɨɬ ɡɧɚɱɟɧɢɹ k0 k 0 ɞɨ k 1 3 2 . ɂɫɯɨɞɹ ɢɡ ɝɟɨɦɟɬɪɢɱɟɫɤɢɯ ɩɨɫɬɪɨɟɧɢɣ ɜ
ɤɪɭɝɟ Ɇɨɪɚ ɧɚ ɫɠɚɬɢɟ (ɪɢɫ. 1), ɜɟɥɢɱɢɧɚ k0 ɨɩɪɟɞɟɥɹɟɬɫɹ ɩɨ ɮɨɪɦɭɥɟ
k0
1 cos2D0
,
sin 2D0
ɝɞɟ ɭɝɨɥ D0 45qE0 ɨɩɪɟɞɟɥɹɟɬɫɹ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ.
112
(9)
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
Ɋɢɫ. 1. Ʉɪɭɝ Ɇɨɪɚ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ
ɇɚ ɷɬɨɦ ɪɢɫɭɧɤɟ Ɉɰ – ɰɟɧɬɪ ɤɪɭɝɚ Ɇɨɪɚ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ;
V , W – ɤɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɤɚɫɚɧɢɹ ɨɝɢɛɚɸɳɟɣ ɤɪɭɝɚ ɧɚ ɨɞɧɨɨɫɧɨɟ ɫɠɚ0
0
ɬɢɟ; Vc – ɩɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ; D0 – ɭɝɨɥ ɫɪɟɡɚ.
ɂɡ ɩɪɢɜɟɞɟɧɧɨɝɨ ɜɵɲɟ ɩɪɟɞɩɨɥɨɠɟɧɢɹ, ɱɬɨ D o 45q ɩɪɢ ɫ 1 3
ɩɨɥɭɱɢɦ, ɱɬɨ ɜ ɩɥɨɫɤɨɫɬɢ ɫɪɟɡɚ ɫɨɨɬɧɨɲɟɧɢɟ ɦɟɠɞɭ ɧɨɪɦɚɥɶɧɵɦ ɢ ɤɚɫɚɬɟɥɶɧɵɦ ɧɚɩɪɹɠɟɧɢɹɦɢ ɜɫɟ ɜɪɟɦɹ ɢɡɦɟɧɹɟɬɫɹ ɫ ɢɡɦɟɧɟɧɢɟɦ ɜɢɞɚ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ. ɗɬɨ ɨɛɫɬɨɹɬɟɥɶɫɬɜɨ ɦɨɠɧɨ ɭɱɟɫɬɶ, ɟɫɥɢ ɞɥɹ ɤɨɷɮɮɢɰɢɟɧɬɚ k ɜɜɟɫɬɢ ɢɧɬɟɪɩɨɥɹɰɢɨɧɧɭɸ ɮɨɪɦɭɥɭ:
k k (c )
1 ªck ( 1 c)k º ,
n
0
»¼
3
1 3 «¬
ɝɞɟ kn ɪɚɜɧɨ ɡɧɚɱɟɧɢɸ, ɩɨɥɭɱɚɟɦɨɦɭ ɞɥɹ ɩɪɟɞɟɥɶɧɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ
ɫɨɫɬɨɹɧɢɹ ( ɫ 1 3 ), ɩɪɢ ɤɨɬɨɪɨɦ ɟɳɟ ɫɩɪɚɜɟɞɥɢɜ ɤɪɢɬɟɪɢɣ Ʉɭɥɨɧɚ–
Ɇɨɪɚ.
ɂɡ ɫɨɨɬɧɨɲɟɧɢɹ (8), ɭɱɢɬɵɜɚɹ ɮɨɪɦɭɥɵ (2) ɢ (3), ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɭɝɨɥ ɫɪɟɡɚ E0 ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɫ:
1 c k 1 k 2 1 c 1 c
2
sin2E0
1 c1 k 2 2
.
(10)
ɍɱɢɬɵɜɚɹ (2),(3) ɢ (7), ɭɫɥɨɜɢɟ (1) ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
113
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
V1s V1s ɫ
2S0
,
1 ccos2E0 k ˜sin 2E0 (11)
ɝɞɟ V1s – ɩɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɧɚɩɪɹɠɟɧɧɨɦ ɫɨɫɬɨɹɧɢɢ
ɞɥɹ c d1 3 .
ȼɟɥɢɱɢɧɚ S0 ɨɩɪɟɞɟɥɹɟɬɫɹ ɢɡ ɫɨɩɨɫɬɚɜɥɟɧɢɹ ɫ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦɢ ɞɚɧɧɵɦɢ. ɉɨɞɨɛɧɨ ɚɧɚɥɨɝɢɱɧɨɦɭ ɦɟɬɨɞɭ [5] ɜɵɪɚɠɟɧɢɹ ɩɪɟɞɟɥɨɜ ɩɪɨɱɧɨɫɬɢ, S0 ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɫɥɟɞɭɸɳɟɦ ɜɢɞɟ:
S0 S00 exp [c Kc2 ,
(12)
ɝɞɟ [ ɢ K ɜ ɨɛɳɟɦ ɫɥɭɱɚɟ ɨɩɪɟɞɟɥɹɸɬɫɹ [6] ɩɨ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ
ɞɚɧɧɵɦ ɞɥɹ ɩɪɟɞɟɥɨɜ ɩɪɨɱɧɨɫɬɢ ɩɪɢ ɤɚɤɢɯ-ɥɢɛɨ ɞɜɭɯ ɜɢɞɚɯ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ.
ɉɪɢ c 0 ɩɚɪɚɦɟɬɪ S0 ɞɨɥɠɟɧ ɛɵɬɶ ɪɚɜɟɧ ɩɪɟɞɟɥɭ ɩɪɨɱɧɨɫɬɢ ɧɚ
ɨɞɧɨɨɫɧɨɟ ɫɠɚɬɢɟ Vɫ . ɋɨɝɥɚɫɧɨ ɷɬɨɦɭ ɭɫɥɨɜɢɸ ɢ ɭɱɢɬɵɜɚɹ (11) ɢ (12),
ɩɨɥɭɱɢɦ
S00 1 Vc cos2E00 k0 sin 2E00 E00 E0 c 0 ,k0 k c
2
0
.
ȼɯɨɞɹɳɢɟ ɜ ɜɵɪɚɠɟɧɢɟ ɞɥɹ S0 ɩɚɪɚɦɟɬɪɵ [ ɢ K ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɜ ɩɟɪɜɨɦ ɩɪɢɛɥɢɠɟɧɢɢ, ɢɫɩɨɥɶɡɭɹ ɬɨɥɶɤɨ ɷɤɫɩɟɪɢɦɟɧɬ ɧɚ ɨɞɧɨɨɫɧɨɟ ɫɠɚɬɢɟ. Ⱦɥɹ ɷɬɨɝɨ ɜɧɚɱɚɥɟ ɧɚɞɨ ɪɚɫɫɦɨɬɪɟɬɶ ɩɨɫɬɪɨɟɧɢɟ ɨɝɢɛɚɸɳɟɣ ɤ ɤɪɭɝɚɦ Ɇɨɪɚ.
ɍɪɚɜɧɟɧɢɟ ɤɪɭɝɨɜ Ɇɨɪɚ ɦɨɠɧɨ ɩɪɟɞɫɬɚɜɢɬɶ ɜ ɜɢɞɟ [6]
M(V, W, ɫ ) V2 W2 (1 c )V1V cV12 0,
(13)
ɝɞɟ ɜ ɤɚɱɟɫɬɜɟ ɩɚɪɚɦɟɬɪɚ ɞɚɧɧɨɝɨ ɫɟɦɟɣɫɬɜɚ ɤɪɭɝɨɜ ɮɢɝɭɪɢɪɭɟɬ ɫ .
ɋɨɝɥɚɫɧɨ ɢɡɜɟɫɬɧɨɣ ɬɟɨɪɟɦɟ [7] ɨɝɢɛɚɸɳɚɹ ɫɟɦɟɣɫɬɜɚ ɜɢɞɚ (13)
ɞɨɥɠɧɚ ɬɚɤɠɟ ɭɞɨɜɥɟɬɜɨɪɹɬɶ ɭɪɚɜɧɟɧɢɸ
ɢɥɢ
Mɫ (V, W, ɫ) 0 ( Mɫ wM / wɫ )
(14)
Mɫ (V, W, ɫ ) (V1 )c (1 c )V V1V V12 2V1c (V1 )c 0.
(15)
ɂɡ ɫɢɫɬɟɦɵ ɭɪɚɜɧɟɧɢɣ (11) ɢ (15) ɨɩɪɟɞɟɥɹɟɬɫɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɬɨɱɟɤ ( V, W ) ɞɚɧɧɨɣ ɨɝɢɛɚɸɳɟɣ:
114
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
V
V1s [V1s 2c(V1s )c ]
(1 c)V1s
,
W
[V1s c(V1s )c ](V1s )c .
s
s
s
s
V1 (1 c)(V1 )c
V1 (1 c)(V1 )c
(16)
ɂɫɩɨɥɶɡɭɹ (11), ɧɚɣɞɟɦ
ɝɞɟ
(V1s )c V1s R(c),
(17)
(k tg2E0 )(sin 2E0 )c kc sin 2E0
.
R(c) [ 2cK 1 1 c
cos2E0 k sin 2E0
(18)
ɋɨɩɨɫɬɚɜɥɹɹ ɜɵɪɚɠɟɧɢɹ (16) ɢ (17), ɩɨɥɭɱɢɦ
V
W
1 2cR
.
(1 c ) (1 cR ) R
ȼ ɫɥɭɱɚɟ ɨɞɧɨɨɫɧɨɝɨ ɫɠɚɬɢɹ ( ɫ
(19)
0 ) ɢɡ (19) ɢɦɟɟɦ
R0 R(0) 1 (V0 / W0 )2 ,
(20)
V0 / W0 (V / W) c 0 .
(21)
ɝɞɟ
ɂɡ (17), ɭɱɢɬɵɜɚɹ (11) ɢ (18), ɩɪɢ ɫ
ɩɚɪɚɦɟɬɪɚ [ :
[ R
0 ɩɨɥɭɱɢɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ
(sin2E0 )c (k tg2E0 ) kc sin2E0 º
w
w
» (sin 2E0 )c wc sin 2E0 ; kc wc k (c) .
cos2E0 k sin2E0
¼
ȼɬɨɪɨɟ ɭɫɥɨɜɢɟ (ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɚɪɚɦɟɬɪɚ K ) ɩɨɥɭɱɢɦ, ɢɫɩɨɥɶɡɭɹ ɫɜɨɣɫɬɜɚ ɨɝɢɛɚɸɳɟɣ ɤ ɤɪɭɝɚɦ Ɇɨɪɚ.
Ʉɨɨɪɞɢɧɚɬɵ ɬɨɱɤɢ ɤɚɫɚɧɢɹ ɨɝɢɛɚɸɳɟɣ ɤɪɭɝɚ ɧɚ ɨɞɧɨɨɫɧɨɟ ɫɠɚɬɢɟ (V0 , W0 ) ɨɩɪɟɞɟɥɹɟɬɫɹ ɞɜɭɦɹ ɦɟɬɨɞɚɦɢ.
1. Ɋɚɫɫɦɨɬɪɢɦ ɝɟɨɦɟɬɪɢɱɟɫɤɨɟ ɩɨɫɬɪɨɟɧɢɟ (ɫɦ. ɪɢɫ. 1).
ɉɪɢ ɢɡɜɟɫɬɧɨɦ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɦ ɡɧɚɱɟɧɢɢ ɭɝɥɚ ɫɪɟɡɚ D0
ɢ ɩɪɟɞɟɥɟ ɩɪɨɱɧɨɫɬɢ ɧɚ ɫɠɚɬɢɟ V c ɫ ɩɨɦɨɳɶɸ ɬɪɢɝɨɧɨɦɟɬɪɢɱɟɫɤɢɯ
ɫɨɨɬɧɨɲɟɧɢɣ ɢɡ ɤɪɭɝɚ Ɇɨɪɚ ɧɚ ɫɠɚɬɢɟ ɨɩɪɟɞɟɥɹɸɬɫɹ V0 , W0 , ɱɬɨ ɞɚɟɬ
V0 / W0
1 cos2D0
.
sin 2D0
(22)
115
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
2. ȼɵɪɚɡɢɦ ɨɬɧɨɲɟɧɢɟ V0 , W0 ɤɚɤ ɮɭɧɤɰɢɸ ɨɬ ɢɫɬɢɧɧɨɣ ɩɨɪɢɫɬɨɫɬɢ (Ɋ) ɪɹɞɚ ɝɨɪɧɵɯ ɩɨɪɨɞ [8]:
V0 , W0
f ( P).
ɉɪɟɞɜɚɪɢɬɟɥɶɧɨ ɨɬɧɨɲɟɧɢɟ V0 , W0 ɞɥɹ ɤɚɠɞɨɣ ɤɨɧɤɪɟɬɧɨɣ ɩɨɪɨɞɵ ɨɩɪɟɞɟɥɹɟɬɫɹ ɦɟɬɨɞɚɦɢ, ɢɡɥɨɠɟɧɧɵɦɢ ɜ [9]:
V0 , W0
2Vɫ Vp
1 (Vɫ Vp )2
,
ɝɞɟ V p – ɩɪɟɞɟɥ ɩɪɨɱɧɨɫɬɢ ɩɪɢ ɪɚɫɬɹɠɟɧɢɢ.
ɉɪɢ ɢɡɜɟɫɬɧɵɯ ɞɚɧɧɵɯ ɩɨɪɢɫɬɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ ɦɨɠɧɨ ɩɨɥɭɱɢɬɶ ɝɪɚɮɢɱɟɫɤɭɸ ɡɚɜɢɫɢɦɨɫɬɶ V0 W0 f ( P) , ɩɪɢ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɤɨɬɨɪɨɣ ɩɨɥɭɱɚɟɦ ɚɧɚɥɢɬɢɱɟɫɤɨɟ ɜɵɪɚɠɟɧɢɟ ɜɢɞɚ
V0 W0 a b ln P ,
ɝɞɟ a ɢ b ɢɦɟɸɬ ɨɩɪɟɞɟɥɟɧɧɵɟ ɡɧɚɱɟɧɢɹ ɞɥɹ ɤɨɧɤɪɟɬɧɵɯ ɝɨɪɧɵɯ ɩɨɪɨɞ.
Ɉɩɪɟɞɟɥɢɜ ɬɟɦ ɢɥɢ ɢɧɵɦ ɫɩɨɫɨɛɨɦ ɜɟɥɢɱɢɧɭ V0 / W0 , ɭɫɥɨɜɢɟ ɞɥɹ
ɩɚɪɚɦɟɬɪɚ K , ɜɯɨɞɹɳɟɝɨ ɜ ɮɨɪɦɭɥɭ (12), ɩɨɥɭɱɢɦ ɫɥɟɞɭɸɳɢɦ ɨɛɪɚɡɨɦ.
Ɇɚɤɫɢɦɚɥɶɧɨɟ ɤɚɫɚɬɟɥɶɧɨɟ ɧɚɩɪɹɠɟɧɢɟ ɩɪɢ ɩɟɪɟɯɨɞɟ ɡɚ ɭɤɚɡɚɧɧɨɟ ɜɵɲɟ ɩɪɟɞɟɥɶɧɨɟ ɫɨɫɬɨɹɧɢɟ cn (ɬ.ɟ. ɩɪɢ c ! c n ) ɨɫɬɚɟɬɫɹ ɩɨɫɬɨɹɧɧɵɦ. ɂɧɚɱɟ ɝɨɜɨɪɹ,
wW
0.
max
wc
c!cn
(23)
(V1s )c V1s 1 ( c t c n ).
1 c
(24)
ɂɡ ɷɬɨɝɨ ɪɚɜɟɧɫɬɜɚ ɜɵɬɟɤɚɟɬ
Ⱦɥɹ ɛɨɥɶɲɢɧɫɬɜɚ ɝɨɪɧɵɯ ɩɨɪɨɞ, ɤɚɤ ɢɡɜɟɫɬɧɨ, c n
1 / 3 . ɉɪɢ ɷɬɨɦ
ɢ ɭɝɨɥ E 0 ɫ ɪɨɫɬɨɦ c ɧɟ ɢɡɦɟɧɹɟɬɫɹ, ɬ.ɟ.
(sin2E0 )c ɫt1/3 0.
(25)
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɢɡ ɭɫɥɨɜɢɹ (24), ɫ ɭɱɟɬɨɦ (17) ɢ (18), ɩɪɢ ɫ 1 3 ɩɨɥɭɱɚɟɦ ɜɵɪɚɠɟɧɢɟ ɞɥɹ ɩɚɪɚɦɟɬɪɚ K :
116
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
K 3 [.
2
Ƚɪɚɮɢɤɢ ɡɚɜɢɫɢɦɨɫɬɟɣ ɪɚɫɱɟɬɧɵɯ (ɫɩɥɨɲɧɚɹ ɥɢɧɢɹ) ɢ ɩɨɥɭɱɟɧɧɵɯ Ⱥ.ɇ. ɋɬɚɜɪɨɝɢɧɵɦ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨ (ɲɬɪɢɯɨɜɚɹ ɥɢɧɢɹ) ɡɧɚɱɟɧɢɣ
ɩɪɟɞɟɥɨɜ ɩɪɨɱɧɨɫɬɢ ɩɪɢɜɟɞɟɧɵ ɞɥɹ ɧɟɤɨɬɨɪɵɯ ɝɨɪɧɵɯ ɩɨɪɨɞ ɜ ɜɢɞɟ
ɩɪɟɞɟɥɶɧɵɯ ɤɪɭɝɨɜ Ɇɨɪɚ ɢ ɨɝɢɛɚɸɳɢɯ ɤ ɷɬɢɦ ɤɪɭɝɚɦ ɧɚ ɪɢɫ. 2.
ɚ
ɛ
ɜ
Ɋɢɫ. 2. ɉɪɟɞɟɥɶɧɵɟ ɤɪɭɝɢ Ɇɨɪɚ ɢ ɨɝɢɛɚɸɳɢɟ ɤ ɧɢɦ ɞɥɹ ɬɚɥɶɤɨɯɥɨɪɢɬɚ (ɚ);
ɦɪɚɦɨɪɚ-II (ɛ); ɤɜɚɪɰɟɜɨɝɨ ɞɢɨɪɢɬɚ Ⱦ-2 (ɜ)
117
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
ɉɪɢɦɟɱɚɧɢɟ. Ʉɨɨɪɞɢɧɚɬɵ ɨɝɢɛɚɸɳɢɯ, ɩɨɥɭɱɟɧɧɵɟ Ⱥ.ɇ. ɋɬɚɜɪɨɝɢɧɵɦ, ɩɨɫɬɪɨɟɧɵ ɫɨɝɥɚɫɧɨ ɩɪɢɥ. 5 [5]. Ʉɚɤɢɦ ɨɛɪɚɡɨɦ ɜɵɱɢɫɥɟɧɵ ɷɬɢ ɤɨɨɪɞɢɧɚɬɵ ɜ ɦɨɧɨɝɪɚɮɢɢ [5], ɧɟ ɭɤɚɡɚɧɨ. ȼɨɡɦɨɠɧɨ, ɜ ɞɚɧɧɨɦ ɩɪɢɥɨɠɟɧɢɢ ɢɦɟɸɬɫɹ ɧɟɤɨɬɨɪɵɟ ɨɲɢɛɤɢ
(ɨɩɟɱɚɬɤɢ), ɱɬɨ ɧɚɝɥɹɞɧɨ ɩɪɨɹɜɢɥɨɫɶ ɜ ɢɡɨɛɪɚɠɟɧɢɢ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɣ ɨɝɢɛɚɸɳɟɣ
ɞɥɹ ɤɜɚɪɰɟɜɨɝɨ ɞɢɨɪɢɬɚ Ⱦ-2.
Ⱥɧɚɥɨɝɢɱɧɨ ɦɨɠɧɨ ɨɩɪɟɞɟɥɢɬɶ ɩɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɨɦ ɧɚɩɪɹɠɟɧɧɨɦ ɫɨɫɬɨɹɧɢɢ, ɡɚɦɟɧɢɜ ɜɨ ɜɫɟɯ ɩɪɟɞɫɬɚɜɥɟɧɧɵɯ ɜɵɲɟ
ɫɨɨɬɧɨɲɟɧɢɹɯ ɩɪɟɞɟɥɵ ɩɪɨɱɧɨɫɬɢ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɩɪɟɞɟɥɵ ɭɩɪɭɝɨɫɬɢ ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɝɨɪɧɵɯ ɩɨɪɨɞ.
Ɉɫɨɛɨɟ ɦɟɫɬɨ ɡɚɧɢɦɚɟɬ ɜɨɩɪɨɫ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɟɞɟɥɚ ɭɩɪɭɝɨɫɬɢ
ɝɨɪɧɵɯ ɩɨɪɨɞ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ. Ʉɪɨɦɟ ɬɨɝɨ, ɧɟɨɛɯɨɞɢɦɨ ɭɫɬɚɧɨɜɢɬɶ, ɤ ɤɚɤɢɦ ɦɚɬɟɪɢɚɥɚɦ ɫɥɟɞɭɟɬ ɢɯ ɨɬɧɟɫɬɢ: ɤ ɢɡɨɬɪɨɩɧɵɦ ɢɥɢ ɚɧɢɡɨɬɪɨɩɧɵɦ.
Ʉɚɤ ɩɨɤɚɡɚɧɨ ɜ [1], ɞɥɹ ɦɧɨɝɢɯ ɝɨɪɧɵɯ ɩɨɪɨɞ ɩɪɢɟɦɥɟɦɨ ɩɪɟɞɩɨɥɨɠɟɧɢɟ ɨɛ ɨɪɬɨɬɪɨɩɧɨɣ ɫɢɦɦɟɬɪɢɢ ɜ ɢɫɯɨɞɧɨɦ ɫɨɫɬɨɹɧɢɢ.
Ɂɚɤɨɧ Ƚɭɤɚ ɞɥɹ ɨɪɬɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ ɩɪɟɞɫɬɚɜɥɹɟɬɫɹ ɜ ɜɢɞɟ
ɟ1
1 V Q12 V Q13 V ,
1
2
3
ȿ1
ȿ2
ȿ3
ɟ2 Q
Q21
V1 1 V2 23 V3 ,
ȿ1
ȿ2
ȿ3
ɟ3 Q31
Q
V1 32 V2 1 V3 ,
ȿ1
ȿ2
ȿ3
ȿi , Qij const; i, j 1, 2, 3.
Ʉɚɤ ɨɩɪɟɞɟɥɢɬɶ ɤɨɧɫɬɚɧɬɵ, ɜɯɨɞɹɳɢɟ ɜ ɷɬɨɬ ɡɚɤɨɧ, ɩɨɤɚɡɚɧɨ
ɜ [1]. ɂɯ ɡɧɚɱɟɧɢɹ ɞɥɹ ɬɚɥɶɤɨɯɥɨɪɢɬɚ ɢ ɦɪɚɦɨɪɚ-II ɜɡɹɬɵ ɡɞɟɫɶ
ɢɦɟɧɧɨ ɢɡ [1], ɚ ɞɥɹ ɤɜɚɪɰɟɜɨɝɨ ɞɢɨɪɢɬɚ Ⱦ-2 [10]. Ⱦɥɹ ɷɬɢɯ ɩɨɪɨɞ
ɤɨɧɫɬɚɧɬɵ ɭɩɪɭɝɨɫɬɢ ɜ ɩɪɟɞɩɨɥɨɠɟɧɢɢ ɢɡɨɬɪɨɩɧɨɫɬɢ ɦɚɬɟɪɢɚɥɚ
ɩɪɢɜɟɞɟɧɵ ɜ [11].
Ⱥ.ɇ. ɋɬɚɜɪɨɝɢɧ [5] ɞɥɹ ɨɩɪɟɞɟɥɟɧɢɹ ɩɪɟɞɟɥɨɜ ɭɩɪɭɝɨɫɬɢ ɩɪɟɞɥɚɝɚɥ ɢɫɩɨɥɶɡɨɜɚɬɶ ɡɚɜɢɫɢɦɨɫɬɶ ɨɛɴɟɦɧɨɣ ɞɟɮɨɪɦɚɰɢɢ ɨɬ ɫɪɟɞɧɟɝɨ ɧɚɩɪɹɠɟɧɢɹ ( T ~ V , ɝɞɟ T H1 2H2 , V (V1 2V2 ) 3 ), ɩɪɢɱɟɦ ɡɚ ɩɪɟɞɟɥɶɧɵɟ
ɭɩɪɭɝɢɟ ɫɨɫɬɨɹɧɢɹ ɩɪɢɧɢɦɚɸɬɫɹ ɬɨɱɤɢ, ɤɨɝɞɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɟ ɤɪɢɜɵɟ ɷɬɨɣ ɡɚɜɢɫɢɦɨɫɬɢ ɨɬɯɨɞɹɬ ɨɬ ɥɢɧɢɢ ɝɢɞɪɨɫɬɚɬɢɱɟɫɤɨɝɨ ɫɠɚɬɢɹ
( ɫ 1 ). ɉɪɢ ɷɬɨɦ ɧɚ ɧɚɱɚɥɶɧɨɦ ɭɱɚɫɬɤɟ ɞɢɚɝɪɚɦɦɵ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ
118
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
ɨɫɟɜɚɹ ɞɟɮɨɪɦɚɰɢɹ ( H1 ) ɞɨɥɠɧɚ ɛɵɬɶ ɩɪɨɩɨɪɰɢɨɧɚɥɶɧɚ ɨɫɟɜɨɦɭ ɧɚɩɪɹɠɟɧɢɸ. Ⱥɧɚɥɢɡɢɪɭɹ ɩɨɥɭɱɟɧɧɵɟ ɬɚɤɢɦ ɨɛɪɚɡɨɦ ɡɧɚɱɟɧɢɹ ɩɪɟɞɟɥɨɜ
ɭɩɪɭɝɨɫɬɢ ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɜɢɞɚɯ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ, ɩɪɢɯɨɞɢɦ ɤ
ɜɵɜɨɞɭ: ɬɨɱɤɢ, ɨɬɜɟɱɚɸɳɢɟ ɡɧɚɱɟɧɢɹɦ ( V1e , H1e ), ɞɟɣɫɬɜɢɬɟɥɶɧɨ ɪɚɫɩɨɥɚɝɚɸɬɫɹ ɧɚ ɫɨɨɬɜɟɬɫɬɜɭɸɳɟɦ ɧɚɱɚɥɶɧɨɦ ɭɱɚɫɬɤɟ ɞɢɚɝɪɚɦɦ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ, ɧɨ ɷɬɢ ɭɱɚɫɬɤɢ ɧɟ ɨɬɜɟɱɚɸɬ ɡɚɤɨɧɭ Ƚɭɤɚ ɧɢ ɞɥɹ ɢɡɨɬɪɨɩɧɨɝɨ,
ɧɢ ɞɥɹ ɨɪɬɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ.
ɂɧɚɱɟ ɝɨɜɨɪɹ, ɞɥɹ ɤɚɠɞɨɝɨ ɧɚɩɪɹɠɟɧɧɨɝɨ ɫɨɫɬɨɹɧɢɹ ɡɧɚɱɟɧɢɹ
ɩɪɟɞɟɥɚ ɭɩɪɭɝɨɫɬɢ ɨɩɪɟɞɟɥɹɥɢɫɶ ɷɦɩɢɪɢɱɟɫɤɢ ɩɪɢ ɢɫɩɨɥɶɡɨɜɚɧɢɢ ɥɢɧɟɣɧɨɣ ɚɩɩɪɨɤɫɢɦɚɰɢɢ ɧɚɱɚɥɶɧɵɯ ɭɱɚɫɬɤɨɜ ɞɢɚɝɪɚɦɦ V1 ~ H1 ɢ ɡɚɜɢɫɢɦɨɫɬɢ T ~ V , ɱɬɨ, ɜ ɫɜɨɸ ɨɱɟɪɟɞɶ, ɡɚɜɢɫɢɬ ɜ ɨɩɪɟɞɟɥɟɧɧɨɣ ɦɟɪɟ ɨɬ ɜɨɥɢ
ɷɤɫɩɟɪɢɦɟɧɬɚɬɨɪɚ. Ʉɪɨɦɟ ɬɨɝɨ, ɨɫɬɚɟɬɫɹ ɧɟɩɨɧɹɬɧɵɦ, ɤɚɤɢɦ ɨɛɪɚɡɨɦ
ɨɩɪɟɞɟɥɹɬɶ ɩɪɟɞɟɥɵ ɭɩɪɭɝɨɫɬɢ ɞɥɹ ɬɟɯ ɧɚɩɪɹɠɟɧɧɵɯ ɫɨɫɬɨɹɧɢɣ, ɞɥɹ
ɤɨɬɨɪɵɯ ɧɟɬ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɯ ɞɚɧɧɵɯ.
ȼ ɞɚɧɧɨɣ ɪɚɛɨɬɟ ɩɪɟɞɥɚɝɚɟɬɫɹ ɫɥɟɞɭɸɳɢɣ ɦɟɬɨɞ ɨɩɪɟɞɟɥɟɧɢɹ ɜɟɥɢɱɢɧ ɩɪɟɞɟɥɨɜ ɭɩɪɭɝɨɫɬɢ.
ɉɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ ɩɪɢ ɨɞɧɨɨɫɧɨɦ ɫɠɚɬɢɢ ɨɩɪɟɞɟɥɹɟɦ ɩɨ ɦɟɬɨɞɢɤɟ Ⱥ.ɇ. ɋɬɚɜɪɨɝɢɧɚ, ɬ.ɟ. ɞɥɹ ɪɚɫɫɦɚɬɪɢɜɚɟɦɵɯ ɦɚɬɟɪɢɚɥɨɜ ɩɪɢɧɢɦɚɟɦ
ɬɟ ɢɯ ɡɧɚɱɟɧɢɹ, ɤɨɬɨɪɵɟ ɞɚɧɵ ɜ [5]. ɇɨ ɬɨɱɤɭ ( V1e , H1e ) ɧɚ ɞɢɚɝɪɚɦɦɟ
V1 ~ H1 ɨɬɤɥɚɞɵɜɚɟɦ ɧɟ ɧɚ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɣ ɤɪɢɜɨɣ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ,
ɚ ɧɚ ɥɢɧɢɢ, ɨɬɜɟɱɚɸɳɟɣ ɡɚɤɨɧɭ Ƚɭɤɚ ɞɥɹ ɨɪɬɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ. Ɂɚɬɟɦ ɬɚɤɢɦ ɠɟ ɨɛɪɚɡɨɦ ɨɬɦɟɱɚɟɦ ɩɪɟɞɟɥɵ ɭɩɪɭɝɨɫɬɢ (ɜɵɱɢɫɥɹɟɦɵɟ ɩɨ
ɭɤɚɡɚɧɧɵɦ ɜɵɲɟ ɡɚɜɢɫɢɦɨɫɬɹɦ) ɩɪɢ ɩɪɨɢɡɜɨɥɶɧɵɯ ɧɚɩɪɹɠɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹɯ.
ɇɢɠɟ, ɧɚ ɪɢɫ. 3, ɩɪɢɜɟɞɟɧɵ ɞɢɚɝɪɚɦɦɵ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ, ɩɨɫɬɪɨɟɧɧɵɟ ɩɨ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɵɦ ɞɚɧɧɵɦ (ɫɩɥɨɲɧɚɹ ɥɢɧɢɹ) [5] ɞɥɹ ɬɚɥɶɤɨɯɥɨɪɢɬɚ, ɦɪɚɦɨɪɚ-II ɢ ɤɜɚɪɰɟɜɨɝɨ ɞɢɨɪɢɬɚ Ⱦ-2, ɥɢɧɟɣɧɵɟ ɡɚɜɢɫɢɦɨɫɬɢ, ɩɨɫɬɪɨɟɧɧɵɟ ɩɨ ɡɚɤɨɧɭ Ƚɭɤɚ ɞɥɹ ɨɪɬɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ (ɲɬɪɢɯɩɭɧɤɬɢɪɧɚɹ ɥɢɧɢɹ) ɢ ɞɥɹ ɢɡɨɬɪɨɩɧɨɝɨ ɦɚɬɟɪɢɚɥɚ (ɲɬɪɢɯɨɜɚɹ ɥɢɧɢɹ),
ɨɬɦɟɱɟɧɵ ɬɚɛɥɢɱɧɵɟ (ɤɪɭɝɨɦ) ɢ ɪɚɫɱɟɬɧɵɟ (ɪɨɦɛɨɦ) ɡɧɚɱɟɧɢɹ ɩɪɟɞɟɥɨɜ ɭɩɪɭɝɨɫɬɢ.
119
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
ɚ
ɛ
Ɋɢɫ. 3. Ⱦɢɚɝɪɚɦɦɵ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ ɬɚɥɶɤɨɯɥɨɪɢɬɚ (ɚ); ɦɪɚɦɨɪɚ-II (ɛ);
ɞɢɨɪɢɬɚ Ⱦ-2 (ɜ) ɩɪɢ ɪɚɡɥɢɱɧɵɯ ɧɚɩɪɹɠɟɧɧɵɯ ɫɨɫɬɨɹɧɢɹɯ (ɫɦ. ɬɚɤɠɟ ɫ. 120)
120
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
ɜ
Ɋɢɫ. 3. Ɉɤɨɧɱɚɧɢɟ
Ȼɢɛɥɢɨɝɪɚɮɢɱɟɫɤɢɣ ɫɩɢɫɨɤ
1. Ɋɵɱɤɨɜ Ȼ.Ⱥ. Ɉ ɞɟɮɨɪɦɚɰɢɨɧɧɨɦ ɭɩɪɨɱɧɟɧɢɢ ɝɨɪɧɵɯ ɩɨɪɨɞ //
ɂɡɜ. ɊȺɇ ɆɌɌ. – 1999. – ʋ 2.
2. Ɋɵɱɤɨɜ Ȼ.Ⱥ. ɍɫɥɨɜɢɟ ɬɟɤɭɱɟɫɬɢ, ɞɢɥɚɬɚɧɫɢɹ ɢ ɪɚɡɪɭɲɟɧɢɟ ɝɨɪɧɵɯ ɩɨɪɨɞ // Ɏɢɡ.-ɬɟɯɧ. ɩɪɨɛɥ. ɩɪɨɱɧɨɫɬɢ ɪɚɡɪɚɛ. ɩɨɥɟɡɧ. ɢɫɤɨɩ. –
2001. – ʋ 1.
3. Ɍɚɪɚɫɨɜ Ȼ.Ƚ. Ɂɚɤɨɧɨɦɟɪɧɨɫɬɢ ɞɟɮɨɪɦɢɪɨɜɚɧɢɹ ɢ ɪɚɡɪɭɲɟɧɢɹ
ɝɨɪɧɵɯ ɩɨɪɨɞ ɩɪɢ ɜɵɫɨɤɢɯ ɞɚɜɥɟɧɢɹɯ: ɚɜɬɨɪɟɮ. ɞɢɫ. … ɞ-ɪɚ ɬɟɯɧ. ɧɚɭɤ. –
ɋɉɛ., 1991.
4. Ɇɨɪ Ɉ. ɑɟɦ ɨɛɭɫɥɨɜɥɟɧɵ ɩɪɟɞɟɥ ɭɩɪɭɝɨɫɬɢ ɢ ɜɪɟɦɟɧɧɨɟ ɫɨɩɪɨɬɢɜɥɟɧɢɟ ɦɚɬɟɪɢɚɥɨɜ. – ɋɉɛ.: Ɉɛɪɚɡɨɜɚɧɢɟ, 1915. – 75 ɫ.
5. ɋɬɚɜɪɨɝɢɧ Ⱥ.ɇ., ɉɪɨɬɨɫɟɧɹ Ⱥ.Ƚ. ɉɥɚɫɬɢɱɧɨɫɬɶ ɝɨɪɧɵɯ ɩɨɪɨɞ. –
Ɇ.: ɇɟɞɪɚ, 1979.
6. Ɋɵɱɤɨɜ Ȼ.Ⱥ., Ʉɨɧɞɪɚɬɶɟɜɚ ȿ.ɂ., Ɇɚɦɚɬɨɜ ɀ.ɕ. Ⱦɢɥɚɬɚɧɫɢɹ
ɝɟɨɦɚɬɟɪɢɚɥɨɜ ɫ ɭɱɟɬɨɦ ɪɚɡɭɩɪɨɱɧɟɧɢɹ // ȼɟɫɬɧɢɤ ɋɚɦɚɪ. ɝɨɫ. ɭɧ-ɬɚ. –
2004. – ɋɩɟɰ. ɜɵɩ.
7. ɉɨɝɨɪɟɥɨɜ Ⱥ.ȼ. Ⱦɢɮɮɟɪɟɧɰɢɚɥɶɧɚɹ ɝɟɨɦɟɬɪɢɹ. – Ɇ.: ɇɚɭɤɚ, 1974.
8. Ɋɵɱɤɨɜ Ȼ.Ⱥ. Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ //
Ⱦɟɮɨɪɦɢɪɨɜɚɧɢɟ ɢ ɪɚɡɪɭɲɟɧɢɟ ɫɬɪɭɤɬɭɪɧɨ-ɧɟɨɞɧɨɪɨɞɧɵɯ ɫɪɟɞ ɢ ɤɨɧ-
121
Ȼ.Ⱥ. Ɋɵɱɤɨɜ, ɇ.Ɇ. Ʉɨɦɚɪɰɨɜ, Ɍ.Ⱥ. Ʌɭɠɚɧɫɤɚɹ
ɫɬɪɭɤɰɢɣ: ɬɟɡ. ɞɨɤɥ. II ȼɫɟɪɨɫ. ɤɨɧɮ., ɇɨɜɨɫɢɛɢɪɫɤ, 10–14 ɨɤɬ. 2007. –
ɇɨɜɨɫɢɛɢɪɫɤ, 2007.
9. Ɉ ɬɟɨɪɟɬɢɱɟɫɤɨɦ ɢ ɷɤɫɩɟɪɢɦɟɧɬɚɥɶɧɨɦ ɩɨɫɬɪɨɟɧɢɢ ɨɝɢɛɚɸɳɟɣ
ɩɪɟɞɟɥɶɧɵɯ ɤɪɭɝɨɜ Ɇɨɪɚ / ȼ.Ɇ. ɀɢɝɚɥɤɢɧ [ɢ ɞɪ.] // Ɏɢɡ.-ɬɟɯɧ. ɩɪɨɛɥ.
ɩɪɨɱɧɨɫɬɢ ɪɚɡɪɚɛ. ɩɨɥɟɡɧ. ɢɫɤɨɩ. – 2010. – ʋ 6. – ɋ. 25–36.
10. Ʉɨɦɚɪɰɨɜ ɇ.Ɇ., Ɋɵɱɤɨɜ Ȼ.Ⱥ. Ⱦɟɮɨɪɦɚɰɢɨɧɧɨɟ ɩɨɜɟɞɟɧɢɟ
ɝɨɪɧɵɯ ɩɨɪɨɞ // ɋɨɜɪɟɦɟɧɧɵɟ ɩɪɨɛɥɟɦɵ ɦɟɯɚɧɢɤɢ ɫɩɥɨɲɧɨɣ ɫɪɟɞɵ:
ɫɛ. ɬɪ. ɦɟɠɞɭɧɚɪ. ɧɚɭɱ. ɤɨɧɮ. – Ȼɢɲɤɟɤ, 2012.
11. ɋɬɚɜɪɨɝɢɧ Ⱥ.ɇ., Ƚɟɨɪɝɢɟɜɫɤɢɣ ȼ.ɋ. Ʉɚɬɚɥɨɝ ɦɟɯɚɧɢɱɟɫɤɢɯ
ɫɜɨɣɫɬɜ ɝɨɪɧɵɯ ɩɨɪɨɞ. – 2-ɟ ɢɡɞ. / ȼɫɟɫɨɸɡ. ɧɚɭɱ.-ɢɫɫɥɟɞ. ɦɚɪɤɲɟɣɞɟɪ.
ɢɧ-ɬ. – Ʌ., 1972. – 267 ɫ.
References
1. Rychkov B.A. O deformatsionnom uprochnenii gornih porod
[About strain hardening of rocks]. Izvestiya Rossiiskoi akademii nauk. Mehanika tverdogo tela, 1999, no. 2.
2. Rychkov B.A. Uslovie tekuchesti, dilatansiya i razrushenie gornih
porod [The yield condition, dilatancy and fracture of rocks]. Phizikotehnicheskie problemi prochnosti razrabotki poleznih iskopaemih, 2001, no. 1.
3. Tarasov B.G. Zakonomernosti deformirovaniya i razrusheniya
gornih porod pri vysokih davleniyah [Logics of deformation and fracture of
rocks under high pressure]. Abstract of the thesis of doctors of technical sciences, St.-Petersburg, 1991.
4. Mohr O. Chem obuslovlen predel uprugosti i vremennoe soprotivlenie materialov [How specified strength limit and temporary material resistance]. St.-Petersburg: Obrazovanie, 1915, 75 p.
5. Stavrogin A.N., Protosenya A.G. Plastichnost’ gornih porod [Plasticity of rocks]. Moskow: Nedra, 1979.
6. Rychkov B.A., Kondrat’eva E.I., Mamatov J.Y. Dilatansiya geomaterialov s uchetom razuprochneniya [Dilatancy of geomaterials inclusive of
strength loss]. Vestnik Samarskogo gosudarstvennogo universiteta, 2004, special issue.
7. Pogorelov A.V. Differentsial’naya geometriya [Differential geometry]. Moscow: Nauka, 1974.
8. Rychkov B.A. O predelah uprugosti i prochnosti gornih porod
[About elastic and strength limits of rocks]. Tezisy doklada II Vserossiiskoi
konferentsii “Deformirovanie i razrushenie strukturno-neodnorodnih sred i
konstruktsii”. Novosibirsk, 2011.
9. Zhigalkin V.M. [et al.] O teoreticheskom i eksperimental’nom postroenii ogibayushei predel’nih krugov Mora [About theoretical and experi122
Ɉ ɩɪɟɞɟɥɚɯ ɭɩɪɭɝɨɫɬɢ ɢ ɩɪɨɱɧɨɫɬɢ ɝɨɪɧɵɯ ɩɨɪɨɞ
mental constructing of extremal Mohr’s circles curve]. Phiziko-tehnicheskie
problemi prochnosti razrabotki poleznih iskopaemih, 2010, no. 6, pp. 25-36.
10. Komartsov N.M., Rychkov B.A. Deformatsionnoe povedenie
gornih porod [Deformationable behavior of rocks]. Trudy mezhdunarodnoi
nauchnoi konferentsii “Sovremennye problemy mehaniki sploshnoi sredy”.
Bishkek, 2012.
11. Stavrogin A.N., Georgievskii V.S. Katalog mehanicheskih svoistv
gornih porod [Catalogue of rocks mechanical properties]. Vsesoyuznii nauchno-issledovatel’skii marksheiderskii institut, 2 edition, Leningrad, 1972, 267 p.
Ɉɛ ɚɜɬɨɪɚɯ
Ɋɵɱɤɨɜ Ȼɨɪɢɫ Ⱥɥɟɤɫɚɧɞɪɨɜɢɱ (Ȼɢɲɤɟɤ, Ʉɵɪɝɵɡɫɬɚɧ) – ɞɨɤɬɨɪ
ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɩɪɨɮɟɫɫɨɪ, ɩɪɨɮɟɫɫɨɪ ɤɚɮɟɞɪɵ ɦɟɯɚɧɢɤɢ Ʉɵɪɝɵɡɫɤɨ-Ɋɨɫɫɢɣɫɤɨɝɨ ɋɥɚɜɹɧɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ (720000, ɝ. Ȼɢɲɤɟɤ, ɭɥ. Ʉɢɟɜɫɤɚɹ, 44, e-mail: rychkovba@mail.ru).
Ʉɨɦɚɪɰɨɜ ɇɢɤɢɬɚ Ɇɢɯɚɣɥɨɜɢɱ (Ȼɢɲɤɟɤ, Ʉɵɪɝɵɡɫɬɚɧ) – ɤɚɧɞɢɞɚɬ ɮɢɡɢɤɨ-ɦɚɬɟɦɚɬɢɱɟɫɤɢɯ ɧɚɭɤ, ɞɨɰɟɧɬ ɤɚɮɟɞɪɵ ɦɚɬɟɦɚɬɢɤɢ Ʉɵɪɝɵɡɫɤɨ-Ɋɨɫɫɢɣɫɤɨɝɨ ɋɥɚɜɹɧɫɤɨɝɨ ɭɧɢɜɟɪɫɢɬɟɬɚ (720000, ɝ. Ȼɢɲɤɟɤ,
ɭɥ. Ʉɢɟɜɫɤɚɹ, 44, e-mail: komartsovnm@mail.ru).
Ʌɭɠɚɧɫɤɚɹ Ɍɚɬɶɹɧɚ Ⱥɥɟɤɫɚɧɞɪɨɜɧɚ (Ȼɢɲɤɟɤ, Ʉɵɪɝɵɡɫɬɚɧ) –
ɚɫɩɢɪɚɧɬ ɤɚɮɟɞɪɵ ɦɟɯɚɧɢɤɢ Ʉɵɪɝɵɡɫɤɨ-Ɋɨɫɫɢɣɫɤɨɝɨ ɋɥɚɜɹɧɫɤɨɝɨ
ɭɧɢɜɟɪɫɢɬɟɬɚ (720000, ɝ. Ȼɢɲɤɟɤ, ɭɥ. Ʉɢɟɜɫɤɚɹ, 44, e-mail: tatianaluzhanskaya@gmail.com).
About the authors
Rychkov Boris Aleksandrovich (Bishkek, Kyrgyzstan) – Doctor of
Physical and Mathematical Sciences, Professor, Department of Mechanics,
Kyrgyz-Russian Slavic University (44, Kievskaya st., 720000, Bishkek,
Kyrgyzstan, e-mail: rychkovba@mail.ru).
Komartsov Nikita Mihailovich (Bishkek, Kyrgyzstan) – Ph. D. in
Physical and Mathematical Sciences, assistant professor, Department of
Mathematics, Kyrgyz-Russian Slavic University (44, Kievskaya st., 720000,
Bishkek, Kyrgyzstan, e-mail: komartsovnm@mail.ru).
Luzhanskaya Tatiana Aleksandrovna (Bishkek, Kyrgyzstan) – postgraduate student, Kyrgyz-Russian Slavic University (44, Kievskaya st.,
720000, Bishkek, Kyrgyzstan, e-mail: tatianaluzhanskaya@gmail.com).
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