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Fbgbkl_jkl\hh[jZah\Zgby JhkkbckdhcN_^_jZpbb
NZdmevl_lijbdeZ^ghcfZl_fZlbdbbf_oZgbdb DZn_^jZl_hj_lbq_kdhcbijbdeZ^ghcf_oZgbdb Iheamq_klvwe_f_glh\dhgkljmdpbc F_lh^bq_kdb_mdZaZgbybjZ[hqbcieZgihdmjkm L_hjbyiheamq_klb ^eyklm^_glh\-]hdmjkZ^g_\gh]hhl^_e_gbyki_p
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F_lh^bq_kdZyjZajZ[hldZij_^gZagZq_gZ^eyklm^_glh\dmjkZ^g_\gh]h hl^_e_gbynZdmevl_lZijbdeZ^ghcfZl_fZlbdbbf_oZgbdb<=M ki_pbZev ghklv-f_oZgbdZ bih^]hlh\e_gZgZhkgh\_^_ckl\mxs_]hmq_[gh]hieZgZ\kh hl\_lkl\bbkdhlhjufjZa^_eIheamq_klvwe_f_glh\dhgkljmdpbcbamqZ_lky\ hk_gg_fk_f_klj_ qZkh\e_dpbcbqZkh\ijZdlbdb JZajZ[hldZkh^_j`bl jZ[hqb_ieZgue_dpbcbijZdlbq_kdboaZgylbckibkhdhkgh\ghcb^hihegb l_evghcebl_jZlmjuh[sb_f_lh^bq_kdb_mdZaZgbydhk\h_gbxl_hj_lbq_kdh]h fZl_jbZeZbZgZebalbih\uomijZ`g_gbcihhkgh\gufl_fZfIjh]jZffZe_d pbhggh]hdmjkZ\dexqZ_lhkgh\gu_j_amevlZluwdki_jbf_glZevgh]hbamq_gby iheamq_klbijbh^ghhkghfjZkly`_gbbl_ogbq_kdb_l_hjbbiheamq_klbf_lh ^uj_r_gbyaZ^ZqmklZgh\b\r_ckyiheamq_klbbaeZ]Z_fu_ihke_^h\Zl_evgh Hkgh\gufbaZ^ZqZfbdmjkZy\eyxlkyhagZdhfe_gb_klm^_glh\kijbjh^hcb \Z`g_crbfbaZdhghf_jghklyfbnbabq_kdboy\e_gbc\we_f_glZodhgkljmdpbc ijhy\eyxsboiheamq_klvkf_lh^Zfbbkke_^h\Zgbcbhkgh\gufbfZl_fZlbq_ kdbfbfh^_eyfbbamqZ_fuoijhp_kkh\P_evxf_lh^bq_kdhcjZajZ[hldby\ey _lkykh^_ckl\b_klm^_glZf\jZpbhgZevghchj]ZgbaZpbbkZfhklhyl_evghcjZ[h lubieZgbjh\Zgby\j_f_gb\qZklghklbiml_f\u^_e_gbydexq_\uo\hijhkh\ bbeexkljZpbchkgh\guoiheh`_gbcl_hjbbgZijbf_j_j_r_gbydhgdj_lguo aZ^Zqbal_ojZa^_eh\dhlhju_h[uqgh\uau\Zxlmklm^_glh\ljm^ghklb
2
JZ[hqbcieZge_dpbhggh]hdmjkZ Iheamq_klvwe_f_glh\dhgkljmdpbc qZkh\ Hkgh\gu_j_amevlZluwdki_jbf_glZevgh]hbamq_gbyiheamq_klbijb h^ghhkghfjZkly`_gbbIheamq_klvbj_eZdkZpbygZijy`_gbc-qZkZ k-244. Djb\u_iheamq_klb-qZkZk-256. L_ogbq_kdb_l_hjbbiheamq_klbHkgh\gu_ihgylby-qZkZk-270. L_hjbyklZj_gby-qZkZk-273, /3/. L_hjbyl_q_gby-qZkZk-277, /5/. L_hjbymijhqg_gby-qZkZk-283,/4/. L_hjbyiheamq_klbkZgbahljhigufmijhqg_gb_f-qZkZk288,/6/. Wdki_jbf_glZevgZyijh\_jdZbZgZebal_hjbciheamq_klb-qZkZ k-295. Hkh[_gghklbdjZldh\j_f_gghciheamq_klb-qZkZk -298. G_mklZgh\b\rZykybmklZgh\b\rZykyiheamq_klv-qZkZk -302. Qbklucba]b[[jmkZ-qZkZk-312. 12. Ihi_j_qgucba]b[[jmkZ–qZkZk-316. Djmq_gb_[jmkZdhevp_\h]hihi_j_qgh]hk_q_gby–qZkZk-318. Djmq_gb_[jmkZg_djm]eh]hihi_j_qgh]hk_q_gby–qZkZk-326. Lhgdhkl_ggu_pbebg^jbq_kdb_ljm[u–qZkZk-327. Lheklhkl_ggu_ljm[u–qZkZk- k-636. Ebl_jZlmjZ Hkgh\gZy 1. FZebgbgGGIjbdeZ^gZyl_hjbyieZklbqghklbbiheamq_klb-F FZrbgh kljh_gb_ – k 2. JZ[hlgh\XGF_oZgbdZ^_nhjfbjm_fh]hl\zj^h]hl_eZ–F Mq_[ihkh[b_ ^ey\mah\-F GZmdZ –k 3. 4. 5. 6.
>hihegbl_evgZy JZ[hlgh\XGJZkqzl^_lZe_cfZrbggZiheamq_klv // –Ba\:GKKKJHl^ l_oggZmd– –K -800. JZ[hlgh\XGIheamq_klvwe_f_glh\dhgkljmdpbc–F GZmdZ –k DZqZgh\EFL_hjbyiheamq_klb–FNbafZl]ba –k B\e_\>>Dl_hjbbg_mklZgh\b\r_ckyiheamq_klb//Ijh[e_fuf_oZgbdb kiehrghckj_^u–F –K-160.
3
JZ[hqbcieZgijZdlbq_kdboaZgylbcihdmjkm Iheamq_klvwe_f_glh\dhgkljmdpbc qZk H^ghhkgh_gZijy`_ggh_khklhygb_L_hjbyklZj_gby-qZk 131, 136, 139 – 141. H^ghhkgh_gZijy`_ggh_khklhygb_L_hjbyl_q_gby-qZk– 135. H^ghhkgh_gZijy`_ggh_khklhygb_L_hjbymijhqg_gby-qZk 138. G_h^ghhkgh_ gZijy`_ggh_ khklhygb_ L_hjby klZj_gby- qZk 142- 145, 148, 152, 154. G_h^ghhkgh_ gZijy`_ggh_ khklhygb_ L_hjby l_q_gby- qZk 146, 150, 151, 155, 156. G_h^ghhkgh_ gZijy`_ggh_ khklhygb_ L_hjby mijhqg_gby- qZk
1.
EBL?J:LMJ: GGFZebgbgDBJhfZgh\::Rbjrh\K[hjgbdaZ^Zqih ijbdeZ^ghcl_hjbbieZklbqghklbbiheamq_klb–F
4
H[sb_f_lh^bq_kdb_mdZaZgby L_hj_lbq_kdbcfZl_jbZekhklZ\eyxsbckh^_j`Zgb_dmjkZ\dexqZ_lke_ ^mxsb_l_fu h[sb_\hijhku e_dpbyb l_hj_lbq_kdb_hkgh\ul_ogbq_ kdbo l_hjbc iheamq_klb e_dpby – j_amevlZlu ijh\_jdb b ZgZeba l_hjbc iheamq_klb e_dpby \uy\e_gb_hkh[_gghkl_cj_r_gbyaZ^ZqdjZldh\j_f_g ghcmklZgh\b\r_ckybg_mklZgh\b\r_ckyiheamq_klb^eyjZaebqguolbih\^_ nhjfZpbc e_dpby – eymklZgh\e_gbyaZ\bkbfhkl_cf_`^m^_nhjfZpby fb gZijy`_gbyfb kdhjhklyfb bo baf_g_gby b \j_f_g_f g_h[oh^bfh fZdkb fZevghh]jZgbqblvqbkehi_j_f_gguob\ukdZaZlvij_^iheh`_gb_hlhff_` ^m dZdbfb ba gbo kms_kl\m_l nmgdpbhgZevgZy aZ\bkbfhklv Wlh ij_^iheh`_ gb_ b ghkbl gZa\Zgb_ l_ogbq_kdhc l_hjbb iheamq_klb Kms_kl\m_l l_ogbq_ kdb_l_hjbbiheamq_klb klZj_gb_l_q_gb_mijhqg_gb_WlbgZa\Zgby\agZqb l_evghc f_j_ y\eyxlky mkeh\gufb Ke_^m_l ihgbfZlv qlh ihkdhevdm ^_nhj fZpbbiheamq_klby\eyxlky\hkgh\ghfg_h[jZlbfufb^eykemqZyh^ghhkgh]h b g_h^ghhkgh]h gZijy`_ggh]h khklhygby ihklmebjm_lky ijbf_gbfhklv hkgh\ guo]bihl_al_hjbbieZklbqghklb:gZeh]bqghZkkhpbbjh\ZgghfmaZdhgml_q_ gby ijbgbfZ_lky kms_kl\h\Zgb_ ihl_gpbZeZ iheamq_klb f l_ ^himkdZ_lky 5
qlh dhfihg_glu kdhjhkl_c ^_nhjfZpbc iheamq_klb hij_^_eyxlky nhjfmehc : ξ ije = λ
∂f MjZ\g_gb_f ij_^klZ\ey_lkh[hcmjZ\g_gb_]bi_jih\_joghklb\ ∂σ ij
ijhkljZgkl\_ dhfihg_glh\ l_gahjZ gZijy`_gbc σ ij d dhlhjhc hjlh]hgZevgu \_dlhju kdhjhkl_c ^_nhjfZpbc iheamq_klb Nmgdpby f ZgZeh]bqgZ khh\_lkl \mxs_cnmgdpbb\l_hjbbieZklbqghklb ?keb ij_^iheh`blv qlh ihl_gpbZe ^_nhjfZpbc iheamq_klb aZ\bkbl hl \lhjh]hbg\ZjbZglZ^_\bZlhjZgZijy`_gbcbgl_gkb\ghklb^_nhjfZpbcb\j_ f_gb l_ f1 = S ij S ij − [Φ 1 (ε i f )]2 = 0 b dZd b \ l_hjbb fZeuo mijm]h – ieZklbq_ 3 2
kdbo^_nhjfZpbcijbgylvqlh ε ij = λ
∂f 1 lhfuihemqZ_fl_hjbxklZj_gbyX ∂σ ij
GJZ[hlgh\ZHij_^_e_gb_gZijy`_gbcb^_nhjfZpbc^eyg_dhlhjh]hagZq_gby \j_f_gb ih l_hjbb klZj_gby wd\b\Ze_glgh j_r_gbx aZ^Zqb ih l_hjbb fZeuo mijm]h–ieZklbq_kdbo^_nhjfZpbc^eyba\_klghc^bZ]jZffu^_nhjfbjh\Zgby ?kebij_^iheh`blvqlhihl_gpbZeiheamq_klbfaZ\bkblhl\lhjh]hbg\Z jbZglZ^_\bZlhjZgZijy`_gbcbgl_gkb\ghklbkdhjhkl_c^_nhjfZpbbiheamq_ klbb\j_f_gbl_ I = 6 LM 6 LM − [Φ ξ LF I ] = lhfuihemqZ_fl_hjbxl_q_gbyE
FDZqZgh\Z ?kebij_^iheh`blvqlhihl_gpbZekdhjhkl_c^_nhjfZpbciheamq_klbaZ \bkblhl\lhjh]hbg\ZjbZglZ^_\bZlhjZgZijy`_gbcbgl_gkb\ghklbkdhjhkl_c
[
]
^_nhjfZpbciheamq_klbbiZjZf_ljZH^d\bklZl_ I = 6 LM 6 LM − Φ ξ LF ∫ Gε LH = ,
lhihemqZ_fl_hjbxbahljhigh]hmijhqg_gby Ijbbamq_gbbl_fuke_^m_lh[jZlblv\gbfZgb_gZlhqlhgZb[he__ijh klufkihkh[hfwdki_jbf_glZevghcijh\_jdbjZaebqguol_hjbciheamq_klby\ ey_lky khihklZ\e_gb_ wdki_jbf_glZevghc djb\hc j_eZdkZpbb ijb ihklhygghc ^_nhjfZpbbkl_hj_lbq_kdbfbihkljh_ggufbihjZaebqgufl_hjbyfiheamq_ klb ?s_h^bgdjm]\hijhkh\lj_[mxsbom]em[e_ggh]hbamq_gbyk\yaZgk\u y\e_gb_f hkh[_gghkl_c aZ^Zq djZldh\j_f_gghc mklZgh\b\r_cky b g_mklZgh \b\r_ckyiheamq_klbLZdwdki_jbf_glZevgu_bkke_^h\ZgbydjZldh\j_f_gghc iheamq_klbiha\hebebmklZgh\blvqlh\hij_^_e_gghf^bZiZahg_l_fi_jZlmjb gZijy`_gbci_j\ZyklZ^bygZdjb\uoiheamq_klbhlkmlkl\m_lbke_^h\Zl_evgh ^ey j_r_gby aZ^Zq g_h[oh^bfh kgylv h]jZgbq_gby h[ hlkmlkl\bb f]gh\_gguo ieZklbq_kdbo^_nhjfZpbcKdhjhklv^_nhjfZpbb\wlhfkemqZ_ij_^klZ\ey_lky \\b^_kmffukdhjhkl_cmijm]hc^_nhjfZpbb ξ e f]gh\_gghcieZklbq_kdhc ξ p b^_nhjfZpbbiheamq_klb ξ c . DZd ihdZaZeb bkke_^h\Zgby g_mklZgh\b\r_cky iheamq_klb gZijy`_gby g_ij_ju\ghbaf_gyxlky\h\j_f_gbb\k_[he__b[he__ijb[eb`Zxlkyd\_eb qbgZfihemq_gguf\j_r_gbbaZ^ZqmklZgh\b\r_ckyiheamq_klbbjZkij_^_
6
e_gb_ gZijy`_gbc ijb mklZgh\b\r_cky iheamq_klb y\ey_lky dZd [u ij_^_ev guf < j_r_gbb aZ^Zq g_mklZgh\b\r_cky iheamq_klb mkeh\b_f kh\f_klghklb ^_nhjfZpbc ^he`gu m^h\e_l\hjylv dhfihg_glu iheguo ^_nhjfZpbc Z \ j_ r_gbb aZ^Zq mklZgh\b\r_cky iheamq_klb mijm]bfb ^_nhjfZpbyfb ih kjZ\g_ gbxk^_nhjfZpbyfbiheamq_klbfh`ghij_g_[j_qvLh]^Zmkeh\byfkh\f_kl ghklb ^_nhjfZpbc ^he`gu m^h\e_l\hjylv dhfihg_glu ^_nhjfZpbc iheamq_ klb
J_r_gb_lbih\uoaZ^Zq AZ^ZqZ IhemqblvmjZ\g_gb_bah]gmlhchkb[Zedbihklhyggh]hihi_j_qgh]hk_q_ gby\mkeh\byomklZgh\b\r_ckyiheamq_klb Bkihevah\ZlvmjZ\g_gb_khklhygby ε = Ωσ 0n .
z
F
l v
J_r_gb_ IjbfZeuoijh]b[Zo^bnn_j_gpbZevgh_mjZ\g_gb_bah]gmlhchkb[Zedb\ mkeh\byomklZgh\b\r_ckyiheamq_klbijbkl_i_gghcaZ\bkbfhklb^_nhjfZpbb iheamq_klbhlgZijy`_gbybf__l\b^ Q
G ϑ 0 Ω ]^_ ϑ − ijh]b[ V = Fz -ba]b[Zxsbcfhf_gl\l_dms_fk_ = GW - Q[
q_gbbhlaZ^ZgghcgZ]jmadb z –jZkklhygb_^hk_q_gbyhlijZ\h]hdhgpZ[Zedb - Q[ -h[h[s_ggucfh f_gl bg_jpbb ihi_j_qgh]h k_q_gby hlghkbl_evgh hkb kbff_ljbb k_q_gby x, i_ji_g^bdmeyjghciehkdhklbba]b[Z Bgl_]jbjmyihemqbf ) Gϑ = & + G] - Q[
Q
Ω] Q + Q + ;
7
) ϑ = 3 + &] + - Q[
Q
Ω] Q + [ Q + Q + ] ,
]^_KbJ–ihklhyggu_hij_^_ey_fu_ba]jZgbqguomkeh\bcijbz=l, v b dv/dz =0. Kmq_lhfgZc^_gguoihklhygguobf__f ) ϑ = - Q[
Q
ΩO Q +
Q+ ] ] − + . Q + O Q + O Q + Q +
Ijh]b[^hklb]Z_lgZb[hevr_c\_ebqbgu\lhqd_ijbeh`_gbykbeuijb ] = ϑ
PD[
) = - Q[
Q
O Q+ Ω . Q +
AZ^ZqZ Klmi_gqZluckl_j`_gvaZdj_ie_gf_`^m^\mfyZ[khexlgh`_kldbfbhih jZfb b gZ]jm`_g ihklhygghc \h\j_f_gbkbehcF>ebgZmqZkldh\– l1bl2Z iehsZ^bihi_j_qguok_q_gbc–:1b:2Hij_^_eblvaZ\bkbfhklvgZijy`_gbc hl \j_f_gb _keb \ gZqZevguc fhf_gl \j_f_gb gZijy`_gby g_ ij_\hkoh^yl ij_^_emijm]hklbBkihevah\Zlvl_hjbxl_q_gby ξ c = Bσ n ]^_ ξ c -kdhjhklv^_ nhjfZpbb iheamq_klb < – nmgdpby \j_f_gb b n – ihklhyggZy fZl_jbZeZ ijb hij_^_e_gghcl_fi_jZlmj_
l1
A1 F
l2
A2
J_r_gb_ MjZ\g_gb_jZ\gh\_kbybf__l\b^ σ 1 + χσ 2 = S ,
(1)
8
A2 F ; S = ; σ 1 b σ 2 -gZijy`_gbyk`ZlbygZi_j\hfbjZkly`_gbygZ A1 A1
]^_ χ =
\lhjhfmqZkldZoKl_j`_gvh^bgjZaklZlbq_kdbg_hij_^_e_gMkeh\b_kh\f_ klghklbkdhjhkl_c^_nhjfZpbc ξ 1 = ξ 2η , (2) ]^_ η =
l2 ; ξ 1 b ξ 2 -kdhjhklb^_nhjfZpbci_j\h]hb\lhjh]hmqZkldh\ l1
Ijbf_fqlhkdhjhklviheghc^_nhjfZpb ξ jZ\gZkmff_kdhjhkl_cmijm ]hc^_nhjfZpbbb^_nhjfZpbbiheamq_klb ξ =ξ e +ξ c. (3) I_j_c^_fhlkdhjhkl_c^_nhjfZpbcdgZijy`_gbyfbkihevam_faZdhg=m dZbmjZ\g_gb_khklhygby ξ c = Bσ n lh]^Zba ihemqbf ξ=
1 dσ + Bσ n E dt
(4)
Ba ke_^m_lqlh
dσ 1 dσ 2 = −χ dt dt
(5)
MjZ\g_gb_ kmq_lhf b ij_h[jZam_fd\b^m dσ 2 , dt f (σ 2 ) = ( s − χσ 2 ) n − ησ 2n .
EBf (σ 2 ) = (η + χ )
(6)
]^_ Ijhbgl_]jbjm_fwlhmjZ\g_gb_mqblu\Zyqlh\gZqZevgucfhf_gl\j_f_ gb σ 2 = σ 2 (0) − gZijy`_gb_ihemqZ_fh_baj_r_gbyaZ^Zqb\ij_^_eZomijm]h klbIjb f (σ 2 ) ≠ 0 bf__f Ω =
σ
η + χ 2 dσ 2 , E σ 2∫( 0) f (σ 2 )
(7)
t
]^_ Ω = ∫ Bdt -fhghlhggh\hajZklZxsZynmgdpby\j_f_gb 0
NhjfmeZ iha\hey_lhij_^_eblvgZijy`_gbygZ\lhjhfmqZkld_\aZ\b kbfhklbhl\j_f_gbGZijy`_gbygZi_j\hfmqZkld_gZoh^ylkybamjZ\g_gby jZ\gh\_kby ihba\_klghcnmgdpbb σ 2 (t ) . Hq_\b^ghqlh bf__lf_klh_keb f (σ 2 ) ≠ 0 Ijb f (σ 2 ) = 0 ba ke_^m_l qlh σ 2 = const l_j_Zebam_lkykhklhygb_mklZgh\b\r_ckyiheamq_klbIjbwlhf ( S − χσ 2 ) n = ησ 2n ,
(8) 1 n
ke_^h\Zl_evgh σ 2 = S (η + χ ) −1 Z σ 1 -hij_^_ey_lkyba AZ^ZqZ Hij_^_eblvaZ\bkbfhklvhl\j_f_gbijh]b[Zk_q_gby:[Zedbihklhyggh]h ihi_j_qgh]hk_q_gby_kebbgl_gkb\ghklvjZ\ghf_jghjZkij_^_e_gghcgZ]jma db q = Ct ]^_ C = const ; t-\j_fyBkihevah\Zlvl_hjbxmijhqg_gby 9
ξ ε β = ασ ν .
(9) t q(t) A
l
J_r_gb_ Bkdhfucijh]b[hij_^_ey_lkykihfhsvxbgl_]jZeZFhjZ W
O
0 ϑ = ∫ ϑ GW ; ϑ = ∫ - Q[
Q
0 .G] F-ba]b[Zxsbcfhf_gl - Q[ -h[h[s_ggucfhf_glbg_jpbbihi_j_qgh]h
k_q_gbyF1-ba]b[Zxsbcfhf_gl\l_dms_fk_q_gbbhl_^bgbqghcgZ]jmadb ijbeh`_gghc\gZijZ\e_gbbbkdhfh]hi_j_f_s_gby\lhclhqd_i_j_f_s_gb_ dhlhjhchij_^_ey_lkybgl_]jbjh\Zgb_ijh\h^blkyih\k_c^ebg_[ZedbD-ih klhyggZyfZl_jbZeZijbhij_^_e_gghcl_fi_jZlmj_<gZr_fkemqZ_ O
ϑ = ∫ χ]G] ,
(10)
]^_ χ -djb\bagZbah]gmlhchkb[Zedbz –dhhj^bgZlZhlkqblu\Z_fZyhl k_q_gby: MklZgh\bfaZ\bkbfhklvdjb\baguhlba]b[Zxs_]hfhf_glZGZhkgh\Zgbb ]bihl_auiehkdbok_q_gbc ε = yχ ; b ξ = y
dχ ]^_y –dhhj^bgZlZhlkqblu\Z_fZy dt
hlhkbx hkvye_`bl\iehkdhklbba]b[Z GZijy`_gb_hij_^_ey_lkyihnhjfme_ 1
dχ χ β ν 1+ β y ν −1 y σ = dt α
(11)
Ba]b[Zxsbcfhf_gl M = ∫ σydA A
kmq_lhf ijbgbfZ_l\b^
Gχ χ - [ , 0 = α GW β
]^_ - [ = ∫ \
ν
+ β +ν ν
(12)
G$ .
$
Ijhbgl_]jbjm_f kmq_lhfgZqZevgh]hmkeh\by ( f = 0 : χ = 0) . 10
W 0 Bf__f χ = α + β ∫ -[
ν
+β
.
<jZkkfZljb\Z_fhcaZ^Zq_ M =
[
χ = α ( + β )(& ′] - [ ) I ν
+ν
+ ν
]
qz 2 = C ′z 2 t ]^_ C ′ = C / 2 ke_^h\Zl_evgh 2
+β
. Ih^klZ\eyyihemq_ggh_\ujZ`_gb_\ bf__f
ν &′ ϑ = α ( + β ) I -[
+ν
+ ν
+β
O
∫]
+β + ν +β
G] ,
hldm^Zihke_bgl_]jbjh\ZgbyhdhgqZl_evghihemqZ_f &′ I ϑ = α + β ν + [ ν
+ν
+β
+ β ⋅ O + β + ν
J_^ZdlhjDmag_ph\ZA?
11
+ β +ν +β
.
