Maziko
Ntchito ya mphamvu ya zinthu ndi kudziwa mphamvu zakunja zomwe thupi lolimba lingathe kupirira (mwachitsanzo, nyumba, gawo la kapangidwe, kapena nthaka). Kuti tiyankhe funsoli, timayambira pa mfundo ziwiri: mbali imodzi, ma naponi sayenera kupitirira malire enaake amene amadalira mtundu wa chinthu, ndipo mbali ina, kusiyana pakati pa mphamvu zakunja ndi zamkati kuyenera kukhala kolimba. Mogwirizana ndi izi, mavuto onse a mphamvu ya zinthu angagawidwe kukhala “Mavuto a ma naponi” ndi “mavuto a kukhazikika.”
Pa ndodo ndi matabwa komanso zonyamulira zomangidwa kuchokera mwa iwo (ma truss, ma frame) kuzindikira ma stress kumachitika m’magawo awiri, moti koyamba potengera mphamvu zakunja zoperekedwa, chigumula cha mphamvu zamkati zomwe zimadutsa kudzera mu cross-section chimawerengedwa, ndiyeno kuchokera ku ma “sechional forces” amenewa ma stress amatsimikiziridwa. Gawo loyamba la njira imeneyi limapanga gawo lapadera, lalikulu, la statics ya zomangamanga, lomwe lili ndi njira zake zokonzedwa bwino ndipo nthawi zambiri limasiyanitsidwa ndi sayansi ya kulimba kwa zinthu, yomwe ntchito yake imangokhala yodziwira ma stress kuchokera ku sectional forces operekedwa.
Mfundo zofunika za kulimba kwa zinthu
Kufotokozera za mikangano kumafika kudzera mu mfundo zotsatirazi za maziko. Tiyeni tiganizire kuti pa thupi pali dongosolo la mphamvu zakunja lofanana. Tiganizire kuti tagawa thupi ndi kudula kulikonse kukhala magawo awiri, ndipo tichotsa limodzi la magawowo pamodzi ndi mphamvu zomwe zimagwira pa ilo; ndiye kuti, kuti kuchita kwake pa gawo lina, kuti kufanana kwa mphamvu ndi kugawa kwa kusintha kusasinthe, kuyenera kusinthidwa ndi mphamvu zowonjezera zomwe zimadutsa kudzera pa pamwamba pa kudula. Mphamvu izi “zamkati” zimagawidwa mosalekeza pa pamwamba pa kudula, moti pa chilichonse cha pamwamba dF pamagwirizana mphamvu dβ. Ngati atchula chiŵerengero dβ dF ndipo apange malire dF -> 0, ndiye chiŵerengero ichi chimayandikira mtengo womaliza umene umatchedwa mikangano. Mikangano pa chigawo china cha pamwamba ndi, monganso dβ, ndipo ingagawanike kukhala zigawo zoyima ndi zogwira limodzi ndi dF, mikangano yachibadwa σ ndi mikangano ya kumeta τ.
Ngati zigawo zodutsira zitengedwa pa mfundo imodzi yomweyi m’njira zosiyanasiyana, ma stress osiyanasiyana adzawagwirizana. Makamaka ngati kuchokera m’thupi atulutsidwa kyubu yoyambira, yosonyezedwa pa chith. 1, pa lililonse la maonekedwe ake asanu ndi limodzi padzachita vector ya stress yomwe ingasiyanitsidwe kukhala zigawo zitatu m’njira ya ma axis a coordinates x, y, z. Ma stress omwe amachita pa mbali zotsutsana za pamwamba amasiyana pakati pawo, ndithudi, pa differential yokha, kotero kuti muyenera kusiyanitsa 3 * 3 = 9 ya ma component stress, omwe amawoneka motere:
Mavuto a wamba σ amapatsidwa index ya axis yomwe ali m’njira yake ndipo amaonedwa ngati abwino ngati amachititsa kukokera, oipa ngati amachititsa kukanikiza. Mavuto a kudula τ amalandira ma index awiri, woyamba mwa iwo ndi wofanana ndi index ya mavuto a wamba ofanana, ndipo wachiwiri umakhudzana ndi axis yomwe m’njira yake mphamvu ya kudula imagwira ntchito. Ngati njira yabwino ya mavuto a wamba ofanana ili m’njira ya axis yolumikizira yabwino (yolakwika), ndiye mavuto a kudula amaonedwanso ngati abwino ngati agwira ntchito m’njira ya axis yolumikizira yabwino (yolakwika).
Chithunzi 1.
Kugwirizana kwa zovuta za shear
Ngati pa elementary cuboid, sl. 1, ayika chofunikira chakuti mphindi ya mphamvu zonse zokhudzana ndi axis x ikhale zero, timapeza τyz dx dz * dy = τzy dx dy * dz Kuchokera apa, ndi kuchokera ku ma equation awiri oyenerera a ma axis y ndi z, zimapezeka kuti zovuta za kukameta ndi ma index omwewo ndi ofanana pakati pawo τxy = τyx, τyz = τzy, τzx = τxz (1)
Kusintha kwa zigawo
Kuti mavuto amene tafotokoza pamwambawa akhale ndi tanthauzo la m’thupi, zimachokera ku kugwirizana kwawo ndi tanthauzo limene thupi lililonse limakhala nalo likakhala pansi pa mphamvu zakunja. Chith. 2 likusonyeza kyubu loyambirira, lomwe lisanapunduke linali la makona anayi ndipo linali ndi utali wa m’mbali dx, dy, dz. Utali wa m’mbali mutapunduka tingawusonyeze ndi dx + Δdx, dy, + Δdy, dz + Δdz; zigawo εx = Δdx|dx, εy =Δdy|dy, εz = Δdz|dz zimatchedwa kutambasuka. Izi ndi kuchuluka kopanda miyeso ndipo, pa zinthu zonse zomwe zingaganizidwe pomanga, ndi zazing’ono poyerekeza ndi 1. Kupatula kutalika kwa m’mbali, kyubu loyambirira dx * dy * dz lingasinthe komanso makona oyambirira olunjika pakati pa mbali zake.
Chithunzi 2. kusintha kwa gawo
Tiyerekeze kuti pambuyo pa kusokonezeka ngodya pakati pa m’mbali zomwe zimakumana m’mphepete OC ndi yofanana ndi 
ndipo mofananamo kuchepa kwa ngodya yolondola m’mphepete AO ndi kofanana ndi γyz, ndipo kuchepa m’mphepete OB ndi kofanana ndi γzx. Kusintha kwa ngodya zitatuzi γxy, γyz, γzx, kumatchedwa kutsetsereka ndipo nthawi zonse kumakhala kochepa poyerekezera ndi 1. Ndi zigawo zisanu ndi chimodzi za ε ndi γ kusokonezeka kwa element ya kvadara kumafotokozedwa kwathunthu “kusokonezeka kakang’ono”. Kusintha kwa element ya voliyumu d_V = dx * dy * dz_ ndi ΔdV = (dx + Δdx) (dy + Δdy) (dz + Δdz) - dx dy dz, pomwe, pakunyalanyaza zopanga za kutambasuka monga zinthu zazing’ono za dongosolo lapamwamba, timapeza kutambasuka kwa cubic ϱ = ΔdV|dV = εx, εy, εz (2)
Kusuntha
Kusintha kwa thupi lolimba kungafotokozedwe kwathunthu mwanjira imeneyi, kuti pa lililonse la malo ake amapatsidwa kusuntha komwe malo amenewo anakumana nako. Ngati kusuntha kwakukulu komwe kumagwirizana ndi kusuntha kwa thupi lolimba kuchotsedwa mu kuyang’ana, mwanjira imeneyi, kuti, ngati kuli kofunikira, mu kuwerengera kuvomerezedwa dongosolo losunthika la ma coordinate, ndiye kuti zigawo za kusuntha μ, ν, ω zimakhala zazing’ono nthawi zonse, ndipo ma derivative awo malinga ndi ma coordinate amakhala ang’ono poyerekeza ndi 1. Pansi pa lingaliro limeneli, malinga ndi chith. 3 kutalika kwa element ya mzere OA = dx pambuyo pa kusintha
motero 
(3)
Mofananamo, kuchokera ku sl. 3 zikuoneka kuti kusuntha kwa sheya γxy γ1 + γ2: ndipo komanso
(4)
Chithunzi 3 kusuntha
(5) 
(6)
Maubwenzi amenewa amatchedwa zinthu zogwirizanitsa ndipo amasonyeza kuti zinthu zosinthika zingagwirizane ngati mozaiki moti malo adzadzazidwa mosalekeza, zomwe sizingatheke ngati kusintha kwa zinthu kuli mwachisawawa. Zinthu zogwirizanitsa ndi ma equation a differential, choncho, n’zoonekeratu, zigawo zonse zisanu ndi chimodzi za kusintha pa khwato lililonse zimakhala zodziyimira pawokha, koma dongosolo lawo m’dera lililonse lofalikira lomalizidwa limamangiriridwa ndi zinthu (5) ndi (6).
Popeza mikhalidweyi yapezeka mwa kusiyanitsa kuchokera mu ma equation (3) ndi (4), pochita kusankhidwa kwake gawo la maubwenzi omwe ali mu ma equation oyambirira limatayika. Choncho sikokwanira kuti gulu limodzi lokha mwa magulu awiriwa a ma equation likwaniritsidwe, koma magulu onse awiri a ma equation ayenera kukwaniritsidwa ngati kufanana kwa kusokonekera kukufuna kutsimikiziridwa.
Ma equation a ma differential a dongosolo la elasticity
Pa kuwerengera nkhawa ndi kusintha kwa thupi pa zinthu zotanuka amagwiritsidwa ntchito ma equation otsatirawa:
Mikhalidwe ya kulingana kwa chinthu cha voliyumu. Mikhalidwe yakuti ma moments ozungulira ma axis atatu akhale ofanana ndi ziro yagwiritsidwa kale ntchito pa kutulutsa equation (1). Mikhalidwe ya kulingana kwa mphamvu imapereka malinga ndi chith. 4
Chithunzi 4
(7a-c)
Ngati ndi X, Y, Z zitalingalira zigawo za mphamvu za voliyumu (mwachitsanzo, kulemera, mphamvu ya centrifugal) pa unit ya voliyumu. Maubwenzi pakati pa kupsinjika ndi kupunduka. Pano kumatanthauza ubwenzi pakati pa component stress ndi component deformation. M’malingaliro a elasticity, nthawi zambiri chimaganiziridwa kokha ubwenzi wa mzere, umene umawonetsedwa ndi lamulo la Hooke:
(8a-c)
(8d-f)
Ngati ma equation (8a-c) athetsedwa malinga ndi kupsinjika, zimapezeka
(8 a)
kso komanso ma equation awiri oyenerana. Lamulo la Hooke lomwe latulutsidwa ndi equation (8) siliri lamulo lachilengedwe lokwaniritsidwa mosamalitsa, koma ndi chithunzithunzi cha khalidwe lenileni, lomwe limasiyana nalo pang’ono kapena kwambiri, kutengera zinthu ndi kukula kwa kupsinjika. Popeza kusanthula kwa masamu kwa vuto la kulimba kwa zinthu zokhala ndi maubwenzi ovuta, osati a mzere pakati pa kupsinjika ndi kusintha kumabweretsa zovuta zazikulu kwambiri ndipo kungachitidwe kwenikweni kokha m’milandu yosavuta kwambiri, ngakhale pa zinthu (konkire!) zomwe khalidwe lazo limasiyana kwambiri ndi lamulo la Hooke, mawerengedwe okhazikitsidwa pa ilo ayenera kugwiritsidwa ntchito monga njira yoyamba, ndipo nthawi zambiri yokhayo yofanana, yomwe koposa apo nthawi zambiri imakhala yothandiza kwambiri komanso imapereka chidziwitso choposa cha khalidwe lokha. G = E|2 (1+μ).
Izi zimachokera mosapeŵeka ku ubale pakati pa zovuta ndi kusintha kwa mawonekedwe (8), kuchokera ku ma equation a kusintha (16) a zovuta ndi dongosolo loyenera la ma equation a kusintha kwa zigawo. E ndi G ali ndi kukula kwa zovuta ndipo pa zinthu zonse za zomangamanga ndi zazikulu poyerekeza ndi zovuta σ ndi τ zomwe zingawoneke. Koma pa mfundo imeneyi pakhazikika lingaliro lovomerezedwa paliponse mu chiphunzitso cha elasticity, loti kusintha kwa mawonekedwe ndi kochepa. Coefficient ya kusintha kwa mbali (nambala ya Poisson) μ ilibe muyezo ndipo nthawi zonse imakhala pakati pa 0 ndi 1/2 .
Relations between component deformations and displacements. These relations are given by equations (3), (4). They are purely geometrical in nature. These three systems give 3+6+6=15 equations for 15 unknowns, that is, for 6 component stresses, 6 component deformations and 3 component displacements. They are therefore sufficient to determine these unknowns. The mathematical handling of such an extensive system of differential equations is, of course, possible only in special cases. If in equation (8) the component deformations are replaced by component displacements according to equations (3) and (4), and then the normal stresses σ and shear stresses τ, according to equations (8 d - f), are introduced into the equilibrium conditions, equation (7), differential equations are obtained that are called the fundamental equations of the theory of elasticity.
(9)
pomwe 
ikuwonetsa Laplace operator. Poganizira (2) awa ndi ma equation atatu a differential okhala ndi kusamutsidwa kwa zigawo zitatu μ, ν, ω; kuchokera mwa awa, pophunzitsa potengera (3), (4) ndi (8), ma stress ndi ma deformation onse a zigawo angapezeke. Ma equation (9) nthawi zambiri amakhala poyambira pa kusanthula milandu yapadera.
Ntchito ya kusintha mawonekedwe
Tanthauzo
Pamene chonyamulira chayikidwa katundu, malo ogwira ntchito a mphamvu zakunja amalandira kusuntha kwa elastic ndipo mphamvu zimagwira ntchito ina. M’njira imeneyi mphamvu yobweretsedwa ingasinthe kukhala mitundu yosiyanasiyana. Ngati katunduyo wayikidwa mwadzidzidzi, kuti njira ya kusokonekera ikhale ndi liwiro lalikulu, gawo lina la mphamvu imeneyi limasanduka mphamvu ya kayendedwe, yomwe m’kupita kwa nthawi imasowa, kaya mwa kusamutsidwa kupita ku nthaka monga mafunde a elastic amene amatayika mopanda malire, kapena mwa kusandulika kukhala kutentha chifukwa cha kukangana kwamkati. Ngati katundu akuyikidwa pang’onopang’ono moti palibe kuchuluka kooneka kwa mphamvu ya kayendedwe komwe kumachitika, ndiye kuti katundu womwe ukukula pang’onopang’ono kuyambira pa ziro umakhala mu mgwirizano wokhazikika ndi mphamvu zamkati, mphamvu imeneyi imangogwiritsidwa ntchito pa kusokoneza thupi ndipo pamenepo imatchedwa ntchito ya kusokoneza.
Kuwerengera ntchito ya kusintha tiwonetsa poyamba pa kagawo komwe kakukokedwa mu njira imodzi ya kutambasula (chith. 5).
Tiyerekeze kuti kupsinjika komwe kulipo σ ndi kutambasuka koyenera ε zikalandire zowonjezeka dσ ndi dε. Ndiye mphamvu σdF pa njira ya dε * dl idzagwira ntchito σdF * dεdl = σdεdV, ngati dV ikutanthauza voliyumu ya chigawocho. Ngati kupsinjika kukwera kuchokera pa ziro kufika pa mtengo wake womaliza ndipo kutambasuka mogwirizana ndi zimenezo kuchoka pa 0 kufika pa ε, ntchito yonse yogwira pa unit ya voliyumu idzakhala:
Momwemonso kwa element yomwe imanyamula kupsinjika kokoka τ (chith. 6) ntchito ya kusintha pa unit ya voliyumu (ntchito ya kusintha yeniyeni, elastic potential):
pomwe mphamvu imodzi yokha mwa zinayi τ * dF imachita ntchito.
Mwachizolowezi, pamene zida zing’onozing’ono zikugwidwa ndi zigawo zonse zisanu ndi chimodzi za kupanikizika, ntchito yeniyeni ya kusokonekera imapezeka pogwiritsa ntchito kuwonjezera ntchito za zigawo za kupanikizika payekha:
(10)
Pogwiritsa ntchito kusiyanitsa pa malire apamwamba a iliyonse mwa ma integral amenewa, timapeza maubale omwe amagwira ntchito pa nkhani yotambalala kwambiri:
(11)
ndipo mofananamo kwa kusokonezeka kwina kwa zigawo. Ngati thupi ndi elastic, pa mkhalidwe uliwonse wodziwika wa kupsinjika pamakhala nthawi zonse mkhalidwe umodzi womwewo wa kusokonezeka, mosasamala kanthu kuti mkhalidwewo wafikiridwa pakunyamula kapena pakutsitsa katundu. Pamenepo pansi pa ma integral pamakhala ma function osaletsa m’modzi ndipo mphamvu yonse yogwiritsidwa ntchito pakunyamula kenaka potsitsa kwathunthu ndi a = 0, chifukwa pa nkhaniyi malire apamwamba a integral amakhala zero. Choncho mphamvu yonse yomwe imaperekedwa ku thupi pakunyamula imabwezedwa pakutsitsa.
Chithunzi 6
Chithunzi 7
Kwa thupi losakwanira elastiki, ubale pakati pa kupsinjika ndi kusintha kwa nthawi yonse ya kutola katundu ndi kutsitsa uli ndi mawonekedwe omwe akusonyezedwa pafupifupi pa chith. 7 ndi mizere O A B. Mphamvu 
yobwera pa kutola katundu ndi yofanana ndi dera OAA, ndipo mphamvu yobweretsedwa pa kutsitsa ndi dera ABA, motero ntchito yofanana ndi dera lozokotedwa imataya (imasanduka kutentha). Ngati mu njira ya oscillation kupsinjika kumasinthasintha nthawi ndi nthawi pakati pa malire abwino 
ndi malire olakwika, 
kusintha kosakwanira elastiki kumayenda mofanana ndi momwe zasonyezedwera ndi mizere ya dashed pa chith. 7 ndipo pa nthawi iliyonse ya oscillation kusanduka kutentha pa unit volume kumakhala mphamvu DABCD. Khalidwe lotere pa kusintha, pomwe kusintha kumatsalira kumbuyo kwa kupsinjika, limatchedwa elastic hysteresis, ndipo curve DABCD ndi loop ya hysteresis.
Ntchito ya kupindika ndi lamulo la Hooke
Ngati muzinthu zophatikizidwa mu equation (10) mulembamo lamulo la Hooke, kungachitike kuphatikiza, ndipo pa ntchito yapadera ya deformation amapezeka mawu awa:
(12a)
(12b)
(12c)
Ngati mawonekedwe achiwiri a awa alowetsedwa mu equation (11), kachiwiri pamapezeka lamulo la Hooke, ma equation (8), mu mawonekedwe athetseka malinga ndi zovuta. Mawu akuti ntchito ya kusintha zimadalira zigawo za kusintha, monga momwe zasonyezedwera ndi equation (12b), motero ndi ofanana ndi lamulo la Hooke. Pamene equation (12c) ikasiyanitsidwa malinga ndi zovuta, kumapezeka
(13)
Maonekedwe amenewa, mosiyana ndi maonekedwe (11), amagwira ntchito kokha kwa zinthu zimene zimachita mogwirizana ndi lamulo la Hooke ndipo sizingagwiritsidwe ntchito pamene maubwenzi pakati pa mikandamizo ndi kusintha ali osiyana. Mwachibadwa kuti kumbali yakumanja ya ma equation (13) muli kusintha kwa zigawo komwe kumayambitsidwa ndi mikandamizo, ndipo zinthu zina zonse zimachotsedwa zimene zikanakhala zitabuka chifukwa cha kusintha kwa kutentha, kuphulika, kutupa, kukonzanso kwa makristalo, ndi zina zotero.
MFUNDO ZA INTEGRAL MU CHIPHUNZITSO CHA ELASTICITY
Kusuntha kwa m’maganizo. Mfundo ya mphamvu yochepera ya potensulo
Monga momwe lamulo la Hooke, ma equation (8) angasinthidwe ndi equation imodzi (12b), momwemonso m’malo mwa mikhalidwe ya kufanana tingayike mfundo ya kusuntha kwa virtual, ngati lamulo lofunika la makina logwira ntchito mwachibadwa. Lamulo ili limanena kuti ma state a kufanana amadziwika ndi mfundo yakuti ntchito yonse yomwe zachitika ndi mphamvu zonse pa kusintha kochepa kwa mkhalidwe wa kusinthika ndi zero. Ngati kusintha kochepa kwa kusuntha kwa elastic titchule δμ δν, δω, ndiye mphamvu za voliyumu zofotokozedwa ndi equation (7) zimagwira ntchito ya virtual
ndipo mphamvu za pamwamba px, py, pz zomwe zimagwira pa chinthu dF cha pamwamba lakunja la thupi ntchito y 
pomwe integral ya katatu ikukhudza voliyumu yonse, ndipo integral ya kawiri ikukhudza pamwamba lonse lakunja la thupi. Pakuwerengera ntchito yochitidwa ndi mphamvu zamkati amagwiritsidwa ntchito equation (12b), choncho pa ntchito pa unit ya voliyumu amapezeka
Mfundo ya virtual displacements pa maziko amenewa imapereka
(14)
Ngati katundu X, Y, Z px, py., pz atengedwa ngati zigawo za mphamvu yokoka (kulemera kwa chinthu chokha ndi katundu wakunja), ma integral ali ku mbali yakumanja akuimira mphamvu ya m’tsogolo yomwe katundu amenewa anataya, pamene mbali yakumanzere ikuimira kuwonjezeka kwa mphamvu ya m’tsogolo (elastiiki) ya thupi, kotero mfundo imeneyi ikusonyeza kuti mphamvu yonse ya m’tsogolo ya onyamulira ndi katundu amene ali pa iwo, pa nkhani ya kufanana, ili ndi mtengo waukulu kwambiri. Ngati mtengo uwu waukulu kwambiri ndi wochepa kwambiri, kufanana kumakhala kokhazikika.
Mfundo ya mphamvu ya potential yochepa, yofotokozedwa ndi equation (14), mawu (12b) a ntchito ya deformation, ndi mikhalidwe ya kugwirizana kwa ma deformation (5), (6) zimapanga dongosolo la ma equation lomwe n’lokwanira pomanga chiphunzitso cha elasticity, ndipo limafanana kwambiri ndi dongosolo la ma equation. Dongosololi lingakhale lothandiza makamaka pamene kukwaniritsa ma differential equations (9) kumabweretsa zovuta zowerengera ndipo chifukwa chake njira yofananira iyenera kufunidwa, mwachitsanzo, kusamutsidwa kumaganiziridwa pasadakhale mu mawonekedwe oyenera okhala ndi ma constants angapo aulere ndipo awa amatsimikiziridwa kuchokera ku chofunikira chakuti equation (14) ikwaniritsidwe. Choncho palibe yankho lolimba la vuto limene limapezeka, koma kungokhala kuyerekezera kwabwino kwambiri komwe kungatheke ndi mawonekedwe a yankho losankhidwa. Kaya kuyerekezako kokwanira kumadalira makamaka mawonekedwe a yankho lomwe latengedwa, ndipo kugwiritsa ntchito bwino njirayi kumafuna luso ndi chidziwitso.
Mfundo ya Castigliano-v
Pamene pa mfundo ya mphamvu yaing’ono kwambiri ya potentiel kufananizidwa mikhalidwe yonse ya deformation yomwe imakwaniritsa zofunikira za kufanana kwa deformation ndipo kuchokera mwa izo kusankhidwa zomwe ma stress ogwirizana nawo amakwaniritsa zofunikira za kusalinganika, pa mfundo ya Castigliano kufananizidwa mikhalidwe yonse ya ma stress yomwe imakwaniritsa zofunikira za kusalinganika ndipo kuchokera mwa izo kusankhidwa zomwe ma deformation ogwirizana nawo amakwaniritsa zofunikira za kufanana kwa deformation. Mfundo imeneyi imati: pa mikhalidwe yonse yotheka ya kusalinganika, yeniyeni yomwe imachitika ndi imene imagwirizana ndi ntchito ya deformation yaying’ono kwambiri. Mfundo ya Castigliano pamodzi ndi equation (12c) ya ntchito ya deformation ndi zofunikira za kusalinganika zimapanga dongosolo la ma equation lomwe likukwanira pomanga kukana kwa zinthu. Popeza mfundo imeneyi ili ndi lamulo la Hooke, mwayi wake wogwiritsidwa ntchito umadalira lingaliro lakuti ma deformation angafotokozedwe mokwanira ndi lamulo limeneli. Izi zimafuna osati khalidwe la linear elastic la zinthu zokha, komanso zimachotsa kusintha kwa kutentha.
Khalidwe la zinthu kupitirira malire a kusinthasintha
Zochitika pa katundu wosasunthika
Kusintha kwa zinthu pansi pa mphamvu ya ma nape ndi chochitika chovuta kwambiri, motero kuyenera kusinthidwa m’njira zosiyanasiyana kuti kumvetsetsedwe. Chimodzi mwa kusinthaku ndi lingaliro lakuti katundu amaikidwa pang’onopang’ono, monga momwe, mwachitsanzo, zimachitidwira ndi zitsanzo poyesa kukokera ndi kukanikiza.
Ngati ndodo yachitsulo yonyamula katundu wokokera iyikidwa pa katundu wowonjezeka pang’onopang’ono ndipo kutambasuka kwake ε kuyezedwa, amapeza tchati chomwe chikusonyezedwa pa chith. 8. Poyamba kutambasukako n’kochepa kwambiri ndipo kumagwirizana ndi kupsinjika, monga momwe lamulo la Hooke limafunira. Kuyambira pa kupsinjika kwina komwe kwatsimikizika, kugwirizanako kumatha ndipo kutambasuka kumawonjezeka mofulumira. Malo enieni a malire a kugwirizanaku n’kovuta kwambiri kuwatsimikiza. Pamene kuyeza kuli kolondola kwambiri, malirewo amapezeka ali otsika kwambiri. Kwa zinthu zambiri amakhala pa σ = 0, ndipo lamulo la Hooke limangokhala kuyerekezera koyamba kwa khalidwe la elastiki, liwu loyamba la kukula mu mndandanda wa Taylor wa ntchito ε = ε (σ) pafupi ndi chiyambi cha ma axisi.
Ngati ndodo itamasulidwa, zikuoneka kuti, ngati naponi sanali wokwera kwambiri, kusokonekera kumathanso, mogwirizana ndi lamulo la Hooke. Koma izi zimasintha ngakhale ngati naponi winawake wapitirira amene ali pafupi ndi malire a proportionality, otchedwa malire a elasticity; ndiye kusokonekera konse kumakhala ndi gawo la elastic (lobwerera) ndi la inelastic (lokhalitsa). Ngakhale malire a elasticity sangathe kuzindikiridwa molondola, amapezeka kutsika kwambiri pamene kulondola kwa kuyeza kukwera. Komabe, popeza ali ndi tanthauzo lalikulu laumisiri, chifukwa amayimira malire a kufunika pa pafupifupi mawerengedwe onse a mphamvu ya zinthu, ndipo nthawi imodzi ndi malire amene pamenepo amawoneka kuwonongeka koyamba kokhalitsa, amatanthauzidwa mosamalitsa pa mayeso a zitsanzo ndi mwambo wina wovomerezedwa, ndipo, mwachitsanzo, monga malire 0.2% amatchulidwa naponi umene pa iwo kutambasuka kokhalitsa kumafika 0.2%.
Mwamsanga malire a elasticity akapitirira, kusokonekera kumachuluka mwadzidzidzi ndipo kungawonedwe ndi maso; pa kupanikizika pafupifupi kosasintha, ndodo imatalikirabe mosalekeza. Kupanikizika kumeneku kumatchedwa malire a kuyenda. Pa makina a hydraulic oyesera zinthu, pambuyo pa kuyamba kwa kuyenda, kungawonedwenso kutsika kwa kupanikizika komwe kumawonekera kwambiri kapena pang’ono; monga momwe zasonyezedwera pa chith. 8 ndi mzere wosweka; pamenepo maximu imatchedwa malire apamwamba a kuyenda, ndipo kupanikizika kosasintha kwa kuyenda komwe kumatsatira kumatchedwa malire otsika a kuyenda.
Pamene kufalikira kwina kwafika, komwe kuli pafupifupi kopanda kusinthika, kupsinjika kumayambanso kukwera, ndipo zinthu zimauma. M’gawo ili la katundu, kuchepa kooneka kwa gawo lodutsa la ndodo kumachitika chifukwa cha kupanikizika kwa mbali (kosasinthika), ndipo kupita patsogolo kwa chithunzi cha kupsinjika ndi kusintha kumadalira makamaka pa funso loti pakawerengedwe kupsinjika σ mphamvu yokoka P yomwe ikugwira ndodoyo igawidwe ndi gawo loyambirira F0 kapena gawo lapano F. Pa zolinga zaukadaulo, kupsinjika P|F0 ndi kofunika. Kupsinjikaku kumafika pa chiwerengero chapamwamba chomwe chimatchedwa mphamvu ya chinthu, ndipo chiwerengero chimenecho chikadutsa, ndodoyo imachepa kwambiri pamalo amodzi ndi kuthyoka. “Kupsinjika kwenikweni” P|F kumakwera kufikira kusweka.
Chithunzi 8.
Khalidwe la zitsulo zina zambiri limasiyana kwambiri ndi lomwe lafotokozedwa pano. Kuyenda kwa pulasitiki nthawi zambiri sikuwoneka bwino, ndipo kuuma kumayamba mwachindunji pambuyo poti malire a elasticity apitirira. Pa zinthu zolimba zosweka mosavuta (mwala, galasi, konkire) kusintha kosakhala kwa elastic kumakhala kochepa kwambiri ndipo ngakhale pa kukulitsidwa pang’ono kumabwera kusweka. Kukulitsidwa konse komwe chinthu chimapirira kufika pa kusweka ndi muyeso wa kutambasuka (mosiyana: kusweka mosavuta) kwa chinthucho, ndipo kwa chitsulo kumafotokozedwa mwatsatanetsatane mu malamulo aukadaulo. Komabe, chifukwa nthawi yomweyo musanathe kusweka, kusintha kwina kumasonkhana pa malo oyandikana a gawo lopapatiza, mtengo wosiyanasiyana wa kukulitsidwa pa kusweka umapezeka malinga ndi ngati pa kutalika kwa muyeso tengedwa gawo lalikulu kapena laling’ono la ndodo, kotero kuti kuti apeze zotsatira zosatsutsika komanso zoyerekezeka pakati pawo, ayenera kuvomerezedwa kale mwambo winawake. Kukulitsidwa pa kusweka, kawirikawiri, kumayesedwa motere: pa ndodo yozungulira yokhala ndi m’mimba mwa d, pamaso pa mayeso amasonyezedwa kutalika kwa muyeso Ι = 10 d ndipo kuchokera pa kutalikira kwake ΔΙ amawerenga kukulitsidwa pa kusweka εs_Ι = ΔΙ|Ι ._ Ngati kuli kofunika kuyesa ndodo za gawo lina, kutalika kwa muyeso kumatsimikiziridwa malinga ndi m’mimba mwa ndodo ya silinda yozungulira yomwe ili ndi dera lomwelo la gawo lolowera.
Katundu wa nthawi yaitali
Pa mayeso a kutambasula sizikhala ndi kusiyana kwakukulu kuti katundu mpaka kusweka ukuwonjezeka mkati mwa 10 m mphindi kapena maola angapo. Komabe, ngati katundu ukugwira ntchito zaka kapena zaka zambiri, pa mikhalidwe ina zimawoneka kuti ngakhale pansi pa malire a kuyenda pulasitiki kupindika pa kupsinjika kosasintha kumawonjezeka pang’onopang’ono pakapita nthawi. Gawo la elastic, ndiko kuti gawo limene limatha pakachepetsedwa katundu kwathunthu, silisintha kwambiri. Kuwonjezeka kumeneku kwa kupindika kosasintha komwe kumachitika nthawi yayitali kumatchedwa kukwawa. Pa zitsulo, kupatula lead, kukwawa kumakhala kofunika kokha pa kutentha kwakukulu; komabe, kwa injiniya wa zomangamanga kumagwira ntchito pa konkire yomwe imamasuka pang’onopang’ono pa kupsinjika, moti kupsinjika kumawonjezeka mu kulimbitsa komwe kukanikizidwa, ngati kulipo.
Katundu wosinthasintha
Kuti makhalidwe a makina a zinthu, omwe amapezeka pa mayeso a kutambasula, athe kugwiritsidwa ntchito monga muyezo wa katundu wololedwa wa zomangamanga, chofunika ndi chakuti chinthucho pa katundu uliwonse wobwerezedwa chipirire kupsinjika komwe chinapirira kale popanda kuwonongeka. Chofunikira ichi sichikukwaniritsidwa ngati chinthucho chikugwiritsidwa ntchito mobwerezabwereza pokweza ndi kutsitsa, monga mmene zimachitikira, mwachitsanzo, pa zonyamula katundu wochititsidwa ndi mphamvu za inertia za makina omwe sali bwino kusanja. Pakubwerezabwereza kwa katundu kokwanira nthawi zambiri, gawo la zomangamanga lingasweke ngakhale pa kupsinjika komwe kuli kutali kwambiri pansi pa mphamvu ya chinthucho yomwe imatsimikiziridwa ndi kuyesa kokhazikika wamba. Chochitika chothyoka cha mtundu umenewu chimatchedwa kuthyoka chifukwa cha kutopa. Chiwerengero cha n cha ma cycle a katundu chimene chinthucho chingapirire chimadalira kukula kwa kupsinjika σ, moti chimakhala chachikulu kwambiri pamene kupsinjika kuli kochepa. Chitsanzo chodziwika cha kudalirana kumeneku chawonetsedwa pa chith. 9, komwe tikuona kuti mphamvu ya chinthucho ili ndi malire apansi, amene chimapirira ngakhale pa chiwerengero chosamveka cha ma cycle a katundu ndipo chimatchedwa mphamvu ya chinthu yolimba ya nthawi yayitali Ma diagram a mtundu umenewu (chith.9) amatchedwa ma curve a Wӧhler.
Chithunzi 9.
Chitetezo cha zomangamanga
Katundu wiri womaliza ndi wofunika pa zomangamanga: katundu umene umayambitsa kuwonongeka kokhazikika koyamba (malire a kufika pa kukwapula), ndi katundu umene mphamvu yonyamula ya zomangamanga yatha, moti imasweka (malire a kusweka). Popeza pamaso pa malire a kufika pa kukwapula zimene zimavomerezedwa ndi sayansi ya kukana kwa zinthu zimagwira ntchito osachepera pafupifupi (kusuntha kochepa, lamulo la Hooke), chitetezo ku kusintha kosatha chingawunikidwe molimba mtima muzochitika zonse zimene zimakwaniritsidwa kugonjetsa mavuto a masamu. Ngati kupsinjika kuli kosiyanasiyana ndi katundu, chiŵerengero cha kupsinjika pa kufika pa kukwapula ndi kupsinjika kwakukulu komwe kumachitika m’chomangamanga chimakhala chofanana ndi chiŵerengero pakati pa katundu umene kufika pa kukwapula kumachitika ndi katundu weniweni ndipo chimayimira koefishenti ya chitetezo pa kufika pa kukwapula. Komabe, pa milandu yambiri chiŵerengero chosavutachi palibe: kupindika pamene kukokera kuchotsedwa, kupsinjika pakati pa malo otukuka, machitidwe amene kapangidwe kake kamasinthasintha ndi, zomwe n’zovuta kwambiri, kupindika kokhala ndi mphamvu ya axial. M’zochitika zonsezi kuwonjezeka kochepa chabe kwa katundu kungabweretse kuwonjezeka kwakukulu kwa kupsinjika, ndipo pamenepo sikokwanira kugwiritsa ntchito koefishenti wamba ya chitetezo pa kupsinjika, koma iyenera kugwiritsidwa ntchito pa katundu.
Zochitika zomwe zimachitika pambuyo pa kupitirira malire a kugwa mpaka kusweka zimatha kutsatiridwa m’milandu yambiri kokha mwa njira yofotokozera. Ngati zinthu zili zotambasuka, magawo omwe malire a kugwa afika koyamba, ndipo omwe amapitiriza kusintha mawonekedwe pa kupsinjika kosasintha kapena kupsinjika komwe kukukwera pang’ono, amatenga nawo mbali pang’ono pa kulandira katundu wina, motero magawo ena a kapangidwe omwe ali ndi kupsinjika kochepa, ngati kuli kotheka potengera kapangidwe kake, amatenga nawo mbali zambiri pa kunyamula. M’kufunitsitsa kumeneku kofananitsa nsonga za kupsinjika muli chitsimikizo china cha chitetezo, chomwe chimatha kugwiritsidwa ntchito powerengera ngati kusintha kosatha kwa mbali zina, komwe kwachitika pa katundu waukulu umodzi wokha, sikuvulaza (njira ya katundu wa malire).
Ngati zinthu zili zosalimba, kuwonongeka koyamba kosatha kumawoneka ngati ming’alu, chifukwa chake pamalo omwe ali pachiwopsezo pamachitika kusonkhanitsidwa kwakukulu kwa kupanikizika, komwe kumapangitsa kusweka nthawi yomweyo. Kusweka chifukwa cha kutopa ngakhale mu zinthu zotambasuka kumakhala ndi makhalidwe a kusweka kosalimba.
Maubwenzi pakati pa ma naponi pa ma ndege osiyanasiyana a gawo
Ma equation a kusintha
Mavuto a zigawo zisanu ndi zinayi pa chith. 1, amalumikizidwa ndi malamulo atatu a kulinganizika. Zinthu zisanu ndi chimodzi zotsalira σx, σy, σz τxy, τyz, τzx zingapeze makhalidwe osiyanasiyana mwachisawawa mosadalirana wina ndi mnzake, monga momwe zimawonekera kuchokera ku kupereka kwa ma equation osiyanasiyana a chiphunzitso cha elasticity. Komabe, makhalidwe amenewa akaperekedwa, mkhalidwe wa kupsinjika pa mfundo ina ya thupi lolimba umakhazikika kwathunthu, ndiko kuti, pamene kupsinjika kwa ndege zitatu zoyang’anizana kuli kodziwika, n’kotheka kudziwa mosiyanasiyana kupsinjika kwa ndege yachinayi yomwe imadutsa pa mfundo yomwe yaperekedwayo ndipo yomwe normal yake ili ndi njira iliyonse.
Pa chith. 10 akuwonetsedwa tetrahedron OABC yomwe mbali zake zitatu zikufanana ndi ndege za dongosolo la ma koordinati x, y, z ndipo yachinayi imayimira chigawo cha pamwamba cholozeredwa mosasinthika ABC = dF. Kuti zisonyeze komwe chigawochi chili ndi komwe zovutazo zimadutsa kudzera mmenemo, limayikidwa dongosolo lina la ma koordinati, lomwe osi ya ξ ndi yopingasa ku dF.
Chithunzi 10
Mfundo yakuti mphamvu zomwe zikugwira tetrahedro iyi m’njira ya olamulira ξ zili pa kusamala kwa mphamvu zimapereka
Ngati tiganizira kuti ndi
zimapezeka
ndipo mofananamo kuchokera ku mikhalidwe ya kufanana m’mbali mwa axis η:
Awa ndi ma equation omwe amagwiritsidwa ntchito powerengera kupsinjika pakusintha kwa ma coordinates. Chofunika kwambiri ndi kuyang’ana zinthu zotere zokha za malo dF zomwe axis ya ξ imagwirizana ndi axis ya z. Mawonekedwe a nkhaniyi tingapeze pochepetsa mawonekedwe am’mbuyomu kapena kuwerenga mwachindunji kuchokera ku chith. 11:
Chithunzi 11
Kuvuta kwa normal kumatenga ma values apamwamba kwambiri pa mayendedwe α = α0 , amene dσξ/dα = 0. Pogwiritsa ntchito kusiyanitsa kuchokera mu equation (16) timapeza
ndipo akafananiza mawuwa ndi ziro
Kupsinjika τξη kudzakhala nthawi yomweyo kupsinjika konse kwa kukameta m’mawonekedwe amenewa, ngati τyz = τzx = 0. Pamenepo, ndi equation (17) awiri olunjika wolunjika pakati pawo amatsimikiziridwa omwe: 1. kupsinjika konse kwa kukameta ndi zero ndipo, 2. kupsinjika kozolowereka kumakhala ndi mfundo zapamwamba kwambiri (maximum ndi minimum). Malangizo amenewa amatchedwa malangizo a kupsinjika kwakukulu, ndipo kupsinjika kogwirizana kumatchedwa kupsinjika kwakukulu. Amasonyezedwa ndi σ1 ndi σ2 ndipo angawerengedwe mwachindunji kuchokera ku chilinganizo
Bwalo la Mohr
Miyeso (16) ingasonyezedwe m’ma graph m’njira yoonekeratu. Kuti muchite zimenezi ayenera kusinthidwa kukhala mawonekedwe:
Mafananowa amayimira mu dongosolo la ma coordinates σ, τ mawonekedwe a parametric a equation ya bwalo, lojambulidwa pa chith. 12, lomwe ma coordinates a pakati pake ndi σ = 1/2 (σx + σy), τ = 0 ndipo radius yake ndi
Bwalo limeneli limatchedwa bwalo la Mohr. Ngati kudzera pa mfundo x atakokedwa mzere wofanana ndi kudula x =const., pomwepo pa mzere umenewo pamadutsa nkhawa σx, τxy, udzadula bwalolo pa mfundo p, yomwe imatchedwa polu ya bwalo la Mohr. Mzere uliwonse pξ wokokedwa kudzera pa poluyi umadula bwalolo pa mfundo ξ yomwe imasankha nkhawa σξ, τξη zomwe zimapatsirana kudzera pa kudula kofanana ndi pξ.
Ponena za chizindikiro cha ma stress a shear pakujambula bwalo la Mohr tiyenera kukumbukira kuti amaikidwa mmwamba ngati ali ndi njira yosonyezedwa pa chith. 12c. Choncho pa ma stress awiri a conjugate τxy, τyx, imodzi nthawi zonse imakhala yabwino, ndipo ina yoipa.
Kwa malire a σ1, σ2 a zovuta za normal amafanana mfundo 1 ndi 2 pa mozungulira, ndipo mizere _p_1 ndi _p_2 imapereka njira za zovuta zazikulu.
Kuwonetsa ma stress omwe ali m’plani ya x-y pogwiritsa ntchito bwalo la Mahr sikumakhudzana ndi lingaliro lililonse lokhudza ma stress σz, τxz, τyz omwe ali perpendicular pa planeyi.
Chithunzi 12
Choncho izi zimagwiranso ntchito pa mkhalidwe wa kupsinjika wa magawo atatu wamba kwambiri, ngati tikuinganiza kokha malo odulirana amene onse amadutsa pa mzere umodzi womwe pano wasankhidwa kukhala olamulira a z ndi ξ. Popeza malinga ndi ma equation a kusintha wamba (15) kupsinjika koyima σ sikungakhale kopanda malire pa malo odulirana aliwonse, kuyenera kukhala ndi pa malo ena chiwongolero chachikulu chomaliza σ1, ndi pa china chochepa kwambiri σ3, zonsezi, ndithu, zingakhalenso zoipa. Ngati pa ma vector σ1 ndi σ3 aperekedwa ndege ndipo bwalo la Mohr lajambulidwa (chith. 13)
Chithunzi 13
za kupsinjika komwe kuli m’njira imeneyi (mwachitsanzo, mayendedwe onse olunjika pa ndondomeko imeneyi), ndiye kuti σ1 ndi σ3 ndi kupsinjika kwakukulu ndi kochepa komwe kungachitike ndipo motero ziyenera kukhala zolunjika pakati pawo. Ngati mayendedwe 1 ndi 3 awonjezedwa ndi mayendedwe 2, kuti mayendedwe atatuwa apange dongosolo la ma coordinates olunjika kumanja, ndipo ngati ma circle a Mahr ajambulidwa pa ma plane 1 ~2 ndi 2 ~ 3, ndiye σ2 pa umodzi mwa ma circle amenewa ndi waukulu kwambiri, ndipo pa wina ndi kupsinjika kochepa kwambiri, ndipo motero pa malo odutsana olunjika ku mayendedwe 1, 2 ndi 3 kupsinjika kwa shearing n’kofanana ndi ziro. Kupsinjika katatu kolunjika pakati pawo σ1, σ2 ndi σ3, komwe kulibe kupsinjika kwa shearing kogwirizana nako, kumatchedwa kupsinjika kwakukulu.
Malire a kutuluka ndi malire a kusweka pa mkhalidwe wa ma nape wa malo
Malo a ma stress amatchedwa a m’modzi, a awiri kapena a atatu-dimensions, mwanjira ina: a mzere, a lathyathyathya kapena a malo, kutengera ngati imodzi, ziwiri kapena ziwiri zonse zazikulu za stress sizili ziro. Pa zigawo zomangira zomwe zimagwiritsidwa ntchito kwambiri, ndiko kuti pa ndodo yotambasulidwa, yopsinjika ndi yopindika, malo a ma stress, osachepera pa malo okhala ndi stress zambiri, ndi a dimension imodzi, chimodzimodzinso pa mayeso a kutambasula, kupsinjika kapena kupindika, omwe pakuyesa zida pafupifupi nthawi zonse amagwiritsidwa ntchito pofotokoza ma stress ololedwa. Mosiyana ndi pamenepo, pa ma slab okhudzidwa m’plane yawo kapena molunjika nayo, komanso pa ma shell, malo a ma stress ndi a dimensions awiri, ndipo pa milandu ina pamapezekanso malo a ma stress a dimensions atatu, choncho ndi kofunika kuti pogwiritsa ntchito malire a kuyenda kwa pulasitiki ndi kusweka, omwe amatsimikiziridwa pa malo a stress a dimension imodzi, tichotse mfundo pa khalidwe la zinthu pa nkhani ya malo a stress a malo.