Ngā tūāpapa
Ko te mahi a te pūkenga o te ātete o ngā rauemi he whakatau ko ēhea ngā kaha o waho e taea e tētahi tinana mārō te kawe (hei tauira, he whare, he wāhanga hanganga, he oneone rānei). Hei whakautu i tēnei pātai ka tīmata mai i ngā paearu e rua: i tētahi taha, me kaua ngā pēhanga e nui ake i ētahi rohe e whakawhirinaki ana ki te momo rauemi, ā, i tētahi taha anō me ū te taurite i waenga i ngā kaha o waho me ngā kaha o roto. E ai ki ēnei, ka taea te wehe i ngā raruraru katoa o te ātete o ngā rauemi ki ngā “Raruraru pēhanga” me ngā “raruraru pūmautanga.”
Mō ngā tokotoko me ngā kurupae me ngā kaikawe i hangaia mai i ēnei (ngā kupenga, ngā anga), ka mahia te whakatau i ngā ahotea i roto i ngā hipanga e rua, arā, i te tuatahi ka tātaihia i runga i ngā kaha o waho kua tohua te huānga o ngā kaha o roto e whakawhitia ana mā te whiti whānui, kātahi ka whakatauria ngā ahotea mai i ēnei “kaha whiti”. Ko te wāhanga tuatahi o te tukanga he marautanga motuhake, whānui hoki, arā, ko te statics o ngā hanganga hanga, he nui ōna tikanga kua whanaketia, ā, ka wehea nuitia i te ātete o ngā rawa, ko tāna mahi he whakatau i ngā ahotea mai i ngā kaha whiti kua hoatu.
Ngā ariā taketake o te kaha o ngā rawa
Ka tae ki te whakamārama o te mānukanuka mā ngā whakaarotanga taketake e whai ake nei. Me whakaaro kei te mahi tētahi pūnaha taurite o ngā kaha o waho ki te tinana. Whakaarotia kua wehea te tinana mā tētahi tapahanga kōwhiri ki ngā wāhanga e rua, ā, kua tangohia tētahi o aua wāhanga me ngā kaha e pā ana ki runga; nō reira, kia noho tonu te taurite o ngā kaha me te tohatoha o ngā panonitanga ki te wāhanga atu, me whakakapi tōna pānga ki ngā kaha tāpiri e whakawhitia ana mā te mata o te tapahanga. Ko ēnei kaha “ā-roto” e tohaina tonutia ana ki te mata o te tapahanga, nō reira ki ia huānga mata dF ka hāngai he kaha dβ. Ki te hangaia te ōwehenga dβ dF, ā, ka mahia te whakawhitinga rohe dF -> 0, katahi ka piki tēnei ōwehenga ki te uara rohe e kīia ana ko te mānukanuka. Ko te mānukanuka i tētahi huānga mata kua whakaritea, pēnei i a dβ, ā, ka taea te wāwāhi ki ngā wāhanga e poutū ana, e tātaka ana ki dF, arā ko te mānukanuka noa σ me te mānukanuka kutikuti τ.
Mēnā ka tāia ngā wāhanga mā roto i te ira kotahi i ngā ahunga rerekē, ka hāngai mai hoki he ngānga rerekē. Ina koa, mēnā ka wehea mai i te tinana tētahi pūroi pūtake, e whakaatuhia ana i te sl. 1, ka mahi i ia o ōna mata e ono tētahi pūoko ngānga ka taea te wetewete ki ngā wāhanga e toru i te ahunga o ngā toki whakarite x, y z. Ko ngā ngānga e mahi ana ki ngā taha taurite o te mata he rerekē i waenganui i a rātou, ā, mō te rerekētanga kōpiko anake, nō reira me wehe 3 * 3 = 9 o ngā ngānga wāhanga, e tohu ana pērā i raro nei:
Ka hoatu ki ngā tōpana noa σ tētahi taupū o te tuaka e takoto ai rātou, ā, ka whakaarotia he pai mēnā ka whakaputa i te kukume, he kino mēnā ka whakaputa i te pēhanga. Ka whiwhi ngā tōpana kutikuti τ i ngā taupū e rua; ko te tuatahi he ōrite ki te taupū o te tōpana noa e hāngai ana, ā, ko te tuarua e pā ana ki te tuaka e mahi ai te tōpana kutikuti. Mēnā kei te ahunga pai o te tōpana noa e hāngai ana te ahunga pai o te tuaka whakarite (kino), ka whakaarotia hoki te tōpana kutikuti he pai mēnā ka mahi i te ahunga pai o te tuaka whakarite (kino).
Whakaahua 1.
Te hono o ngā pēhanga kutikuti
Ki te whakatakotoria, mō te paraka māmā, tirohia 1, te tikanga kia kore te momeniti o ngā kaha katoa e pā ana ki te tuaka x, ka riro te τyz dx dz * dy = τzy dx dy * dz. Nā konei, ā, mai i ngā whārite e rua e hāngai ana mō ngā tuaka y me z, ka kitea he ōrite tahi ngā pēhanga kutikuti me ōrite anō ngā taupū, arā τxy = τyx, τyz = τzy, τzx = τxz (1)
Component deformations
Ko te whai tikanga ā-ā-tinana o ngā taumahatanga kua tautuhia i mua ake nei, ka takea mai i tō rātou hononga ki te whakamāramatanga e pāngia ai ia tinana e ngā kaha o waho. E whakaatu ana a Whaka. 2 i tētahi poraka taketake, i mua i te huringa he tapawhā, ā, he roa ngā tapa dx, dy, dz. Ka taea te tohu i ngā roa o ngā tapa i muri i te huringa hei dx + Δdx, dy + Δdy, dz + Δdz; ko ngā ōwehenga εx = Δdx|dx, εy =Δdy|dy, εz =Δdz|dz e kīia ana he toronga. He rahinga kore-āhuatanga ēnei, ā, mō ngā rauemi katoa e whai wāhi ana ki te hanganga whare, he iti nei i te 1. I tua atu i te roa o ngā tapa, ka taea hoki e te poraka taketake dx * dy * dz te pāngia e te huringa o ngā koki tōtika taketake i waenga i ōna taha.
Whakaahua 2. rerekētanga wāhanga
Kia rite te koki i waenga i ngā papa e tūtaki ana i te tapa OC i muri i te whakarerekētanga ki te 
, ā, pēnei anō, kia rite te hekenga o te koki tika i te tapa AO ki te γyz, me te hekenga i te tapa OB ki te γzx. Ko ēnei huringa koki e toru, γxy, γyz, γzx, e kīia ana he paheke, ā, he iti tonu hoki i ngā wā katoa ki te 1. Mā ngā wāhanga e ono ε me γ ka tino whakatauria te whakarerekētanga o te pūtaparanga tauwhāiti, arā, te “whakarerekētanga i te iti”. Ko te huringa o te huānga rōrahi d_V = dx * dy * dz_ ko ΔdV = (dx + Δdx) (dy + Δdy) (dz + Δdz) - dx dy dz, ā, mā konei, ina waiho ngā hua whakahekenga hei rahinga iti o te raupapa teitei ake, ka whiwhi te whakahekenga pūtoru ϱ = ΔdV|dV = εx, εy, εz (2)
Nekenekanga
Ka taea te whakaahua katoa i te rerekētanga o te tinana mārō mā tēnei ara anō, arā, ka hoatu ki ia o ōna ira te nekehanga i pā ki taua ira. Mēnā ka wehea atu i te tirohanga ngā nekehanga rahi mutunga e hāngai ana ki te nekehanga o te tinana mārō, mā te whakamahi, mēnā e tika ana, i tētahi pūnaha ruruku nekeneke i roto i ngā tatauranga, ka noho tonu ngā nekehanga wāhanga μ, ν, ω hei rahinga iti, ā, he iti ō rātou tāruarua ki ngā ruruku i runga i te 1. I raro i taua whakapae, e ai ki te whakak. 3 ko te roa o te huānga rārangi OA = dx i muri i te rerekētanga
no reira 
(3)
Ka kitea anō i te āhua pērā i te wh. 3 ko te kutikuti γxy _γ1 + _γ2: me te hoki
(4)
Whakaahua 3 retiretanga
(5) 
(6)
Ko ēnei herenga e kīia ana ko ngā tikanga hototahi, ā, e whakaatu ana ka taea ngā wāhanga kua huri te āhua te whakakotahi pēnei i te kōrero mosaic kia kī tonu ai te wāhi, kāore e tutuki mēnā he noa iho ngā huringa āhua o ngā wāhanga. He whārite takirua-ira ngā tikanga hototahi, nō reira, ka mārama, he motuhake ngā wāhanga whakakē katoa e ono mō ia tapawhā motuhake, engari ko tō rātou whakaritenga i ia rohe kua horapa mutunga e herea ana e ngā tikanga (5) me (6).
I te mea i whiwhia ēnei tikanga mā te pārōnaki mai i ngā whārite (3) me (4), i te whakaputanga ka ngaro tētahi wāhanga o ngā hononga kei roto i ngā whārite taketake. Nō reira kāore e ranea kia whakatutukihia kotahi anake o ēnei pūnaha whārite e rua; me tutuki ngā pūnaha whārite e rua, mēnā e hiahia ana kia ū ai te hototahitanga o te huringa.
Ngā whārite pārōnaki o te pūnaha kōaro-kiko
Hei tātai i ngā taumahatanga me ngā whakarerekētanga i ngā tinana tōrū, ka whakamahia ēnei whārite e whai ake nei:
Ngā tikanga taurite o te huānga rōrahi. Ko ngā tikanga kia ōrite ki te kore ngā taipana huri noa i ngā tuaka e toru kua whakamahia kē i te whakaputanga o te whārite (1). Ko ngā tikanga taurite o ngā kaha e ai ki te wh. 4
Whakaahua 4
(7a-c)
Mēnā ka tohua ngā wāhanga o ngā kaha rōrahi (hei tauira, te taumaha, te kaha tōpana) ki te ia waeine rōrahi ki te X, Y, Z. Ngā hononga i waenga i ngā ahotea me ngā panoni ā-āhua. E kōrerohia ana i konei ko te hononga i waenga i ngā ahotea wāhanga me ngā panoni wāhanga. I te ariā tōtika, i te nuinga o te wā ko te hononga rārangi anake e whakaarotia ana, ā, e whakaaturia ana e te ture a Hooke:
(8a-c)
(8d-f)
Ki te oti ngā whārite (8a-c) mā te whakatau mō ngā āheinga, ka puta
(8 a)
kso me ngā whārite e rua e hāngai ana. Ko te ture a Hooke i whakahuatia mā te whārite (8) ehara i te ture taiao e tino whakatutukihia ana, engari he whakatauira o te whanonga tūturu, e kotiti atu ana i taua ture, nui ake, iti iho rānei, i runga i te rauemi me te rahi o te kōpikopiko. I te mea he tino nui ake ngā uauatanga o te maimoatanga pāngarau o te raruraru ātete rauemi me ngā hononga uaua, kāore e rārangi, i waenga i te tō me te kōpiko, ā, ka taea kē te whakatinana i roto i ngā take māmā rawa anake, me whakamahi hoki mō aua rauemi (raima!) e tino kotiti ana tā rātou whanonga i te ture a Hooke, ngā tātaitanga e tūāpapa ana ki runga hei otinga tata tuatahi, ā, te nuinga hoki ko te otinga tata anake, ā, he mea tino whaihua noa atu i te nuinga, ā, he māramatanga ake i te mea he kounga noa iho. G = E|2 (1+μ).
Ka hua mai tēnei i te hononga i waenga i te pēhanga me te huringa (8), i ngā whārite whakawhiti (16) mō ngā pēhanga me te pūnaha whārite e hāngai ana mō ngā huringa wāhanga. He āhua pēhanga tō E me G, ā, i roto i ngā rauemi hanga katoa he nui ake ēnei i ngā pēhanga σ me τ ka puta pea. Ki tēnei meka e ū ai te whakapae kua whāia whānuitia i roto i te ariā elastik, arā he iti ngā huringa. Ko te tauwehe toronga taha (tau Poisson) μ kāore he ahu, ā, kei waenganui tonu i 0 me 1/2 .
Ngā hononga i waenganui i ngā huringa ā-tikanga me ngā nekehanga. Ko ēnei hononga e hoatu ana e ngā whārite (3), (4). He tino āhuahanga noa ēnei. Mā ēnei pūnaha e toru e puta ai 3+6+6=15 whārite mō 15 ngaro, arā, mō 6 ngā pēhanga ā-tikanga, 6 ngā huringa ā-tikanga me 3 ngā nekehanga ā-tikanga. Nō reira, e ranea ana ēnei hei whakatau i aua mea ngaro. Ko te whakatau pāngarau i tētahi pūnaha nui pēnei o ngā whārite rerekētanga, he mea taea, māori noa iho, i ngā wā motuhake anake. Mēnā i roto i te whārite (8) ka whakakapia ngā huringa ā-tikanga e ngā nekehanga ā-tikanga kia rite ki ngā whārite (3) me (4), ā, kātahi ka whakaurua ngā pēhanga noa σ me ngā pēhanga kutikuti τ, e ai ki ngā whārite (8 d - f), ki roto i ngā tikanga taurite, wh. (7), ka puta ngā whārite rerekētanga e kīia ana ko ngā whārite taketake o te ariā pānoni.
(9)
ko te mea nei 
e tohu ana i te kaiwhakamāhorahora Laplace. I runga i (2) he toru ēnei whārite rerekētanga me ngā nekehanga wāhanga e toru μ, ν, ω; mā te rerekētanga i ēnei, i runga i (3), (4) me (8), ka taea te whiwhi i ngā ahotea me ngā kōpikopiko wāhanga katoa. Ko ngā whārite (9) he tīmatanga pai i te nuinga o te wā mō te rangahau i ngā take motuhake.
Te mahi whakarereke
Whakamārama
Ina utaina te tautoko, ka pāngia ngā wāhi mahi o ngā kaha o waho e ngā nekehanga rapa, ā, ka mahia hoki e ngā kaha tētahi mahi. Mā konei ka taea te huri o te pūngao kua tukuna ki ngā momo āhua rerekē. Mēnā ka tukuna ohorere te kawenga, kia nui ai te tere o te tukanga whakarerekē ā-āhua, ka huri tētahi wāhanga o tēnei pūngao hei pūngao nekehanga, ā, i te roanga o te wā ka memeha atu, ahakoa mā te whakawhiti ki te whenua hei ngaru rapa ka ngaro atu ki te mutunga kore, ahakoa mā te hurihanga nā te waku ā-roto hei wera. Ki te tukuna ngā kawenga kia māmā rawa, kia kore ai e puta he pikinga kitea o te pūngao nekehanga, arā, kia noho ngā kawenga e piki āta ake ana mai i te kore ki te taurite pūmau me ngā kaha ā-roto, e mārama ana ka pau noa iho te pūngao hei whakarerekē i te tinana, ā, ka kīia taua mea ko te mahi whakarerekē ā-āhua.
Ka whakaaturia tuatahitia te tātai i te mahi huringa mā tētahi wāhanga e utaina ana ki te tōtika i tētahi ahunga (āhua 5).
Me tuku te ngānati kei reira σ me te kume hāngai ε i ngā pikinga dσ me dε. Ka mahi te kaha σdF i runga i te ara dε * dl i te mahi σdF * dεdl = σdεdV, mēnā ko te dV te rōrahi o te wāhanga iti. Mēnā ka piki te ngānati mai i te kore ki tōna uara whakamutunga, ā, ka piki hoki te kume e ai ki tēnā mai i 0 ki te ε, ko te katoa o te mahi i mahia mō ia wae rōrahi ka ōrite ki:
Waihoki, mō te huānga e kawengia ana e te ahotea kutikuti τ (sl. 6) ko te mahi whakakōpiko mō ia waeine rōrahi (mahi whakakōpiko motuhake, pūngao tāwari):
i reira kotahi anake o ngā kaha e whā τ * dF e mahi ana.
I te take whānui, ina pā mai ngā wāhanga e ono katoa o ngā pēhanga wāhanga ki tētahi wāhanga pūtake, ka whiwhia te mahi deformation motuhake mā te tapiri i ngā mahi o ia pēhanga wāhanga takitahi:
(10)
Mā te rerekē ki te rohe o runga o ia o ēnei whakauru, ka whiwhi ngā hononga e whai mana ana i te take whānui rawa:
(11)
ā, pēnei anō mō ērā atu whakarerekētanga wāhanga. Mēnā he tōrū te tinana, ka ōrite tonu te āhua taumahatanga kua whakatauria ki te āhua whakarerekētanga kotahi tonu, ahakoa i tutuki i te utanga, i te whakangāwari rānei. Nō reira, kei raro i ngā whakauru he mahi kotahi-uara, ā, ko te pūngao katoa i whakapaua i te wā o te utanga, ā, kātahi ka tino whakangāwaritia, ko te a = 0, i te mea i tēnei take ka huri ngā rohe runga o ngā whakauru ki te kore. Nō reira, ka whakahokia mai te pūngao katoa i tukuna ki te tinana i te utanga, i te whakangāwari atu anō.
Whakaahua 6
Whakaahua 7
Mō te tinana kāore i tino rapa, he āhua tata te hononga i waenganui i te taumahatanga me te huringa āhua mō te huringa katoa o te kawenga me te whakangā, ā, e whakaaturia ana i te āhua 7 mā ngā rārangi O A B. Ko te pūngao 
i kawea mai i te wā e utaina ana he ōrite ki te horahanga OAA, ā, ko te pūngao i whakahokia i te wā e whakangāhia ana he ōrite ki te horahanga ABA, nō reira ka ngaro (ka huri hei wera) te mahi e rite ana ki te horahanga kua whakāwhiotia. Mēnā i roto i te tukanga wiriwiri ka huri kau te taumahatanga i waenganui i te uara rohe pai 
me te uara rohe kino, 
ka rere te huringa kāore i te rapa pēnei i te mea e whakaaturia ana e ngā rārangi kōpikopiko i te āhua 7, ā, i ia wāhanga o te wiriwiri ka huri hei wera, mō ia wae rōrahi, te pūngao DABCD. Ko taua whanonga i te wā o te huringa āhua, e tōmuri ai te huringa āhua i muri i te taumahatanga, ka kīia ko elastični histeresis, ā, ko te āwhio DABCD he koropiko histeresisa.
Mahi whakarerekē me te ture a Hooke
Ki te whakaurua te ture a Hooke ki ngā tāpirihanga o te whārite (10), ka taea te whakatinana i te tātai, ā, ka puta ngā kīanga mō te mahi whakarerekē motuhake:
(12a)
(12b)
(12c)
Ki te whakaurua te āhua tuarua o ēnei ki te whārite (11), ka puta anō te ture a Hooke, ngā whārite (8), i te āhua kua oti te whakaoti mā ngā pēhanga. Ko te kīanga e whakawhirinaki ana te mahi deformation ki ngā wāhanga deformation, pēnei i te mea e whakaaturia ana e te whārite (12b), nō reira he ōrite ki te ture a Hooke. Ina whakatātaritanga te whārite (12c) mā ngā pēhanga, ka puta
(13)
Ko ēnei tauira, ka rerekē i ngā tauira (11), e hāngai ana anake ki ngā tinana e whai ana i te ture a Hooke, ā, kāore e taea te whakamahi ina rerekē ngā hononga i waenga i ngā taumahatanga me ngā rerekētanga āhua. E mārama ana kei te taha matau o ngā whārite (13) ko ngā rerekētanga āhua anake nā ngā taumahatanga i hua ake, ā, ka whakakorehia katoatia ērā atu i puta pea nā te huringa o te pāmahana, te ngokingoki, te pupuhi, te whakahounga kōata, me ērā atu.
NGĀ WHAKAPUTA WHAKAURU I ROTO I TE ARIĀ WHAKARĪRĀTANGA
Nekehanga mariko. Te mātāpono o te pūngao tōpū iti rawa
Pērā i te ture a Hooke, ka taea te whakakapi i ngā whārite (8) ki tētahi whārite kotahi (12b), ā, ka taea hoki te whakakapi i ngā tikanga taurite ki te mātāpono o ngā nekehanga ā-whakaahua, hei ture taketake whānui o te mīkanika. E whakaatu ana tēnei ture, ka motuhake ngā āhuatanga taurite i te mea, i reira, ko te mahi katoa i mahia e ngā kaha katoa i te panonitanga iti o te āhua whakakino he ōrite ki te kore. Mēnā ka tohu mātou i ngā panonitanga iti o ngā nekehanga rapa hei δμ δν, δω, kātahi ka mahia e ngā kaha rōrahi kua tautuhia e te whārite (7) he mahi ā-wā.
a ngā kaha mata px, py, pz e mahi ana ki te wāhanga dF o te mata o waho o te tinana mahi 
where the triple integral refers to the entire volume, and the double integral to the entire outer surface of the body. To calculate the work done by internal forces, equation (12b) is used, so for the work per unit volume one obtains
Nā reira ka puta i te mātāpono o ngā nekehanga ā-wāwāhi
(14)
Mēnā ka whakaarotia ngā kawenga X, Y, Z px, py., pz hei wāhanga o te kaha tō (te taumaha ake me te kawenga o waho), ko ngā whakakotinga i te taha matau e whakaatu ana i te pūngao pea kua ngaro i ēnei kawenga, ā, ko te taha mauī e whakaatu ana i te pikinga o te pūngao pea (pūngao māro) o te tinana, nō reira e kī ana tēnei mātāpono ko te pūngao pea katoa o te kaikawe me ngā kawenga e takoto ana ki runga i a ia, i te wā o te taurite, he uara tino. Mēnā ko taua uara tino he iti rawa, he pūmau te taurite.
Ko te mātāpono o te iti rawa o te pūngao kōkiritanga e whakaaturia ana mā te whārite (14), ko te kīanga (12b) mō te mahi kōpikopiko me ngā tikanga hototahitanga o ngā kōpikopiko (5), (6) he pūnaha whārite e ranea ana hei hanga i te ariā o te hāungatara, ā, he rite tonu tōna pūtake ki te pūnaha whārite. Ka tino whaihua tēnei pūnaha i ngā wā ka uaua te whakatutuki i ngā whārite rerekētanga (9) i runga i te tātai, nō reira me kimi he otinga tata; hei tauira, mō ngā nekehanga ka whakaarohia i mua he āhua e tika ana, e mau ana i ētahi pūmau herekore, ā, ka whakatauria ēnei i runga i te whakaritenga kia tutuki te whārite (14). Mā konei kāore e whiwhi he otinga tino pū o te rapanga, engari ko te whakatata pai rawa ka taea mā te āhua otinga kua kōwhiria. Ko te pātai mēnā he rawaka te whakatata, e whakawhirinaki ana tuatahi ki te āhua o te otinga kua tangohia, ā, me whai pūkenga me te wheako kia angitū ai te whakamahi i tēnei tikanga.
Tumuaki a Castigliano
Ahakoa i te mātāpono mō te iti rawa o te pūngao hanga e whakatauritea ana ngā āhua kōpikopiko katoa e tutuki ana i ngā here taurite o ngā kōpikopiko, ā, ka wehea mai i roto i a rātou ko ēnā e tutuki ai ngā ngānga e hāngai ana i ngā here taurite, i te mātāpono a Castigliano ka whakatauritea ngā āhua ngānga katoa e tutuki ana i ngā here taurite, ā, ka wehea mai i roto i a rātou ko ēnā e tutuki ai ngā kōpikopiko e hāngai ana i ngā here taurite o ngā kōpikopiko. E pērā ana te kōrero o tēnei mātāpono: mai i ngā āhua taurite ka taea katoa, ko te āhua tūturu ko taua mea e hāngai ana ki te iti rawa o te mahi hanga. Ko te mātāpono a Castigliano me te whārite (12c) mō te mahi hanga me ngā here taurite ka hanga i tētahi pūnaha whārite e ranea ana hei hanga i te ātete o ngā rauemi. Nā te mea kei roto i tēnei mātāpono te ture a Hooke, ka herea te āhei o tana whakamahinga ki te whakapae ka taea te whakaahua tino i ngā kōpikopiko mā tēnei ture. E tono ana tēnei ehara i te whanonga rapa rārangi anake o te rauemi, engari ka whakakore hoki i ngā panonitanga pāmahana.
Te whanonga o ngā rauemi i runga ake i ngā rohe ā-kōtē
Ngā āhuatanga i raro i te utanga pateko
He tino uaua te āhuatanga o te rerekētanga o te rauemi i raro i te awe o ngā taumahatanga, nō reira me whakamāmā ki ngā huarahi rerekē kia mārama ai. Ko tētahi o ēnei whakamāmā ko te whakapae ka utaina ngā kawenga āta, pēnei i te mahi ki ngā tauira i te whakamātautau tō me te pēhi, hei tauira.
Ki te tukuna tētahi tokotoko maitai kua utaina ki te tō hiki ki tētahi kawenga e piki haere ana āta āta, ā, ka inehia tōna roherohe ε, ka puta te hoahoa e whakaaturia ana i te whaka. 8. I te tīmatanga he tino iti te roherohe, ā, he taurite ki te taumahatanga, pēnei i tā te ture a Hooke e tono ai. Mai i tētahi taumahatanga motuhake ka mutu te taurite, ā, ka tere ake te piki o te roherohe. He uaua te whakatau tika i te tūranga o te rohe o tēnei tauritetanga. Ka tika ake te ine, ka iti ake te wāhi ka puta ai. Mō te nuinga o ngā rawa kei te σ = 0, ā, kātahi ka noho te ture a Hooke hei whakatau tata tuatahi noa iho mō te whanonga rapa, ko te wāhanga tuatahi o te whanaketanga i te raupapa Taylor o te mahi ε = ε (σ) huri noa i te takenga o ngā tuaka.
Ki te whakangāwaritia te tokotoko, ka kitea, mēnā kāore i tino nui te taumahatanga, ka ngaro hoki te whēru, i runga i te ture a Hooke. Engari ka rerekē anō tēnei mēnā ka hipa tētahi taumahatanga e takoto tata ana ki te paearu ōritenga, arā te paearu whakatōtanga; kātahi ka whai wāhi te whēru katoa ki tētahi wāhanga māngohe (pūhou) me tētahi wāhanga kāore i te māngohe (mau tonu). Kāore hoki e taea te tautuhi tika i te paearu whakatōtanga; ka noho ki raro ake i te mea ka piki ake te tika o te inenga. Heoi anō, nā te mea he nui tōna hiranga hangarau, nā te mea ko ia te rohe whai tikanga i roto i te nuinga o ngā tatauranga mō te ātete o ngā rauemi, ā, i te wā kotahi ko te rohe hoki e tīmata ai ngā kino mau tonu tuatahi, ka tino tautuhia mō te whakamātau tauira e tētahi tikanga kua whakaaetia, ā, hei tauira, ka tohua hei paearu 0.2% te taumahatanga e tae ai te toro mau tonu ki te 0.2%.
I muri tonu iho i te pānuitanga o te rohe hāngū, ka tino piki te huringa, ā, ka kitea mā te kanohi tahanga; i te pēhanga tata pūmau ka toro tonutia te tokotoko. E kīia ana tēnei pēhanga ko te rohe rerenga. I ngā mīhini wai mō te whakamātau rawa ka kitea anō, i muri i te tīmatanga o te rerenga, he hekenga atu, he iti noa rānei, he kaha rānei o te pēhanga; pēnei i te whakaaturanga i te āhua 8 mā te ānau whati; kātahi ka kīia te tāupe nui ko te rohe rerenga runga, ā, ko te pēhanga rerenga pūmau ka whai mai ko te rohe rerenga raro.
Ina tae ki tētahi roha motuhake, e tata ana ki te kore-rapa katoa, ka tīmata anō te piki o te taumahatanga, arā ka whakapakari te rawa. I tēnei rohe utanga ka tino heke te wāhanga-whiti o te tokotoko nā te kōpeketanga taha (kore-rapa), ā, ko te rere anō o te hoahoa taumahatanga me te whakarerekētanga ka whakawhirinaki tuatahi ki te pātai mēnā, i te tātai i te taumahatanga σ, ka wehehia te kaha tō P e mahi ana ki te tokotoko ki te wāhanga taketake F0 rānei, ki te wāhanga ināianei F rānei. Mō ngā kaupapa hangarau he mea nui te taumahatanga P|F0 . Ka tae tēnei taumahatanga ki tētahi mōrahi e kīia ana ko te kaha o te rawa, ā, ka hipa i taua mōrahi, ka tino whāiti te tokotoko i tētahi wāhi, ka whati. Ka piki te “taumahatanga tūturu” P|F kia tae noa ki te hinganga.
Whakaahua 8.
He nui te rerekētanga o te whanonga o ērā atu konganuku i tō tā konei i whakaahuatia. He maha ngā wā kāore e tino kitea te rere, ā, ka tīmata te whakapakari tonu i muri tonu i te whiti atu i te pae tāupe. Ki ngā rauemi pakarukaru (te kōhatu, te karaehe, te raima) he iti noa iho ngā kōpiritanga koretāupe, ā, ahakoa he iti noa te whakanekehanga ka puta te whati. Ko te whakanekehanga katoa ka pāngia e te rauemi kia whati rā anō he inenga nō te torohū (he ritenga kē: nō te pakarukaru) o te rauemi, ā, mō te rino ka whakaritea tikatia i roto i ngā tikanga hangarau. Heoi anō, i te mea i mua tata i te whati ka kukū te kōpiritanga anō ki ngā wāhi tata o te wāhanga kua whāiti, ka puta he uara rerekē o te whakanekehanga i te wā ka whati, i runga i te whānui ake, whāiti ake rānei o te wāhanga o te tokotoko e tangohia ana hei roa ine, nō reira kia whiwhi ai i ngā hua kotahi te tikanga, ā, ka taea te whakataurite ki ētahi atu, me whakatū i mua tētahi tikanga motuhake. Ko te whakanekehanga i te whati, i te nuinga o te wā, ka inehia mā te tohu i te roa ine Ι = 10 d ki runga i tētahi tokotoko porohita o te whānui d i mua i te whakamātautau, ā, mā te toronga ΔΙ ka tatauria te whakanekehanga i te whati εs_Ι = ΔΙ|Ι ._ Mēnā e tika ana kia whakamātauria ngā tokotoko o tētahi wāhanga rerekē, ka whakatauria te roa ine e ai ki te whānui o te tokotoko porotakaroa porowhita he ōrite tōna horahanga wāhanga whakawhiti.
Uta roa
Mō te takanga o te whakamātau tōtanga, kāore i tino whai take mēnā ka whakanuia te kawenga tae noa ki te whati i roto i te 10 mmeneti, i ētahi hāora rānei. Heoi anō, mēnā ka mahi te kawenga mō ngā tau, mō ngā tekau tau rānei, i raro i ētahi tikanga ka kitea, ahakoa kei raro i te rohe rerenga, ka piki māori haere te huringa āhua i te wā e noho pūmau ana te taumahatanga. I konei, kāore e tino rerekē te wāhanga rapa, arā te wāhanga e ngaro ana i te wā e tino whakawātea ana. Ko tēnei pikinga o te huringa kore-rapa e puta ana mō te wā roa ka kīia he ngokingoki. Ki ngā konganuku, hāunga te mata, he mea nui te ngokingoki i ngā pāmahana teitei anake; heoi, mō te mātanga hanganga, ka whai wāhi tēnei ki te raima e paku tuku ana i raro i te pēhanga, ā, ka piki te taumahatanga i roto i te rino whakakaha kōpeke, mēnā kei reira taua rino.
Te kawenga rerekē
Kia taea ai ngā āhuatanga pūkaha o te rauemi, i whiwhi ai i te whakamātau tō, te whakamahi hei paearu mō te kawenga whakaaetia o ngā hanganga, me ū te tikanga kia āhei te rauemi ki te kawe anō i ia kawenga anō i te pēhanga i kawea e ia kotahi, kāore he kino. Kāore tēnei tikanga e tutuki mēnā ka pāngia te rauemi e te tukurua auau o te kawenga me te wetewete, pēnei i te, hei tauira, ngā kaikawe e utaina ana e ngā kaha inercia o ngā mīhini kāore i tino taurite. Mā te tukurua nui o te kawenga ka taea te kawe i tētahi wāhanga hanga ki te pakaru ahakoa i te pēhanga e tino raro iho ana i te kaha o te rauemi i whakatauria mā te whakamātau pateko noa. Ka kīia tēnei momo pakaru he pakarutanga nā te ngenge. Ko te maha n o ngā huringa kawenga ka taea e te rauemi te kawe e whakawhirinaki ana ki te rahi o te pēhanga σ, ā, ka nui ake i te iti iho o te pēhanga. E whakaaturia ana tētahi tauira angamaheni o tēnei whakawhirinakitanga i te āhua 9, e kitea ai he paepae raro tō te kaha o te rauemi, ka taea hoki te kawe ahakoa he tino nui rawa te maha o ngā huringa kawenga, ā, e kīia ana ko te kaha ngenge o te rauemi Ko ngā hoahoa o tēnei momo (āhua 9) e kīia ana ko ngā ānau Wӧhler.
Whakaahua 9.
Haumarutanga o ngā hanganga
E rua ngā kawenga pae he mea nui mō ngā hanganga: te kawenga ka tīmata ai ngā pakarutanga pūmau tuatahi (te pae tuku), me te kawenga ka pau katoa te kaha kawe o te hanganga, ā, ka whati (te pae whati). I te mea, i mua i te pae tuku, e tata ana te tika o ngā whakapae pūtaiao o te ātete rauemi (ngā nekehanga iti, te ture a Hooke), ka taea te aromātai pūmau i te haumaru ki ngā whakakēnga pūmau i ngā āhuatanga katoa e taea ai te hinga i ngā uauatanga pāngarau. Ki te ōrite te pēhanga ki te kawenga, ka ōrite te ōwehenga o te pēhanga tuku ki te pēhanga nui rawa e puta ana i te hanganga ki te ōwehenga i waenga i te kawenga ka puta ai te tuku me te kawenga tūturu, ā, ka tohu i te tau whakamarumaru ki te tukunga. Heoi anō, i te tini o ngā take kāore tēnei hononga māmā e tū: te piko ina whakakorehia te kukume, te pēhanga i waenga i ngā mata kōpiko, ngā pūnaha he rerekē tō rātou hanganga, ā, ko te mea tino uaua, ko te piko me te kaha ā-tuaka. I ēnei āhuatanga katoa ka taea e te pikinga āhua iti o te kawenga te whakapiki nui i te pēhanga, ā, kāore e ranea kia whakamahia noa te tau whakamarumaru ki ngā pēhanga; me whakamahi kē ki te kawenga.
Ko ngā āhuatanga ka puta i muri i te whitinga o te pae rerenga tae noa ki te pakarutanga, i te nuinga o ngā wā, ka taea anake te whai mā te āhua kounga. Ki te tōpuhi te rauemi, ko ngā wāhanga i tae tuatahi ai te pae rerenga, ā, e tāpuputu tonu ana i raro i te nuku pūmau, i te nuku rānei e piki paku ana, he iti noa tā rātou whai ki te kawe i te kawenga anō; nā reira ko ētahi atu wāhanga o te hanganga, he iti ake te pēhanga ki runga i a rātou, ki te taea i runga i tōna hanganga, ka nui ake tā rātou whai ki te kawe. Kei roto i tēnei wawata kia taurite ngā tihi o te pēhanga tētahi rahui haumaru, ka taea te whakamahi ā-tatau mēnā kāore e kino ana te deformity pūmau o ētahi wāhanga, i puta mai i tētahi kawenga nui kotahi anake (tikanga kawenga rohe).
Mēnā he pakarukaru te rauemi, ko te kino pūmau tuatahi ka kitea i roto i ngā kapiti, ā, nā reira ka kaha ake te kukū o te pēhanga ki te wāhi e mōrea ana, ka puta tonu te whati. Ko te whati nā te ngenge, ahakoa i ngā rauemi tōtoro, he āhua pēnei i te whati pakarukaru.
Ngā hononga i waenga i ngā pēhanga mō ngā papa whiti rerekē
Ngā whārite whakawhiti
Ko ngā ahotea ā-wāhanga e iwa i sl. 1, e herea ana ki ngā tikanga taurite e toru. Ko ngā rahinga e ono e toe ana σx, σy, σz τxy, τyz, τzx ka taea, takitahi i tētahi i tētahi, te whiwhi i ngā uara kōwhiri noa, pēnei i te kitenga mai i te whakaputanga o ngā whārite rerekētanga o te ariā māhorahora. Heoi, ina hoatu ēnei uara, kua tino oti te āhua ahotea i tētahi wāhi o te tinana mārō, arā, ina mōhiotia ngā ahotea mō ngā paparangi e toru e pīkau kōtika ana tētahi ki tētahi, ka taea hoki te whakatau mārama i te ahotea mō tētahi paparangi tuawhā e haere ana mā te wāhi kua homai, ā, he ahunga kōwhiri noa tō tōna noa.
I te wh 10 ka whakaaturia te tetrahedron OABC e toru ōna mata e whakarara ana ki ngā paparangi o te pūnaha taunga x, y, z, ā, ko te tuawhā he huānga mata whai tūnga noa ABC = dF. Kia tohu ai i tōna ahunga me te ahunga o te ngānati e whakawhitia ana mā reira, ka whakaurua he pūnaha taunga tuarua, ko tōna tuaka ξ he poutū ki dF.
Whakaahua 10
Ko te herenga taurite o ngā kaha e tau mai ana ki tēnei tetrahedra i te ahunga o te tuaka ξ e hoatu ana
Ki te whakaarohia ko te
ka whiwhi
ā, pēnei anō hoki i raro i ngā tikanga taurite i te ahunga o te tuaka η:
Koinei ngā whārite hei tātai i ngā pēhanga i te hurihanga o ngā taunga. He mea nui rawa kia tirohia anake ngā huānga pēnei o te mata dF e hāngai ana te tuaka ξ ki te tuaka z. Ka taea ngā tauira mō tēnei take te whiwhi mā te whakangāwari i ngā tauira o mua, ka pānui tika rānei i te whika 11:
Whakaahua 11
Ka tae te ngānati noa ki ngā uara tino mō ngā ahunga α = α0 , mō ēnei dσξ/dα = 0. Mā te whakahere i te taurite ka puta mai i te whārite (16)
ā, nō reira, ina whakatauria tēnei kīanga ki te kore
Ko te pēhanga τξη ka noho hoki hei pēhanga kutikuti katoa i ēnei rererangi, mēnā ko τyz = τzx = 0. Kātahi mā te whārite (17) ka tautuhia ngā ahunga e rua e takatū ana ki a rāua anō e whai ake nei: 1. he kore te pēhanga kutikuti katoa, ā, 2. he uara tino nui me te tino iti ō te pēhanga ā-poutū. Ka kīia ēnei ahunga ko ngā ahunga o ngā pēhanga matua, ā, ko ngā pēhanga e hāngai ana he pēhanga matua. Ka tohuia ēnei ki te σ1 me σ2 ā, ka taea te tātai tika mai i te tauira
Porowhita a Mohr
Ka taea te whakaatu ā-whakaahua ngā whārite (16) i runga i te āhua mārama. Hei tēnei me huri ki te āhua:
Ko ēnei whārite e whakaatu ana i roto i te pūnaha tairitenga σ, τ i te āhua tawhā o te whārite o te porowhita, kua tuhia ki te fig. 12, ā, ko ngā whakatāunga o tōna pokapū ko σ = 1/2 (σx + σy), τ = 0, ā, ko tōna hātepe irirangi
E kīia ana tēnei porowhita ko te porowhita a Mohr. Mēnā ka tuhia mā te wāhi x he rārangi whakarara ki te wāhanga x = const., e whakawhitia ai ngā taumahatanga σx, τxy, ka motu taua rārangi i te porowhita ki te ira p, e kīia ana ko te pou o te porowhita a Mohr. Ko ia rārangi pξ ka tuhia mā tēnei pou ka motu i te porowhita ki te ira ξ, e whakatau ana i ngā taumahatanga σξ, τξη e whakawhitia ana mā te wāhanga whakarara ki pξ.
Mō te tohu o ngā ahotea kutikuti i te tānga o te porowhita a Mohr me mātua whakaarohia ka tāia whakarunga rātou mēnā he ahunga tō rātou e whakaaturia ana i te whakaahua 12c. Nō reira, o ngā ahotea takirua τxy, τyx, ko tētahi ka pai tonu, ko tētahi ka kino.
Ki ngā uara pae o σ1, σ2 o ngā ahotea noa ka ōrite ngā ira 1 me 2 i runga i te porowhita, ā, mā ngā rārangi _p_1 me _p_2 e whakaatu ngā ahunga o ngā ahotea matua.
Ko te whakaatu i ngā ahotea e takoto ana i te papa x-y mā te porowhita a Mahr kāore e pātata ki tētahi whakapae mō ngā ahotea σz, τxz, τyz e poupou ana ki tēnei papa.
Whakaahua 12
Nō reira ka mana hoki tēnei mō te āhua ahu-toru whānui rawa o te āhua tō, mēnā ka whakatauritea ko ngā rererangi wāhanga anake e haere katoa ana mā te rārangi kotahi, kua kōwhiria nei hei tuaka z me ξ. Nā runga i ngā whārite whānui o te huringa (15), kāore e taea e te tō tika σ te riro mure ore mō tētahi rererangi wāhanga, nō reira me whai ia i tētahi rererangi he mōrahi mutunga σ1, ā, mō tētahi atu he kōpakatanga iti rawa σ3, ā, e rua ēnei, e mārama ana, ka āhei hoki kia tōraro. Mēnā ka whakatakotoria he rererangi mā ngā pūoko σ1 me σ3, ā, ka tāia te porowhita a Mohr (wh. 13)
Whakaahua 13
mō ngā pēhanga e takoto ana i tēnei rererangi (arā, ngā ahunga katoa e poutū ana ki tēnei rererangi), ko ngā σ1 me σ3 ngā pēhanga tōtika nui rawa, iti rawa hoki e taea ana te puta, nō reira me ūpoko-taha rāua. Ki te tāpiritia ngā ahunga 1 me 3 ki tētahi ahunga 2, kia noho ēnei ahunga e toru hei pūnaha ruruku koki-māro, ā, ki te tuhia ngā porowhita a Mahr mō ngā rererangi 1 ~2 me 2 ~ 3, kātahi ko σ2 mō tētahi o ēnei porowhita ko te pēhanga tōtika nui rawa, ā, mō tētahi atu ko te iti rawa, nō reira i ngā wāhanga e poutū ana ki ngā ahunga 1, 2 me 3 he kore te pēhanga kutikuti. Ko ngā pēhanga tōtika e toru, e ūpoko-taha ana ki a rātou anō σ1, σ2 me σ3, kāore ō rātou pēhanga kutikuti e hāngai ana, ka kīia ko ngā pēhanga matua.
Te rohe tuku me te rohe pakaru i te āhua ā-mokowā o ngā taumahatanga
Ka kīia te āhua o ngā ahotea he kotahi-, rua-, toru-ā-rua rānei, arā anō: he rārangi, he papa, he ātea rānei, i runga i te mea mēnā kotahi, e rua, e toru katoa rānei o ngā ahotea matua kāore i te ōrite ki te kore. I ngā wāhanga hanga e tino whakamahia ana, arā i te kōpae tō, kōpae pēhi me te aho piko, he kotahi-ā-rua te āhua o ngā ahotea, i te iti rawa ki ngā wāhi tino utaina; he pēnei anō i ngā whakamātautau tō, pēhi, piko rānei, e whakamahia nuitia ana i te whakamātau rauemi hei whakatau i ngā ahotea whakaaetia. He rerekē, i ngā papa e utaina ana i roto i tō rātou papa ake, ūpoko hoki ki taua papa, pēnei anō i ngā anga, he rua-ā-rua te āhua o ngā ahotea, ā, i ētahi wā ka puta hoki ngā āhua toru-ā-rua o ngā ahotea, nō reira me unu ngā whakatau mō te whanonga o te rauemi i te take o te āhua ātea o ngā ahotea, i runga anō i te rohe o te rerenga me te whati, i whakatauria i raro i te āhua kotahi-ā-rua o ngā ahotea.