Xeùt 1 maët nghieâng so vôùi 3 phöông chính 1 2 3 Phaùp tuyeán cuûa maët

Xeùt 1 maët nghieâng so vôùi 3 phöông chính 1

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: ñöôïc goïi laø 3 baát bieán (khoâng phuï thuoäc heä toaï ñoä) cuûa tensor öùng suaát. Xeùt 1 maët nghieâng so vôùi 3 phöông chính 1, 2, 3. Phaùp tuyeán cuûa maët nghieâng laø x. ÖÙng suaát phaùp cuûa maët nghieâng: 2 3 x 3 2 2 x 2 2 1 x 1 x n n . n . (2.22) Vôùi 3 x 2 x 1 x x n n n n : phaùp tuyeán ñôn vò 1 n n n 2 3 x 2 2 x 2 1 x Hình 2.4 -Töông töï xeùt 1 maët nghieâng caùc phaùp tuyeán ñôn vò: ) 1 n n n ( n n n n n 2 3 y 2 2 y 2 1 y x 3 y 2 y 1 y y (2.23) 2 3 y 3 2 2 y 2 2 1 y 1 y n . n . n . (2.24) -Vaø maët nghieâng coù phaùp tuyeán ñôn vò 3 z 2 z 1 z z n , n , n n vuoâng goùc vôùi 2 maët treân 1 2 3 2 x x y n x n y z z n 1 3
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 75 TS. Vuõ Coâng Hoøa ) 1 n n n ( 2 3 z 2 2 z 2 1 z (2.25) 2 3 z 3 2 2 z 2 2 1 z 1 z n . n . n . (2.26) Do ñoù: ) n . n n ( ) n . n n ( ) n . n n ( 2 3 z 2 3 y 2 3 x 3 2 2 z 2 2 y 2 2 x 2 2 1 z 2 1 y 2 1 x 1 z y x (2.27) z y x n , n , n laø 3 phaùp tuyeán ñôn vò vuoâng goùc vôùi nhau töøng ñoâi moät, neân: 1 n n n ; 1 n n n ; 1 n n n 2 3 z 2 3 y 2 3 x 2 2 z 2 2 y 2 2 x 2 1 z 2 1 y 2 1 x (2.28) Vaø do (x, y, z) laø 1 heä toaï ñoä vuoâng goùc baát kyø, neân: (2.29) Phöông trình (2.18) coù 3 nghieäm soá, coù theå xaûy ra 2 tröôøng hôïp: hoaëc caû 3 nghieäm ñeàu thöïc hay 1 nghieäm thöïc vaø 2 nghieäm phöùc lieân hieäp. Xeùt tröôøng hôïp coù 3 nghieäm thöïc: 3 2 1 , , . Caùc phaùp tuyeán ñôn vò töông öùng vôùi 3 nghieäm ñoù: 3 z 3 y 3 x 3 2 z 2 y 2 x 2 1 z 1 y 1 x 1 n , n , n n ; n , n , n n ; n , n , n n (2.30) Töø (5.16): 0 n . n . n . 0 n . n . n . 0 n . n . n . 1 z 1 z 1 y yz 1 x xz 1 z zy 1 y 1 y 1 x xy 1 z zx 1 y yx 1 x 1 x (5.31) Nhaân phöông trình thöù nhaát vôùi 2 x n , phöông trình thöù 2 vôùi 2 y n vaø phöông trình thöù 3 vôùi 2 z n roài coäng laïi: 0 n ]. n . n . n . [ n ]. n . n . n . [ n ] n . n . n [ 2 z 1 z 1 z 1 y yz 1 x xz 2 y 1 z zy 1 y 1 y 1 x xy 2 x 1 z zx 1 y yz 1 x 1 x (2.32) Töông töï: const I z y x 3 2 1 1  
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 76 TS. Vuõ Coâng Hoøa 0 n ]. n . n . n . [ n ]. n . n . n . [ n ] n . n . n [ 1 z 2 z 2 z 2 y yz 2 x xz 1 y 2 z zy 2 y 2 y 2 x xy 1 x 2 z zx 2 y yz 2 x 2 x (2.33) Tröø (5.33) vôùi (5.32) . & 0 n . n n . n n . n 2 1 2 z 1 z 2 y 1 y 2 x 1 x 2 x (2.34) Vaäy: 0 n . n n . n n . n 2 z 1 z 2 y 1 y 2 x 1 x
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