Hình 27 Phöông trình vi phaân caân baèng theo x y z Z z y x Y z y x X z y x z

Hình 27 phöông trình vi phaân caân baèng theo

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Hình 2.7 Phöông trình vi phaân caân baèng theo x, y, z: 0 Z z y x 0 Y z y x 0 X z y x z zy zx yz y yx xz xy x (2.39) 3. TRAÏNG THAÙI ÖÙNG SUAÁT PHAÚNG: 3.1. Ñònh nghóa vaø caùch bieåu dieãn: Ña soá baøi toaùn thöôøng gaëp laø nhöõng baøi toaùn ñaõ xaùc ñònh moät maët chính vaø treân maët chính ñoù öùng suaát baèng khoâng, luùc ñoù ta coù traïng thaùi öùng suaát phaúng. Vaäy traïng thaùi öùng suaát phaúng khi, taïi 1 ñieåm, vector öùng suaát (toång) luoân naèm trong cuøng moät maët phaúng, vôùi moïi maët vi phaân khaûo saùt. Tensor cuûa traïng thaùi öùng suaát phaúng: 2 dy y y 1 1 3 z x y dx dz dy dy y yz yz dy y yx yx dz z z z 1 dz z zx zx dz z zy zy dx x xz xz dx x x x dx x xy xy
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 79 TS. Vuõ Coâng Hoøa 0 0 0 0 0 T y yx xy x (2.40) Hay 0 0 0 0 0 0 0 T 2 1 (2.41) Traïng thaùi öùng suaát phaúng cuûa moät phaân toá coù theå ñöôïc bieåu dieãn: (hình2.8 ) Hình 2.8 3.2. ÖÙng suaát treân maët caét nghieâng baát kyø: Goïi ) n , x ( Ñieàu kieän caân baèng: 0 p sin . cos . p sin . cos . p z y xy y yx x x (2.42) , p hay p , p p y x (2.43) z ) a x y y x xy yx 1 2 ) b 1 2 y x z ) c 2 1
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 80 TS. Vuõ Coâng Hoøa Hình 2.9 Chieáu p x , p y leân phöông phaùp tuyeán n vaø tieáp tuyeán t cuûa maët nghieâng: cos sin 2 sin . cos . sin . p cos . p xy 2 y 2 x y x (2.44) 2 sin . 2 cos 2 2 xy y x y x (2.45) 2 cos 2 sin 2 cos . p sin p xy y x y x (2.46) 3.2.1. Ñieàu kieän xaùc ñònh phöông chính: y x xy 2 2 tg 0 (2.47) 2 2 arctg . 2 1 y x xy (2.48) 3.2.2. Caùc öùng suaát chính: Ta coù moät öùng suaát chính: Hai öùng suaát chính coøn laïi ñöôïc xaùc ñònh: 2 tg 1 2 tg 2 sin ; 2 tg 1 1 2 cos 2 2 (2.49) 0 i y t y y x yx x yx xy n y P x P x xy 0
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Giaùo Trình CÔ CHÖÔNG 2: ÖÙNG SUAÁT VAØ BIEÁN DAÏNG 81 TS. Vuõ Coâng Hoøa 2 xy 2 y x y x k , j 2 2 (2.50)
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