References
85
u =
1
2E
(σ
2
1 + σ
2
2 − 2νσ 1 σ 2 );
u vol =
1 − 2ν
6E
(σ 1 + σ 2 )
2
;
u f =
1 + ν
3E
(σ
2
1 + σ
2
2 − σ 1 σ 2 ).
(8.23)
In arbitrary axes x, y, formulas (8.23) will be
u =
1
2E
(σ
2
x + σ
2
y − 2νσ x σ y ) +
τ 2
2G
;
u vol =
1 − 2ν
6E
(σ x + σ y )
2
;
u f =
1 + ν
3E
(σ
2
x + σ
2
y − σ x σ y ) +
τ 2
2G
,
(8.24)
where the shear modulus G is related to the Young modulus E and Poisson
coefficient ν by the dependency (1.10).
References
1. M. Leonov, Osnovy mekhaniki uprugogo tela (Fundamentals of elastic body mechanics).
(Frunze, Izd-vo AS Kirg. SSR, 1963)
2. V. Molotnikov, Osnovy teoreticheskoi mekhaniki (Fundamentals of theoretical mechanics).
(Feniks Publ., Rostov on Don, 2004)
3. V. Molotnikov, Mekhanika konstruktsii (Mechanics of structures). (SPb., Moscow, Krasnodar,
Lan’ Publ., 2012)
4. E. Nikolai, Teoreticheskaya mezanika. Ch. 2. Dinamika. Izd. 13-e. (Theoretical mezanika. Part
2. Dynamics. Prod. The 13th.). (Fizmatlit Publ., Moscow, 1938)