398
30 Complex Strain of Soils
˙
G +
1
τ
G =
1
τ
exp
⎡
⎣ −aa
t
t o
v(ξ )dξ
⎤
⎦
2
.
(30.16)
The solution to Eq. (30.16) at v = const, t o = 0 and the initial condition G(0) =
G o is the function [3]
G(u) = G o exp
−
u
vτ
Q
√
π exp(Q
2 ) ((au − Q) − (−Q)) + 1
,
(30.17)
where the constant value Q is defined by the formula
Q =
1
2aτ τv
,
(30.18)
and (x) is the probability integral [3]:
=
2
√
π
x
0
exp(−t
2 )dt.
(30.19)
The found function (30.17) equally well describes strains in any monotonous
hardening of material and is universal in this sense. Now let us go over to
considering partial types of loadings at constant rates.
References
1. P. Bakhtin, Trudy Stavropol’skogo NII s.-kh. (Proceedings of the Stavropol research Institute of agriculture), Frunze, Izd-vo Ilim Publ., chapter Znachenie fiziko-mekhanicheskikh i
tekhnologicheskikh svoistv osnovnykh tipov pochv (Significance of physical, mechanical and
technological properties of the main types of soils) (1971), pp. 333–342
2. S. Batdorf, B. Budyanskii, Mekhanika: sb. perev., no 5(33) (Mechanics: a collection
of translations, no 5(33)). (Moscow, Inostrannaya literatura Publ., chapter Zavisimost’
mezhdu napryazheniyami i deformatsiyami dlya uprochnyayushchegosya metalla pri slozhnom
napryazhennom sostoyanii (The relationship between stress and deformations for the hardening
of metals under complex stress state). Mekhanika: sb. perev., no 5(33) [Mechanics: a collection
of translations] (1955) pp. 120–127
3. I. Gradshtein, I. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii (Tables of integrals,
sums, series, and works). (Nauka Publ., Moscow, 1971)
4. A. Il’yushin, Plastichnost’. (Osnovy obshchei matematicheskoi teorii) (Plasticity. Fundamentals of General mathematical theory). (Izd-vo AS USSR Publ., Moscow, 1963)
5. A. Ioffe, I. Revut, Osnovy agrofiziki / red. Ioffe A.F., Revut I.B. (Fundamentals of agrophysics).
ed. by A.F. Ioffe, I.B. Revut (Fizmatlit Publ., Moscow, 1959)
6. M. Leonov, Mekhanika deformatsii i razrusheniya (Deformation and fracture mechanics) (Izdvo Ilim Publ., Frunze, 1981)
30 Complex Strain of Soils
˙
G +
1
τ
G =
1
τ
exp
⎡
⎣ −aa
t
t o
v(ξ )dξ
⎤
⎦
2
.
(30.16)
The solution to Eq. (30.16) at v = const, t o = 0 and the initial condition G(0) =
G o is the function [3]
G(u) = G o exp
−
u
vτ
Q
√
π exp(Q
2 ) ((au − Q) − (−Q)) + 1
,
(30.17)
where the constant value Q is defined by the formula
Q =
1
2aτ τv
,
(30.18)
and (x) is the probability integral [3]:
=
2
√
π
x
0
exp(−t
2 )dt.
(30.19)
The found function (30.17) equally well describes strains in any monotonous
hardening of material and is universal in this sense. Now let us go over to
considering partial types of loadings at constant rates.
References
1. P. Bakhtin, Trudy Stavropol’skogo NII s.-kh. (Proceedings of the Stavropol research Institute of agriculture), Frunze, Izd-vo Ilim Publ., chapter Znachenie fiziko-mekhanicheskikh i
tekhnologicheskikh svoistv osnovnykh tipov pochv (Significance of physical, mechanical and
technological properties of the main types of soils) (1971), pp. 333–342
2. S. Batdorf, B. Budyanskii, Mekhanika: sb. perev., no 5(33) (Mechanics: a collection
of translations, no 5(33)). (Moscow, Inostrannaya literatura Publ., chapter Zavisimost’
mezhdu napryazheniyami i deformatsiyami dlya uprochnyayushchegosya metalla pri slozhnom
napryazhennom sostoyanii (The relationship between stress and deformations for the hardening
of metals under complex stress state). Mekhanika: sb. perev., no 5(33) [Mechanics: a collection
of translations] (1955) pp. 120–127
3. I. Gradshtein, I. Ryzhik, Tablitsy integralov, summ, ryadov i proizvedenii (Tables of integrals,
sums, series, and works). (Nauka Publ., Moscow, 1971)
4. A. Il’yushin, Plastichnost’. (Osnovy obshchei matematicheskoi teorii) (Plasticity. Fundamentals of General mathematical theory). (Izd-vo AS USSR Publ., Moscow, 1963)
5. A. Ioffe, I. Revut, Osnovy agrofiziki / red. Ioffe A.F., Revut I.B. (Fundamentals of agrophysics).
ed. by A.F. Ioffe, I.B. Revut (Fizmatlit Publ., Moscow, 1959)
6. M. Leonov, Mekhanika deformatsii i razrusheniya (Deformation and fracture mechanics) (Izdvo Ilim Publ., Frunze, 1981)