14.7 Invariant Form of Hooke’s Law
191
≈
σ 0
K
or σ 0 = 3Kε 0 .
(14.24)
When the principal stresses are not equal between each other, as the experiments
of Davidenkov [2] et al. showed, the volumetric elasticity law (14.23) (14.24) also
remains true in the general case of stressed state.
By noting that K σ 0 for the stress σ 0 of about σ s , the ratio σ 0 /K equals several
fractions of a percent. Therefore, in plasticity theory (especially in the case of
developed plastic strains), volumetric compression of materials is usually neglected
and it is assumed as follows:
ε x − ε 0 ≈ ε x , . . . .
Moreover, let us also recall (p. 170) that all-around compression slightly affects the
plasticity condition and ratios between stresses and strains beyond the yield limit.
However, the effect of hydrostatic pressure on material plasticity is rather high. For
example, brittle bodies such as hard rocks (marble, granite, etc.) exposed to high
all-around pressure by additional forces can acquire high residual deformations.
14.7 Invariant Form of Hooke’s Law
Previously, we gave (p. 14) two forms of the mathematical representation of Hooke’s
law. The given formulas are
σ x = λλ + 2Gε x ,
σ y = λλ + 2Gε y ,
σ z = λλ + 2Gε z ;
τ xy = Gγ xy ,
τ yz = Gγ yz ,
τ zx = Gτ zx ,
(14.25)
which designates
λ =
2Gν
1 − 2ν
.
The shift modulus G included in the last formulas is sometimes denoted by the
letter μ, and [6, 9] used the parameters λ, μ name “Lame constants.”
By summing the three first ratios (14.25), we obtain
σ 0 = Kε 0 ,
K =
E
1 − 2ν
,
191
≈
σ 0
K
or σ 0 = 3Kε 0 .
(14.24)
When the principal stresses are not equal between each other, as the experiments
of Davidenkov [2] et al. showed, the volumetric elasticity law (14.23) (14.24) also
remains true in the general case of stressed state.
By noting that K σ 0 for the stress σ 0 of about σ s , the ratio σ 0 /K equals several
fractions of a percent. Therefore, in plasticity theory (especially in the case of
developed plastic strains), volumetric compression of materials is usually neglected
and it is assumed as follows:
ε x − ε 0 ≈ ε x , . . . .
Moreover, let us also recall (p. 170) that all-around compression slightly affects the
plasticity condition and ratios between stresses and strains beyond the yield limit.
However, the effect of hydrostatic pressure on material plasticity is rather high. For
example, brittle bodies such as hard rocks (marble, granite, etc.) exposed to high
all-around pressure by additional forces can acquire high residual deformations.
14.7 Invariant Form of Hooke’s Law
Previously, we gave (p. 14) two forms of the mathematical representation of Hooke’s
law. The given formulas are
σ x = λλ + 2Gε x ,
σ y = λλ + 2Gε y ,
σ z = λλ + 2Gε z ;
τ xy = Gγ xy ,
τ yz = Gγ yz ,
τ zx = Gτ zx ,
(14.25)
which designates
λ =
2Gν
1 − 2ν
.
The shift modulus G included in the last formulas is sometimes denoted by the
letter μ, and [6, 9] used the parameters λ, μ name “Lame constants.”
By summing the three first ratios (14.25), we obtain
σ 0 = Kε 0 ,
K =
E
1 − 2ν
,