16
1 Summary of Elasticity Theory: Basic Concepts
In a similar way, the average of linear relative deformations (ε 0 ) in the directions of
normal lines to these areas will be called the hydrostatic strain:
ε 0 =
1
3
(ε x + ε y + ε z ) =
3
.
(1.15)
By directly checking, we can define that the dependencies (1.11) and (1.12),
taking into account the designations (1.14) and (1.15), are equivalent to the
following:
ε 0 =
σ 0
K
, D ε =
1
2G
D σ ,
(1.16)
where
K =
E
1 − 2ν
,
(1.17)
and D σ , D ε are deviators of stresses and strains, respectively, which are defined
using tables:
D σ =
⎛
⎝
σ x − σ 0 τ xy
τ xz
τ yx σ y − σ 0 τ yz
τ zx
τ zy σ z − σ 0
⎞
⎠ ,
D ε =
⎛
⎝
ε x − ε 0 γ xy
γ xz
γ yx ε y − ε 0 γ yz
γ zx
γ zy ε z − ε 0
⎞
⎠ .
(1.18)
1.15 Plane Stress-Strain State
Assume that relative elongation in the direction of one of the axes equals zero (for
example, ε z = 0). In this case, movements of all body points in the case of its
deformations occur in parallel to the plane z = 0. Such deformation is referred to as
plane strain.
In the assumption ε z = 0, the last formula (1.11) gives
σ z = ν(σ x + σ y ),
(1.19)
and the two first formulas can be written as
ε x =
1
E ∗ (σ x − ν
∗ σ y ); ε y =
1
E ∗ (σ y − ν
∗ σ x ),
(1.20)