170
13 Initial Concepts of Plasticity Theory
This hypothesis propagates phenomena of disappearance of elastic deformation and
preservation of non-elastic deformations observed in uniaxial elongation experiments in the case of full unloading after pre-elongation beyond the elastic limit.
Let us designate the components of plastic deformation as e ij , assuming for
example:
it i = j then e xx = ε e
x ,
if i = j then e xy =
1
2
γ
e
xy .
Since the components of elastic deformation are unambiguously defined through
stresses under Hooke’s law, the fundamental problem of plasticity theory is defining
the link between the components of plastic deformation (the rate of plastic deformation, to be more precise), the stressed state at this point of time, and loading
history.
Let us represent the stress tensor T σ as the sum of two tensors:
T σ = S σ + D σ ,
(13.17)
where
S σ =
⎛
⎝
σ 0 0 0
0 σ 0 0
0 0 σ 0
⎞
⎠ ,
σ 0 =
1
3
(σ x + σ y + σ z )
(13.18)
is referred to as spherical stress tensor, and D σ is the stress deviator earlier found
from the first formula (1.18), (p. 16):
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
⎞
⎠ .
To designate them, the following writing will be also used:
S σ = {σ 0 δ ij }, D σ = {σ
ij } = {σ ij − σ 0 δ ij }.
(13.19)
The experiments of Bridgman [1], Nadai [3] et al. proved that for plastic materials,
hydrostatic stress (spherical tensor) did not affect the conditions of occurrence
and development of plastic deformation, and plastic deformation did not affect the
volume change. This means that first of all, the volumetric deformation of plastic
materials within a wide range of pressures is elastic and related to the stress of the
first dependency (1.16):
ε 0 =
σ 0
K
,
13 Initial Concepts of Plasticity Theory
This hypothesis propagates phenomena of disappearance of elastic deformation and
preservation of non-elastic deformations observed in uniaxial elongation experiments in the case of full unloading after pre-elongation beyond the elastic limit.
Let us designate the components of plastic deformation as e ij , assuming for
example:
it i = j then e xx = ε e
x ,
if i = j then e xy =
1
2
γ
e
xy .
Since the components of elastic deformation are unambiguously defined through
stresses under Hooke’s law, the fundamental problem of plasticity theory is defining
the link between the components of plastic deformation (the rate of plastic deformation, to be more precise), the stressed state at this point of time, and loading
history.
Let us represent the stress tensor T σ as the sum of two tensors:
T σ = S σ + D σ ,
(13.17)
where
S σ =
⎛
⎝
σ 0 0 0
0 σ 0 0
0 0 σ 0
⎞
⎠ ,
σ 0 =
1
3
(σ x + σ y + σ z )
(13.18)
is referred to as spherical stress tensor, and D σ is the stress deviator earlier found
from the first formula (1.18), (p. 16):
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
⎞
⎠ .
To designate them, the following writing will be also used:
S σ = {σ 0 δ ij }, D σ = {σ
ij } = {σ ij − σ 0 δ ij }.
(13.19)
The experiments of Bridgman [1], Nadai [3] et al. proved that for plastic materials,
hydrostatic stress (spherical tensor) did not affect the conditions of occurrence
and development of plastic deformation, and plastic deformation did not affect the
volume change. This means that first of all, the volumetric deformation of plastic
materials within a wide range of pressures is elastic and related to the stress of the
first dependency (1.16):
ε 0 =
σ 0
K
,