168
13 Initial Concepts of Plasticity Theory
where ˙
ε ij are the components of the strain rate tensor
˙
ε i,j =
1
2
˙
u i,j + ˙
u j,i
,
˙
u =
du
dt
,
and then ˙
σ ij = dσ ij /dt is the stress rate, whereas t is the time or any other
monotonously increasing parameter. Below we will adopt that a point over the value
designation means a time derivative unless indicated otherwise.
Due to the law of twoness of tangential stresses (p. 12) σ ij = σ ji , e.g. the stress
tensor T σ is symmetric. As applicable to the stressed state, formula (13.3) yields
σ
ij = β ik β js σ ks ,
or in an expanded view
σ
ij = β i1 β js σ 1s + β i2 β js σ 2s + β i3 β js σ 3s
= β i1 β j 1 σ 11 + β i1 β j 2 σ 12 + β i1 β j 3 σ 13 + . . . + β i3 β j 3 σ 33
=
we give similar terms, considering that σ ij = σ ji and β ij = β ji
= β i1 β j 1 σ 11 + β i2 β j 2 σ 22
+ β i3 β j 3 σ 33 + 2(β i1 β j 2 σ 12 + β i2 β j 3 σ 23 + β 13 β j 1 σ 31 ).
(13.9)
In the principal axes, the stress tensor will be
T σ =
⎛
⎝
σ 1 0 0
0 σ 2 0
0 0 σ 3
⎞
⎠ .
The principal values (σ 1 , σ 2 , σ 3 ) of the stress tensor are referred to as principal
stresses. They are defined as roots of a characteristic equation [2]
σ x − σ τ xy
τ xz
τ yx σ y − σ τ yz
τ zx
τ zy σ z − σ
= 0.
(13.10)
Equation (13.10) can also be written as follows:
[σ ij − σ δ ij ] = 0,
(13.11)
where δ ij is defined by the formula
δ ij =
1 for i = j,
0 for i = j,
13 Initial Concepts of Plasticity Theory
where ˙
ε ij are the components of the strain rate tensor
˙
ε i,j =
1
2
˙
u i,j + ˙
u j,i
,
˙
u =
du
dt
,
and then ˙
σ ij = dσ ij /dt is the stress rate, whereas t is the time or any other
monotonously increasing parameter. Below we will adopt that a point over the value
designation means a time derivative unless indicated otherwise.
Due to the law of twoness of tangential stresses (p. 12) σ ij = σ ji , e.g. the stress
tensor T σ is symmetric. As applicable to the stressed state, formula (13.3) yields
σ
ij = β ik β js σ ks ,
or in an expanded view
σ
ij = β i1 β js σ 1s + β i2 β js σ 2s + β i3 β js σ 3s
= β i1 β j 1 σ 11 + β i1 β j 2 σ 12 + β i1 β j 3 σ 13 + . . . + β i3 β j 3 σ 33
=
we give similar terms, considering that σ ij = σ ji and β ij = β ji
= β i1 β j 1 σ 11 + β i2 β j 2 σ 22
+ β i3 β j 3 σ 33 + 2(β i1 β j 2 σ 12 + β i2 β j 3 σ 23 + β 13 β j 1 σ 31 ).
(13.9)
In the principal axes, the stress tensor will be
T σ =
⎛
⎝
σ 1 0 0
0 σ 2 0
0 0 σ 3
⎞
⎠ .
The principal values (σ 1 , σ 2 , σ 3 ) of the stress tensor are referred to as principal
stresses. They are defined as roots of a characteristic equation [2]
σ x − σ τ xy
τ xz
τ yx σ y − σ τ yz
τ zx
τ zy σ z − σ
= 0.
(13.10)
Equation (13.10) can also be written as follows:
[σ ij − σ δ ij ] = 0,
(13.11)
where δ ij is defined by the formula
δ ij =
1 for i = j,
0 for i = j,