2.14 Principal Stress in Three Dimensions
23
where
I 1 = σ x + σ y + σ z
(2.31a)
I 2 = σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
xz
(2.30b)
I 3 =
σ x τ xy τ xz
τ xy σ y τ yz
τ xz τ yz σ z
(2.31c)
The three roots of Eq. (2.31) are the principal stresses, corresponding to which are
three sets of direction cosines that establish the relationship of the principal planes
to the origin of the non-principal axes.
2.15 Stress Invariant
Invariant means those quantities that are permanent, unexchangeable and do not
vary under different conditions. In the context of stress tensor, invariants are such
quantities that do not change with rotation of axes or which remain unaffected under
transformation, from one set of axes to another. Therefore, the combination of stresses
at a point that do not change with the orientation of co-ordinate axes is called stress
invariants. Hence, the definition from Eq. (2.31)
σ x + σ y + σ z = I 1 = First invariant of stress
σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
zx = I 2 = Second invariant of stress
σ x σ y σ z − σ x τ
2
yz − σ y τ
2
xz − σ z τ
2
xy + 2τ xy τ yz τ xz = I 3 = Third invariant of stress
2.16 Equilibrium of a Two-Dimensional or Plane Element
Differential Element
When a body is in equilibrium, any isolated part of the body is acted upon by an
equilibrium set of forces. The small element with unit thickness shown in Fig. 2.11
represents part of a body and therefore must be in equilibrium if the entire body is
to be in equilibrium.
It is to be noted that the components of stress generally vary from point to point
in a stressed body. These variations are governed by the conditions of equilibrium
23
where
I 1 = σ x + σ y + σ z
(2.31a)
I 2 = σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
xz
(2.30b)
I 3 =
σ x τ xy τ xz
τ xy σ y τ yz
τ xz τ yz σ z
(2.31c)
The three roots of Eq. (2.31) are the principal stresses, corresponding to which are
three sets of direction cosines that establish the relationship of the principal planes
to the origin of the non-principal axes.
2.15 Stress Invariant
Invariant means those quantities that are permanent, unexchangeable and do not
vary under different conditions. In the context of stress tensor, invariants are such
quantities that do not change with rotation of axes or which remain unaffected under
transformation, from one set of axes to another. Therefore, the combination of stresses
at a point that do not change with the orientation of co-ordinate axes is called stress
invariants. Hence, the definition from Eq. (2.31)
σ x + σ y + σ z = I 1 = First invariant of stress
σ x σ y + σ y σ z + σ z σ x − τ
2
xy − τ
2
yz − τ
2
zx = I 2 = Second invariant of stress
σ x σ y σ z − σ x τ
2
yz − σ y τ
2
xz − σ z τ
2
xy + 2τ xy τ yz τ xz = I 3 = Third invariant of stress
2.16 Equilibrium of a Two-Dimensional or Plane Element
Differential Element
When a body is in equilibrium, any isolated part of the body is acted upon by an
equilibrium set of forces. The small element with unit thickness shown in Fig. 2.11
represents part of a body and therefore must be in equilibrium if the entire body is
to be in equilibrium.
It is to be noted that the components of stress generally vary from point to point
in a stressed body. These variations are governed by the conditions of equilibrium