This is a cubic equation for the unknown λ. If the stress matrix is symmetric and
has real stress values (opposed to imaginary values), the three roots of the cubic
equation λ
3 + aλ
2 + bλ + c ¼ 0 are all real numbers.
The three roots of the cubic equation λ 1 , λ 2 , λ 3 are the three principal stresses
λ 1 ¼ σ 1 , λ 2 ¼ σ 2 , and λ 3 ¼ σ 3 .
In order to find the principal directions, we can use
n
f g σ
½ À λ I
½
½
¼0
If we substitute σ 1 for λ in the above equation, these three equations reduce to
only two linearly independent equations. These two equations can be solved with the
help of the n
2
1 þ n
2
2 þ n
2
3 ¼ 1 requirement, in order to determine the direction cosines
of the normal n
1
ð Þ
È É ¼ n
1
ð Þ
X n
1
ð Þ
Y n
1
ð Þ
X
n
o
to the plane which σ 1 acts on.
When we are solving for direction cosines, since one of the equations
n
2
1 þ n
2
2 þ n
3
1 ¼ l
Â
Ã
is quadratic, two solutions will be found representing two
oppositely directed normals to the same plane. The choice of which one is the
positive is arbitrary.
In the same manner, the process is repeated for σ 2 to find n
2
ð Þ
X , n
2
ð Þ
Y , n
2
ð Þ
Z
n
o
and σ 3
to find n
3
ð Þ
X , n
3
ð Þ
Y , n
3
ð Þ
Z
n
o
.
The third principal direction can also be found by taking direction orthogonal to
the first two directions.
When all the roots of the characteristic equation are different, then these are three
unique orthogonal principal directions.
However, when we find the roots of the characteristic equation, if two of them are
equal and one is different, then the direction of the different root is unique; however,
the other two must be arbitrarily chosen as any two perpendicular axes to each other
and perpendicular to the unique axis. Three axes must define a right-handed system.
If all three principal stresses are equal, the state of stress is said to be hydrostatic state
of stress because this is only the state of stress that can exist in fluids.
2.6 Stress Tensor Invariants
The determinant of the characteristic equation:
σ XX À λ
σ XY
σ XZ
σ YX
σ YY À λ
σ YZ
σ ZX
σ ZY
σ ZZ À λ
¼ 0
λ
3
À I 1 λ
2
À I 2 λ À I 3 ¼ 0
ð3:4Þ
24
2 Stress and Strain in Continuum
has real stress values (opposed to imaginary values), the three roots of the cubic
equation λ
3 + aλ
2 + bλ + c ¼ 0 are all real numbers.
The three roots of the cubic equation λ 1 , λ 2 , λ 3 are the three principal stresses
λ 1 ¼ σ 1 , λ 2 ¼ σ 2 , and λ 3 ¼ σ 3 .
In order to find the principal directions, we can use
n
f g σ
½ À λ I
½
½
¼0
If we substitute σ 1 for λ in the above equation, these three equations reduce to
only two linearly independent equations. These two equations can be solved with the
help of the n
2
1 þ n
2
2 þ n
2
3 ¼ 1 requirement, in order to determine the direction cosines
of the normal n
1
ð Þ
È É ¼ n
1
ð Þ
X n
1
ð Þ
Y n
1
ð Þ
X
n
o
to the plane which σ 1 acts on.
When we are solving for direction cosines, since one of the equations
n
2
1 þ n
2
2 þ n
3
1 ¼ l
Â
Ã
is quadratic, two solutions will be found representing two
oppositely directed normals to the same plane. The choice of which one is the
positive is arbitrary.
In the same manner, the process is repeated for σ 2 to find n
2
ð Þ
X , n
2
ð Þ
Y , n
2
ð Þ
Z
n
o
and σ 3
to find n
3
ð Þ
X , n
3
ð Þ
Y , n
3
ð Þ
Z
n
o
.
The third principal direction can also be found by taking direction orthogonal to
the first two directions.
When all the roots of the characteristic equation are different, then these are three
unique orthogonal principal directions.
However, when we find the roots of the characteristic equation, if two of them are
equal and one is different, then the direction of the different root is unique; however,
the other two must be arbitrarily chosen as any two perpendicular axes to each other
and perpendicular to the unique axis. Three axes must define a right-handed system.
If all three principal stresses are equal, the state of stress is said to be hydrostatic state
of stress because this is only the state of stress that can exist in fluids.
2.6 Stress Tensor Invariants
The determinant of the characteristic equation:
σ XX À λ
σ XY
σ XZ
σ YX
σ YY À λ
σ YZ
σ ZX
σ ZY
σ ZZ À λ
¼ 0
λ
3
À I 1 λ
2
À I 2 λ À I 3 ¼ 0
ð3:4Þ
24
2 Stress and Strain in Continuum