Manipulations of Stress Tensor Components
Manipulations of stress tensor components
We have adopted the tensor notation for global stresses :
| [S] = |
|
(Symmetric: Upper triangle = lower triangle) |
Where
XX
,
YY
,
ZZ
are the "direct" stresses, and
XY
,
YZ
,
XZ
the shear stresses.
By convention the ij suffices on these terms mean "stress in the direction i on a plane of constant j", so:
-
XX
is "direct" X stress, that is stress in the X direction on
a plane of constant X.
-
YZ
is "shear" YZ stress, that is stress in the Y direction on
a plane of constant Z.
When a stress tensor is described as symmetric this is because opposite off-diagonal terms are the same, that is:
XY
==
YX
,
YZ
==
ZY
,
XZ
==
ZX
Rotating a tensor to give element local stresses
If we have a set of direction cosines in the 3x3 matrix [R] then we can rotate a tensor thus:
| [S'] = [R][S][R] T | where [S'] = |
|
This is how the global to local transformation of stresses and strains is carried
out when the
is set to
: the
[R]
matrix is formed from
the local axes of the element. The prime " ' " notation is used to signify that
the component is in the local (as opposed to global) coordinate system.
Computing von Mises Stress: The deviatoric stress
| This is given by: |
|
Computing signed von Mises Stress
| This is given by: |
SGN *
vonMises
|
SGN is calculated either as the sign of the principal stress with the greatest magnitude or as the sign of the first stress invariant (I1 =
XX
+
YY
+
ZZ
). The method is selected in the preferences file (see
Appendix B).
Computing PRESSURE: The hydrostatic pressure
| This is given by: |
-(
XX
+
YY
+
ZZ
)
/ 3.0.
|
(Note: compression +ive.) |
Computing PRINCIPAL stresses
The principal stresses (maximum, middle, minimum) are the three roots ( P ) of the cubic:
| P MAX' P MID' P MIN solve |
|
| where P MAX > P MID > P MIN . | |
Computing MAX_SHEAR_STRESS
| This is given by: | (P MAX - P MIN ) / 2.0. |
Computing DEVIATORIC PRINCIPAL stresses
Deviatoric principal stresses ( ) are given as the deviation from the hydrostatic pressure, (recall compression is +ive).
| They are given by: | [P MAX + PRESSURE], [P MID + PRESSURE] , and [P MIN + PRESSURE]. |
Computing 2D PRINCIPAL stresses
2D in-plane principal stresses are computed for shell, thick shell, SPH and DES elements from the element local stresses.
|
They are given by: |
|
| Equivalently, these are the two roots (P) of the quadratic equation: |
|
Note!!
The in-plane computation ignores any out of plane stresses, for
thin shells that is
Y'Z'
,
X'Z
'
If these are significant the in-plane principal stresses
do not represent the true stress state in the element, so these data components
should only be used in plane stress situations.
Computing TRIAXIALITY
Triaxiality is the ratio of hydrostatic pressure and von Mises stress.
| This is given by: | - PRESSURE / VON MISES. |
Computing LODE ANGLE
| This is given by: |
|
Computing LODE PARAMETER
| This is given by: |
|
Computing ALTERNATIVE LODE PARAMETER
| This is given by: |
|
Computing SHELL TENSION/COMPRESSION
This component shows whether a shell element is in tension, compression, zero stress or not computed on both sides of the shell. A shell in bending will show compression (blue) on one side and tension (red) on the other side. The contour colour will depend on which side of the shell is currently visible.
A surface is defined as being in tension if the magnitude of the maximum principal stress is greater than the magnitude of the minimum principal stress and vice versa for compression. The value on a surface may not be computed if, for example, the top surface integration point is not included in the results.
Computing YIELD_UTILISATION_FACTOR
| This is given by: |
|
Computing YIELD_UTILISATION_PERCENTAGE
| This is given by: |
|