Library / Mechanical / MEC-60
Mechanical
First stress invariant (trace)
Sum of the three normal stresses.
- Ref
- MEC-60
- Rev
- v1
- Field
- Mechanical
- Result
- I1 [MPa]
I1, the trace of the stress tensor, is the sum of the three normal stresses. It has the same value in every set of axes, so it equals the sum of the principal stresses. One third of it is the mean (hydrostatic) stress, which changes the volume of the material but not its shape. The shear stresses do not enter at all. The lecture check: the trace of the three-dimensional example is 180 MPa before and after transformation.
Sum of the three normal stresses. It is the same in every set of axes, so it equals the sum of the three principal stresses, and one third of it is the mean (hydrostatic) stress.
Calculator
- Mean (hydrostatic) stress, I1/360 MPa
- Deviatoric part of σx, σx − I1/320 MPa
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The lecture state has I1 = 180 MPa, which equals the sum of its principal stresses 110 + 50 + 20. Turning the element by 30° about z changes σx and σy to 106 and 54.02 MPa, but their sum is still 160 and σz is unchanged, so I1 is still 180 MPa.
- A hydrostatic state with H in all three directions has I1 = 3H and no distortion: both Tresca and von Mises are zero. The mean stress therefore controls volume change, and the part of the stress left over controls yield.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (MPa) | Local step | Change in result |
|---|---|---|---|---|---|---|
| σx Normal stress in x | −200 to 300 MPa | −100 | 400 | 500 | +10 MPa | +5.56 % |
| σy Normal stress in y | −200 to 300 MPa | −100 | 400 | 500 | +10 MPa | +5.56 % |
| σz Normal stress in z | −200 to 300 MPa | −40 | 460 | 500 | +10 MPa | +5.56 % |
Sweep of Normal stress in x
| Normal stress in x (MPa) | First invariant (MPa) |
|---|---|
| −200 | −100 |
| −137.5 | −37.5 |
| −75 | 25 |
| −12.5 | 87.5 |
| 50 | 150 |
| 112.5 | 212.5 |
| 175 | 275 |
| 237.5 | 337.5 |
| 300 | 400 |
Sweep one variable
Variables and how each one contributes
First invariant
What the equation gives you. Everything else feeds it.
Normal stress in x
Normal stress on the x face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I1 by the same amount. Raising it by 10 MPa (to 90 MPa) moves the result by +5.56 %, from 180 to 190 MPa.
Normal stress in y
Normal stress on the y face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I1 by the same amount. Raising it by 10 MPa (to 90 MPa) moves the result by +5.56 %, from 180 to 190 MPa.
Normal stress in z
Normal stress on the z face, in MPa. Tension is positive and compression negative, so the slider runs from −200 to +300 MPa. Raises I1 by the same amount. Raising it by 10 MPa (to 30 MPa) moves the result by +5.56 %, from 180 to 190 MPa.
Worked example with the values above
- Add the normal stresses: 80 + (80) + (20).
- I1 = 180 MPa.
- Mean stress: I1/3 = 60 MPa.
Assumptions and limits
- The three normal stresses act on mutually perpendicular faces at one point.
- Tension positive, compression negative.
Common mistakes
- Dropping the sign of a compressive stress when summing.
- Expecting the shear stresses to change I1. They do not.
Statistics and uncertainty
The trace is a plain sum, so independent errors of ±5 MPa in each normal stress give about ±8.7 MPa in I1 (5 times √3) and ±2.9 MPa in the mean stress. A common bias, for example from a zero offset in the gauge amplifier, adds three times over in I1.
Where it is used
- Checking a principal-stress calculation by comparing sums.
- Separating the mean stress from the shape-changing part of the stress.
Maths you need
- Sign convention: tension positive
- Principal stresses: the stresses on planes with no shear