Analyze volumetric stress state with clear inputs quickly. Compare tension and compression conventions easily here. Export results for reports, checks, and design reviews fast.
Hydrostatic stress is the mean normal stress. It represents the purely volumetric part of a stress state. With three mutually orthogonal normal stresses, the first invariant is I1 = σ1 + σ2 + σ3.
The hydrostatic (mean) stress is σh = I1 / 3. The equivalent pressure is reported as p = −σh when tension is positive.
For additional insight, the deviatoric part is s′i = σi − σh, and J2 = 1/6[(σ1−σ2)²+(σ2−σ3)²+(σ3−σ1)²], giving σv = √(3J2).
| Case | σx (MPa) | σy (MPa) | σz (MPa) | σh (MPa) | p (MPa) |
|---|---|---|---|---|---|
| A | 50 | 30 | 10 | 30 | -30 |
| B | 80 | 20 | -40 | 20 | -20 |
| C | -60 | -60 | -60 | -60 | 60 |
Example pressure uses tension-positive convention, so p = −σh.
Hydrostatic stress (also called mean stress) is the average of the three normal stresses acting on a material point. It captures the volumetric, or pressure-like, part of the stress state. When the mean stress is large in compression, voids tend to close; when it is large in tension, voids can grow and damage can accelerate.
The calculator evaluates the first stress invariant I1 = σ1 + σ2 + σ3 and then computes σh = I1/3. It also reports deviatoric stresses s′i = σi − σh, the second deviatoric invariant J2, and an equivalent von Mises stress σv = √(3J2) for context.
Many mechanics texts use tension-positive stresses, while several geomechanics and pressure applications use compression-positive stresses. This tool lets you select the convention so your reported pressure p aligns with your workflow. With tension-positive stresses, a positive pressure magnitude produces negative σh.
Hydrostatic stress is commonly expressed in kPa for soils and fluids, MPa for machine elements and polymers, and GPa for extreme confinement studies. For reference, 1 MPa equals 1000 kPa, and 1 ksi is about 6.895 MPa. Using consistent units is essential when comparing mean stress to yield or cavitation thresholds.
A mean stress near zero indicates a primarily distortional loading, often dominated by shear. A strongly compressive σh indicates confinement, which can raise apparent ductility and delay crack growth. A tensile σh can increase stress triaxiality and promote void growth, especially in ductile metals.
Engineers use mean stress in pressure vessel assessments, underground excavation analysis, polymer creep and crazing studies, and compacted soil performance. It also appears in damage models and fracture criteria where hydrostatic tension influences void nucleation. In testing, triaxial fixtures deliberately vary σh.
Before calculating, confirm whether your stresses are principal or merely component values. If significant shear exists, compute principal stresses first, then use the principal mode. Also verify whether compressive stresses are entered as negative or positive based on the chosen convention to avoid a sign-flipped mean stress.
For reports, include σh, the equivalent pressure, and the unit system used. If you are comparing cases, the CSV export supports quick sorting and plotting. The PDF option is designed for clean printing, keeping the computed blocks readable and consistent across design reviews.
Q1: Is hydrostatic stress the same as pressure?
Hydrostatic stress is pressure-like and equals the mean normal stress. Depending on sign convention, the reported
pressure may be p = −σh or p = σh. The tool shows both to avoid confusion.
Q2: Can I use σx, σy, σz directly if shear exists?
If shear stresses are nonzero, σx, σy, σz are not principal stresses. The mean stress is still (σx+σy+σz)/3, but
deviatoric metrics like J2 and σv require principal or full tensor input.
Q3: Why does confinement change material behavior?
A compressive mean stress suppresses void growth and reduces crack opening. Many ductile materials show delayed
damage under confinement, while tensile mean stress can accelerate void growth and fracture under triaxial tension.
Q4: What does a zero deviatoric stress mean?
If s′1, s′2, and s′3 are all zero, the stress state is purely hydrostatic.
In that case J2 and σv become zero, indicating no distortional loading.
Q5: Which unit should I choose for output?
Choose the unit that matches your design documents. MPa is common for structural and machine parts, kPa for soils
and fluids, and psi/ksi for imperial reports. The calculator converts all outputs consistently to your selection.
Q6: Why is von Mises shown in a hydrostatic calculator?
Hydrostatic stress alone does not predict yielding in ductile metals. von Mises stress provides a quick view of
the distortional intensity, helping you separate “pressure effects” from “shape-changing” effects in the same run.
Q7: Does hydrostatic stress affect brittle fracture?
It can. Tensile mean stress increases crack-tip triaxiality and can reduce apparent toughness, while compressive
mean stress can inhibit crack opening. Always combine σh with appropriate fracture or strength criteria.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.