Formula Used
Under hydrostatic pressure, the normal stresses are equal in all directions: σx = σy = σz = −p. For a linear, isotropic solid, the normal strain in each direction becomes:
εx = εy = εz = −(p/E) · (1 − 2ν)
The volumetric strain is the sum of the three normal strains:
εv = εx + εy + εz = −3p(1 − 2ν)/E
Using the bulk modulus K = E / [3(1 − 2ν)], the volumetric strain simplifies to: εv = −p/K and εx = εv/3.
How to Use This Calculator
- Enter the hydrostatic pressure magnitude and choose its unit.
- Select a mode: use E and ν or use K.
- Fill in the required material properties with correct units.
- Click Calculate Strain to view results above the form.
- Use the CSV or PDF buttons to export the computed summary.
Example Data Table
| Pressure (MPa) | E (GPa) | ν | Linear Strain (ε) | Volumetric Strain (εv) |
|---|---|---|---|---|
| 50 | 200 | 0.30 | -1.00e-4 | -3.00e-4 |
| 120 | 70 | 0.33 | -5.83e-4 | -1.75e-3 |
| 10 | 3.0 | 0.45 | -3.33e-4 | -1.00e-3 |
Examples assume uniform compression and small-strain linear elasticity.
Understanding Strain from Hydrostatic Pressure
Hydrostatic pressure loads a material equally in all directions. Unlike uniaxial stress, it does not create shear. In linear elasticity, this uniform pressure primarily changes volume, making it ideal for estimating volumetric strain in fluids, polymers, and metals under confinement.
Hydrostatic Pressure Basics
Hydrostatic pressure is an isotropic compressive stress state: σx = σy = σz = −P. Because all principal stresses match, the deviatoric stress is zero, so distortion is minimal and volumetric response dominates.
Bulk Modulus and Compressibility
The bulk modulus K measures resistance to uniform compression. Its inverse, compressibility (β = 1/K), indicates how easily volume changes with pressure. Higher K means smaller strain for the same pressure. Typical engineering solids have K in the tens to hundreds of gigapascals.
Pressure–Strain Relationship
For small strains, volumetric strain is εv = −P/K. If the material is isotropic, the equal principal (linear) strains are ε = εv/3 = −P/(3K). This calculator reports both linear and volumetric strain for quick interpretation.
Engineers often report these values as percent strain or microstrain (με). For example, a linear strain of −50 με corresponds to a very small contraction of 50 parts per million. Converting to microstrain can make comparisons easier when you are working with high-stiffness materials and moderate pressures in practice.
Unit Handling and Magnitude Checks
Consistent units are essential. Enter pressure and bulk modulus in compatible units (for example, Pa with Pa, or MPa with MPa). As a sanity check, 10 MPa on a 50 GPa material gives εv ≈ −2×10−4, a small but measurable compression.
Reference Bulk Modulus Data
Approximate K values: water ~2.2 GPa, rubbery polymers ~1–3 GPa, aluminum ~70–80 GPa, steel ~150–170 GPa, and fused silica ~35–40 GPa. Use supplier datasheets when accuracy matters because temperature, porosity, and microstructure can shift K significantly.
Where Engineers Use This
Volumetric strain from pressure is used in deep-sea housings, high-pressure pipelines, pressure vessel components, soil consolidation models, and acoustic wave speed estimates. It is also helpful when converting pressure changes into density or volume changes in nearly incompressible media.
Assumptions and Limits
This model assumes linear, elastic, isotropic behavior and small strains. At very high pressures, materials may yield, densify, or exhibit nonlinear compressibility. If you expect plasticity, viscoelastic effects, or large deformations, use a nonlinear material model and validated experimental data.
FAQs
1) What is the difference between linear strain and volumetric strain?
Volumetric strain measures total fractional volume change. Linear strain is the equal principal strain in each direction under hydrostatic loading. For isotropic materials, linear strain equals one-third of the volumetric strain.
2) Why is the strain negative in compression?
By common sign convention, compressive pressure reduces dimensions and volume, producing negative strains. If you prefer magnitudes only, interpret the absolute value as the amount of compression.
3) Can I use Young’s modulus instead of bulk modulus?
Not directly. Hydrostatic compression is governed by bulk modulus. If you only know Young’s modulus and Poisson’s ratio, you can compute bulk modulus using K = E / [3(1 − 2ν)].
4) What values of bulk modulus are reasonable for metals?
Many metals fall between about 50 and 200 GPa. Aluminum alloys often sit near 70–80 GPa, while steels are commonly around 150–170 GPa. Always confirm with a datasheet for your exact grade.
5) Does hydrostatic pressure cause shear strain?
Ideal hydrostatic loading produces no shear stress and therefore no shear strain in linear elasticity. Any measured shear typically comes from non-uniform loading, geometry constraints, friction, or material anisotropy.
6) How accurate is the small-strain formula at high pressure?
Accuracy decreases as pressure increases and nonlinear compressibility, plasticity, or phase changes appear. For high-pressure regimes, use pressure-dependent material properties and compare results with experiments or validated models.
7) What should I do if my inputs produce extremely large strain?
Check unit consistency first. Then verify that the bulk modulus is realistic and that the loading remains elastic. Large strains may indicate that the material model is outside its valid range.