Calculator Inputs
Enter displacement gradients for small-strain analysis. The form uses a three-column layout on large screens, two on medium screens, and one on mobile.
Formula Used
The calculator uses the infinitesimal strain tensor. It is appropriate when displacement gradients are small and rotations remain modest.
Tensor definition
εij = 1/2 ( ∂ui/∂xj + ∂uj/∂xi )
Expanded 3D strain tensor
[ ε ] = [[εxx, εxy, εxz], [εxy, εyy, εyz], [εxz, εyz, εzz]]
- εxx = ∂u/∂x, εyy = ∂v/∂y, εzz = ∂w/∂z
- εxy = 1/2(∂u/∂y + ∂v/∂x)
- εxz = 1/2(∂u/∂z + ∂w/∂x)
- εyz = 1/2(∂v/∂z + ∂w/∂y)
- Volumetric strain = εxx + εyy + εzz
- Principal strains are the eigenvalues of the symmetric tensor.
How to Use This Calculator
- Enter the nine displacement-gradient terms.
- Choose a display unit for reporting values.
- Select the decimal precision you want.
- Click Calculate Strain Tensor.
- Review the tensor matrix, principal strains, and summary metrics.
- Inspect the Plotly graph for quick comparison.
- Export the result set as CSV or PDF.
Example Data Table
This example uses the same format as the calculator. Values are rounded for readability.
| ∂u/∂x | ∂u/∂y | ∂u/∂z | ∂v/∂x | ∂v/∂y | ∂v/∂z | ∂w/∂x | ∂w/∂y | ∂w/∂z | ε1 | ε2 | ε3 | εv |
|---|---|---|---|---|---|---|---|---|---|---|---|---|
| 0.0120 | 0.0040 | -0.0010 | 0.0060 | 0.0150 | 0.0030 | -0.0020 | 0.0050 | 0.0090 | 0.0193 | 0.0118 | 0.0049 | 0.0360 |
Frequently Asked Questions
1. What does this calculator produce?
It calculates the full small-strain tensor, engineering shear strains, principal strains, volumetric strain, invariants, and maximum shear measures from displacement gradients.
2. Which inputs should I enter?
Enter the nine spatial derivatives of the displacement field: ∂u/∂x, ∂u/∂y, ∂u/∂z, ∂v/∂x, ∂v/∂y, ∂v/∂z, ∂w/∂x, ∂w/∂y, and ∂w/∂z.
3. When is the small-strain model valid?
Use it when deformations and rotations are small. For large deformation problems, a finite-strain measure is usually more appropriate.
4. What is the difference between εxy and γxy?
εxy is the tensor shear component. γxy is the engineering shear strain. Engineering shear is twice the tensor shear for the same plane.
5. Why do principal strains matter?
Principal strains identify the pure extension and compression directions where shear vanishes. They are useful for interpretation, failure checks, and reporting.
6. Can I display results in percent or microstrain?
Yes. The calculator can display strain outputs as unitless values, percent, or microstrain for easier reporting.
7. What does volumetric strain represent?
Volumetric strain is the trace of the strain tensor. It estimates relative volume change for small deformations.
8. Why are invariants included?
Invariants remain unchanged under coordinate rotation. They help compare states objectively and support constitutive modeling or advanced post-processing.