Analyze constants, invert matrices, and inspect tensor consistency. Useful for research, engineering, classes, and verification. See matrix behavior before fabrication, simulation, optimization, or publication.
The default values below are prefilled in the calculator.
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 16.40 | -5.74 | -7.22 | 0.00 | 0.00 | 0.00 |
| 2 | -5.74 | 16.40 | -7.22 | 0.00 | 0.00 | 0.00 |
| 3 | -7.22 | -7.22 | 18.80 | 0.00 | 0.00 | 0.00 |
| 4 | 0.00 | 0.00 | 0.00 | 43.50 | 0.00 | 0.00 |
| 5 | 0.00 | 0.00 | 0.00 | 0.00 | 43.50 | 0.00 |
| 6 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 44.60 |
| 1 | 2 | 3 | 4 | 5 | 6 | |
|---|---|---|---|---|---|---|
| 1 | 0.00 | 0.00 | 0.00 | 0.00 | 584.00 | 0.00 |
| 2 | 0.00 | 0.00 | 0.00 | 584.00 | 0.00 | 0.00 |
| 3 | -171.00 | -171.00 | 374.00 | 0.00 | 0.00 | 0.00 |
| Term | Value |
|---|---|
| εr11 | 1700.00 |
| εr22 | 1700.00 |
| εr33 | 1470.00 |
1. Stiffness matrix at constant electric field: cE = (sE)-1
2. Piezoelectric stress matrix: e = d × cE
3. Estimated coupling factors: k ≈ |e| / √(c × ε)
4. Permittivity conversion: ε = εr × ε0, where ε0 = 8.8541878128 × 10-12 F/m
5. Stability screen: all leading principal minors of the symmetrized stiffness matrix should remain positive.
Enter the six by six compliance matrix values in ×10-12 m²/N.
Enter the three by six piezoelectric strain matrix values in pC/N.
Enter the relative permittivity terms for the main material axes.
Press Calculate Matrix to invert the compliance matrix.
Review the result block above the form. Check symmetry, determinants, condition number, and stability.
Use the CSV or PDF buttons to save the generated output.
Piezoelectric materials convert mechanical strain into electric response. They also create strain under an applied field. Accurate matrix data is essential. This calculator helps you move from compliance data to stiffness constants quickly.
The stiffness matrix links stress and strain in compact tensor form. It supports constitutive modeling. It also supports finite element work, ultrasonic design, actuator development, and sensor validation. Small coefficient errors can distort simulation outputs. Reliable inversion is important.
You can enter the full 6×6 compliance matrix. You can also add the 3×6 piezoelectric strain matrix. Optional dielectric inputs extend the analysis. The tool inverts compliance values to produce the stiffness matrix at constant electric field. It also estimates stress coefficients and coupling indicators.
Physical datasets should be nearly symmetric. They should also remain numerically stable during inversion. This page checks matrix symmetry, determinant behavior, condition number, and leading principal minors. These checks help you spot noisy lab data, unit mistakes, and nonphysical entries before deeper modeling.
Use this calculator when comparing ceramics, crystals, polymers, or composites. It works well for research notes, coursework, lab reports, and engineering review. You can export results for documentation. You can also keep a repeatable workflow when screening multiple materials.
Higher stiffness terms usually indicate stronger resistance to deformation along specific directions. Off diagonal terms show coupling between axes. The piezoelectric stress matrix adds another layer of interpretation. It shows how electrical and mechanical domains interact inside anisotropic media.
Start with measured coefficients from a trusted source. Keep symmetry wherever crystal class requires it. Recheck signs on shear terms. Compare inverted values against published ranges. Large condition numbers may signal sensitive data. In that case, even small rounding changes can shift the final stiffness matrix.
Use exported tables when writing reports or comparing batches. Saved output reduces transcription errors. It also makes peer review and lab audits easier. Always confirm units before interpreting results. Correct units keep the matrix physically meaningful and ready for simulation, design, and publication.
It returns the six by six stiffness matrix, the three by six stress matrix, symmetry checks, determinants, condition number, principal minors, and estimated coupling indicators.
Symmetry is a physical expectation for many valid material tensors. Strong asymmetry often suggests entry errors, unit mistakes, or inconsistent experimental processing.
Enter compliance in ×10-12 m²/N, piezoelectric strain in pC/N, and relative permittivity as dimensionless values. The page converts them internally.
The page stops the inversion and shows an error. A singular or near singular matrix cannot produce a reliable stiffness tensor.
No. They are quick estimates based on representative terms. They help screening, but final material characterization should use the full constitutive framework.
Yes. The full six by six compliance input supports anisotropic behavior. Results are most meaningful when the entered tensor follows the correct material symmetry.
They show how loading in one direction influences response in another. This matters in crystals, transducers, layered composites, and directional device design.
Check symmetry, compare magnitudes with literature, inspect the condition number, and confirm that the leading principal minors stay positive after symmetrization.
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.