PSI - Issue 81

Andrejs Kovalovs et al. / Procedia Structural Integrity 81 (2026) 388–395

394

3. Results and Analysis The identified elastic constants for the unidirectional and twill-weave laminates obtained from the inverse procedure are summarized in Table 4. For the principal in-plane moduli ( E 1 , G 12 ), deviations from the reference values of the virtual experiment do not exceed 1%, confirming the accuracy of the proposed identification methodology. Noticeable discrepancies greater than 5% were observed for the out-of-plane shear modulus G 23 in the unidirectional laminate and for the through-thickness modulus E 3, as well as Poisson’s ratios ν 13 and ν 23 in the twill-weave laminate. These differences arise from the limited sensitivity of the selected bending-dominated vibration modes to through-thickness deformation and reflect the inherent constraints of the thin-plate assumption adopted in the FE model. The comparison betw een “measured” (virtual experiment) and predicted natural frequencies demonstrates almost perfect correlation, with all points lying on the line of equality. The maximum relative discrepancy does not exceed 0.06%, which is within the numerical precision of the FE solution. The ratio E 1 / E 2 = 16 and G 12 / E 1 = 0.035 for the unidirectional laminate corresponds closely to the theoretical predictions of classical laminate theory. In contrast, the twill-weave laminate shows E 1 = E 2 , indicating quasi-isotropic in-plane behavior.

Table 4. Identified elastic constants of unidirectional and twill-weave CFRP laminates.

Elastic constants Experimental data

Identified value

Error (%)

Elastic constants

Experimental data

Identified value

Error (%)

E 1 (GPa)

139.37

139.59

-0.16 0.00 4.10 4.62 -0.25 22.42

E 1 = E 2 (GPa)

73.87

73.89

-0.03 42.33

E 2 = E 3 (GPa)

8.74

8.74

E 3 (GPa)

8.74

5.04

ν 12 = ν 13

0.268 0.325 4.858 1.289

0.257 0.310

0.029 0.286

0.029 0.319

0.0

ν 12

ν 13 = ν 23

-19.03

ν 23

G 12 = G 13 (GPa)

4.87 1.00

G 12 (GPa)

6.19

6.19 4.57

0.0

G 23 (GPa)

G 13 = G 23 (GPa) 4.86

5.97

Unidirectional CFRP

Twill-weave CFRP

The influence of stiffness anisotropy on the modal spectrum is evident: as the stiffness contrast increases, torsional modes separate more distinctly from bending modes. This behavior agrees with analytical solutions for orthotropic plates, where frequency ratios depend mainly on E 1 / E 2 and G 12 / E 2 . The method thus not only identifies the numerical values of elastic constants but also provides insight into how anisotropy influences the global dynamic response of composite laminates. Table 5 presents a detailed comparison of the natural frequencies obtained from the FE model and from the RSM-based identification for both materials. Although certain discrepancies were observed in the identified elastic constants (Table 4), the agreement between the identified and reference frequencies remains excellent: the average relative error is below 0.1%, and the coefficient of determination ( R 2 ) exceeds 0.98 for all modes for the unidirectional laminate and 0.99 for the twill-weave CFRP plate. These results indicate that the spectrum of natural frequencies is only weakly sensitive to specific elastic constants. To improve the identifiability of these less influential parameters, it is likely necessary to analyse specimens under alternative boundary conditions that more strongly excite through-thickness or shear-dominated deformation modes.

Table 5. Comparison of measured and predicted modal frequencies for unidirectional and twill-weave CFRP plates.

No.

Mode (m,n)

Frequency (Hz)

FEM Frequency (Hz)

Error (%)

R 2 (%)

Mode (m,n)

Frequency (Hz)

FEM Frequency (Hz)

Error (%)

R 2 (%)

1. 2. 3. 4. 5. 6. 7. 8. 9.

(1,1) (0,2) (1,2) (2,0) (2,1) (2,2) (0,3) (1,3) (3,0) (2,3) (3,1) (3,2)

30.138 62.410 87.209 97.596 114.54 169.10 172.34 194.16 268.61 270.44 283.13 330.05

30.15 62.41 87.23 97.60

0.04 100 0.01 100 0.02 100 0.00 100 0.02 100 0.04 98.98 0.02 98.82 0.03 100 0.00 98.61 0.04 96.10 0.01 98.32 0.03 98.96

(1,1) (0,2) (1,2) (2,0) (2,1) (2,2) (0,3) (1,3) (3,0) (2,3) (3,1) (3,2)

47.85 99.25

47.887 99.256 138.92 254.07 269.99 274.81 308.98 474.67 535.67 567.36 699.28 335.5

0.07 100 0.01 100 0.06 100 0.02 100 0.04 99.99 0.04 99.99 0.06 100 0.07 100 0.09 100 0.07 100 0.09 100 0.09 99.84

138.84 254.01 269.89 274.71 308.80 335.27 474.23 535.29 566.87 698.68

114.56 169.17 172.38 194.21 268.61 270.56 283.16 330.16

10. 11. 12.

Unidirectional CFRP

Twill-weave CFRP

Sensitivity coefficients derived from the response surfaces make it possible to evaluate how frequency perturbations affect identified material constants. For the current virtual experiment, a ±0.1% deviation in modal frequency corresponds to an