Exploring buckling and post-buckling behavior of incompressible hyperelastic beams through innovative experimental and computational approaches

Figures & data

Figure 1. Raw rubber compound.

Table 1. Natural rubber ingredients.

Material No.	Material	Value (PHR)
1	Butadiene rubber (PBR-1220)	45
2	Natural rubber (SMR-20)	55
3	Carbon Black (N660)	50
4	Aromatic oil	7
5	Sulfur	19
6	Accelerator (TBBS)	8
7	Antioxidants	15
8	Zinc oxide	15
9	Stearic acid	15

Figure 2. Rubber rheometer test diagram.

Figure 3. (a) Metal mold design and (b) steel mold and compound rubber.

Figure 4. Steps of making a hyperelastic beam.

Figure 5. The uniaxial tensile test based on the ASTM D412 standard.

Figure 6. Overlap of experimental tensile test outcomes with strain energy functions.

Table 2. Fixed coefficients of Yeoh and Neo-Hookean model (Khaniki et al. 2021; Khaniki et al. 2022b; Khaniki et al. 2022c).

Fixed coefficient	C10 (pa)	C20 (pa)	C30 (pa)
Yeo: $U=C_{10}\left(I_1-3\right)+C_{20}{\left(I_1-3\right)}^2$ + $C_{30}{\left(I_1-3\right)}^3$	708663	19707	−149
Neo-Hookean: $U=C_{10}\left(I_1-3\right)$	207843	–	–

Figure 7. Experimental test steps of DIC.

Figure 8. Experimental test of hyperelastic beams before and after loading.

Figure 9. Boundary and loading conditions in the finite element model.

Figure 10. First three buckling modes.

Figure 11. The X-axis displacement contour of the (a) experimental test (b) finite element of the beam.

Figure 12. The Y-axis displacement contour of the (a) experimental test (b) finite element of the beam.

Figure 13. The force-displacement diagram of the experimental test and the FEM.

Figure 14. Strain contour of the (a) experimental test (b) finite element of the structure.

Figure 15. Variations of critical buckling load versus the material hardness coefficient.

Figure 16. Variations of critical buckling load versus cross-section thickness.

Figure 17. Variations of critical buckling load versus different lengths.

Figure 18. Variations of critical buckling force versus the eccentric loads.

Table 3. Variations of linear eigenvalue versus geometry, material properties and eccentric loads.

Eigenvalue Number	Eigenvalue 1	Eigenvalue 2	Eigenvalue 3	Eigenvalue 4
The length of the beam
H = 300 mm	20.586	177.16	469.92	859
H = 250 mm	28.174	259.02	670.11	1138.3
H = 200 mm	48.62	411.28	1004.3	1278
The material properties
Rockwell 50	33.229	291.34	753.67	1280.7
Rockwell 40	28.174	259.02	670.11	1138.3
Rockwell 70	38.611	325.13	837.82	1423.01
The cross-section thickness
w = 15 mm	9.6	84.53	225.94	419.4
w = 20 mm	28.174	259.02	670.11	1138.3
w = 25 mm	71.779	608.37	1392.6	1540.7
The eccentric loads
e = 0 mm	28.174	259.02	670.11	1138.3
e = 1 mm	30.63	267.13	697.66	1280.7
e = 2 mm	38.611	325.13	837.82	1423

Figure 19. The training, validation, test, and whole model performance evolution versus the Epoch number.

Figure 20. The regression schemes for four subsets: (a) training, (b) validation, (c) test, and (d) full evaluation.

Khaniki, H. B., M. H. Ghayesh, R. Chin, and M. Amabili. 2021. Large amplitude vibrations of imperfect porous-hyperelastic beams via a modified strain energy. Journal of Sound and Vibration 513:116416. doi: 10.1016/j.jsv.2021.116416.

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Khaniki, H. B., M. H. Ghayesh, R. Chin, and L.-Q. Chen. 2022b. Experimental characteristics and coupled nonlinear forced vibrations of axially travelling hyperelastic beams. Thin-Walled Structures 170:108526. doi: 10.1016/j.tws.2021.108526.

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Khaniki, H. B., M. H. Ghayesh, R. Chin, and S. Hussain. 2022c. Nonlinear continuum mechanics of thick hyperelastic sandwich beams using various shear deformable beam theories. Continuum Mechanics and Thermodynamics 34 (3):781–827. doi: 10.1007/s00161-022-01090-y.

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