PSI - Issue 36

Iakov Lyashenko et al. / Procedia Structural Integrity 36 (2022) 394–400 Iakov Lyashenko, Vadym Borysiuk / Structural Integrity Procedia 00 (2021) 000 – 000

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Fig. 3. (a) Dependencies, similar to the shown in Fig. 2 a , during the indentation of the cylinders of the various diameters into the rubber layer (the magnitudes of the diameter are shown in the figure for all dependencies in millimetres); (b) Dependencies, shown in panel ( a ) (solid lines) with additional theoretical solution for half-space approximation (dotted lines) and results of the simulations (dashed lines) for the rubber layer with thickness h = 15 mm with parameters E = 50 kPa and ν = 0.49. Curves obtained from theory and simulations are shown only for d > 0 mm; (c) Results of simulations of the indentation of the cylindrical indenter with diameter 15 mm into the elastic layer with parameters E = 50 kPa and ν = 0.49. Dashed line shows theoretical solution for the half-space approximation (elastomer with infinite thickness h ). Upper line is related to the thickness of the rubber h = 15 mm, every next dependence from top to bottom relates to the increasing of the thickness of the rubber by 5 mm, with exception for the last bolded line, related to h = 150 mm. Let us first analyse data shown in the Fig. 3 a . From the formula (1), for the indenters with different diameters we can estimate (within half-space approximation) following parameters: D = 4 mm: E * = 56.6 kPa, γ 12 = 0.18 J/m 2 ; D = 7 mm: E * = 65.3 kPa, γ 12 = 0.23 J/m 2 ; D = 10 mm: E * = 74.6 kPa, γ 12 = 0.24 J/m 2 ; D = 15 mm: E * = 92 kPa, γ 12 = 0.28 J/m 2 . These experiments were performed after several weeks from the experiment, shown in Fig. 2 a . However, for the indenter with diameter D = 10 mm the results almost the same as were obtained earlier (see Fig. 2), which shows good repeatability of experiments. For other indenters with different radius the E * values are varying in a wide range. From this we can conclude that rubber substrate with such geometric configuration cannot be considered as a half-space. It is worth to note that elastic modulus of the half space E * is a parameter of a material and do not depend on the radius of the indenter. To investigate in details the disagreement between the theory and experiment, the additional simulations within boundary elements method were performed. The full description of the simulation technique that also takes into account the thickness of the elastomer can be found in (Li et al. (2020)). The results of the performed simulations are shown in Fig. 3 b by dashed lines. Here, the results of the experiment from Fig. 3 a are also shown for comparison. Theoretical curves, calculated according to equation (1) within half-space approximation, are shown by dotted lines. Dashed lines relate to the simulations of the sample with thickness h = 15 mm, elastic modulus E = 50 kPa and Poisson ratio ν = 0.49. Dependencies related to the same diameter of the indenter (experiment, simulations and theoretical calculations within half-space approximation) are denoted by the same symbols. In simulations we used the elastic modulus of the rubber from its specification and reference value of Poisson ratio for the elastomers. The thickness of the rubber is the same as in experiment. In simulations elastomer was located on the absolutely rigid substrate, in experiments it was located on the glass plate. Nevertheless, the deformation of the glass substrate is significantly small comparing to the deformation of the elastomer. As it follows from the figure, the results of the simulations are in good enough agreement with theory only for the indenter with smallest diameter D = 4 mm. Therefore, we can conclude that half-space approximation is valid if the thickness of the rubber is at least 15/4 = 3.75 times larger than the diameter of the indenter. According to all estimated dependencies, the rigidity in simulations is always larger than in experiments, even though the thickness of the rubber is the same in both cases. The reason for this is the boundary limits for rubber sample in experiment, while in simulations within half-space approximation, rubber sample considered as an infinite in tangential directions. To estimate the sizes of the elastomer, needed for half-space approximation to be valid we performed additional computer simulations. The results of the simulation of indentation of the cylinder with diameter D = 15 mm into the