Soft elastic composites: Microstructure evolution, instabilities and relaxed response by domain formation

• Published in: European journal of mechanics, A, Solids Vol. 100; p. 105033
• Main Author: Ponte Castañeda, Pedro
• Format: Journal Article
• Language: English
• Published: Elsevier Masson SAS 01-07-2023
• Subjects: Composite materials; Homogenization; Relaxation; Variational methods; Homogenization; Relaxation; Composite materials; Variational methods
• ISSN: 0997-7538; 1873-7285
• DOI: 10.1016/j.euromechsol.2023.105033

This work provides a review of several complementary analytical methods for characterizing the macroscopic response of ‘soft’ elastic composites subjected to large deformations. For this purpose, we first recall the variational definitions of the homogenized response for hyperelastic composites with periodic and random microstructures. We distinguish between the principal solution for the ‘mesoscopic’ stored-energy function, which is based on one-cell-periodic solutions for the periodic microstructures and on the macroscopically uniform solution for the random microstructures, and the ‘relaxed’ or ‘macroscopic’ homogenized stored-energy function, which is obtained after the onset of ‘microscopic’ (periodic on multiple cells) or ‘macroscopic’ (long wavelength) instabilities. The various techniques include variational bounds of the Voigt, and generalized Reuss and Hashin–Shtrikman types, variational estimates based on the use of suitably optimized ‘linear comparison composites,’ and bounds based on the relaxation of the principal solution for the mesoscopic energy by means of a generalized Maxwell procedure consisting in the quasiconvexification of the energy and leading to domain formation. Although the methods are completely general, for simplicity, we review here illustrative results for neo-Hookean laminates and fiber-reinforced elastomers subjected to plane strain loading conditions. Among the main results, it was found that the predicted formation of twins or lamellar domains and associated soft modes of deformation for the laminates is consistent with experimental observations for thermoplastic elastomers with highly ordered lamellar microstructures. In addition, it was found that the predictions for the macroscopic instabilities by means of the relaxation of the mesoscopic energy are generally different (more conservative) and more consistent with the variational definitions of the macroscopic homogenized energy than those based on loss of ellipticity of the incremental elasticity homogenization problem.
•Review of several methods to estimate macroscopic response of soft elastic composites.•Bounds for the macroscopic stored-energy function.•Variational estimates using “linear comparison composites”.•Macroscopic instabilities and relaxed response by domain formation or twinning.•Applications to neo-Hookean laminates and particle reinforced elastomers.