A third gradient-based continuum model for the mechanics of continua reinforced with extensible bidirectional fibers resistant to flexure

초록

The mechanics of fiber-reinforced composite materials is presented within the framework of the second strain gradient theory. As such, a continuum-based model is developed for the analysis of elastic materials reinforced with bidirectional fibers and subjected to finite plane deformations. The obtained model is subsequently applied to the cases of the unidirectional fiber composites for the purpose of model implementation. The Euler equilibrium equations and the associated boundary conditions are obtained via the variational principle and iterative integration by parts. In particular, we formulate the complete expressions of Piola-type triple stress and its coupled triple force arising in the third gradient of continuum deformations, which, in turn, yield the unique deformation maps in the presence of admissible boundary conditions of higher order. The solutions of the resulting systems of differential equations are obtained via the custom-built numerical scheme from which smooth and dilatational shear angle distributions are predicted throughout the entire domain of interest. It is also observed that the third gradient constitutive parameter is associated with the volume dilatation of third-gradient continua, which may be appeared in the form of shear band inclination angle. © 2023, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

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