Abstract
A comprehensive linear model for an elastic solid reinforced with fibers resistant to extension and flexure is presented. This includes the analysis of both unidirectional and bidirectional fiber-reinforced composites subjected to in-plane deformations. Within the prescription of the superposed incremental deformations, the fiber kinematics are approximated and used to determine the Euler equilibrium equations. The constraints of bulk incompressibility and admissible boundary conditions are also discussed for completeness. In particular, the complete systems of differential equations are obtained for the cases of Neo-Hookean and Mooney–Rivlin types of materials from which analytical solutions can be obtained.
Highlights
The second gradient theory of elasticity for the mechanics of meshed structures is presented in [23,24,25]
The authors in [26,27,28,29] developed continuumbased models in the analysis of fiber-reinforced composites, where the extension and bending resistance of fibers are incorporated via the computations of the first and second gradient of deformations. e majority of the aforementioned studies address nonlinear continuum theory of fiber composites so that little has been devoted to the development of compatible linear models describing the mechanics of an elastic medium reinforced with fibers
Mooney–Rivlin types of materials are elaborated where we show that the resulting Euler equilibrium equations are of the same form as those obtained from the NeoHookean model. e corresponding Piola stresses, on the other hand, are distinguished in that they demonstrate clear dependency on both the first and second invariant of the deformations gradient tensor. e obtained model can be adopted in the study of composite structures subjected to small in-plane deformations
Summary
A considerable amount of attention is committed to the development of continuum models and analyses in an effort to predict the mechanical responses of fiber-matrix systems subjected to external forces and/or induced deformations (see, for example, [6, 7] and references therein). For the estimation of resultant properties, one may consider multiscale modeling method which integrates the material properties of continua assigned in different length scales by means of the Cauchy–Born rule. Examples of such practice can be found in the works of Shahabodini et al [8, 9].
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