Mechanics of third-gradient continua reinforced with fibers resistant to flexure in finite plane elastostatics
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O R I G I NA L A RT I C L E
Chun Il Kim
· Suprabha Islam
Mechanics of third-gradient continua reinforced with fibers resistant to flexure in finite plane elastostatics
Received: 13 June 2019 / Accepted: 21 January 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract A third-gradient continuum model is developed for the deformation analysis of an elastic solid, reinforced with fibers resistant to flexure. This is framed in the second strain gradient elasticity theory within which the kinematics of fibers are formulated, and subsequently integrated into the models of deformations. By means of variational principles and iterated integrations by parts, the Euler equilibrium equation is obtained which, together with the constraints of bulk incompressibility, compose the system of the coupled nonlinear partial differential equations. In particular, a rigorous derivation of the admissible boundary conditions arising in the third gradient of virtual displacement is presented from which the expressions of the triple forces are derived. The resulting triple forces are, in turn, coupled with the Piola-type triple stress and are necessary to determine a unique deformation map. The proposed model predicts smooth and dilatational shear angle distributions, as opposed to those obtained from the first- and second-gradient theory where the resulting shear zones are either non-dilatational or non-smooth. Keywords Finite plane deformations · Fiber-reinforced materials · Flexure · Second strain gradient theory · Triple force 1 Introduction Problems involving the mechanics of an elastic solid, reinforced with embedded fibers, have received a considerable amount of attention mainly because of their fundamental importance in materials science and engineering in general. Fiber-reinforced elastic solids, also known as fiber composites, are a special class of materials where the microstructure (fibers) dominates the general responses of the composites [1–4]. Traditional approaches to examining these microstructured materials include the direct estimations of an individual fiber–matrix system (see, for example, [5,6]). Such local analyses are an effective means of characterizing the intrinsic properties of composite materials. However, they often rely heavily upon computationally expensive identification processes when predicting the mechanical responses of the materials subjected to certain types of boundary conditions (e.g., external loadings, edge conditions, etc.). Instead, a continuum description can be considered a promising alternative in a sense that, in most cases, fibers are densely distributed, so as to render the idealization of ‘continuous’ distribution. Within this prescription, the kinematics of fibers are mapped into the model of the continuum deformation. Since the continuum-based models offer the advantages of a compact mathematical frame work and the associated analyses, they have been adopted in a number of pertinent problems (see, for example, [7–9] and the references therein). However, as the
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