A Cosserat Model of Elastic Solids Reinforced by a Family of Curved and Twisted Fibers

Abstract

<jats:p>A Cosserat theory for fiber-reinforced elastic solids developed in Steigmann (2012) is generalized to accommodate initial curvature and twist of the fibers. The basic variables of the theory are a conventional deformation field and a rotation field that describes the local fiber orientation. Constraints on these fields are introduced to model the materiality of the fibers with respect to the underlying matrix deformation. A variational argument delivers the relevant equilibrium equations and boundary conditions and furnishes the interpretation of the Lagrange multipliers associated with the constraints as shear tractions acting on the fiber cross sections. Finally, the theory of material symmetry for such solids is developed and applied to the classification of some explicit constitutive functions.</jats:p>

A numerical investigation on impulse-induced nonlinear longitudinal waves in pantographic beams

“…It is for this reason that the obtained results also suggest that the homogenized continuum equations must be studied in order to determine with analytical or semi-analytical methods whether propagating wave profiles could exist and possibly exhibit a solitary-like pattern. Indeed, exploring the multitude of possible situations through numerical computations guided by a trial-and-error method is much more difficult than being guided by the understanding that one might get with a synthetic continuum model [64][65][66][67].…”

Buckling of chiral rods due to coupled axial and rotational growth

“…Similarly, in chiral rods where material symmetry arises from fibre reinforcement [36], rod growth is a result of individual fibre growth and it is the growth pattern of these fibres that dictates whether the rod’s cross-sections must rotate, as they are translated during growth. Consider a rod that is composed of fibres twisted helically in the unstressed reference state; if the cross-sections are not allowed to rotate during growth, it would have an unwinding or over-winding effect on the fibres, thus generating stresses.…”

Cosserat Elasticity of Lattice Solids

“…-appropriate for an initially straight fiber of circular cross section composed of an isotropic material [7,17] -in which T and F respectively are the constant torsional and flexural stiffnesses. As explained in [7,17] this furnishes…”

“…The kinematics of a fiber family are controlled by a fiber-specific rotation field and its gradient in addition to a conventional underlying continuum deformation. Accordingly, three-dimensional Cosserat elasticity, suitably extended, furnishes the appropriate conceptual framework [2,[5][6][7]. In contrast to conventional Cosserat elasticity [8][9][10][11][12][13][14], the fibers induce a directional Cosserat effect and convect with the continuum deformation as material curves.…”