Abstract
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.
Highlights
In the present, work we generalize a theory for fiber-reinforced elastic solids proposed in [1,2]that accounts for the intrinsic flexural and torsional elasticities of the fibers, regarded as continuously distributed spatial rods of the Kirchhoff type in which the kinematics are based on a position field and an orthonormal triad field [3,4,5]
That accounts for the intrinsic flexural and torsional elasticities of the fibers, regarded as continuously distributed spatial rods of the Kirchhoff type in which the kinematics are based on a position field and an orthonormal triad field [3,4,5]. This model is a special case of the Cosserat theory of nonlinear elasticity [6,7,8,9,10,11,12]. We extend this theory to accommodate initially curved and twisted fibers and develop an associated framework for the characterization of material symmetry
The fibers confer anisotropy to the composite but their instrinsic flexural and torsional elasticities are not taken into account. The latter can be expected to play a significant role in local fiber buckling and kink-band failure due to the length scale inherent in the flexural and torsional stiffnesses of the fibers [15]
Summary
Work we generalize a theory for fiber-reinforced elastic solids proposed in [1,2]. That accounts for the intrinsic flexural and torsional elasticities of the fibers, regarded as continuously distributed spatial rods of the Kirchhoff type in which the kinematics are based on a position field and an orthonormal triad field [3,4,5] This model is a special case of the Cosserat theory of nonlinear elasticity [6,7,8,9,10,11,12]. We use standard notation such as At , A−1 , SkwA, det A and trA These are respectively the transpose, the inverse, the skew part, the deteminant and the trace of a tensor A, regarded as a linear transformation from a three-dimensional vector space to itself. Bold subscripts are used to denote derivatives of scalar functions with respect to their vector or tensor arguments
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