Theory of elastic solids reinforced with fibers resistant to extension, flexure and twist

https://doi.org/10.1016/j.ijnonlinmec.2012.04.007Get rights and content

Abstract

A model of non-linearly elastic solids reinforced by continuously distributed embedded fibers is formulated in which elastic resistance of the fibers to extension, bending and twist is taken into account explicitly. This generalizes the conventional theory in which the solid is modeled as a transversely isotropic simple material.

Highlights

► Resistance to flexure and twist are modeled in elastic fiber-reinforced materials. ► Non-linear Cosserat elasticity theory is the natural setting for such materials. ► Constraints on the deformation of the embedded fibers and the continuum are used.

Introduction

The mechanics of fiber-reinforced solids is a well established subject with a long history [1], [2] that has significantly enriched and advanced continuum mechanics in general. It has been based almost entirely on the concept of a simple anisotropic material in which the response functions depend on the conventional deformation gradient, possibly augmented by constraints such as bulk incompressibility or fiber inextensibility. In the latter case the deformation is often so constrained as to be essentially kinematically determinate. The associated theory also exhibits a number of novel features such as the hyperbolicity of the equilibrium equations wherein the fiber trajectories emerge as characteristic curves of the associated differential equations [1]. The continuum theory presumes the fibers to be so densely distributed as to render meaningful the idealization of a continuous distribution, and purports to describe homogenized fiber–matrix composites.

Recently a significant advance in the continuum theory of fiber-reinforced solids was achieved by introducing the bending resistance of the fibers explicitly [3]. This is framed in the setting of the non-linear strain-gradient theory [4], [5], [6] of anisotropic elasticity in which elastic resistance is assigned to changes in curvature (flexure) of the fibers. The latter is calculated from the second gradient of the continuum deformation in which the fibers are regarded as convected curves. The model also accounts for additional effects associated with the gradients of the fiber stretches.

In the present work we develop a model in which the fibers offer elastic resistance to twist in addition to flexure and stretch. Thus the fibers are 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 [7], [8], [9]. Variation of the triad along the length of a fiber accounts for flexure and twist, while the position field generates the fiber stretch. We seek a model that accounts for these effects but which does not attribute elastic resistance to the gradient of fiber stretch. This restriction is in accord with established rod theories.

The present model is a special case of the Cosserat theory of non-linear elasticity [4], [10], [11], [12], which has received renewed attention in recent years. Current applications of the general theory are treated in [13], [14], [15], [16], for example, and mathematical aspects of the subject are addressed in [17]. This framework may be combined with strain-gradient theory, which is also in a period of active development [18], [19], to obtain a model in which bending and twisting effects are combined with fiber stretch gradients. We do not consider such a possibility here, however, as we are concerned with the simplest generalization of the conventional theory. For the same reason we suppress transverse shearing of the fiber cross sections and thus work in the setting of the constrained Cosserat theory in which the directors are described by a finite rotation-tensor field. This is decoupled, at the local level, from the deformation of the homogenized continuum.

The basic elements of Kirchhoff rod theory are summarized in Section 2. This is followed, in Section 3, by a survey of non-linear Cosserat elasticity [11], [12], [17]. There we also introduce a kinematic constraint on the rotation and deformation fields to ensure that fibers are convected as material curves. We focus on a special case of the general theory in which Cosserat effects are attributed exclusively to the fibers. The resulting model is similar in structure to the Kirchhoff theory, with the effects of fiber–matrix interaction manifesting themselves as forces and couples distributed along the lengths of the embedded fibers. The theory of material symmetry is developed in Section 4, and specialized to the case of transverse isotropy in which the fibers are normal to the planes of isotropy in a reference configuration. This discussion leads to a non-standard problem in representation theory which may be of independent interest. Rather than pursue the general solution to this problem, we simply record some example solutions in the form of scalars that automatically satisfy the relevant invariance requirement. An example is used, in Section 5, to solve the problem of finite torsion of an elastic cylinder in which the fibers are aligned with the generators of the cylinder prior to deformation.

We use standard notation such as At,A1,A,SymA, SkwA and tr A. These are, respectively, the transpose, the inverse, the cofactor, the symmetric part, the skew part and the trace of a tensor A, regarded as a linear transformation from a three-dimensional vector space to itself. We also use Sym and Skw to denote the linear subspaces of symmetric and skew tensors and Orth+ to identify the group of rotation tensors. The tensor product of three-vectors is indicated by interposing the symbol , and the Euclidean inner product of tensors A,B is denoted and defined by A·B=tr(ABt); the associated norm is |A|=A·A. The symbol |·| is also used to denote the usual Euclidean norm of three-vectors. Latin and Greek indices take values in {1,2,3} and {2,3}, respectively, and, when repeated, are summed over their ranges. Finally, the notation FA stands for the tensor-valued derivative of a scalar-valued function F(A).

Section snippets

Kirchhoff rod theory

In the present theory we regard the embedded fibers as continuously distributed spatial Kirchhoff rods [7], [8], [9], [20], [21]. Configurations of a spatial rod are described by a position field r(S), where S measures arclength along the rod in a reference placement, and a field {di(S)} of orthonormal vectors in which d1 is everywhere tangent to the space curve defined by r(S); dα; α=2,3 are vectors embedded in the rod cross section. Thus,r(S)=λd1whereλ=|r(S)|is the local stretch of the rod.

Kinematics

To model the kinematics of the embedded fibers, we assume the body, regarded as a homogenized continuum consisting of matrix material and fibers together, to be endowed with a rotation field R(X) in addition to the usual deformation χ(X). To exhibit the main ideas as simply and clearly as possible, we confine attention here to materials that are reinforced by a single family of fibers.

Drawing on the structure of rod theory with axial extension, we further assume the existence of a referential

General considerations

In this section we develop the theory of material symmetry for elastic Cosserat materials subject to the constraint (14). Our development borrows from that of Noll for conventional elasticity [27]. We first describe the manner in which the constitutive function for the strain energy may be computed for any choice of reference when that pertaining to any particular choice is given. We then derive a restriction on the constitutive function pertaining to any given choice of reference following

Example: torsion of a cylinder

We illustrate the theory by the simple example of finite torsion of a right circular cylinder. The reference placement ξ of the body is the region defined by 0ra, 0θ<2π, 0zL in a cylindrical polar coordinate system (r,θ,z). Position of a material point in this region is given byX=rer(θ)+zk,where er is the radial unit vector at azimuth θ, directed away from the cylinder axis, k is the fixed unit vector along the axis and eθ=k×er. We pursue a standard semi-inverse strategy and seek a

Acknowledgment

The support of the Powley Fund for Ballistics Research is gratefully acknowledged. This work was inspired by a comment made by Q.-S. Zheng following a presentation by A.J.M. Spencer on the work described in [3], in the course of a symposium dedicated to R.S. Rivlin which formed part of the 2006 meeting of the Society of Engineering Science at Pennsylvania State University.

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