Pure bending of helically wound ideal fibre-reinforced cylinders

Abstract

An ideal fibre-reinforced material is defined to be inextensible in one or more directions through each point of the material. It has been found to be a good model for predicting the deformation and stresses produced in real materials which are very much stronger in resisting extension in those directions than in any other possible mode of deformation.We consider a circular cylindrical tube of incompressible ideal fibre-reinforced material in which the reinforcement throughout its thickness is directed along two families of helices making angles +05 with the axis of the cylinder. It is shown that the kinematic constraints allow pure bending deformations, and the solution for a linearly elastic reinforced cylinder is explicitly determined. This kinematically admissible deformation is also statically admissible, and can be supported (or produced) solely by the action of bending couples on the ends of the cylinder. The bending moment-curvature relation is deduced, and compared with the EulerBernoulli equation for prismatic beams. The effect of different values of the winding angle 05 is discussed, and we show that the case tan 05 = ~/2 is of particular importance.

Investigation of the optimal collagen fibre orientation in human iliac arteries

“…This value is surprisingly similar to the adventitial fibre angle measured in experiments ( Holzapfel et al, 2000 ; Schriefl et al, 2012 ). However, this should be viewed with caution since the presence of fibres is not exactly equivalent to the inextensible case of Spencer et al (1975) , and hence the agreement on the value of β A could be a coincidence.…”

“…At this point, it is worth noting that Spencer et al (1975) considered a circular cylindrical tube of incompressible ideal fibre-reinforced material in which the reinforcement throughout its thickness is directed along two families of helices making angles of ± β A . Of particular importance, they argued that , so that β A = 54.7° in order to avoid the narrow bands of stress concentration near the surfaces at the inner and outer radii.…”

Magic angles for fibrous incompressible elastic materials

“…The magic angle is the optimal winding angle for the design of filament-wound structures and, as mentioned earlier, is often derived in the composite structures literature by netting analysis (see, for example, [18]). An application of the idealized linear theory to bending of helically wound tubes is carried out in Spencer et al [12], where again it is shown that reinforcement at the magic angle (1.3) leads to considerable simplification. A review of such results and extensions to large elastic and plastic deformations is given by Rogers [19].…”

“…The first papers in which the magic angle occurs in the linear idealized theory appear to be those of Spencer et al [11,12]. In these papers, the problem of a circular cylindrical tube reinforced by a double family of helically wound fibres is considered.…”

“…In §2, we briefly describe how the magic angle concept arises in a special linearized theory of elasticity. This was first demonstrated in papers by Spencer et al [11,12] using an idealized theory pioneered by Spencer [13,14]. This theory assumes that the fibres are perfectly inextensible and the bulk material is incompressible.…”