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To cite this version:A.J. M. Spencer, K.P. Soldatos. Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness. International Journal of Non-Linear Mechanics, Elsevier, 2007, 42 (2) This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. In memory of Ronald RivlinAbstract.In the conventional theory of finite deformations of fibre-reinforced elastic solids it is assumed that the strain-energy is an isotropic invariant function of the deformation and a unit vector A that defines the fibre direction and is convected with the material. This leads to a constitutive equation that involves no natural length . To incorporate fibre bending stiffness into a continuum theory, we make the more general assumption that the strain-energy depends on deformation, fibre direction, and the gradients of the fibre direction in the deformed configuration. The resulting extended theory requires, in general, a non-symmetric stress and the couple stress. The constitutive equations for stress and couple-stress are formulated in a general way, and specialized to the case in which dependence on the fibre direction gradients is restricted to dependence on their directional derivatives in the fibre direction. This is further specialized to the case of plane strain, and finite pure bending of a thick plate is solved as an example. We also formulate and develop the linearized theory in which the stress and couple-stress are linear functions of the first and second spacial derivatives of the displacement. In this case for the symmetric part of the stress we recover the standard equations of transversely isotropic linear elasticity, with five elastic moduli, and find that, in the most general case, a further seven moduli are required to characterize the couple-stress.

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The accurate discrimination of the aerodynamic parameters affecting the flight of sports balls is essential in the product development process. Aerodynamic studies reported to date have been limited, primarily because of the inherent difficulty of making accurate measurements on a moving or spinning ball. Manufacturers therefore generally rely on field trials to determine ball performance, but the approach is time-consuming and subject to considerable variability. The current paper describes the development of a method for mounting stationary and spinning footballs in a wind tunnel to enable accurate force data to be obtained. The technique is applied to a number of footballs with differing constructions and the results reported. Significant differences in performance are noted for both stationary and spinning balls and the importance of the ball orientation to the flow is highlighted. To put the aerodynamic data into context the results are applied in a flight model to predict the potential differences in the behaviour of each ball in the air. The aerodynamic differences are shown to have a considerable effect on the flight path and the effect of orientation is shown to be particularly significant when a ball is rotating slowly. Though the techniques reported here are applied to a football they are equally applicable to other ball types.