Finite deformations of fibre-reinforced elastic solids with fibre bending stiffness

Abstract

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.

On small azimuthal shear deformation of fibre-reinforced cylindrical tubes

“…When the perfectly flexible fibres considered in the previous sections are replaced with fibres possessing bending stiffness, the theory is required to account further for possible action of couple-stress and therefore asymmetric stress [Spencer and Soldatos 2007]. The linearized version of the relevant hyper elasticity theory presented in that paper is considered for a study of the effects that fibre bending stiffness has on the azimuthal shear strain problem discussed in the preceding section.…”

Application of Polar Elasticity to the Problem of Pure Bending of a Thick Plate

“…The properties of both homogeneous models are set as effective (averaged) properties of the PS model. In addition, the bending stiffness parameter in the EP model is set The motivation for such comparison sprung from chapter 4 in [1] where the authors consider Euler-Bernoulli beam with the cosinusoidal Young's modulus distribution to point out the inaccuracy of the conventional theory of fibre-reinforced materials [3] in cases where the fibres are not infinitesimally thin.…”

“…The recently developed polar elasticity theory [1,2] deals with fibre-reinforced materials containing fibres that possess bending stiffness. The contribution of the fibre bending stiffness is accounted for by employing couple stresses and introducing additional elastic parameters to the constitutive equations.…”

Euler-Bernoulli Beam Theory in the Presence of Fiber Bending Stiffness

“…Quantities with superscript "k" are due to the ℎ layer of a cross-ply laminate. In the presence of the fiber bending stiffness, the elasticity theory requires to include the couple-stress and consequently asymmetric stress [1]. The anti-symmetric part of the associated shear stress tensor component is denoted by symbol with indices in square brackets.…”

“…A non-symmetric stress and the couple-stress are requiredinasymmetric elasticity theory presented in [1]. Relatively, a version of asymmetric linear elasticity theory has been produced, as a particular case, by Spencer and Soldatos [1].A generalized, 6-degree-of-freedom, 2D model for plates with fibres resistant to bending has been developed by Soldatos in [2] where one extra elastic modulus, which has dimensions of force, is involved in each layer of the laminate. Consequently, some intrinsic material area or length parameter involved in the analysis that may be associated, for instance, with fiber thickness of fiber spacing [3].…”