An analysis of a fibrous assembly at large deformations has been explored as a means of predicting the compressional behavior, especially the compression hysteresis. The essential principle is to classify all the fiber contact points in the assembly as either slipping or nonslipping, so they can be dealt with separately. The compressional modulus and Poisson's ratio are derived and shown to be dominated by the mechanical properties and the directional distribution of the individual fibers within the assembly. An iterative algorithm, in which the system geometry is updated on successive incre ments, is developed to cope with large and nonlinear deformations. A comparison between the theoretical prediction and the experiment has indicated reasonable agree ment. An improvement could probably be achieved by including the contribution of fiber viscoelasticity to the total hysteresis.
The initial response of the unit fibrous cell to an externally applied shear stress is assumed to involve both the bending of fiber sections and slippage at contact points. The criterion used to determine whether a contacting fiber will bend the contacted fiber section or whether it will slip along it depends on the relative angles of the two fiber sections to the external stresses. The total proportion of slipping and nonslipping contact points is thus derived using the density function of fiber orientations within the unit fibrous cell. The derived modulus values for shear behavior are related to the moduli derived earlier for compression behavior. The symmetry rules for selected idealized orientation distributions ( e.g., random) of the fibers in the unit cell provide a check on the validity of the derivations.The use of continuum mechanics to model the behavior of assemblies of fibers remains restricted by inadequate knowledge of the constitutive properties of the unit fibrous cell. In the last three years since we first raised this issue [4], a number of papers have appeared that have improved our understanding of how the unit cell responds in tension [ 2 ] and compression [ 3,8 ] . This understanding includes a prediction [ 3,8 ] of the various Poisson's ratio terms so that algorithms are now available for estimating many of the tangent compliance terms in the general material properties matrix.While these recent analyses of tension and compression have been able to draw on a long established legacy of published research dating back at least to the theories of Grosberg [ 5 ] and van Wyk ( I 1 ] , respectively, the same does not apply to shear behavior. Indeed while it has long been recognized that shear between layers of fibers in yarns [ I ] or fabrics [ 5 ] does occur, the theoretical treatment of shear as a continuum strain has received only minimal attention. Where it has been modeled, the shear deformation has been treated as frictional slippage between layers of fibers. As such there has been no consideration of the low-strain response of the assembly under shear stress, no elastic strain energy due to shear has been calculated, and only the inelastic mechanisms associated with catastrophic shear failure ( i.e., massive slippage) have been dealt with.It is cert.ainly true that such simple textile deformations as twisting a yarn [9] or bending a fabric [ 5 ] involve almost immediate shear failure, because the ' essentially parallel fibers slide past each other with almost no restraint in accommodating large deformations of the yarn or fabric. The simplest way of treating this problem is to assume a high initial linear shear stiffness [ 5 J ( e.g., equal to the fiber modulus) up to a small critical shear stress. Thereafter, the assembly may be considered to shear via frictional slippage with no increase in shear stress. The work done against friction must be accounted for in the energy equations, but the tangent compliance in shear for initial shear stresses above the threshold level may in fact be rega...
The compressional mechanism of a random fiber assembly is analyzed by an energy method. An infinitesimal fiber segment, which is bounded by two neighboring contact points, is chosen as the unit bending element. The geometry of this element is characterized by its arc length, curvature, and orientation. The change in bending energy of each fiber segment due to the compression of the assembly is derived in terms of the compressional strain and the Poisson's ratio of the assembly. The summation of each energy contribution is done using a continuous joint probability density function of the length and orientation of the segments.
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