Ideal fibers

The resistance to plastic flow can be schematically illustrated by dashpots with characteristic viscosities. The resistance to deformations within the elastic regions can be characterized by elastic springs and spring force constants. In real fibers, in contrast to ideal fibers, the mechanical behavior is best characterized by simultaneous elastic and plastic deformations. Materials that undergo simultaneous elastic and plastic effects are said to be viscoelastic. Several models describing viscoelasticity in terms of springs and dashpots in various series and parallel combinations have been proposed. The concepts of elasticity, plasticity, and viscoelasticity have been the subjects of several excellent reviews (21,22). 

The Ideal Fiber-Reactive Dye Profile. Eigure 3 shows the general profile for the apphcation of a reactive dye. In addition to showing the rate profile of fixation between dye and fiber, three other practical parameters (A—C) are noted. 

Fig. 6. Idealized fiber diffiactogram firom X-ray or neutron scattering. An assembly of partially ordered blocks of microcrystallites diffracts to produce diffraction spots that are recorded on a flat film or an area detector. The periodicity of the macromolecular constituents within the microcrystallites is indicated by a series of diffracting lines having a regular spacing. The equator corresponds to the layer line 0, intersecting the nondiffracted central beam. The meridian is perpendicular to the equator and it lies parallel to the fiber axis. The spacing along the meridian provides information about the periodicity of the macromolecule and its hehcal symmetry. The so-called helical parameters, n and h, are directly related to the syrmnetry of the macromolecular chain n is the number if residues per turn, and h the projection of the residue on the helical axis.

We will examine what happens when a load is applied to an ideal fiber composite in which the matrix material is reinforced by fibers which are uniform, continuous, and arranged uniaxially, as shown in Figure 3.47a. 

Equation 3.127 and Equation 3.129 apply to ideal fiber composites having uni-axial arrangement of fibers. In practice, however, not all the fibers are aligned in the direction of the load. This practice reduces the efiiciency of the reinforcement, so Equation 3.127 and Equation 3.129 are modified to the forms... 

An analysis of the theoretical methods of calculation of the ideal fiber birefringence has been presented in the literature [308], Using a modified Lorentz-Lorenz equation, theoretical birefringence was calculated by considering intermolecular interactions. The calculations showed considerable discrepancies between the theoretical and the experimental values. 

FIGURE 8.1 Idealized fiber diffraction pattern of a semicrystalline material with reflexions on layer lines. High intensity is represented by dark color. The fiber axis is vertical. 

Note It has been assumed that the ideal fiber contains no free dislocations or porosity. If it is fully graphitized, it may be expected to contain dislocations which increase Sn, S12, and S44 by a factor 3, with corresponding reductions in Cn, C,2 and C44. 3, S13, C33 and C13 would not be affected in this case. 

O Bradaigh and Pipes originally studied the plane stress flows of ideal fiber reinforced fluids (IFRF) and have used a penalty method to impose the fiber inextensibility constraint with biquadratic velocity/bilinear discontinuous tension elements. A linear and quasistatic scheme has been used to calculate instantaneous velocities which are multiplied by the time step to displace the mesh. Such an approach involves the buildup of considerable error unless time steps are extraordinarily small. Recently, this model has been improved by developing a mesh updating scheme that incorporates finite incompressibility and inextensibility constraints [8]. The new model also uses large displacement contact/friction elements to model tool contact and interply slip between layers of IFRF, in plane strain. 

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