Shear deformation parameter

Based on the same notations of Fig. 10.11, the general solution for the adhesive shear stress governing equation (i.e. equation [10.13] below) is identieal to that of Hart-Smith s general solution. However, the exponent of the elastie shear stress distribution (A) in Hart-Smith s (1973) theory is ealled the elongation parameter in Tsai et a/. s (1998) theory, although its mathematical formula is identical (refer to equation [10.7]). It appears in the general equation for Hart-Smith s solution instead of the current solution s P parameter of equation [10.15], whieh is the eore difference between both theories. The parameter P is redefined by two parameters X and o) (refer to equation [10.15]). The latter is the shear deformation parameter which accounts for the shear deformations of the adherend in its mathematical formula (i.e. equation [10.14]). 

Material parameters defined by Equations (1.11) and (1.12) arise from anisotropy (i.e. direction dependency) of the microstructure of long-chain polymers subjected to liigh shear deformations. Generalized Newtonian constitutive equations cannot predict any normal stress acting along the direction perpendicular to the shearing surface in a viscometric flow. Thus the primary and secondary normal stress coefficients are only used in conjunction with viscoelastic constitutive models. 

Once values for R , Rp, and AEg are calculated at a given strain, the np product is extracted and individual values for n and p are determined from Eq. (4.19). The conductivity can then be calculated from eq. (4.18) after the mobilities are calculated. The hole mobility is the principal uncertainty since it has only been measured at small strains. In order to fit data obtained from elastic shock-loading experiments, a hole-mobility cutoff ratio is used as a parameter along with an unknown shear deformation potential. A best fit is then determined from the data for the cutoff ratio and the deformation potential. 

As already noted, the main merit of fibers used as a filler for conducting composite materials is that only low threshold concentrations are necessary to reach the desired level of composite conductivity. However, introduction of fiber fillers into a polymer with the help of ordinary plastic materials processing equipment presents certain difficulties which are bound up mainly with significant shearing deformations entailing fiber destruction and, thereby, a decrease of parameter 1/d which determines the value of the percolation threshold. 

The strength of a fibre is not only a function of the test length, but also of the testing time and the temperature. It is shown that the introduction of a fracture criterion, which states that the total shear deformation in a creep experiment is bounded to a maximum value, explains the well-known Coleman relation as well as the relation between creep fracture stress and creep fracture strain. Moreover, it explains why highly oriented fibres have a longer lifetime than less oriented fibres of the same polymer, assuming that all other parameters stay the same. 

Other important parameters for the correlation between GJJ and GJ include the ductility or the failure strain, particularly the non-linear strain (Jordan and Bradley, 1988 Jordan et al., 1989) of the matrix resin, the bond strength of the fiber-matrix interface (Jordan and Bradley, 1987 Bradley 1989a, b), and the fiber V and their distributions in the composites (Hunston et al., 1987). A high failure strain promotes the intrinsic capacity of the resin to permit shear deformation, and is shown to increase the G and G. values almost linearly, the rate of increase being steeper for G j than for Gf. ... 

In terms of tonnage, polyolefins are by far the most important polymeric materials for structural applications, and there is consequently enormous interest in optimising their fracture properties. A rational approach to this requires detailed understanding of the relationships between macroscopic fracture and molecular parameters such as the molar mass, M, and external variables such as temperature, T, and test speed, v. Considerable effort is therefore also devoted to characterising the irreversible processes (crazing and shear deformation) that accompany crack initiation and propagation in these polymers, some examples of which will given. 

Figure 7.21 Temperature profiles from T2 parameter images of SBR cylinders with different carbon-black filler contents undergoing oscillatory shear deformation (a) Temperature calibration curves, (b) Temperature profiles across the cylinders...

In a subsequent investigation, with Roos and Kampschreur (1989), Northolt extended the modified series model to include viscoelasticity. For that an additional assumption was made, viz. that the relaxation process is confined solely to shear deformation of adjacent chains. The modified series model maybe applied to well-oriented fibres having a small plastic deformation (or set). In particular it explains the part of the tensile curve beyond the yield stress in which the orientation process of the fibrils takes place. The main factor governing this process is the modulus for shear, gd, between adjacent chains. At high deformation frequencies yd attains its maximum value, ydo at lower frequencies or longer times the viscoelasticity lowers the value of gd, and it becomes a function of time or frequency. Northolt s relations, that directly follow from his theoretical model for well-oriented fibres, are in perfect agreement with the experimental data if acceptable values for the elastic parameters are substituted. 

Elastic effects in polymer melts, 578 Elastic moduli of some materials, 732 Elastic parameters, 383,386,391 Elastic shear deformation, 500, 531 Elastic shear quantities, 556 Electret, 329,331... 

When the stress is decomposed into two components the ratio of the in-phase stress to the strain amplitude (j/a, maximum strain) is called the storage modulus. This quantity is labeled G (co) in a shear deformation experiment. The ratio of the out-of-phase stress to the strain amplitude is the loss modulus G"(co). Alternatively, if the strain vector is resolved into its components, the ratio of the in-phase strain to the stress amplitude t is the storage compliance J (m), and the ratio of ihe out-of-phase strain to the stress amplitude is the loss compliance J"(wi). G (co) and J ((x>) are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. Tlie loss parameters G" w) and y"(to) on the other hand reflect the nonrecoverable use of applied mechanical energy to cause flow in the specimen. At a specified frequency and temperature, the dynamic response of a polymer can be summarized by any one of the following pairs of parameters G (x>) and G" (x>), J (vd) and or Ta/yb (the absolute modulus G ) and... 

Fig. 4.1. Energy as a function of shear deformation in A1 (adapted from Mehl and Boyer (1991)). The lattice parameter a is fixed during the deformation and hence the energy characterizes a one-parameter family of deformations of the fee lattice, with the members of the family parameterized by b.

During flow the initially spherical drop deforms into a prolate ellipsoid with the long axis, a, and two orthogonal short axes, a The drop deformability parameter, D, is a complex function. At low shear stress it can be expressed as [Taylor, 1934] ... 

It is evident from the above description that G (oj) and are associated with the periodic storage and complete release of energy in the sinusoidal deformation process. The loss parameters G" oj) and on the other hand, reflect the nonrecoverable use of applied mechanical energy to cause viscous flow in the material. At a specified frequency and temperature, the dynamic response of a polymer in shear deformation can be summarized by any one of the following pairs of parameters G ( w) and G"( w), J ioj) and or absolute modulus G and tan 6. 

The molecular organization in the banded texture is understood in some detail as a result of optical microscopy [49, 50, 83, 110-112, 115-122, 126], transmission electron microscopy [49, 76, 89, 90, 115, 127] and scanning electron microscopy [50]. It has been shown that the orientation variation of the molecules in this texture is near sinusoidal about the shear direction with most of the misorientation within the shear plane. There is a high degree of register of the molecules between each of the layers [50]. The influence of a number of parameters on the formation of bands has been studied the molecular weight [96], the sample thickness [110,120-122], the shear rate [109], the time during which the shear stress was applied [ 122] and the total shear deformation [121]. 

If we visualize a very small volume of material between two parallel plates, as in Figure 13.1, we can define the parameters of shear deformation and flow. The plates are a very small distance dy, apart and the space betweai th is completely filled with the matoial being investigated. If a fcvce F is applied to... 

Assuming that the relaxation process with the constant distribution of activation energies is confined to the shear deformation, a relaxation function for the orientation parameter u(t) = is introduced having the form... 

In Equation (3) the first and third terms in the square brackets, when multiplied by the factor outside these brackets, represent respectively the flexural and shear deflections of a simply supported beam carrying a point load at its centre. The second term accounts for the fact that the beam is not simply supported but forms part of a frame with identical semi-rigid beam-column Joints. Also in Equation (3), the a (= EI/GA) and p (= EI/K, where K is the joint stiffness) parameters reflect the contributions of shear deformation and semi-rigidity respectively to the mid-span deflection. Similar expressions have been derived for the beam translation under sway mode loading and the corresponding rotations at the beam-column joints and pinned bases. Some of these expressions have been used to predict the frame deformations described in the next section. 

Where a is the longitudinal stress, e is corresponding strain, and E is called Young s modulus (or the modulus of elasticity). Similarly, in shear deformation, the modulus is called the shear modulus or the modulus of rigidity (G). When a hydrostatic force is applied, a third elastic modulus is used the modulus of compressibility or bulk modulus (K). It is defined as the ratio of hydrostatic pressure to volume strain. A deformation (elongation or compression) caused by an axial force is always associated with an opposite deformation (contraction or expansion) in the lateral direction. The ratio of the lateral strain to the longitudinal strain is the fourth elastic constant called Poisson s ratio (v). For a small deformation, elastic parameters can be correlated in the following way ... 

---