Stiffnesses shear

Suppose we want to analyze the stresses in the two stiffeners. The geometry of the sandwich-blade stiffener is actually more complicated and less amenable to analysis than is the hat-shaped stiffener. Issues that arise in the analysis to determine the influence of the various portions of the stiffeners include the in-plane shear stiffness. In the plane of the vertical blade is a certain amount of shear stiffness. That is, the shear stiffness is necfessary to transfer load from the 0° fibers at the top of the stiffener down to the panel. In hat-shaped stiffeners, that shear stiffness is the only way that load is transferred from the 0° fibers at the top of the stiffener down to the panel. Thus, shear stiffness is the dominant issue in the design. And that is why we typically put 45° fibers in the web of the hat-shaped stiffener. 

Another issue that turns out to be very important for the sandwich-blade stiffener, but not at all important for the hat-shaped stiffener, is shear in the vertical web. Not shear in the plane of the web, but shear in the plane perpendicular to the web. This transverse shear stiffness turns out to dominate the behavior or be very important in the behavior of the sandwich blade, but simply is not addressed at all in the hatshaped stiffener. You can imagine that the transverse shearing stiffness would be more important in the sandwich blade when you consider the observation that the sandwich blade is a thick element and the hatshaped stiffener is a thin element. That is, bending and in-plane shear would dominate this response, whereas transverse shear, because the sandwich blade is thick, can very easily be an important factor in the sandwich blade. For both stiffeners, appropriate analyses and design rationale have been developed to be able to make an optimally shaped stiffener. 

Two simple invariants, U, and U5, were shown in the previous subsubsection to be the basic indicators of average laminate stiffnesses. For isotropic materials, these invariants reduce to U. =Qi. and U5 = Qqq, the extensional stiffness and shear stiffness. Accordingly, Tsai and Pagano suggested the orthotopic invariants U., and U5 be called the isotropic stiffness and isotropic shear rigidity, respectively [7-16 and 7-17]. They observed that these isotropic properties are a realistic measure of the minimum stiffness capability of composite laminates. These isotropic properties can be compared directly to properties of isotropic materials as well as to properties of other orthotropic laminates. Obviously, the comparison criterion is more complex than for isotropic materials because now we have two measures, and U5, instead of the usual isotropic stiffness or E. Comparison of values of U., alone is not fair because of the degrading influence of the usually low values of U5 for composite materials. 

The next problem area is transverse shearing effects. There are some distinct characteristics of composite materials that bear very strongly on this situation because for a composite material the transverse shearing stiffness, i.e., perpendicular to the plane of the fibers, is considerably less than the shear stiffness in the plane of the fibers. There is a shear stiffness for a composite material in a plane that involves one fiber direction. Shear involves two directions always, and one of the directions in the plane is a fiber direction. That shear stiffness is quite a bit bigger than the shear stiffness in a plane which is perpendicular to the axis of the fibers. The shear stiffness in a plane which is perpendicular to the axis of the fibers is matrix-dominated and hardly fiber-influenced. Therefore, that shear stiffness is much closer to that of the matrix material itself (a low value compared to the in-plane shear stiffness). 

Under transverse loading, bending moment deflection is proportional to the load and the cube of the span and inversely proportional to the stiffness factor, El. Shear deflection is proportional to the load and span and inversely proportional to shear stiffness factor N, whose value for symmetrical sandwiches is ... 

The load on the wheel produces a contact area of finite length a. The distortion caused by the load is ignored in the brush model. This means that the above relation is really a shear relation. The fibers have a large compression stiffness and a small shear stiffness, which in fact is true for rubbers. The large contact length is created by the air inflation chamber of the tire. Solid mbber wheels bulge out. 

G = (3q/4ji)2(l/R)4. Note that Equation 3.16 shows that the shear stiffness is inversely proportional to the polarizability. This is confirmed in Figure 3.8 and is an important aspect for understanding hardness. 

The glide planes on which dislocations move in the diamond and zincblende crystals are the octahedral (111) planes. The covalent bonds lie perpendicular to these planes. Therefore, the elastic shear stiffnesses of the covalent bonds... 

A plot of them (Figure 5.6) shows that they are proportional to the bond moduli. Thus the bond moduli are fundamental physical parameters which measure shear stiffness, and vice versa. Also, it may be concluded that hardness (and dislocation mobility) depends on the octahedral shear stiffnesses of this class of crystals (see also Gilman, 1973). 

Figure 5.6 Correlation of octahedral shear stiffnesses with bond moduli for Group IV crystals. The octahedral stiffnesses measure the elastic shear resistances of the covalent bonds across the (111) planes.

For covalent crystals temperature has little effect on hardness (except for the relatively small effect of decreasing the elastic shear stiffness) until the Debye temperature is reached (Gilman, 1995). Then the hardness begins to decrease exponentially (Figure 5.14). Since the Debye temperature is related to the shear stiffness (Ledbetter, 1982) this softening temperature is proportional to C44 (Feltham and Banerjee, 1992). 

The Ni octahedra derive their stability from the interactions of s, p, and d electron orbitals to form octahedral sp3d2 hybrids. When these are sheared by dislocation motion this strong bonding is destroyed, and the octahedral symmetry is lost. Therefore, the overall (0°K) energy barrier to dislocation motion is about COCi/47r where = octahedral shear stiffness = [3C44 (Cu - Ci2)]/ [4C44 + (Cu - C12)] = 50.8 GPa (Prikhodko et al., 1998), and the barrier = 4.04 GPa. The octahedral shear stiffness is small compared with the primary stiffnesses C44 = 118 GPa, and (Cn - C12)/2 = 79 GPa. Thus elastic as well as plastic shear is easier on this plane than on either the (100), or the (110) planes. 

By means of hardness studies, Chin et al. (1978) investigated the type of bonding in three of these compounds. They determined the Chin-Gilman ratios (the hardness number divided by the shear stiffness).Table 8.2 lists their results. The small ratios and the moderately large hardness values indicate that... 

Chin, et al. (1972) measured the hardnesses of Na and K halides (Cl, Br, and I) containing various additions of Ca++, Sr++, or Ba++. Then they extrapolated the measurements back to zero additions to get values for the pure crystals. They found that the latter depended linearly on the Young s moduli of their crystals. Gilman (1973) found an equally good correlation with the shear stiffnesses, where FI = 1.37 x 10 2 C44 (d/cm2) in excellent agreement with Equation 9.1. A comparison of the data and the theory is given in Figure 9.5. 

The hardness of A1203 is VHN = 2700kg/mm2 and its rms. shear stiffness is 366 GPa so its Chin-Gilman parameter is 0.074. This suggests that its chemical bonding is a combination of covalent and ionic bonding. 

The Group IV elements also show a linear correlation of their octahedral shear moduli, C44(lll) with chemical hardness density (Eg/2Vm).This modulus is for for shear strains on the (111) planes. It is a measure of the shear stiffnesses of the covalent bonds. The (111) planes lie normal to the bonds that connect the atoms in the diamond (or zinc blende) structure. In terms of the three standard moduli for cubic symmetry (Cn, Q2, and C44), the octahedral shear modulus is given by C44(lll) = 3CV1 + [4C44/(Cn - Ci2)]. Since the (111) planes have three-fold symmetry, they have only one shear modulus. The bonds across the octahedral planes have high resistance to shear which probably results from electron correlation in the bonds (Gilman, 2002). 

Another property that is related to chemical hardness is polarizability (Pearson, 1997). Polarizability, a, has the dimensions of volume polarizability (Brinck, Murray, and Politzer, 1993). It requires that an electron be excited from the valence to the conduction band (i.e., across the band gap) in order to change the symmetry of the wave function(s) from spherical to uniaxial. An approximate expression for the polarizability is a = p (N/A2) where p is a constant, N is the number of participating electrons, and A is the excitation gap (Atkins, 1983). The constant, p = (qh)/(2n 2m) with q = electron charge, m = electron mass, and h = Planck s constant. Then, if N = 1, (1/a) is proportional to A2, and elastic shear stiffness is proportional to (1/a). 

Among the metals of the Periodic Table, osmium has the highest bulk modulus (412 GPa), and shear stiffness constants of C44 = 270GPa and C66 = 268 GPa. (Pantea et al., 2008).The corresponding values for diamond are B = 433 GPa and C44 (111) = 507 GPa. Although the bulk modulus of Os is about 95% that of diamond, the indentation hardness is only about 3% of diamond s hardness. In other words dislocations move readily in Os but not in diamond. 

Another analysis method was based on the local wave vector estimation (LFE) approach applied on a field of coupled harmonic oscillators.39 Propagating media were assumed to be homogeneous and incompressible. MRE images of an agar gel with two different stiffnesses excited at 200 Hz were successfully simulated and compared very well to the experimental data. Shear stiffnesses of 19.5 and 1.2 kPa were found for the two parts of the gel. LFE-derived wave patterns in two dimensions were also calculated on a simulated brain phantom bearing a tumour-like zone and virtually excited at 100-400 Hz. Shear-stiffnesses ranging from 5.8 to 16 kPa were assumed. The tumour was better detected from the reconstructed elasticity images for an input excitation frequency of 0.4 kHz. 

In-Plane Shear Properties. The basic lamina in-plane shear stiffness and strength is characterized using a unidirectional hoop-wound (90°) 0.1 -m nominal internal diameter tube that is loaded in torsion. The test method has been standardized under the ASTM D5448 test method for in-plane shear properties of unidirectional fiber-resin composite cylinders. D5448 provides the specimen and hardware geometry necessary to conduct the test. The lamina in-plane shear curve is typically very nonlinear [51]. The test yields the lamina s in-plane shear strength, t12, in-plane shear strain at failure, y12, and in-plane chord shear modulus, G12. 

For Newtonian liquids, X = R and therefore shear stiffness G, by Equation 1, is zero. Viscosity is then given by the function 2R2/[Pg.164]

This unusual effect of increased high frequency shear stiffness on initial contact with water was evident even when the water was applied as vapor. With thin films of PVA, the A db increases rapidly with even a single moist breath blown on the sample and then returns to its initial value slowly as the sorbed water diffuses out of the film. By blowing moist air continuously over a dried PVA film, the A db can be made to pass through a maximum just as it does when liquid water is applied. These responses are shown schematically in Figure 2a. 

Because methanol is not actually a solvent for PVA, some interesting sorption/desorption cycling experiments can be conveniently run on PVA films without disrupting the film integrity. As noted above, methanol produces a slow increase of shear stiffness (Adh) when layered on PVA. The time scale and curve shape are suggestive of a diffusion process (liquid into polymer). By removing the liquid layer, we should permit methanol to evaporate from the film and thus reverse the effect. The... 

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