Shear mode

Until now we have restricted ourselves to consideration of simple tensile deformation of the elastomer sample. This deformation is easy to visualize and leads to a manageable mathematical description. This is by no means the only deformation of interest, however. We shall consider only one additional mode of deformation, namely, shear deformation. Figure 3.6 represents an elastomer sample subject to shearing forces. Deformation in the shear mode is the basis... 

Returning to the Maxwell element, suppose we rapidly deform the system to some state of strain and secure it in such a way that it retains the initial deformation. Because the material possesses the capability to flow, some internal relaxation will occur such that less force will be required with the passage of time to sustain the deformation. Our goal with the Maxwell model is to calculate how the stress varies with time, or, expressing the stress relative to the constant strain, to describe the time-dependent modulus. Such an experiment can readily be performed on a polymer sample, the results yielding a time-dependent stress relaxation modulus. In principle, the experiment could be conducted in either a tensile or shear mode measuring E(t) or G(t), respectively. We shall discuss the Maxwell model in terms of shear. 

We shall follow the same approach as the last section, starting with an examination of the predicted behavior of a Voigt model in a creep experiment. We should not be surprised to discover that the model oversimplifies the behavior of actual polymeric materials. We shall continue to use a shear experiment as the basis for discussion, although a creep experiment could be carried out in either a tension or shear mode. Again we begin by assuming that the Hookean spring in the model is characterized by a modulus G, and the Newtonian dash-pot by a viscosity 77. ... 

A crack in a body may grow as a result of loads appHed in any of the three coordinate directions, lea ding to different possible modes of failure. The most common is an in-plane opening mode (Mode I). The other two are shear loading in the crack plane (Mode II) and antiplane shear (Mode III), as defined in Figure I. Only Mode I loading is considered herein. 

Fig. 9. Schematic view of cylindrical shear modes for a nested tubule telescope mode (u,-) and rotary mode (wr).

Figure 3-55 Extensional Mode and Shear Mode of Fiber Buckling...

Rgure 3-59 Rber Deformations During Shear Mode Buckling... 

Dow and Rosen s results are plotted in another form, composite material strain at buckling versus fiber-volume fraction, in Figure 3-62. These results are Equation (3.137) for two values of the ratio of fiber Young s moduius to matrix shear modulus (Ef/Gm) at a matrix Poisson s ratio of. 25. As in the previous form of Dow and Rosen s results, the shear mode governs the composite material behavior for a wide range of fiber-volume fractions. Moreover, note that a factor of 2 change in the ratio Ef/G causes a factor of 2 change in the maximum composite material compressive strain. Thus, the importance of the matrix shear modulus reduction due to inelastic deformation is quite evident. 

The symmetric stress-intensity factor k, is associated ith the opening mode of crack extension in Figure 6-10. The skew/-symmetric stress-intensity factor l<2 is associated ith the fonward-shear mode. These plane-stress-intensity factors must be supplemented by another stress-intensity factor to describe the parallel-shear mode. The stress-intensity factors depend on the applied loads, body geometry, and crack geometry. For plane loads, the stress distribution around the crack tip can always be separated into symmetric and skew-symmetric distributions. 

Other researchers have substantially advanced the state of the art of fracture mechanics applied to composite materials. Tetelman [6-15] and Corten [6-16] discuss fracture mechanics from the point of view of micromechanics. Sih and Chen [6-17] treat the mixed-mode fracture problem for noncollinear crack propagation. Waddoups, Eisenmann, and Kaminski [6-18] and Konish, Swedlow, and Cruse [6-19] extend the concepts of fracture mechanics to laminates. Impact resistance of unidirectional composites is discussed by Chamis, Hanson, and Serafini [6-20]. They use strain energy and fracture strength concepts along with micromechanics to assess impact resistance in longitudinal, transverse, and shear modes. 

Bonded-bolted joints generally have better performance than either bonded or bolted joints. The bonding results in reduction of the usual tendency of a bolted joint to shear out. The bolting decreases the likelihood of a bonded joint debonding in an interfacial shear mode. The usual mode of failure for a bonded-bolted joint is either a tension failure through a section including a fastener or an interlaminar shear failure in the composite material or a combination of both. 

An analysis of loading mode effects has also provided evidence of the critical role of hydrogen. A stress-intensity factor (K) can be achieved in either a tensile loading mode (mode I) or a shearing mode (mode III) (Section 8.9). Under mode I conditions the volume of metal immediately in... 

The shear mode involves the application of a load to a material specimen in such a way that cubic volume elements of the material comprising the specimen become distorted, their volume remaining constant, but with opposite faces sliding sideways with respect to each other. Shear deformation occurs in structural elements subjected to torsional loads and in short beams subjected to transverse loads. 

AT-cut, 9 MHz quartz-crystal oscillators were purchased from Kyushu Dentsu, Co., Tokyo, in which Ag electrodes (0.238 cm2) had been deposited on each side of a quartz-plate (0.640 cm2). A homemade oscillator circuit was designed to drive the quartz at its resonant frequency both in air and water phases. The quartz crystal plates were usually treated with 1,1,1,3,3,3-hexamethyldisilazane to obtain a hydrophobic surface unless otherwise stated [28]. Frequencies of the QCM was followed continuously by a universal frequency counter (Iwatsu, Co., Tokyo, SC 7201 model) attached to a microcomputer system (NEC, PC 8801 model). The following equation has been obtained for the AT-cut shear mode QCM [10] ... 

The resonator is usually made of quartz and both longitudinal and shear modes can be used. As to the quartz, crystallographic cuts showing a highly stable temperature operation dependence are carefully selected in order to improve the possibility of obtaining satisfactory resolution values. 

Some attempts to exploit sensor dynamics for concentration prediction were carried out in the past. Davide et al. approached the problem using dynamic system theory, applying non-linear Volterra series to the modelling of Thickness Shear Mode Resonator (TSMR) sensors [4], This approach gave rise to non-linear models where the difficulty to discriminate the intrinsic sensor properties from those of the gas delivery systems limited the efficiency of the approach. 

As an example of the use of array methodology to study chemical sensor properties let us consider the thirteen molecular structures reported in Figure 5. To investigate the sensing properties of these molecules we studied the behaviour of the response of thickness shear mode resonators (TSMR) sensors, each coated with a molecular film, to different concentration of various volatile compounds (VOC). Analyte compounds were chosen in order to have different expected interaction mechanisms. 

Slip is not always a purely dissipative process, and some energy can be stored at the solid-liquid interface. In the case that storage and dissipation at the interface are independent processes, a two-parameter slip model can be used. This can occur for a surface oscillating in the shear direction. Such a situation involves bulk-mode acoustic wave devices operating in liquid, which is where our interest in hydrodynamic couphng effects stems from. This type of sensor, an example of which is the transverse-shear mode acoustic wave device, the oft-quoted quartz crystal microbalance (QCM), measures changes in acoustic properties, such as resonant frequency and dissipation, in response to perturbations at the surface-liquid interface of the device. 

The shear-mode acoustic wave sensor, when operated in liquids, measures mass accumulation in the form of a resonant frequency shift, and it measures viscous perturbations as shifts in both frequency and dissipation. The limits of device operation are purely rigid (elastic) or purely viscous interfaces. The addition of a purely rigid layer at the solid-liquid interface will result a frequency shift with no dissipation. The addition of a purely viscous layer will result in frequency and dissipation shifts, in opposite directions, where both of these shifts will be proportional to the square root of the liquid density-viscosity product v Pifti-... 

To model this, Duncan-Hewitt and Thompson [50] developed a four-layer model for a transverse-shear mode acoustic wave sensor with one face immersed in a liquid, comprised of a solid substrate (quartz/electrode) layer, an ordered surface-adjacent layer, a thin transition layer, and the bulk liquid layer. The ordered surface-adjacent layer was assumed to be more structured than the bulk, with a greater density and viscosity. For the transition layer, based on an expansion of the analysis of Tolstoi [3] and then Blake [12], the authors developed a model based on the nucleation of vacancies in the layer caused by shear stress in the liquid. The aim of this work was to explore the concept of graded surface and liquid properties, as well as their effect on observable boundary conditions. They calculated the hrst-order rate of deformation, as the product of the rate constant of densities and the concentration of vacancies in the liquid. 

Fig. 1.11 Deformation of an ideal material under normal (tension, compression) forces and shear forces exerted by balls. The positions of balls are shown for impact and shear mode of milling...

One of the most important properties which control the damage tolerance under impact loading and the CAI is the failure strain of the matrix resin (see Fig. 8.8). The matrix failure strain influences the critical transverse strain level at which transverse cracks initiate in shear mode under impact loading, and the resistance to further delamination in predominantly opening mode under subsequent compressive loading (Hirschbuehler, 1987 Evans and Masters, 1987 Masters, 1987a, b Recker et al., 1990). The CAI of near quasi-isotropic composite laminates which are reinforced with AS-4 carbon fibers of volume fractions in the range of 65-69% has... 

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