Transverse shearing effects

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). 

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Note that no assumptions involve fiber-reinforced composite materials explicitly. Instead, only the restriction to orthotropic materials at various orientations is significant because we treat the macroscopic behavior of an individual orthotropic (easily extended to anisotropic) lamina. Therefore, what follows is essentially a classical plate theory for laminated materials. Actually, interlaminar stresses cannot be entirely disregarded in laminated plates, but this refinement will not be treated in this book other than what was studied in Section 4.6. Transverse shear effects away from the edges will be addressed briefly in Section 6.6. 

HOLES IN LAMINATES TRANSVERSE SHEAR EFFECTS POST-CURING LAMINATE SHAPE ... 

The preceding subsection was devoted to a comparison of a special exact elasticity solution with classical lamination theory results. The importance of transverse shear effects was clearly demonstrated. However, that demonstration was for a special problem of rather narrow interest. The objective of this subsection is to display approaches and results for the approximate consideration of transverse shear effects for general laminated plates. 

Luo Q, Tong L (2009b) Energy release rates for interlaminar delamination in laminates considering transverse shear effects. Compos Struct... 

In order to check the influence of transverse shear effects on the flexural vibrations, all flexural data were also evaluated with the simple Bernoul1i-Euler frequency equations (1), which, as already described above, apply only if these effects are negligible. The resulting Young s moduli, referred to as are also given in Tables 2 and 3. 

Composite materials typically have a low matrix Young s modulus in comparison to the fiber modulus and even in comparison to the overall laminae moduli. Because the matrix material is the bonding agent between laminae, the shearing effect on the entire laminate is built up by summation of the contributions of each interlaminar zone of matrix material. This summation effect cannot be ignored because laminates can have 100 or more layersi The point is that the composite material shear moduli and G are much lower relative to the direct modulus than for isotropic materials. Thus, the effect of transverse shearing stresses. 

Study of transverse shearing stress effects is divided in two parts. First, some exact elasticity solutions for composite laminates in cylindrical bending are examined. These solutions are limited in their applicability to practical problems but are extremely useful as checl oints for more broadly applicable approximate theories. Second, various approximations for treatment of transverse shearing stresses in plate theory are discussed. 

The treatment of transverse shear stress effects in plates made of isotropic materials stems from the classical papers by Reissner [6-26] and Mindlin [6-27. Extension of Reissner s theory to plates made of orthotropic materials is due to Girkmann and Beer [6-28], Ambartsumyan [6-29] treated symmetrically laminated plates with orthotropic laminae having their principal material directions aligned with the plate axes. Whitney [6-30] extended Ambartsumyan s analysis to symmetrically laminated plates with orthotropic laminae of arbitrary orientation. 

Whitney and Pagano [6-32] extended Yang, Norris, and Stavsky s work [6-33] to the treatment of coupling between bending and extension. Whitney uses a higher order stress theory to obtain improved predictions of a, and and displacements at low width-to-thickness ratios [6-34], Meissner used his variational theorem to derive a consistent set of equations for inclusion of transverse shearing deformation effects in symmetrically laminated plates [6-35]. Finally, Ambartsumyan extended his treatment of transverse shearing deformation effects from plates to shells [6-36]. 

Eric Reissner, The Effect of Transverse Shear Deformation on the Bending of Elastic Plates, Journal of Applied Mechanics, June 1945, pp. A-69-77. 

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. 

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. 

For materials with a strong bond between the matrix and the fiber, models for steady transverse creep are available. The case of a linear matrix is represented exactly by the effect of rigid fibers in an incompressible linear elastic matrix and is covered in texts on elastic materials.7,11,12 For example, the transverse shear modulus, and therefore the shear viscosity, of a material containing up to about 60% rigid fibers in a square array is approximated well... 

In a first approach to the study of beam bending, it is convenient to make some hypotheses (1). The first of these hypotheses is that the sections that are flat before flexion remain flat after flexion. For slender beams—that is, for beams whose transverse dimensions are small in comparison with their length—this hypothesis is substantially correct. In this case, the shear effects in the cross sections are relatively negUgible. It will be further assumed that the inertial forces arising from the rotation of each element around its center of mass can be ignored. This is, in fact, the second hypothesis. 

In shear bands chain conformations are induced which are likely to be strained in comparison to the ones in the undeformed material. To detect shear bands by NMR the transverse relaxation effective under OW4 irradiation has been recorded as a function of 2D space with the pulse sequence of Fig. 10.3.6 [Wei5, Wei7, Tral]. In the imaging experiment, the sample was oriented in such a way that the drawing direction was perpendicular to the static magnetic field. 

For the transverse shear modulus, the approach designated self-consistent was based on the formula obtained by the self-consistent method for the plane-strain bulk modulus (11.61), on the transverse modulus calculated using the Chamis approach (11.49b) and the in-plane Poisson s ratio given by the rule of mixtures. Except when used to predict the axial modulus and the major Poisson s ratio, the rule of mixtures underestimates the remaining composite elastic properties. The Bridging Model proved to be a very effective theory to account for all five elastic properties for unidirectional composites that are transversely isotropic. 

Coefficients d, d2i and d-ij, describe the longitudinal piezoelectric effect (see symbol L in Table 5.1). The normal mechanical stress component causes piezoelectric polarization parallel to it in such case. Second possibihty is the piezoelectric polarization perpendicular to the applied normal mechanical stress. Such piezoelectric effect is so called transversal effect (see symbol T in Table 5.1) and it is characterized by one of the coefficients di2, d -i, J21, dj2, dn or J32. Application of shear mechanical stress might result in the piezoelectric polarization perpendicular to the plane of applied shear. Such shear piezoelectric effect is called longitudinal shear (see symbol 5l in Table 5.1) and it is characterized by one of the piezoelectric coefficients du, d25 or d e- Second possibility of shear piezoelectric effect is the piezoelectric polarization parallel to the plane of the applied shear stress. Such effect is called transversal shear (see symbol in Table 5.1 and in Fig. 5.2). This effect is related to one of the piezoelectric coefficients J15, di, d24, d26, d- orc 35. 

Table 5.1 Four possibilities of piezoelectric effect - longitudinal, transversal, longitudinal shear and transversal shear...

Fig. 6.12. Definition of the axes in piezo materials, a The digits 4, 5 and 6 indicate the shear on the axes 1, 2 and 3 b longitudinal (dss) effect, c transversal (dsi) effect...

Since the relaxation times t< are very long, it is relatively easy to realize shear rates (.r) larger than 1 /t( and to measure the resulting properties. It is convenient to work in situations of permanent ( ), where the molecules reach a steady state with finite distortions. This can be achieved only in transverse shears. For dilute solutions (Chapter VII) we discarded the transverse shear situations because they led to small (and thus complex) effects. For concentrated solutions, the distortions become strong even in transverse shear flows, and we can restrict our attention to this more common case. 

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