Stiffness in shear

The properties desired of an ideal spacer layer are that it be stiff in shear, but that the spacer itself contribute minimally to the bending stiffness of the base structure, shifting the neutral plane as little as possible. We note that for the spaced constrained layer, the combined function of the viscoelastic layer and spacer is to provide a thick, dissipative and appropriately stiff (in shear) layer between the constraining and base layers. Therefore the order of the viscoelastic and spacer elements is arbitrary and they may be subdivided as long as the desired properties are preserved. These possibilities give additional freedom in adapting viscoelastic materials for effective damping. 

Viscosity is a distinguishing feature of fluids. As a fluid becomes more viscous, the less fluid-like it is. A related property in solids is the shear modulus or stiffness in shear. We previously found that the shear stress and shear strain in solids are related as follows ... 

Accordingly, we have supposedly found the shear modulus G.,2. However, a relationship such as Equation (2.107) does not exist for strengths because strengths do not transform like stiffnesses. Thus, this experiment cannot be relied upon to determine S, the ultimate shear stress (shear strength), because a pure shear deformation mode has not been excited with accompanying failure in shear. Accordingly, other approaches to obtain S must be used. 

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. 

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). 

The moduli of elasticity, G for shear and E for tension, are ratios of stress to strain as measured within the proportional limits of the material. Thus the modulus is really a measure of the rigidity for shear of a material or its stiffness in tension and compression. For shear or torsion, the modulus analogous to that for tension is called the shear modulus or the modulus of rigidity, or sometimes the transverse modulus. 

The second major assumption is that the material is elastic, meaning that the strains are directly proportional to the stresses applied and when the load is removed the deformation will disappear. In engineering terms the material is assumed to obey Hooke s Law. This assumption is probably a close approximation of the material s actual behavior in direct stress below its proportional limit, particularly in tension, if the fibers are stiff and elastic in the Hookean sense and carry essentially all the stress. This assumption is probably less valid in shear, where the plastic carries a substantial portion of the stress. The plastic may then undergo plastic flow, leading to creep or relaxation of the stresses, especially when the stresses are high. 

In most applications where bending apparently takes place, the rubber is also deformed in shear, tension or compression, for example in a shaped door seal, when the test for stiffness would be a compression test on the actual part. Generally, rubbers are not stiff enough in flexure to support appreciable loads so that there is not much need for flexural tests and, at the same time, the lack of stiffness makes such tests a little difficult to carry out with precision. There are, however, some cases where stiffness in bend is of interest, for example with thin sheet and coated fabrics as a measure of... 

The viscoelastic behaviour of rubbers is not linear stress is not proportional to strain, particularly at high strains. The non-linearity is more pronounced in tension or compression than in shear. The result in practice is that dynamic stiffness and moduli are strain dependent and the hysteresis loop will not be a perfect ellipse. If the strain in the test piece is not uniform, it is necessary to apply a shape factor in the same manner as for static tests. This is usually the case in compression and even in shear there may be bending in addition to pure shear. Relationships for shear, compression and tension taking these factors into account have been given by Payne3 and Davey and Payne4 but, because the relationships between dynamic stiffness and the basic moduli may be complex and only approximate, it may be preferable for many engineering applications to work in stiffness, particularly if products are tested. 

There was previously a separate ISO standard for adhesion in shear but this was withdrawn in favour of extending the standard for shear modulus to allow the test to be continued to the failure point, i.e. the two methods have been combined. The composite method is contained in ISO 182715 and uses the same quadruple element test piece as did the separate adhesion standard. The double sandwich construction is intended to provide a very stiff test piece which will remain in alignment under high stresses. The present standard quadruple test piece uses rubber elements 4 1 mm thick and 20 5 mm long and these tolerances are much less tight than previously. The measured adhesion strength in shear is less affected by the test piece shape factor then tension tests8 and the wider tolerances should be perfectly satisfactory. The test piece is strained at a rate of 50 mm/min, in line with the speed for most other adhesion to metal tests, and the result expressed as the maximum force divided by the total bonded area of one of the double sandwiches. The British equivalent BS 903 Part A 1416 is identical. 

For an ideal solid, Hooke s law holds the stress, cr, applied is proportional to the deformation, e, and the proportionality constant is the modulus of elasticity E, so a = E e. Besides E also other quantities play a role, such as the shear modulus, G, in a shearing deformation or torsion, which is related to E. For the sake of simplicity we shall mainly use as a representative quantity for the elastic stiffness in any geometry of loading. 

Materials selection process can be depicted in terms of Figure 1.40. Materials selection involves many factors that have to be optimized for a particular application. The foremost consideration is the cost of the material and its applicability in the environmental conditions so that integrity can be maintained during the lifetime of the equipment. When the material of construction is metallic in nature, the chemical composition and the mechanical properties of the metal are significant. Some of the important mechanical properties are hardness, creep, fatigue, stiffness, compression, shear, impact, tensile strength and wear. 

The sequence of events which we have detected at ultrasonic frequencies when hydrophilic polymers are contacted by strongly hydrogen bonding solvents involves a sharp increase in shear stiffness followed by a more or less rapid decrease to attenuation levels close to that of the solvent. 

No universal standard appears to exist for the selection of specimen height in relation to its thickness. Adkins (2) attributes about a 1% decrease in shear stiffness to en3 effects at a height-to-thickness ratio of 24. Parin, et al (10) recommends a height-to-thickness ratio greater than 4. The values of this ratio were 20 and 500 in the present tests on the Soundcoat and the 3M materials, respectively. 

In spite of the above drawbacks. Equations 6.1 and 6.2 are popular mainly due to their simple form and have been used to estimate Me of protein (e.g., egg, gelatin) and other gels from shear modulus-concentration data (Table 6-2) (Fu, 1998). Its successful application to protein gels initially has been attributed to the greater flexibility of polypeptide chains in comparison to polysaccharide chains that are relatively stiff. In addition, the inability to employ low strain rates in early experimental studies on polysaccharide gels could be another reason. 

The data for the modulus-temperature curve are most often gathered in the dynamic mode at a fixed frequency of around 1 rad/s, either in shear or flex, depending on the stiffness range of the test material over the desired temperature range. See Appendix 3 of Chapter 2. 

In order to find a beam s torsional stiffness, the shear stress distribution in the cross-sectional plane (the c, y plane) must be determined. This requires the solution of Poisson s equation for the cross section... 

It includes the two coefficients yield limit, Xo and Bingham s viscosity, Tjpi = const, also referred to in the literature as stiffness. In this particular case, X becomes the measured wall shear stress, Xr, and T is the true shear rate in the shear fraction of flow. 

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