Shear stress-strains

Shear deformation occurs in structural elements such as those subjected to torsional loads and in short beams subjected to transverse loads. Shear S-S data can be generated by twisting (applying torque) a specimen at a specified rate while measuring the angle of twist between the ends of the specimen and the torque load exerted by the specimen 

Another couple must counter these two stresses. If the small element is taken as a differential one, the magnitude of the horizontal stresses must have the value of the two vertical stresses. This principle is sometimes phrased as cross-shears are equal that refers to a shearing stress that cannot exist on an element without a like stress being located 90 degrees around the corner. 

G is direcdy comparable to the modulus of elasticity used in direct-stress applications. Only two material constants are required to characterize a material if one assumes the material to be linearly elastic, homogeneous, and isotropic. However, three material constants exist the tensile modulus of elasticity (E), Poisson s ratio (v), and the shear modulus (G). An equation relating these three constants, based on engineering s elasticity principles is as follows  

This calculation that is true for most metals, is generally applicable to 

The shear modulus of a material can be determined by a static torsion test or by a dynamic test employing a torsional pendulum or an oscillatory rheometer. The maximum short-term shear stress (strength) of a material can also be determined from a punch shear test. 

Unlike the methods for tensile, flexural, or compressive testing, the typical procedure used for determining shear properties is intended only to determine the shear strength. It is not the shear modulus of a material that will be subjected to the usual type of 

Because strain measurements are difficult if not impossible to measure, few values of yield strength can be determined by testing. It is interesting to note that tests of bolts and rivets have shown that their strength in double shear can at times be as much as 20% below that for single shear. The values for the shear yield point (kPa or psi) are generally not available however, the values that are listed are usually obtained by the torsional testing of round test specimens. 

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Fig. 12. Adhesive thick adherend shear stress/strain. Reproduced by permission of the Boeing Company.

The curves for 3M XP251S fiberglass-epoxy are shown in Figures C-1 through C-5 [C-1]. Curves are given for both tensile and compressive behavior of the direct stresses. Note that the behavior in the fiber direction is essentially linear in both tension and compression. Transverse to the fiber direction, the behavior is nearly linear in tension, but very nonlinear in compression. The shear stress-strain curve is highly nonlinear. The Poisson s ratios (not shown) are essentially constant with values v.,2 =. 25 and V21 =. 09. 

Figure C-5 Shear Stress-Strain Curve for 3M XP251S Fiberglass-Epoxy (Adapted from [C-1])...

There are a number of different modes of stress-strain that can be taken into account by the designer. They include tensile stress-strain, flexural stress-strain, compression stress-strain, and shear stress-strain. 

Table 2-2 Examples of specific room temperature shear stress-strain data and Poisson s ratio for several plastics and other materials...

Tear strength is only applicable to flexible materials and is very little used to monitor ageing simply because tensile strength will serve perfectly well. There are circumstances where compression stress-strain properties would be relevant but the relatively bulky test pieces will be subject to the limitation of oxygen diffusion in any accelerated tests and changes can probably be estimated from tensile measurements. Similarly, shear stress-strain is very rarely used for monitoring ageing. 

In the [ 45]j tensile test (ASTM D 3518,1991) shown in Fig 3.22, a uniaxial tension is applied to a ( 45°) laminate symmetric about the mid-plane to measure the strains in the longitudinal and transverse directions, and Ey. This can be accomplished by instrumenting the specimen with longitudinal and transverse element strain gauges. Therefore, the shear stress-strain relationships can be calculated from the tabulated values of and Ey, corresponding to particular values of longitudinal load, (or stress relations derived from laminated plate theory (Petit, 1969 Rosen, 1972) ... 

ASTM D 3518 (1991). Practice for in-plane shear stress-strain response of unidirectional reinforced plastics. 

Petit, P.H. (1969). A simplified method of determining the in-plane shear stress-strain response of unidirectional composites. In Composite Materials Testing and Design, ASTM STP 460, ASTM, Philadelphia, PA, pp. 83-93. 

Figure 6-1. Shear stress/strain rate curves...

The inplane shear stress-strain tests reported here have been well demonstrated to be a reliable test for matrix-dominated properties in composites 141). For the selected mechanical properties that were monitored, their sensitivity to the thermal history was well demonstrated. In particular, the embrittlement process during the sub-Tg annealing or physical aging has been clearly observed. This decrease in molecular mobility, which gives rise to an increase in relaxation time and hence a decrease in toughness, can be rationalized as a decrease in free volume in an approach towards the equilibrium glassy state. 

FIG. 13.72 Shear stress-strain curves for PMMA at 22 °C under different pressures at a strain rate of approximately 4 x 10-4 s. The filled circles connect all fracture points in a fracture envelope. 

The shear behavior also involves matrix cracking and fiber failure.26 However, the ranking of the shear stress-strain curves between materials (Fig. 1.6) differs appreciably from that found for tension (Fig. 1.5). Preliminary... 

Fig. 1.6 Shear stress-strain curves measured for 2-D CMCs.

Fig.l. 37 Normalized in-plane shear stress-strain curves with the non-dimensional parameter W indicated. 

The constitutive equation for a dry powder is a governing equation for the stress tensor, t, in terms of the time derivative of the displacement in the material, e (= v == dK/dt). This displacement often changes the density of the material, as can be followed by the continuity equation. The constitutive equation is different for each packing density of the dry ceramic powder. As a result this complex relation between the stress tensor and density complicates substantially the equation of motion. In addition, little is known in detail about the nature of the constitutive equation for the three-dimensional case for dry powders. The normal stress-strain relationship and the shear stress-strain relationship are often experimentally measured for dry ceramic powders because there are no known equations for their prediction. All this does not mean that the area is without fundamentals. In this chapter, we will not use the approach which solves the equation of motion but we will use the friction between particles to determine the force acting on a mass of dry powder. With this analysis, we can determine the force required to keep the powder in motion. 

FIGURE 8.5 Sample shear flow definition of shear stress, strain, and shear rate. 

FIGURE 4.2 Simple shear flow deflnition of shear stress, strain, and shear rate Xyy, and are the Cartesian coordinates and the direction of normal stresses in shear flow. 

The shear stress-strain relationship is in many cases linear to greater strains than in tension or compression. 

When analyzing the results, it is important to determine the type of error. For example, the pressure loss error may be in the thin, thick, or virtually all sections. If the error appears in all, it may mean that there is simply an offset that is caused by a difference between the viscosity data used in the flow analysis and the actual viscosity during processing. If this is true, changing the processing speed should allow the flow-analysis data to be duplicated with a different fill or exit time. If, however, the flow analysis overstated thick sections and understated thin sections, there could be a serious problem with the mathematics used. See shear stress-strain. 

Mohr circle of stress A graphical representation of the components of stress and strain that acts across various planes at a given point and is drawn with reference to axes of normal stress (strain) and shear stress (strain). See shear strain stress. 

Figure 6.5 Schematic of shear stress—strain plot.

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