Shear stress and strain

The shearing force S causes the beam to tilt at an angle y, called the shear strain, where 

The stimulus-response relationship between the shear stress and shear strain is given by the shear modulus G  

It can be seen that there is a correspondence between the axial stress and strain and the shear stress and strain  

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The convention normally used is that direct stresses and strains have one suffix to indicate the direction of the stress or strain. Shear stresses and strains have two suffices. The first suffix indicates the direction of the normal to the plane on which the stress acts and the second suffix indicates the direction of the stress (or strain). Poisson s Ratio has two suffices. Thus, vi2 is the negative ratio of the strain in the 2-direction to the strain in the 1-direction for a stress applied in the 1-direction (V 2 = — il for an applied a ). v 2 is sometimes referred to as the major Poisson s Ratio and U2i is the minor Poisson s Ratio. In an isotropic material where V21 = i 2i. then the suffices are not needed and normally are not used. 

Let us first consider the case of an isotropic material, then simplify it for the case of an orthotropic material (same properties in the two directions orthogonal to the fiber axis—in this case, directions 2 and 3), snch as a nnidirectionally reinforced composite lamina. Eqnation (5.128) can be written in terms of the strain and stress components, which are conpled dne to the anisotropy of the material. In order to describe the behavior in a manageable way, it is cnstomary to introdnce a reduced set of nomenclature. Direct stresses and strains have two snbscripts—for example, an, 22, ti2, and Y2i, depending on whether the stresses and strains are tensile (a and s) or shear (t and y) in natnre. The modnli should therefore also have two subscripts En, E22, and G 2, and so on. By convention, engineers nse a contracted form of notation, where possible, so that repeated snbscripts are reduced to just one an becomes a, En becomes En but Gn stays the same. The convention is fnrther extended for stresses and strains, such that distinctions between tensile and shear stresses and strains are... 

In the ideal case of a Hookean body, the relationship between stress and strain is fully linear, and the body returns to its original shape and size, after the stress applied has been relieved. The proportionality between stress and strain is quantified by the modulus of elasticity (unit Pa). The proportionality factor under conditions of normal stress is called modulus of elasticity in tension or Young s modulus E), whereas that in pure shear is called modulus of elasticity in shear or modulus of rigidity (G). The relationships between E, G, shear stress, and strain are defined by ... 

A fluid in which the shear stress is proportional to the shear velocity, corresponding to this law, is called an ideal viscous or Newtonian fluid. Many gases and liquids follow this law so exactly that they can be called Newtonian fluids. They correspond to ideal Hookeian bodies in elastomechanics, in which the shear strain is proportional to the shear. A series of materials cannot be described accurately by either Newtonian or Hookeian behaviour. The relationship between shear stress and strain can no longer be described by the simple linear rule given above. The study of these types of material is a subject of rheology. 

Equivalent stresses (a) and strains (e) were derived from surface shear stresses and strains by means of the Von Mises yield criterion ... 

II. Viscous behavior (viscous flow) is characterized by proportionality between the shear stress and strain rate, i.e. by linear dependence between t and the rate of shear, y =dy/dt, given by Newton s law ... 

It can be very gentle on sensitive or fragile fluids that are affected by shear stress and strain. 

The complex shear modulus, G, defines the relationship between shear stress and strain for a linear viscoelastic material. 

ESDU, Inelastic shear stresses and strains in the adhesives bonding lap Joints in tension or shear, Engineering Science Data Unit Report No, 79016, London, 1979. 

Figure 2. Shear stress and strain of surimi gels heated by various heating methods. Abbreviations indicated in Figure 1. Adapted from Yongsawatdigul et al. (1995).

An ideal fluid (referred to as a Newtonian fluid) is defined as one that conforms to Newton s law of viscosity, in which the viscosity is independent of the strain rate. Newton s law states that the shear stress and strain rate are proportional, with the constant of proportionality, called the viscosity, p ... 

Figure 5.7. Illustration of two parallel plates separated by a fluid under shear, indicating the corresponding definitions of shear stress and strain rate.

Figure 10.3 Periodic shear stress and strain rate vectors.

Herein, L is the length of the cylinder, T is the appUed torque, r is the radial distance, J is the polar second moment of area and G is the shear modulus. These equations are developed assuming a linear relation between shear stress and strain as well as homogeneity and isotropy. With these assumptions, the shear stress and strain vary linearly with the radius and a pure shear stress state exists on any circumferential plane as shown on the surface at point A in Fig. 2,2. The shear modulus, G, is the slope of the shear stress-strain curve and may be found from. 

Shear Modulus Because only shear stresses and strains exist for the case of pure shear, the shear modulus can easily be determined from a torsion test by measuring the angle of twist over a prescribed length under a known torque, i.e.. 

The shear stresses and strains in the planes between polarization and transverse directions are coupled to flux density and field strength along the respective transverse axis ... 

Fig. 6 indicates how the adherend stresses drop to zero at one or other end of the bonded overlap and that, as a consequence of this, there are differential movements between the adherends, across the bond line, that result in adhesive shear stresses, and strains, that peak at the ends and are reduced throughout the elastic trough in the interior. If the load is high enough, the adhesive will go plastic in the load transfer zones at the ends. These zones are shown by... 

If the bar is simply twisted, there is no volume change ( = 0), only distortion. (Lines scratched along the sides of the bar would be converted to helices by the strain.) This type of deformation is called shear, and the constant of proportionality between the instantaneous elastic shear stress and strain is the shear modulus, G, The shear modulus is related to Young s modulus by... 

This is the viscosity that controls the rate of change of the volume of the porous network (with no liquid in the pores) when it is subject to a hydrostatic stress. As the porosity decreases, TV approaches 1/2 and the network becomes incompressible. Shear stresses and strains are related by the shear viscosity, >... 

Figure 2.6 Comparison of axial stress and strain (left) with shear stress and strain (right). (Reprinted with permission from Prof. Hiroshi Toshiyoshi, Institute of Industrial Science (IIS), The University of Tokyo, Japan.)...

FIGURE 15.12 Boltzmann superposition principle (a) applied strain history (b) resulting stress history. The experiment can also be reversed, with an applied stress history causing a strain history. The shear stresses and strains shown can also be replaced with tensile stresses and strains. 

Lhuger J, Wollny K, Huck S (2002) Direct strain oscillation a new oscillatory method enabling measurements at very small shear stresses and strains. Rheol Acta41 356-361... 

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