Shear strain, defined

The ratio of the shear stress, T, to the shear strain, % defines the shear modulus,... 

Here, the symbol 7 stands for the shear strain, defined as... 

The response of the material, in terms of the relationship between the shear stress and shear strain, defines the mechanical properties by which the material may be classified. If the material between the plates is a perfectly rigid solid, it will not move at all no matter how much force is applied (unless it breaks). Thus, Eq. 3.3 can be used to define this material ... 

If the maximum resolved shear stress r and the plastic shear strain rate y are defined according to (it is assumed that the Xj and Xj directions are equivalent)... 

A shear stress induces a shear strain. If a cube shears sideways by an amount w then the shear strain is defined by... 

Glasses, like metals, are formed by deformation. Liquid metals have a low viscosity (about the same as that of water), and transform discontinuously to a solid when they are cast and cooled. The viscosity of glasses falls slowly and continuously as they are heated. Viscosity is defined in the way shown in Fig. 19.7. If a shear stress is applied to the hot glass, it shears at a shear strain rate 7. Then the viscosity, ij, is defined by... 

For small shear strains we can define a time-dependent compliance (reciprocal modulus) by the equation... 

Other anisotropic elasticity relations are used to define Chentsov coefficients that are to shearing stresses and shearing strains what Poisson s ratios are to normal stresses and normal strains. However, the Chentsov coefficients do not affect the in-plane behavior of laminaeS under plane stress because the coefficients are related to S45, S46, Equation (2.18). The Chentsov coefficients are defined as... 

For a monolayer film, the stress-strain curve from Eqs. (103) and (106) is plotted in Fig. 15. For small shear strains (or stress) the stress-strain curve is linear (Hookean limit). At larger strains the stress-strain curve is increasingly nonlinear, eventually reaching a maximum stress at the yield point defined by = dT Id oLx x) = 0 or equivalently by c (q x4) = 0- The stress = where is the (experimentally accessible) static friction force [138]. By plotting T /Tlx versus o-x/o x shear-stress curves for various loads T x can be mapped onto a universal master curve irrespective of the number of strata [148]. Thus, for stresses (or strains) lower than those at the yield point the substrate sticks to the confined film while it can slip across the surface of the film otherwise so that the yield point separates the sticking from the slipping regime. By comparison with Eq. (106) it is also clear that at the yield point oo. 

The mechanical behavior of a material, and its corresponding mechanical or rheological properties, can be defined in terms of how the shear stress (tyx) (force per unit area) and the shear strain (yyx) (which is a relative displacement) are related. These are defined, respectively, in terms of the total force (Fx) acting on area Ay of the plate and the displacement (Ux) of the plate ... 

The manner in which the shear strain responds to the shear stress (or vice versa) in this situation defines the mechanical or rheological classification of the material. The parameters in any quantitative functional relation between the stress and strain are the rheological properties of the material. It is noted that the shear stress has dimensions of force per unit area (with units of, e.g., Pa, dyn/cm2, lbf/ft2) and that shear strain is dimensionless (it has no units). 

The functions t(x) and (x) are depicted in Fig. 20. The maximum shear stress rm, which occurs for x=pl4 or at a (maximum) shear strain ofp/(4dc) is defined as the ultimate shear strength r0... 

As shown in Figure 1.10(b), the horizontal displacement of the solid is proportional to the distance from the fixed plate. If the upper plate is displaced a distance and the solid has a thickness h then the shear strain y is defined as... 

We now use this to calculate the stress in the melt after the retraction has occurred. The deformation is described by the tensor E defined so that an arbitrary vector V in the material is deformed affinely into the vector E.v. For example, in simple shear of shear strain 7, and in uniaxial extension of strain e, the tensor E takes the forms... 

Figure 2.6 shows that the element distorts (shears) as well as dilates. The next task is to develop expressions for the shear strain rates, rz and ezr. By convention, the definition of the two-dimensional shear strain rate is taken as the average rate at which the angles defining the element sides decrease. Thus... 

The viscosity of a fluid, rj, is defined in terms of a test in which it is sheared. The viscosity is the ratio of the shear stress to the shearing strain rate y,r] = x/y. The strain rate, y, is the rate of shearing between two planes divided by the distance between them. Determine the SI units for viscosity. 

Batch Mixers In a batch mixer the shear rates throughout the volume are not uniform, and neither are the residence times of various fluid particles in the various shear-rate regions. Consequently, after a given time of mixing, different fluid particles experience different strain histories and accumulate different shear strains y. The SDF, g(y) dy, is defined as the fraction of the fluid in the mixer that has experienced a shear strain from y to y + dy. Alternatively, it is the probability of a fluid particle fed to the mixer to accumulating a shear strain of y in time t. By integrating g(y) dy, we get ... 

Many of the comments in the previous chapter about the selection of grade, additives and mixing before moulding apply equally in preparation for extrusion. It is important of course that the material should be appropriate for the purpose, uniform, dry, and free from contamination. It should be tested for flow and while many tests have been devised for this it is convenient to classify them as either for low or high rates of shear. The main terms used in such testing ( viscosity , shear rate , shear strain , etc.) are defined in words and expressed as formulae in ISO 472, and it is not necessary to repeat them here. Viscosity may be regarded as the resistance to flow or the internal friction in a polymer melt and often will be measured by means of a capillary rheometer, in which shear flow occurs with flow of this type—one of the most important with polymer melts—when shearing force is applied one layer of melt flows over another in a sense that could be described as the relationship between two variables—shear rate and shear stress.1 In the capillary rheometer the relationship between the measurements is true only if certain assumptions are made, the most important of which are ... 

All real materials fall Theologically between two extremes the perfectly elastic Hookean solid, for which stress is directly proportional to strain, and the Newtonian liquid, for which (shear) stress is directly proportional to (shear) strain rate. Strain can be defined as deformation relative to a reference length, area or volume (Barnes et al., 1989) it is dimensionless. Strain... 

The diagonal elements of Eq. 10.3 are the stretches or tensile strains. The nondiagonal elements are the shear strains. The variation of the displacement vector, u, with the position vector, d, for a point in the solid is used to define the nine tensor components in Eq. 10.3, as follows ... 

The constants s and c ( = 1 /s) are known as the elastic compliance constant and the elastic stiffness constant, respectively. The elastic stiffness constant is the elastic modulus, which is seen to be the ratio of stress to strain. In the case of normal stress-normal strain (Fig. 10.3a) the ratio is the Young s modulus, whereas for shear stress-shear strain the ratio is called the rigidity, or shear, modulus (Fig. 10.36). The Young s modulus and rigidity modulus are the slopes of the stress-strain curves and for nonHookean bodies they may be defined alternatively as da-/ds. They are requited to be positive quantities. Note that the higher the strain, for a given stress, the lower the modulus. 

In the text, we defined the shear stress and shear strain in a dynamic-mechanical experiment in terms of sinusoidal functions. 

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