Shear strain definition

By analogy with Eq. (3.1), we seek a description for the relationship between stress and strain. The former is the shearing force per unit area, which we symbolize as as in Chap. 2. For shear strain we use the symbol y it is the rate of change of 7 that is involved in the definition of viscosity in Eq. (2.2). As in the analysis of tensile deformation, we write the strain AL/L, but this time AL is in the direction of the force, while L is at right angles to it. These quantities are shown in Fig. 3.6. It is convenient to describe the sample deformation in terms of the angle 6, also shown in Fig. 3.6. For distortion which is independent of time we continue to consider only the equilibrium behavior-stress and strain are proportional with proportionality constant G ... 

The critical gel equation is expected to predict material functions in any small-strain viscoelastic experiment. The definition of small varies from material to material. Venkataraman and Winter [71] explored the strain limit for crosslinking polydimethylsiloxanes and found an upper shear strain of about 2, beyond which the gel started to rupture. For percolating suspensions and physical gels which form a stiff skeleton structure, this strain limit would be orders of magnitude smaller. 

The original length 6f the specimen Is LO and its stretched length is L. At very small deformations, all the strain definitions of Table 2 are equivalent, For shear tests (see Figure 2)... 

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 shear-strain rate r = e r is given by the average rate at which the vertex angle decreases, which is the same definition as in the r-z plane ... 

An elastic solid has a definite shape. When an external force is applied, the elastic solid instantaneously changes its shape, but it will return instantaneously to its original shape after removal of the force. For ideal elastic solids, Hooke s Law implies that the shear stress (o force per area) is directly proportional to the shear strain (7 Figure H3.2.1A) ... 

The equation leads to the definition of a time and strain-dependent memory fimction which can be further factorized into a time-dependent part (the linear memory function) and a strain-dependent damping function. Though on one hand, there is some experimental evidence for this in limited time ranges, on the other hand, a few experiments might question this strong hypothesis since, for example, the damping function obtained fi um step shear rate data is found to be different from that in step shear strain. 

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

As most readers may not be familiar with this formulation, let us illustrate the approach by considering a specific situation, namely that of a homogeneous bulk fluid. By definition, at thermodynamic equilibrium, bulk fluids cannot sustain any shear strain, that is, Tao = 0 for all a /3. Thius,... 

It is interesting to note the similarity between the definitions of the shear viscosity (Equation 13.1) and the shear modulus (Equation 13.2), which leads to a relationship (Equation 13.3) between r and G in terms of the ratio of the shear strain (y) divided by the shear rate (y). 

In the above considerations, a sinusoidal shear strain is applied to the sample. It should be clear that a sinusoidal shear stress could also be applied resulting in corresponding compliance functions J and J". The former results from the deformation in phase with the stress, while the latter corresponds to the out-of-phase deformation. The value of tan 5 remains the same, as can be seen from the curves in Figure 2-13, where we can easily imagine the stress as the applied variable and strain as the measured variable. Tensile stress is equally applicable and definitions of E (co), E" (o), D"(co), D co), etc. are completely analogous to the derived shear parameters. At a given frequency, the value of tan 8 is always the same for any of these quantities, i.e., tan 8 = E"/E = D"/D . 

This is expressed in one of Isaac Newton s many laws, which states that the shear stress Tis directly proportional to the shear rate, that is, to the rate at which the fluid is strained. The shear rate by definition is dy/dt, where yis the shear strain. Using the symbol yfor dyidt, we obtain... 

The plastic hardening law of material is derived from the definition of plastic energy in thermodynamic potential. We have chosen the generalised plastic shear strain as the plastic... 

Erom the definition of the shear spring constant k of a laminated rubber/ steel spring, expressing the force F and deflection x in terms of shear stress and shear strain of the composite, we obtain... 

To represent the above-named Bingham viscosity in the simulation model the possibility to characterize Non-Newtonian fluids in the CFD-code is necessary. Most commercial CFD-Software allocate a so-called Power-Law-Model whereby (26) may be possibly specified. Otherwise the Bingham model has to be implemented in the actual internal expression language, e.g. an executive Fortran-Routine in ANSYS/FLOTRAN or a definition in the CFX-Expression Language (CEL). In either case it is inevitable for the CFD-code to have dynamic access to the (system-) variable for the shear strain rate y. 

Before proceeding, some definitions are useful. Stress is the ratio of the force on a body to the cross-sectional area of the body. The true stress refers to the infinitesimal force per (instantaneous) area, while the engineering stress is the force per initial area. Strain is a measure of the extent of the deformation. Normal strains change the dimensions, whereas shear strains change the angle between two initially perpendicular lines. In correspondence with the true stress, the Cauchy (or Euler) strain is measured with respect to the deformed state, while the Green s (or Lagrange) strain is with respect to the undeformed state. 

There is, however, a problem with our definition of the shear strain, which is illustrated in Fig. 2.17. In particular, e-j does not differentiate between rotations and distortions. For example, if e 2— 2v shown in Fig. 2.17(b), the values of are non-zero even though this only represents a rotation. In order to overcome this problem, it is possible to separate any general shear distortion into a combination of pure shear and rotation components by writing... 

From these definitions, e j represents normal strains when i=j and pure shear strains when i j. Moreover, so that only six of the components are inde-... 

We see that the Coulomb yield criterion therefore defines both the stress condition required for yielding to occur and the directions in which the material will deform. Where a deformation band forms, its direction is one that is neither rotated nor distorted by the plastic deformation, because its orientation marks the direction that establishes material continuity between the deformed material in the deformation band and the undistorted material in the rest of the specimen. If volume is conserved, the band direction denotes the direction of shear in a simple shear (by the definition of a shear strain). Thus for a Coulomb yield criterion the band direction is defined by Equation (11.6). 

For compliance constants the substitution is based on the above conversion for stiffness constants, but additional rules apply because of the factor 2 difference between the definition of engineering and tensor shear strains ... 

Because the material properties are direction-dependent in a cubic crystal, they have to be stated together with the corresponding direction. According to the definition, the load direction has to be stated for Young s modulus Ei. Because the shear stress Tij and shear strain -y j have two indices, two indices are needed for the shear modulus Gij. Poisson s ratio relates strains in two directions. Here the second index j denotes the direction of the strain that causes the transversal contraction in the direction marked by the first index i eu = If the coordinate system is aligned with the axes... 

For example, in Chapter 6, to begin with three parameters, p (shear stress), e (shear strain), and E (modulus or rigidity), are introduced to define viscosity and viscoelasticity. With respect to viscosity, after the definition of Newtonian viscosity is given, a detailed description of the capillary viscometer to measure the quantity t follows. Theories that interpret viscosity behavior are then presented in three different categories. The first category is concerned with the treatment of experimental data. This includes the Mark-Houwink equation, which is used to calculate the molecular weight, the Flory-Fox equation, which is used to estimate thermodynamic quantities, and the Stockmayer-Fixman equation, which is used to... 

In most treatises,"- 3 the strain tensor is defined with all components smaller by a factor of 2 than inequation 3, so that 711 = dui/dxi and 721 = du2/bx + bui/bx ). However, such a definition makes discussion of shear or shear flow somewhat clumsy either a practical shear strain and practical shear rate must be introduced which are twice 721 and 721 respectively, or else a factor of 2 must be carried in the constitutive equations. Since most of the discussion in this book is concerned with shear deformations, we use the definition of equation 3 which follows Bird and his school" and Lodge. - This does cause a slight inconvenience in the discussion of compressive and tensile strain, where a practical measure of strain is subsequently introduced (Section F below). In older treatises on elasticity, strains are defined without the factor of 2 appearing in the diagonal components of equation 3, but with the other components the same. 

---