Shear modulus

The shear strain (y is defined as the deformation ( 5) parallel to the direction of the shear force, divided by the original length (/ perpendicular to this direction  

Shear modulus is the ratio of shear stress to shear strain  

The shear modulus is defined as the ratio of shear stress to shear strain. For cubic symmetry, the off-diagonal coefficients vanish, G — C44. 

Applications where there is compression will often have a shear component, even if not by design. The following methods detail the determination of shear modulus. 

Determination of modulus in shear or adhesion to rigid plates Quadruple shear method 

The shear stress results are expressed in Pascals (N/m2). The calculation is obtained by dividing the applied force indicated by the tensile machine by twice the actual bonded area of one block in square meters (nominally 20 x 25 x 10 The shear strain is obtained by dividing half the actual deformation by the actual thickness of one block. All measurements must be in the same units. 

---

Restored parameters for the evaluation of PDSM, may be different PMF of material tensor of stresses or its invariants, spatial gradients of elastic features (in particular. Young s modulus E and shear modulus G), strong, technological ( hardness HRC, plasticity ), physical (density) and others. 

Using the fluctuation-dissipation theorem [361, which relates microscopic fluctuations at equilibrium to macroscopic behaviour in the limit of linear responses, the time-dependent shear modulus can be evaluated [371 ... 

The shear modulus for an ideal elastomer in a perfect network is not difficult to derive ... 

A general relationship between Young s modulus and the shear modulus is E... 

Equation (3.58) can be written in terms of the shear modulus by dividing both sides of the equation by the constant strain ... 

Figure 3.8 Time-dependent shear modulus [as G(t)/Gol versus time (as t/r) (a) linear coordinates and (b) log-log coordinates.

The dynamic viscosity is related to the loss component of the shear modulus through the result 77= G"/co As co 0, the dynamic viscosity approaches the zero shear viscosity of an ordinary Uquid, 77 -... 

A value one-third this large applies to the shear modulus. We can eliminate N, from this last result by observing that the limit of Eq. (3.90) as t 0 is... 

We observed above that the Rouse expression for the shear modulus is the same function as that written for a set of Maxwell elements, except that the summations are over all modes of vibration and the parameters are characteristic of the polymers and not springs and dashpots. Table 3.5 shows that this parallel extends throughout the moduli and compliances that we have discussed in this chapter. In Table 3.5 we observe the following ... 

The procedure described above is an application of the time-temperature correspondence principle. By shifting a set of plots of modulus (or compliance) versus time (or frequency) at any temperature (subscript 1) along the log t axis, we obtain the value of that mechanical property at another time and temperature (subscript 2). Using the shear modulus as an example, the time-temperature correspondence principle states... 

Superabsorbents. Water-sweUable polymers are used extensively in consumer articles and for industrial appUcations. Most of these polymers are cross-linked acryUc copolymers of metal salts of acryUc acid and acrylamide or other monomers such as 2-acrylamido-2-methylpropanesulfonic acid. These hydrogel forming systems can have high gel strength as measured by the shear modulus (134). Sometimes inorganic water-insoluble powder is blended with the polymer to increase gel strength (135). Patents describe processes for making cross-linked polyurethane foams which contain superabsorbent polymers (136,137). 

Rheology. The rheology of foam is striking it simultaneously shares the hallmark rheological properties of soHds, Hquids, and gases. Like an ordinary soHd, foams have a finite shear modulus and respond elastically to a small shear stress. However, if the appHed stress is increased beyond the yield stress, the foam flows like a viscous Hquid. In addition, because they contain a large volume fraction of gas, foams are quite compressible, like gases. Thus foams defy classification as soHd, Hquid, or vapor, and their mechanical response to external forces can be very complex. 

Fig. 4. Schematic representation of a two-dimensional model to account for the shear modulus of a foam. The foam stmcture is modeled as a coUection of thin films the Plateau borders and any other fluid between the bubbles is ignored. Furthermore, aH the bubbles are taken to be uniform in size and shape.

When shear is appHed, the total area of the thin films increases, and the surface tension results in a restoring force, providing the shear modulus of the... 

Although aH these models provide a description of the rheological behavior of very dry foams, they do not adequately describe the behavior of foams that have more fluid in them. The shear modulus of wet foams must ultimately go to zero as the volume fraction of the bubbles decreases. The foam only attains a solid-like behavior when the bubbles are packed at a sufficiently large volume fraction that they begin to deform. In fact, it is the additional energy of the bubbles caused by their deformation that must lead to the development of a shear modulus. However, exactly how this modulus develops, and its dependence on the volume fraction of gas, is not fuHy understood. 

Fig. 7. Relations between elastic constants and ultrasonic wave velocities, (a) Young s modulus (b) shear modulus (c) Poisson s ratio and (d) bulk...

Fig. 2. Effect of temperature on the shear modulus of dry nylon-6,6 (—) and nylon-6,6 plus 30% glass fiber (-). To covert MPa to psi, multiply by 145.

Material Young s modulus, E, GPa Proportionality limit d, % extension Shear modulus, G, GPa Poisson s ratio, P Bulk modulus, B, GPa "... 

With appropriate caUbration the complex characteristic impedance at each resonance frequency can be calculated and related to the complex shear modulus, G, of the solution. Extrapolations to 2ero concentration yield the intrinsic storage and loss moduH [G ] and [G"], respectively, which are molecular properties. In the viscosity range of 0.5-50 mPa-s, the instmment provides valuable experimental data on dilute solutions of random coil (291), branched (292), and rod-like (293) polymers. The upper limit for shearing frequency for the MLR is 800 H2. High frequency (20 to 500 K H2) viscoelastic properties can be measured with another instmment, the high frequency torsional rod apparatus (HFTRA) (294). 

The RMS-800 provides steady-shear rotational rates from 10 to 100 rad/s and oscillatory frequencies from 10 to 100 rad/s. An autotension device compensates for expansion or contraction. With the standard 25- and 50-mm parallel plates, the viscosity range is 50-10 mPa-s, and the shear modulus range is 8 x 10 to 10 N/m. These ranges can be expanded with nonstandard plates, cones, and a Couette system. The temperature range is 20-350°C (-150 0 optional). 

Moduli and Poisson s Ratio. The Young s modulus of vitreous sihca at 25°C is 73 GPa (<1.06 x 10 psi), the shear modulus is 31 GPa (<4.5 X 10 psi), and the Poisson s ratio is 0.17. Minor differences in values can arise owing to density variations. The elastic modulus decreases with increasing density and Poisson s ratio increases (26). 

---