Modulus of rigidity

The coefficient Tj is termed the modulus of rigidity. The viscosities of thixotropic fluids fall with time when subjected to a constant rate of strain, but recover upon standing. This behavior is associated with the reversible breakdown of stmctures within the fluid which are gradually reestabflshed upon cessation of shear. The smooth sprea ding of paint following the intense shear of a bmsh or spray is an example of thixotropic behavior. When viscosity rises with time at constant rate of strain, the fluid is termed rheopectic. This behavior is much less common but is found in some clay suspensions, gypsum suspensions, and certain sols. 

Fig. 3. Effect of density on compressive modulus of rigid cellular polymers. A, extmded polystyrene (131) B, expanded polystyrene (150) C-1, C-2, polyether polyurethane (151) D, phenol—formaldehyde (150) E, ebonite (150) E, urea—formaldehyde (150) G, poly(vinylchloride) (152). To convert...

Tensile strength and modulus of rigid foams have been shown to vary with density in much the same manner as the compressive strength and modulus. General reviews of the tensile properties of rigid foams are available (22,59,60,131,156). 

Those stmctural variables most important to the tensile properties are polymer composition, density, and cell shape. Variation with use temperature has also been characterized (157). Flexural strength and modulus of rigid foams both increase with increasing density in the same manner as the compressive and tensile properties. More specific data on particular foams are available from manufacturers Hterature and in References 22,59,60,131 and 156. Shear strength and modulus of rigid foams depend on the polymer composition and state, density, and cell shape. The shear properties increase with increasing density and with decreasing temperature (157). 

Magnesium alloys have a Young s modulus of elasticity of approximately 45 GPa (6.5 x 10 psi). The modulus of rigidity or modulus of shear is 17 GPa (2.4 X 10 psi) and Poisson s ratio is 0.35. Poisson s ratio is the ratio of transverse contracting strain to the elongation strain when a rod is stretched by forces at its ends parallel to the rod s axis. 

The constant G, called the shear modulus, the modulus of rigidity, or the torsion modulus, is directly 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, follows ... 

The moduli of elasticity, G for shear and E for tension, are ratios of stress to strain as measured within the proportional limits of the material. Thus the modulus is really a measure of the rigidity for shear of a material or its stiffness in tension and compression. For shear or torsion, the modulus analogous to that for tension is called the shear modulus or the modulus of rigidity, or sometimes the transverse modulus. 

The shearing stress should therefore be proportional to the shear strain, i.e., the rubber should obey Hookers law in shear whereas this is not true of elongation. The quantity RTve/V is the modulus of rigidity. 

Byrex Chemical Resistance Glass has a Young s modulus of 6-1 x 10 dynes/cm , a modulus of rigidity of 2-5 x 10 dynes/cm and a Poisson s ratio of 0-22. Similar values are found for other glasses. 

Figure 7.6 Models fitted for the effect of the number of freeze/thaw cycles on the apparent moduli of elasticity in the compression (Ec) a-tid tensile tests (Et), and on the modulus of rigidity (Cs) in the shear test.

The reduced dynamic modulus of rigidity, obtained from the dispersion viscosity, is of the form (10) ... 

For each type of sirain and stress there is a modulus, which is ihe ratio of the stress lo the corresponding strain. In the case of elongation or linear compression, it is commonly called Young s modulus we also have the hulk modulus and the shear modulus Of rigidity. See Table I. 

Poisson s ratio-ratio of lateral strain to axial strain in an axial loaded specimen. It is a constant that relates the modulus of rigidity to Young s modulus. 

The addition of carbon black to ABS resin improves its hardness, modulus of rigidity, heat deflection temperature, and ultraviolet stability but reduces its ultimate strength, particularly its impact strength. Fine particle size and high structure carbon blacks have the greatest effects. 

Flexural strength and modulus of rigid foams increase with increasing density in the same manner as the compressive and tensile properties. 

The modulus of rigidity, or shear modulus (G), is defined as the ratio of shear stress to strain. It is a measure of a gel s ability to resist deformation. The minimum rigidity for a strong gel to resist deformation under its own weight is equal to about gpl which is the product of the acceleration due to gravity (g), density (p), and a linear dimension (/) of the sample. Therefore, the minimum rigidity is about 100 Pa (lO dyn/cm) for a gel sample 1 cm long. 

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 ... 

This equation describes Hookean elasticity, and Po = G (G is the modulus of rigidity). In Fig. 9, the classical mechanical spring model representing Eq. (14) is illustrated. If, however, it is assumed that jSi is the only nonzero constant in Eq. (13), then ... 

Materials can show linear and nonlinear viscoelastic behavior. If the response of the sample (e.g., shear strain rate) is proportional to the strength of the defined signal (e.g., shear stress), i.e., if the superposition principle applies, then the measurements were undertaken in the linear viscoelastic range. For example, the increase in shear stress by a factor of two will double the shear strain rate. All differential equations (for example, Eq. (13)) are linear. The constants in these equations, such as viscosity or modulus of rigidity, will not change when the experimental parameters are varied. As a consequence, the range in which the experimental variables can be modified is usually quite small. It is important that the experimenter checks that the test variables indeed lie in the linear viscoelastic region. If this is achieved, the quality control of materials on the basis of viscoelastic properties is much more reproducible than the use of simple viscosity measurements. Non-linear viscoelasticity experiments are more difficult to model and hence rarely used compared to linear viscoelasticity models. 

The two main elastic properties are modulus of elasticity, which describes the relationship of load (stress) to deformation (strain), and modulus of rigidity or shear modulus, which describes the internal distribution of shearing stress to shear strain or, more precisely, angular displacement within a material. 

Rheological aspects must surely have a bearing on the question of rates of volume regulation by the cell, both via the viscosity (of the aqueous part of the cytoplasm) and via the effect of viscosity on diffusion coefficients. Figure 3 shows the modulus of rigidity of a minced fish paste as a function of temperature. This... 

Figure 3. Modulus of rigidity during heating of fish protein concentrate as a function of temperature. Note minima at 28° and 45°C. (From Wu et al., 1985a).

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