Torsion modulus

Young s modulus Torsion modulus Poisson s ratio Compressibility (dv/Vf dp) ... 

Modulus of elasticity Shear modulus, torsional modulus Torsional modulus... 

Modulus The ratio of stress to strain in a material over the range for which this value is constant. The type of modulus, which is measured, depends on the method of measurement, e.g., dynamic modulus, compressive modulus, elastic or tensile (Young s) modulus, shear modulus, torsion modulus, sonic modulus. 

Schergradient shear loss modulus (90°, out-of-phase modulus/ viscous modulus) Scherverlustmodul shear modulus (torsion modulus/ modulus of rigidity)... 

Figure 7.46 Shear modulus (torsion pendulum data at 1 Hz) of linear polyethylenes as a function of crystallinity at different temperatures. From Boyd (1979) who used the data of Illers (1973).

Much more information can be obtained by examining the mechanical properties of a viscoelastic material over an extensive temperature range. A convenient nondestmctive method is the measurement of torsional modulus. A number of instmments are available (13—18). More details on use and interpretation of these measurements may be found in references 8 and 19—25. An increase in modulus value means an increase in polymer hardness or stiffness. The various regions of elastic behavior are shown in Figure 1. Curve A of Figure 1 is that of a soft polymer, curve B of a hard polymer. To a close approximation both are transpositions of each other on the temperature scale. A copolymer curve would fall between those of the homopolymers, with the displacement depending on the amount of hard monomer in the copolymer (26—28). 

Free- Vibration Methods. Free-vibration instmments subject a specimen to a displacement and allow it to vibrate freely. The oscillations are monitored for frequency and damping characteristics as they disappear. The displacement is repeated again and again as the specimen is heated or cooled. The results are used to calculate storage and loss modulus data. The torsional pendulum and torsional braid analy2er (TBA) are examples of free-vibration instmments. 

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). 

Mechanical Properties. Although wool has a compHcated hierarchical stmcture (see Fig. 1), the mechanical properties of the fiber are largely understood in terms of a two-phase composite model (27—29). In these models, water-impenetrable crystalline regions (generally associated with the intermediate filaments) oriented parallel to the fiber axis are embedded in a water-sensitive matrix to form a semicrystalline biopolymer. The parallel arrangement of these filaments produces a fiber that is highly anisotropic. Whereas the longitudinal modulus of the fiber decreases by a factor of 3 from dry to wet, the torsional modulus, a measure of the matrix stiffness, decreases by a factor of 10 (30). 

T and are the glass-transition temperatures in K of the homopolymers and are the weight fractions of the comonomers (49). Because the glass-transition temperature is directly related to many other material properties, changes in T by copolymerization cause changes in other properties too. Polymer properties that depend on the glass-transition temperature include physical state, rate of thermal expansion, thermal properties, torsional modulus, refractive index, dissipation factor, brittle impact resistance, flow and heat distortion properties, and minimum film-forming temperature of polymer latex... 

Flexibility Stresses Bending and torsional stresses shall be computed using the as-instaUed modulus of elasticity E and then combined in accordance with Eq. (10-100) to determine the computed displacement stress range Sg, which shah not exceed the allowable stress range [Eqs. (10-93) and (10-94).]... 

The section modulus. Z, of the box should be 2 / times greater than the section modulus, z, of the pin in a drill collar connection. On the right side of the connection are the spots at which the critical area of both the pin (Ap) and box (A ) should be measured for calculating torsional strength. 

The shear modulus of a material can be determined by a static torsion test or by a dynamic test employing a torsional pendulum or an oscillatory rheometer. The maximum short-term shear stress (strength) of a material can also be determined from a punch shear test. 

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 ... 

Torsion property As noted, the shear modulus is usually obtained by using pendulum and oscillatory rheometer techniques. The torsional pendulum (ASTM D 2236 Dynamic Mechanical Properties of Plastics by Means of a Torsional Pendulum Test Procedure) is a popular test, since it is applicable to virtually all plastics and uses a simple specimen readily fabricated by all commercial processes or easily cut from fabricated products. 

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. 

Some important conditions concerning the estimation of error should be pointed out. First, modulus measurements of rectangular bars are made in torsion and the calculations contain assumptions that may depend on geometry. How this influences error, particularly at low torque levels is not known. Second, the strains were kept constant at 0.1% other strains might not yield the same results. On the other hand one would expect an inverse proportionality to exist between the magnitudes of error and strain. Thirdly, these errors were estimated for a frequency of 1Hz. 

Relative energy part of the modulus at T = 298K, from stress-strain measurements with X = 1.02-1.04 (ored u red and from torsional vibration experiments (G y/G 5... 

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