L. H. Sperling

Departments of Chemical Engineering and Materials Science and Engineering Materials Research Center,

Center for Polymer Science and Engineering, and Polymer lnterfaces Center

Lehigh University Bethlehem, PA 18015-3194

Recently, chemists and chemical engineers interested in polymers have become interested in mechanical behavior. This Back Page will be devoted to non-fracture mechanical terminology, The subject has been reviewed many times.^1-3

The first term most people need is the modulus. The most common modulus is Young’s modulus, E, in elongation or compression, defined by the ratio stress, s, to strain, e,

E = s / e

A major consideration involves the time required for the experiment, or the inverse, the frequency of the measurement. All materials exhibit creep and relaxation, commonly called viscoelasticity in polymers.

The standard measurement is the ten second measurement or its equivalent, a tenth of a Hertz. The modulus goes up if the experiment is done faster, or at higher frequency. In important frequency and temperature ranges, the Williams-Landel-Ferry (WLF) equation along with the time-temperature superposition principle can make an excellent correction.

For dynamic experiments, the favorite today for many purposes, a sign wave is imposed on the sample. The wave may be of constant or variable frequency. Then, the modulus is split into two quantities, a storage modulus, E’, a measure of the energy stored during a cycle, and the loss modulus, E’’, a measure of the energy lost. The quantity E’’ has a maximum at the glass transition of the polymer, where it softens from a glassy to a rubbery polymer. The quantity tand is the ratio of E’’ to E’.

The quantities E’ and E’’ are related to the complex Young’s modulus via

E* = E’ + iE’’

where i represents the square root of rninus one. The absolute value of E* is Young’s modulus.

While the above Young’s modulus relations refer to elongation and compression, another favorite involves shearing. A simple modern experiment has a polymer placed between two circular plates, one of which is rotated or oscillated with respect to the other. As before, principle variables can be temperature and/or frequency. The shear modulus, G, is related to E through Poisson’s ratio, n,

E=2(1 + n)G

Poisson’s ratio is a measure of the decrease of the thickness of a body relative to its increase in length on extension, good for only very small strains. Poisson’s ratio varies from a maximum of nearly 1/2 to a minimum of about 0.2. Thus, for many measurements on plastics, G is about 1/3E. The dynamic quantities G’ and G’’ and tand based on the ratio of G’’ to G’ have the corresponding meanings as those based on Young’s modulus.

Another quantity of interest is the bulk modulus, B, defined as the resistance of the material’s volume to hydrostatic pressure. It is related to Young’s modulus by

E=3B(1 - 2n)

Thus, Poisson’s ratio can approach 0.5, but never quite equal it. The inverse of the bulk modulus, and more often seen in the literature, is the compressibility, b. The inverse of Young’s modulus is called the compliance, J.

Two expansion coefficients are in wide use, the linear coefficient of expansion and the volume coefficient of expansion, a, the latter being three times the magnitude of the former. The quantity a is defined as the change in volume with temperature,

a = (1/V) (dV/dT) P

with a similar definition holding for the linear coefficient of expansion.

Polymers exhibit live distinct regions of viscoelasticity, see Figure 1. By number, they are 1, the glassy region, where the polymer is stiff, 2, the glass transition region, where the polymer softens, 3, the rubbery plateau region, where the polymer behaves rubbery, the rubbery flow region, where the polymer exhibits both rubbery and flow characteristics, and finally, 5, the liquid flow region, where the polymer can flow. If the polymer is crosslinked sufficiently, it doesn’t soften, but follows the dotted line. All the above refers to amorphous polymers. If the polymer is semi-crystalline, it may follow the dashed line. At the melting temperature, the modulus falls off rapidly, usually into the rubbery flow or liquid flow regions.

1. A. Kaye, R. F. T. Stepto, W. J. Work, J. V. Aleman, and A. Ya. Malkin, Pure Appl. Chem., 70, 701 (1998).
2. L. E. Nielsen and R. F. Landel, Mechanical Properties of Polymers and Composites, Dekker, New York, 1994.
3. L. H. Sperling, Introduction to Physical Polymer Science, 3^rd Ed., Wiley, New York, 2001.

Figure 1. The five regions of viscoelasticity in amorphous polymers, plus the crosslinked behavior, dotted line, and semi-crystalline behavior, dashed line.³

First published: ACS Division of Polymeric Materials: Science and Engineering (PMSE), 84 (Fall, 2000).