This paper is concerned with the phenomenology of visco-elasticity approached by means of differential equations. Three results are given which are new and not generally known to workers in the field. 1. The geometrical meaning of 1st order differential equations is given in the context of mechanical behavior of fibers. This frees the researcher of an obligation to interpret his equations in terms of springs and dashpots. 2. Operational methods are given for determining if any 1st order differential equation (linear or not) can provide an adequate description for a particular fiber. 3. For cases where a differential equation is satisfactory an operational method is given for constructing it directly in the laboratory rather than attempting to fit experimental trajectories to solutions of assumed differential equations.
A PROBLEM IN PHENOMENOLOGY
When a material exhibits elastic, plastic and fluid properties simultaneously (as most textile fibers do) its specification in succinct form becomes an intriguing exercise in applied mathematics and physical intuition. The great variety of behavior possible and the almost complete absence of a technically useful description of all that is possible has provided the stimulus for a great deal of research into this problem (1). 5 The problem of subsequent interpretation of mechanical behavior in terms of molecular theories is quite different and will not be considered here.
A GRAPHICAL SOLUTION At present, the most satisfactory and trustworthy description is pictorial, for example (2)--a line diagram is made of the force response to different types of extension as determined by a suitable testing machine. This method of description could be used in subsequent stress analysis if the loading pattern exerted by the testing machine bears some resemblence to the loads subsequently encountered by the material in use. But if not, then a second diagram must be made to cover the extended range of load patterns. And, in general, if the material is to be subjected to many different types of load a series of pictures will be needed for an adequate description. This is the situation with tire cord, for example, which is commonly subjected to steady forces, forces containing a few low frequencies, and impact forces containing many high frequencies. This is an engineer's solution to the problem of description--a representation of the behavior in functional form. To a American Viscose Corporation, Marcus Hook, Pa.
The boldface numbers in parentheses refer to the references appended to this paper.
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F. AKUTOWICZ [J. F. I.
be sure, the function is given graphically on several sheets of graph paper but still a function.
SOLUTION BY DIFFERENTIAL EQUATIONS
Another method of specifying mechanical behavior of fibers is in terms of differential equations relating force, elongation, and their time derivatives. Such an equation is
where a, fl and ~ are constants, F is force, D is elongation and t is time. This is an underdetermined equation which can be made determinate only by fixing either F(t) or D(t). Let D = it. Then the solution of Eq. 1 becomes F--1-1 /~-t-~ D=O. ot This is shown in Fig. 1 for different values of k. Such an equation de-V l" ;'" k' ~ k, D Fro. I. Some possible solutions to Eq. I.
scribes a rate sensitive material since k is the rate of extension of the specimen. Equation 1 by itself thus represents an infinity of Force-Elongation diagrams.
The capabilities of an underdetermined differential equation for describing variegated behavior have been appreciated for a long time. And as a result many mathematical models have been produced intending to describe visco-elastic behavior. A summary of some of the more successful models is given in (3).