Toward an equation of state for solid materials with memory by use of the half-order derivative

The time derivative operator does not depend upon the difference between the current time and the past times; however, the fractional time derivative operator does. Thus, it is reasonable to expect that the fractional derivative would be useful in describing the mathematical theory of the behavior of materials with memory. An equation of state is proposed for solid materials with memory by introducing the half-order fractional calculus derivative in order to relate to the empirical expression used in the fundamental work of Tobolsky and Catsiff. This theory replaces the three empirical functions used by Tobolsky and Catsiff in reducing their experimental data for the low temperature glassy region, the transition region and the quasi-static rubbery plateau region. The square root differential operator with respect to time, D-1/2, has built in memory since the kernel of this operator depends upon the difference between the current time and the past time. D-1/2 is a special case of the Abel operator, which is used in the theory of integral equations. The present theory introduces integrals into the standard linear solid resulting in an integral differential equation governing the stress, strain and temperature. It is shown that this proposed linear equation of state for a solid material, which undergoes a second order transition, requires only four phenomenological constants to completely determine the behavior of the solid material. These four phenomenological constants are two relaxation times and two creep times, both of which are functions of the temperature.