Evaluating the second coefficient of viscosity from sound dispersionor absorption data

T HERMAL response of a viscoelastic rod subject to high- frequency cyclic loading is studied. The material con- sidered models a solid rocket propellant with temperature- dependent thermal conductivity. The mathematical model takes the form of a set of nonlinear, coupled partical differen- tial equations. An algorithm based on an iterative finite- difference method and an averaging technique have been developed to solve these coupled equations directly. The results indicate that the temperature of a material with temperature-dependent thermal conductivity is higher than that of one with constant thermal conductivity. This shows that noticeable errors can occur, especially at the thermal reso- nant frequencies where thermal conductivity is considered constant. Viscoelastic materials are dissipative in nature. They dissipate large amounts of mechanical energy in the form of heat when subjected to high-frequency loading. Solid pro- pellant, which is used as the rocket fuel, is also a viscoelastic material. Due to the high speed of the rocket, the solid fuel is subjected to high-frequency vibration. Heating due to vibra- tion near a resonance frequency may lead to melting or material failure. Tormey and Britton conducted vibration tests on solid fuel. They found that the heating due to vibraReceived Feb. 1, 1988; revision received Feb. 20, 1989. Copyright © 1989 by the American Institute of Aeronautics and Astronautics, Inc. All rights reserved. *Assistant Professor, Mechanical Engineering Department. Member AIAA. tGraduate Student, Mechanical Engineering Department. tion increased the material temperature significantly to the ex- tent that it even flowed out of the motor. Henter derived a set of coupled partial differential equations for the propagation of stress, strain, and temperature distribution in a viscoelastic media. Huang and Lee studied longitudinal oscillations of viscoelastic rods. They solved the nonlinear coupled equations by iteration, thus determining stress and temperature distribu- tions along the rod. Mukherjee solved the same problem by a simpler method. He substituted a partially linear equation for the nonlinear differential equation and then solved this equa- tion by the finite-difference method. In all the papers cited certain parameters such as thermal conductivity and specific heat, which are weak functions of temperature, are assumed to be constants. This assumption, however, can lead to signifi- cant errors, especially at higher temperatures and critical fre- quencies.' This is due to the strong temperature dependence of the mechanical properties, which makes the system of equa- tions very sensitive to temperature variations. In this paper the effects of temperature-dependent thermal conductivity on the temperature distribution of a viscoelastic rod subjected to cyclic loading are discussed. Numerical solutions are obtained by the finite-difference method over a wide range of frequen- cies. The effects of frequency on the temperature and dis- sipated energy are studied. This leads to a critical frequency at which the temperature is maximum. Heating due to vibration near a critical frequency may cause the solid fuel to soften and melt the material. Therefore, accurate evaluation of these critical frequencies is vital. To simulate the solid rocket pro- pellant, a viscoelastic rod of length 1 insulated on its lateral surface, as shown in Fig. 1, is considered. The left end is free, while the right end is attached to a vibrator that has a pre- scribed stress given by a = a0 cosotf where a0 is the stress amplitude, co the frequency, and / the time. The temperature of the vibrator is assumed constant at T0, a convective boun- dary condition is assumed at x = 0, H is the surface conduc- tance, and K is the thermal conductivity of the material. It is also assumed that the solid rocket propellant is a ther- mohelogically simple material. The objective is to find the temperature distribution along the rod. Solving the governing equations (the energy balance equation, equation of motion, and stress-strain relationship), they reduce to the form