Irreversible thermodynamics and variational principles with applications to viscoelasticity

• Main Author: Schapery, Richard Allan
• Format: Others
• Published: 1962
• Online Access: https://thesis.library.caltech.edu/6576/1/Schapery_ra_1962.pdf; Schapery, Richard Allan (1962) Irreversible thermodynamics and variational principles with applications to viscoelasticity. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/QEB0-N308. https://resolver.caltech.edu/CaltechTHESIS:08092011-111909246 <https://resolver.caltech.edu/CaltechTHESIS:08092011-111909246>

A unified theory of the thermo-mechanical behavior of viscoelastic media is developed from studying the thermodynamics of irreversible processes, and includes discussions of the general equations of motion, crack propagation, variational principles, and approximate methods of stress analysis. The equations of motion in terms of generalized coordinates and forces are derived for systems in the neighborhood of a stable equilibrium state. They represent a modification of Biot's theory in that they contain explicit temperature dependence, and a thermodynamically consistent inclusion of the time-temperature superposition principle for treating media with temperature-dependent viscosity coefficients. The stress-strain-temperature and energy equations for viscoelastic solids follow immediately from the general equations and, along with equilibrium and strain-displacement relations, they form a complete set for the description of the thermomechanical behavior of media with temperature-dependent viscosity. In addition, an energy equation for crack propagation is derived and examined briefly for its essential features by applying it to a specific problem. The thermodynamic equations of motion are then used to deduce new variational principles for generalized coordinates and forces, employing convolution-type functionals. Anticipating various engineering applications, the formulation is phrased alternately in terms of mechanical displacement, stresses, entropy displacement, and temperature in thermally and mechanically linear solids. Some special variational principles are also suggested for applications wherein the nonlinear thermal effects of temperature dependent viscosity and dissipation may be important. Building upon the basic variational formulation, it is next shown that when these convolution functionals are Laplace-transformed with respect to time, some convenient minimum principles result which can be employed for the approximate calculation of transformed, viscoelastic responses. The characteristic time dependence of exact and approximate solutions is then derived and used in relating error in approximate viscoelastic solutions to error in the associated elastic solutions. The dissertation is concluded with a study of some approximate methods of viscoelastic analysis. First, the important problem of inverting complicated Laplace transforms to physical time-dependent solutions is resolved by advancing two easily applied, approximate methods of transform inversion. These inversion methods and variational principles are then used in some illustrative, numerical, examples of stress and heat conduction analysis.