Analytical solution method for rheological problems of solids

Abstract

In classical continuum theory, Volterra's principle [1, 2] is a long-known method to solve linear rheological (viscoelastic) problems derived from the corresponding elastic ones. Here, we introduce and present another approach that is simpler to apply (no operator inverse is required to compute but only linear ordinary differential equations to solve). Our method starts with the known elastic solution, replaces the elasticity coefficients with time dependent functions, derives differential equations on them, and determines the solution corresponding to the initial conditions. We present several examples solved via this new method, like tunnels and spherical hollows opened in various initial stress states, and pressurizing of thick-walled tubes and spherical tanks. These examples are useful for applications and, in parallel, are suitable for testing and validating numerical methods of various kinds.