Computational protocols for viscoelastic Prony series hereditary convolution integrals and for variable coefficient integral–differential equations

  • Harry H. Hilton University of Illinois at Urbana-Champaign
  • Germain Sessou Universit ́e de Technologie de Belfort-Montb ́eliard (UTBN)


The hereditary and/or convolution integrals associated with linear viscoelastic material constitutive relations based on Prony series characterization are recast into or- dinary non-convolution time integrals, which can be more efficiently evaluated analytically and numerically. Application of this protocol greatly reduces computational time, CPU us- age and memory requirements used to solve linear temperature dependent and/or indepen- dent viscoelastic problems involving integral-differential equations with variable coefficients. The formulation includes temperature dependent materials with time and space dependent temperatures as well as stresses due to thermal expansions. Approximate approaches for dealing with time dependent temperatures are derived and solutions to non-convolution in- tegral equations as well as to differential equations with variable coefficients are formulated. Relaxation time consistent relations are derived for isotropic viscoelastic materials. Appli- cations of Galerkin and Runge-Kutta methods to viscoelastic solutions are discussed and evaluated. These protocols include solutions to IODEs and IPDEs with variable coefficients. An illustrative algorithm to be used in conjunction with differential equation solvers such as MATLAB’sTM ODE45 has been developed which allows for numerical solutions simultane- ously in both real and reduced time spaces without approximations of linear and nonlinear integral differential equations with variable or without coefficients. Similar protocols could be readily extended to other software such as MATHEMATICATM, MAPLETM, etc.