A Rheological Model of Nonlinear Viscoplastic Solids

Abstract

A general rheological model of a nonlinear viscoplastic solid is developed, affording better quantitative prediction of behavior of viscoplastic materials. Rheological properties are quantified in terms of three types of curves obtained by tests, commonly used for description of mechanical material properties: constant rate, force deformation, creep, and relaxation curves. The mathematical model is based on a characteristic nonlinear differential equation describing the mechanical properties of a material. This equation defines the force F acting on a test specimen as composed of a cubic elasticity force K0x+rx3, viscous damping force Cẋ, and internal friction force Ff (sgn ẋ). The elastic parameter K0 quantifies linear elasticity while the strain hardening (or softening when negative) parameter r affords prediction of nonlinear behavior of the material. The viscous and Coulomb damping forces properly account for both rate‐dependent and rate‐independent energy dissipative properties of the material. Local linearization of the characteristic equation permits a closed‐form solution for creep at constant loading force and relaxation at constant deformation. The parameters of the nonlinear model can be read off or computed from practical curves, whereby a realistic link may be established between the mathematical model and real viscoplastic materials. The properties of a nonlinear viscoplastic solid are further clarified in terms of a time‐ and deformation‐dependent relaxation modulus. By setting the nonlinear parameters to zero the deformation‐independent linear relaxation modulus of a Boltzmann solid is obtained. It is believed that this model will afford better quantification of rheological properties of materials and improved interpretation of experimental data.