Construction of Nonlinear Monotonic Functions With Selectable Intervals of Almost Constant or Linear Behavior

Abstract

A rather general method is given to construct classes of functions with an arbitrary almost constant (linear) initial interval followed by a nonprescribed interval of monotonic nonlinear behavior. This region of nonlinear behavior is succeeded by an unbounded interval of almost constant (linear) behavior. The functions contain not more than four selectable parameters and are synthesized from analytic, monotonic, normalized, and bounded base functions through the introduction of two separate kernel sets, subsequent addition, and integration. As examples we give the special functions based on the error, the hyperbolic tangent, the inverse tangent, a rational, and the incomplete gamma function. Limiting function forms, such as the bilinear form, are derived for limiting values of the parameters. The functions developed herein are primarily used as coefficient functions in a differential constitutive equation for viscoplasticity developed elsewhere. They are indispensible tools for the numerical modeling of stress-strain diagrams, strain (stress)-rate effects, creep and relaxation curves for monotonic, and cyclic loading both in one and three dimensions.