Exact Analytic Solutions of the Plastic Spin Equations in Simple Shear

Abstract

We prove that the first-order nonlinear differential system governing the plastic spin response in simple shear (Dafalias’s equations) is reduced to an equivalent equation of the Abel normal form by means of admissible functional transformations. In similar Abel equations result also the original and the generalized Volterra differential systems, describing the problem of two populations conflicting with one another. The above reduced Abel equations do not admit exact analytic solutions in terms of known (tabulated) functions, since only very special cases of these types of equations can be analytically solved in parametric form. We provide a mathematical solution methodology leading to the construction of exact implicit analytic solutions of the above-mentioned type of equations. Since the plastic spin nonlinear differential system results in a special unsolvable form of Abel’s equation of the normal form, we perform the exact implicit analytic solution of this system too.