Analytical investigation of the unloading wave: PMM vol. 40, n≗ 2, 1976, pp. 327–336

Abstract

In the general case, the determination of the unloading wave shape [1] in the theory of elastic plastic wave propagation is reduced to the solution of a functional equation of complex structure. A characteristics method [2] is proposed for the approximate construction of the unloading wave, in particular loading cases formulas are obtained to determine its initial slope [3] and the next derivatives at the initial point [4–6]. An investigation of the general properties of an unloading wave is given in [7]. It is shown that as the load tends to zero asymptotically, the unloading wave at the end of a semi-infinite bar has an asymptote with a slope determined by the elastic wave velocity. An investigation of the functional equation is given in this paper and a method of solution of this equation in the form of a power series is proposed. This approach to the problem permits obtaining both known and some new results. In the general loading case, formulas are obtained to determine the initial slope of the unloading wave and a method of determining the next derivatives at the initial point is indicated. Conditions are found for linear hardening for which the unloading wave is a straight line. The existence of an asymptote different from those mentioned in [7] is proved; it is shown how to continue the solution to adjacent sections by means of some known section, and an unloading wave in a material with delayed yielding is investigated.