The Hodograph Transformation in Plastic Waves With Discontinuous Loading Conditions

[+] Author and Article Information
L. Bevilacqua

Division of Graduate Studies, COPPE-UFRJ, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil

J. Appl. Mech 39(2), 407-415 (Jun 01, 1972) (9 pages) doi:10.1115/1.3422693 History: Received January 15, 1971; Revised April 12, 1971; Online July 12, 2010


In this paper the theory of the hodograph transformation is developed in a systematic way for one-dimensional wave propagation in elastic-plastic solids. The material is assumed to have isotropic strain-hardening characteristics with no Bauschinger effect. It is shown that in the hodograph plane it is possible to obtain a Riemann solution for the Cauchy problem. The mappings of different kinds of shock fronts and the elastic-plastic boundary onto the hodograph plane are derived. Two problems are solved using the hodograph method. The first one is related to a long rod subjected to a square pulse loading in stress at the free end. Although this problem is well known it provides a good illustration for the method. The second is related to a square pulse loading in stress followed by loading in reversal at the free end of a long rod. It is shown that under certain conditions the shock wave originated by loading in reversal penetrates the progressing plastic waves indefinitely. The region behind the shock front is plastic again but does not belong to the class of simple waves as the region ahead of the shock. The characteristics with positive slope in the region behind the shock in the X-t plane are concave toward the X-axis. The hodograph method is used to investigate if the condition of plastic loading is satisfied behind the shock and to find the first part of the elastic-plastic boundary.

Copyright © 1972 by ASME
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