Vol. 10, No. 2, 2015

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ISSN: 1559-3959
Nonuniqueness and instability of classical formulations of nonassociated plasticity, I: Case study

Thomas Pučik, Rebecca M. Brannon and Jeffrey Burghardt

Vol. 10 (2015), No. 2, 123–148
Abstract

A dynamic instability, which is relatively unexplored in the literature despite having been long ago previously asserted to exist for any conventional nonassociated plastic flow model, is illustrated by means of an example problem. This instability is related to a condition known as achronicity, in which the wave speed in plastic loading is greater than that in elastic unloading (making it absolutely not related to the well-studied phenomena of localization and flutter). The one-dimensional example problem initializes an elastic-plastic half-space to a initially quiescent uniform state of prestress that places it in an achronic condition if a nonassociated flow rule is used. The initial stress state is perturbed by an axial stress pulse applied at the surface. The problem is first solved analytically for the case of constant wave speeds, and it is shown to possess a two-parameter family of nonunique solutions. These solutions are unstable in that both the amplitude and the width of the propagating pulse increase linearly with time. The case-study problem technically represents spontaneous motion from a quiescent state, but it does not violate thermodynamics (as the energy driving the instability is available from elastic stored energy of the initial prestress). The example problem is additionally solved numerically for both constant and nonconstant plastic wave speeds, where the instability is observed in either case. Furthermore, it is shown that neither the constant nor the nonconstant wave speed solution converges with mesh refinement, which therefore represents a numerical inadmissibility associated with the underlying loss of uniqueness of solution. It is the nonuniqueness of the unstable solution, not the existence of the instability itself, that is of primary concern. Unlike conventional localization instability, this achronic instability is not yet known to be a real phenomenon. This case-study problem illustrates the need for novel laboratory testing methods sufficient to determine if the instability is truly physical, or merely an anomalous shortcoming of classical nonassociated plasticity formulations. Some guidance for appropriate laboratory testing is presented, with emphasis on why such testing is highly nontrivial as a result of irreducible uncertainty in direct validation of a regular flow rule.

Keywords
plasticity, flow rule, nonassociated flow rule, instability, incremental nonlinearity
Milestones
Received: 2 July 2014
Revised: 7 March 2015
Accepted: 4 April 2015
Published: 5 August 2015
Authors
Thomas Pučik
(Deceased) Pučik Consulting
2057 Brixham Dr.
Roseville, CA 95747
United States
Rebecca M. Brannon
University of Utah
50 S. Campus Dr.
Salt Lake City, UT 84108
United States
Jeffrey Burghardt
Schlumberger
1935 S. Fremont Dr.
Salt Lake City, UT 84104
United States