Hysteresis in kinking nonlinear elastic solids and the Preisach-Mayergoyz model A. G. Zhou (周爱国, 1, * S. Basu, 1 G. Friedman, 2 P. Finkel, 1,† O. Yeheskel, 1,‡ and M. W. Barsoum 1 1 Department of Materials Science and Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA 2 Department of Electrical and Computer Engineering, Drexel University, Philadelphia, Pennsylvania 19104, USA Received 14 April 2010; published 13 September 2010 Herein we show that the stress-induced, dislocation-based, elastic hysteric loops of kinking nonlinear elastic solids—polycrystalline cobalt, 10 vol % porous Ti 2 AlC, and fully dense Ti 3 SiC 2 —obey the scalar Preisach- Mayergoyz phenomenological model because they exhibit wiping out and congruency, two necessary and sufficient tenets of the model. We also demonstrate the power of the model in predicting the response of these materials to complex stress histories, as well as, determining the distributions of the threshold and friction stresses associated with the incipient kink bands—the fundamental microscopic units responsible for kinking nonlinear elasticity. DOI: 10.1103/PhysRevB.82.094105 PACS numbers: 62.20.-x, 02.90.+p I. INTRODUCTION Mechanical hysteresis is common in solids. Granular sol- ids, such as rocks, exhibit hysteresis that has been related to the internal friction of cracks that are common in such solids. 1–3 One of the hallmarks of this mechanism is a reduc- tion in modulus with cycling. 4–6 This paper does not deal with this type hysteresis. As shown below, the hysteresis de- scribed herein is due to the reversible motion of dislocations; intergranular friction does not play a role. Probably the most convincing evidence for this state of affairs are the following facts: i at a given stress, fine-grained samples, with their much higher number of intergranular contacts dissipate less energy than their coarse-grained counterparts; 7–9 ii no cy- clic softening is observed even after 100 cycles at 700 MPa in fine-grained Ti 3 SiC 2 . 10 Instead, after deformation at higher temperatures, cyclic hardening is observed. 7,10 iii Mg and Co—clearly not granular solids—were shown to follow the same relationships as Ti 3 SiC 2 . The remainder of this paper will thus focus on kinking and how it relates to hysteresis. Recently we classified a large class of solids as kinking nonlinear elastic, KNE, for which the only requirement for belonging is plastic anisotropy. 7–9 This class is quite large and includes layered solids such as mica, 11 MAX phases, 7–9,12 and their solid solutions 13 as well as hexagonal solids such as graphite, 14 titanium, 15 magnesium, 16 cobalt, 17 sapphire, 18 and LiNbO 3 , 19 among many others. When loaded, KNE solids outline fully reversible, reproducible, stress- strain hysteretic loops. This response has been attributed to the formation of dislocation-based incipient kink bands IKBs 7,8 comprised of multiple parallel dislocation loops in which dislocation segments, on either side, are of opposite signs. As shown by Frank and Stroh, 20 the shape of the IKBs endows them with, first, a threshold stress needed to nucleate them and, second, a driving force that results in their shrink- age, or elimination, when the load is reduced below a certain threshold. Initially, our motivation was to try to obtain an appropri- ate model that described hysteresis and end-point memory of KNE solids. Following the lead of the geologists, 21–23 we tested the Preisach model. First developed to describe ferro- magnetic hysteresis, 24,25 the Preisach model is based on the idea that macroscopically observed irreversible processes can be decomposed into independent switching events described by independent bistable relays. Mayergoyz, 26,27 recognizing that the Preisach model offered a general mathematical framework for the description of hysteresis of different physical origins, derived the necessary and sufficient condi- tions for representation of any given hysteresis by the Prei- sach model. These conditions are: first, that each local stress maximum wipes out the effect of other local stress maxima below it, and second, congruency of the hysteresis loops ob- tained via cycles with the same end points of input, but dif- ferent prehistories. Mayergoyz called these properties wiping out and congruency, respectively. Thereafter the Preisach model was renamed the Preisach-Mayergoyz, or PM, model. Before the PM model can be used it is essential to estab- lish that wiping out and congruency are indeed valid. It should be noted that Guyer et al. 21,28 and Ortín 29 successfully applied this model to describe the nonlinear elastic response of granular geological materials and a shape memory alloy, respectively. And while previous work has clearly shown wiping out, as far as we are aware, dislocation-based congru- ency has never been reported in mechanical systems. In this paper, we present the experimental evidence that verifies that the PM model can indeed be used to describe the response of KNE solids to stress, , and consequently illustrate the pre- dictive power of this conclusion. Most importantly, we show that the model can be used to calculate the distributions of the onset and friction stresses associated with the IKBs—the fundamental microscopic units responsible for kinking non- linear elasticity. Before presenting the experimental evidence it is important to summarize our KNE model and how it relates to the PM model and to briefly explain kink band formation. Kink bands have been invoked to explain the deformation of numerous materials and structures including organic crystals, 30 card decks, 31 rubber laminates, 32 oriented polymer fibers, 32–36 wood, 37 graphite fibers, 38,39 laminated C-C and C-epoxy composites, 40–42 among others. However, outside geology, 43–45 the formation of kink bands in crystalline solids, has been for the past 70 years, and since first reported by Orowan, 46 more of an after- thought. Orowan induced kink band formation when he com- PHYSICAL REVIEW B 82, 094105 2010 1098-0121/2010/829/09410510 ©2010 The American Physical Society 094105-1