Consistency for two multi-mechanism models in isothermal plasticity

Abstract

A new class of constitutive equations (the multi-mechanism models) devoted to the prediction of ratcheting and of complex material behavior, for instance of steel, has developed during the last years. The main difference between these models and the “Chaboche” type models is the representation of different mechanisms of plastic strain with a possible interaction between them. This sophistication has a cost related to the complexity to understand well how the various parameters act. The identification process thus becomes more difficult as it will be difficult to give reasonable bounds. Two multi-mechanism models have been proposed during the last decade, see Cailletaud and Sai [Cailletaud, G., Sai, K., 1995. Study of plastic/viscoplastic models with various inelastic mechanisms. International Journal of Plasticity 11, 991], Taleb et al. [Taleb, L., Cailletaud, G., Blaj, L., 2006. Numerical simulation of complex ratcheting tests with a multi-mechanism model type. International Journal of Plasticity, 22, 724], and Sai and Cailletaud [Sai, K., Cailletaud, G., 2007. Multi-mechanism models for the description of ratcheting: effect of the scale transition rule and of the coupling between hardening variables. International Journal of Plasticity 23, 1589]. In this paper, we would like to ensure a kind of “after-sales service” by exploring the thermodynamic way in order to see if there exist any restrictions on the parameter values. We prove thermodynamic consistency of some two-mechanism models under suitable conditions. As a result, we obtain conditions for the parameters which ensure the non negativity of the free energy and the fulfilling of the Clausis–Duhem inequality. Besides this, we perform numerical simulations, considering classical histories (monotonic tension, cyclic strain controlled, uniaxial ratcheting, biaxial ratcheting). These simulations show that the violation of the established restrictions may lead to anomalous predictions.

Introduction

The understanding of ratcheting is one of the more complex tasks in mechanical engineering. This complexity appears through the recent experimental observations made by several authors: Aubin et al., 2003, Aubin and Degallaix, 2006, Bari and Hassan, 2000, Bari and Hassan, 2001, Bari and Hassan, 2002, Basuroychowdhury and Voyiadjis, 1998, Bocher et al., 2001, Chen and Jiao, 2004, Chen et al., 2005, Corona et al., 1996, Dieng et al., 2005, Feaugas and Gaudin, 2004, Hassan and Kyriakides, 1992, Hassan and Kyriakides, 1994a, Hassan and Kyriakides, 1994b, Hassan et al., 1992, Johansson et al., 2005, Kang et al., 2002, Kang and Gao, 2002, Manonukul et al., 2005, Ohno et al., 1998, Colak, 2004, Portier et al., 2000, Taleb and Hassan, 2006, Vincent et al., 2004, Voyiadjis and Abu Al-Rub, 2003, Voyiadjis and Basuroychowdhury, 1998, Yaguchi and Takahashi, 2005a, Yaguchi and Takahashi, 2005b, Yoshida, 2000 and others. Significant efforts have been made by many researchers, especially in the last two decades, in order to describe the observed phenomena by adequate phenomenological constitutive equations correctly; in particular by Abdel-Karim, 2005, Abdel-Karim and Ohno, 2000, Bari and Hassan, 2002, Basuroychowdhury and Voyiadjis, 1998, Bocher et al., 2001, Burlet and Cailletaud, 1987, Chaboche, 1991, Chaboche, 1994, Delobelle et al., 1995, Jiang and Sehitoglu, 1996a, Jiang and Sehitoglu, 1996b, Johansson et al., 2005, McDowell, 1995, Ohno and Wang, 1993, Ohno et al., 1998, Colak, 2004, Taheri and Lorentz, 1999, Taleb et al., 2006, Vincent et al., 2004, Voyiadjis and Abu Al-Rub, 2003, Voyiadjis and Basuroychowdhury, 1998, Yaguchi and Takahashi, 2005a, Yaguchi and Takahashi, 2005b, Yoshida, 2000 and others.

Of course, for the complexity of these models one has a certain cost related to the number and the complexity of the material parameters. The “direct” experimental identification of these parameters is sometimes very difficult and/or expensive and sometimes impossible as some parameters may represent some physical phenomena difficult to see macroscopically. Therefore, more and more researchers and engineers use “automatic” identification based on some optimization processes. This procedure saves time and money but may be dangerous as we can lose the physical meaning of the parameters. Hence, if the identified set of parameters may be adequate for the identification conditions this is not necessarily true for any other conditions. In order to ensure a safe use of the models, we need some kind of “protection” avoiding the “inappropriateness” of the identified set of parameters. Such a role may be ensured by thermodynamic considerations for the models based on this approach.

In this paper, we try to do this work by considering some multi-mechanism models. These models differ from the classical one (represented for instance by the Chaboche model) by taking into account different possible mechanisms for plastic strain with a possible interaction between them. This class of models is attractive as for a reasonable number of material parameters; a wide range of phenomena may be represented.

We will consider successively two versions of multi-mechanism models with one criterion. The first one, here named as 2M1C-Ia, was proposed by Cailletaud and Sai in 1995. The second one was proposed in Taleb et al. (2006), it is named as 2M1C-Ib. We choose this notation having in mind planned future investigations.

The objective of the present paper is not to develop new models, but to give some complementary aspects helping the use of already published models. For some modifications of two-mechanism models we refer to the recent paper by Sai and Cailletaud (2007). An application of two-mechanism models with two criteria to inelastic behavior of a nickel alloy can be found in Sai et al. (2004).

For each of the models under consideration, we will establish sufficient conditions in order to ensure both the non-negativity of the free energy and the validity of the Clausius–Duhem inequality. In order to focus, and as essential issues of two-mechanism models can be already shown in iso-thermal case, we neglect thermal effects. To deal with them in the context of multi-mechanism models is a topic of future investigations.

Concerning two-mechanism models, the problem of thermodynamic consistency has been addressed (cf. Sai (1993)), but not solved in detail. Of course, in the case of so-called “generalized standard models” (cf. Besson et al. (2001)), thermodynamic consistency is easily ensured by general considerations. But modifications arising in certain applications (cf. Taleb et al., 2006, Sai and Cailletaud, 2007) lead to models non being “generalized standard”. These cases require additional specific efforts. For investigations of thermodynamic consistency of complex material behavior we refer to Haupt, 2000, Lion, 2000, Helm, 2006 e.g.

Besides this, the complex material behavior of steel can be modeled in the framework of two-mechanism models. We refer to Videau et al., 1994, Wolff et al., 2008 for a macroscopic approach, and to Aeby-Gautier and Cailletaud, 2004, Sai and Cailletaud, 2007 for a mesoscopic–macroscopic approach.

Finally, we note that the complex behavior of important materials (such as visco-plastic materials, shape-memory alloys, soils, granular materials, composites, biological tissues) being intensively investigated may lead to multi-mechanisms in a “natural way”. However, the multi-mechanism approach is not directly used. Exemplarily, we refer to Fang, 2004, Saleeb and Arnold, 2004, Wulandana and Robertson, 2005, Nguyen, 2006, Anandarajah, 2007, Helm, 2007, Reese and Christ, 2007.

The next section presents some notations and definitions used in this paper while Section 3 is devoted to a short presentation of the two multi-mechanism models considered in this study. In the following Section 4, the conditions ensuring the thermodynamic consistency are investigated for both models. In Section 5, we present numerical simulations, and show that data not fulfilling the established restrictions may lead to anomalous predictions. A summary is given in the last Section 6.