Modeling of transformation-induced plasticity and its effect on the behavior of porous shape memory alloys. Part I: constitutive model for fully dense SMAs
Introduction
Since the discovery of the shape memory alloys (SMAs) in the 1960s researchers have been investigating both experimental aspects of their behavior as well as their constitutive modeling. While most of the issues associated with the thermomechanical response of SMAs have been addressed, there are still questions that remain open. One such open question is associated with the development of plastic strains during cyclic phase transformation. As indicated in the literature review below, this problem has been addressed for the case of one-dimensional SMA actuators (wires) undergoing cyclic temperature-induced phase transformation. However, a three-dimensional formulation is still lacking.
This problem resurfaces when dealing with porous SMAs undergoing cyclic phase transformation. It has been experimentally observed that large amount of plastic strain accumulates during such loading and it is very desirable to have a model, which is able to capture this effect. When modeling porous SMAs using micromechanical methods, one has to first establish an accurate model for the dense SMA matrix, which forms the walls and the connecting struts of the porous SMA. Due to the inherent three-dimensional stress state that develops during loading of porous SMAs, their successful modeling requires a three-dimensional model for the dense SMA matrix, that accounts for plastic strain development during cyclic loading.
Therefore, the focus of the current research effort will be to establish a three-dimensional model for fully dense SMAs with simultaneous evolution of transformation and plastic strains. Special attention will also be paid to the estimation of the material parameters required for the modeling of stress-induced phase transformation. The development of this model for dense SMAs will be presented in the first part of this two-part paper. It will then be used in the second part (Entchev and Lagoudas, to be published) to obtain the effective properties of porous SMAs using micromechanical averaging techniques.
SMAs are metallic alloys which can recover permanent strains when they are heated above a certain temperature. The key characteristic of all SMAs is the occurrence of a martensitic phase transformation. The martensitic transformation is a shear-dominant diffusionless solid-state phase transformation occurring by nucleation and growth of the martensitic phase from a parent austenitic phase. When an SMA undergoes a martensitic phase transformation, it transforms from its high-symmetry, usually cubic, austenitic phase to a low-symmetry martensitic phase, such as the monoclinic variants of the martensitic phase in a NiTi SMA.
The SMAs can be fabricated either as single crystals, such as some of the copper-based SMAs, or polycrystal, such as NiTi-based, iron-based and copper-based SMAs. While single crystal SMAs can be used for detailed analysis of the nature of the martensitic phase transformation, their use in applications is limited by the high manufacturing costs. Thus, since most practical applications involve polycrystalline SMAs, the current work will focus on the description of polycrystalline SMAs.
The key effects of SMAs associated with the martensitic transformation, which are observed according to the loading path and the thermomechanical history of the material are: pseudoelasticity, one-way shape memory effect and two-way shape memory effect.
An SMA exhibits the Shape Memory Effect (SME) when it is deformed while in the martensitic phase and then unloaded while still at a temperature below M0f. If it is subsequently heated above A0f it will regain its original shape by transforming back into the parent austenitic phase. The pseudoelastic behavior of SMAs is associated with recovery of the transformation strain upon unloading and encompasses both superelastic and rubberlike behavior (Otsuka and Wayman, 1999). The superelastic behavior is observed during loading and unloading above A0s and is associated with stress-induced martensite and reversal to austenite upon unloading. When the loading and unloading of the SMA occurs at a temperature above A0s, partial transformation strain recovery takes place. When the loading and unloading occurs above A0f, full recovery upon unloading takes place.
The superelastic behavior described above constitutes an approximation to the actual behavior of SMAs under applied stress. In fact, only a partial recovery of the transformation strain induced by the applied stress is observed. A small residual strain remains after each unloading. Experimental results on the behavior of SMAs undergoing cyclic loading have been presented by McCormick and Liu (1994), Strnadel et al., 1995a, Strnadel et al., 1995b, Lim and McDowell, 1994, Lim and McDowell, 1999, Bo and Lagoudas (1999b), Kato et al. (1999) and Sehitoglu et al. (2001), among others. The thermomechanical cycling of the SMA material results in what is called training process. Different training sequences can be used, i.e., by inducing a non-homogeneous plastic strain at a martensitic or austenitic phase; by aging under applied stress, in the austenitic phase, in order to stabilize the parent phase, or in the martensitic phase, in order to create a precipitant phase (Ni–Ti alloys); by thermomechanical, either superelastic or thermal cycles (Manach and Favier, 1993; Hebda and White, 1995; Liu and McCormick, 1990; Miller and Lagoudas, 2001). The main result of the training process is the development of Two-Way Shape Memory Effect (TWSME). In the case of TWSME, a shape change is obtained both during heating and cooling. Another effect of the training cycle is the development of macroscopically observable plastic strain. The magnitude of this strain is comparable to the magnitude of the recoverable transformation strain. The training also leads to secondary effects, like change in the transformation temperatures, change in the hysteresis size and decrease in the macroscopic transformation strain. These effects are similar to those observed during thermomechanical fatigue tests (Rong et al., 2001). It is important to define optimal conditions of training, because an insufficient number of training cycles produces a non-stabilized two-way memory effect and over-training generates unwanted effects that reduce the efficiency of training.
During the last two decades the area of constitutive modeling of SMAs has been a topic of many research publications and significant advancements have been reported. The majority of the constitutive models reported in the literature can be formally classified to belong to one of the two groups: micromechanics-based models and phenomenological models. Representative works from both of these groups are reviewed below.
The essence of the micromechanics-based models is in the modeling of a single grain and further averaging of the results over a representative volume element (RVE) to obtain a polycrystalline response of the SMA. Such models have been presented in the literature by different researchers. As an example, the micromechanics-based model based on the analysis of phase transformation in single crystals of copper-based SMAs has been presented by Patoor et al. (1996). The behavior of a polycrystalline SMA is modeled by utilizing the model for single crystals and using the self-consistent averaging method to account for the interactions between the grains. A micromechanical model for SMAs which is able to capture different effects of SMA behavior such as superelasticity, shape memory effect and rubber-like effect has been presented by Sun and Hwang, 1993a, Sun and Hwang, 1993b. In their work, the evolution of the total transformation strain in each grain is considered. Then the evolution of the martensitic volume fraction is obtained by balancing the internal dissipation during the phase transformation with the external energy output. One of the recent micromechanical models for SMA has been presented by Gao et al., 2000a, Gao et al., 2000b, where the evolution of each martensitic variant has been considered. The advantage of the micromechanics models is their ability to predict the effective material response using only crystal lattice parameters and information from the martensitic transformation at the crystalline and grain levels. Thus, their use provides valuable insight on the phase transformation process at the crystal level. Their disadvantage, however, is in the large number of numerical computations required to be performed, and the complexity of the micro-to-macro transition. Thus the use of such models for directly modeling the overall response of SMA structures is rather limited.
In the case of the phenomenological models, a macroscopic free energy function that depends on internal variables and their evolution equations are usually postulated and used in conjunction with the second law of thermodynamics to derive constraints on the constitutive behavior of the SMA material. Thus the resulting models do not directly depend on the behavior of the material at the microscopic level, but the effective behavior of the polycrystalline SMA. These models have the advantage of being easily integrated into a structural analysis computational framework, e.g., using the finite element method.
Some of the early three-dimensional models from this group were derived by generalizing one-dimensional results, such as the models by Liang and Rogers (1992), Brinson (1993), Boyd and Lagoudas (1994) and Tanaka et al. (1995). In a publication by Lagoudas et al. (1996) it has been shown that the various phenomenological models can be unified under a common thermomechanical framework. The differences among the different models arise from the specific choices of the transformation hardening function. More recent phenomenological models have also been presented by Leclercq and Lexcellent (1996), Auricchio et al. (1997), Reisner et al. (1998), Rengarajan et al. (1998), Levitas (1998), Govindjee and Hall (1999) and Rajagopal and Srinivasa (1999). In a recent work Qidwai and Lagoudas (2000b) presented a general thermodynamic framework for phenomenological SMA constitutive models, which for different choice of the transformation function can be tuned to capture different effects of the martensitic phase transformation, such as pressure dependance and volumetric transformation strain. It should be mentioned, however, that the majority of the models are not able to handle the case of detwinning of the martensitic variants. Notable exceptions are the works by Lagoudas and Shu (1999) and Brinson (1993) where the martensitic volume fraction is decomposed into two parts: twinned and detwinned martensite.
In addition to modeling of the development of transformation strains under pseudoelastic or SME conditions, during martensitic phase transformations, several other modeling issues have also been topics of intensive research. One of the important problems recently addressed by the researchers is the behavior of SMAs under cycling loading. During cycling phase transformation a substantial amount of plastic strains is accumulated. In addition, the transformation loop evolves with the number of cycles and TWSME is developed. Based on the experimental observations researchers have attempted to create models able to capture the effects of cycling loading. One-dimensional models for the behavior of SMA wires under cycling loading have been presented by Tanaka et al. (1995), Lexcellent and Bourbon (1996), Lexcellent et al. (2000) and Abeyaratne and Kim (1997), among others. A three-dimensional formulation is given by Fischer et al. (1998). Their model defines a transformation function to account for the development of the martensitic phase transformation and a separate yield function to account for the development of plastic strains. However, neither the identification of the material parameters nor implementation of the model is presented in that work. One of the most recent works on the cyclic behavior of SMA wires has been presented in a series of papers by Bo and Lagoudas, 1999a, Bo and Lagoudas, 1999b, Bo and Lagoudas, 1999c and Lagoudas and Bo (1999). In that work most of the issues regarding behavior of SMA wires under cycling loading, including the development of TWSME, have been addressed. The calibration of the model has been performed using experimental data for SMA wires undergoing thermally induced phase transformation under applied load. The dependance of the transformation strain on the level of the applied stress has also been addressed. The modeling results have been compared with experimental data for the case of SMA wire actuators thermally activated under load and have been found to be in good agreement.
As described in the literature review above, the experimental observations for SMAs undergoing cyclic loading in the pseudoelastic regime have shown that a significant part of the developed strain is not recovered upon unloading. This effect has been attributed to the development of plastic strains during the thermomechanical cycling of dense SMAs undergoing phase transformation. Similar effects have also been observed during cyclic mechanical loading of porous SMAs.
The majority of the models found in the literature describing the development of plastic strains in SMAs have one-dimensional formulation. Since the ultimate goal of the current study is to be able to accurately model the behavior of porous SMAs, where three-dimensional effects must be taken into account, successful modeling requires a three-dimensional model for the dense SMA matrix undergoing simultaneous transformation and plasticity. In addition to modeling porous SMAs, a three-dimensional formulation is also needed for modeling various other SMA components, such as torque tubes. In contrast to the earlier work by Bo and Lagoudas (1999a), which considered thermally-induced phase transformation, the current paper will focus on the modeling of stress-induced transformation, where both transformation and plastic strains occur simultaneously as a result of the applied stress.
Therefore, the main objective of the first part of this two-part work will be to develop a three-dimensional constitutive model for fully dense SMAs which is able to account for simultaneous development of transformation and plastic strains during phase transformation under applied loads and evolution of the material behavior during cyclic loading. The three-dimensional model development will follow the methodology presented for the one-dimensional case by Bo and Lagoudas, 1999a, Bo and Lagoudas, 1999b, Bo and Lagoudas, 1999c and Lagoudas and Bo (1999), of transformation induced plasticity. Since the above-mentioned works describe the behavior of SMAs undergoing thermally-induced transformation, the necessary modifications to adapt the formulation for the case of stress-induced martensitic transformation will be made in this work.
The remainder of the paper is organized as follows: in Section 2 the derivations of the fully dense SMA constitutive model are given. The material parameters for the model are identified in Section 3 and a procedure for their estimation is presented. The results of the model and their comparison with the available experimental data are presented in Section 4. A summary of the current research effort and conclusions are presented in Section 5.
The direct notation is adopted in this work. Capital bold Latin letters represent fourth-order tensors (stiffness S, etc.) while bold Greek letters are used to denote second-order tensors (stress σ, strain ε). Regular font is used to denote scalar quantities as well as the components of the tensors. Multiplication of two fourth-order tensors A and B is denoted by AB=(AB)ijkℓ≡AijpqBpqkℓ, while the operation “:” defines contraction of two indices when a fourth-order tensor acts on a second-order one, A:E≡AijkℓEkℓ.
Section snippets
Thermomechanical constitutive modeling of fully dense polycrystalline shape memory alloys
In this section the derivation of a three-dimensional thermomechanical constitutive model for SMAs undergoing cyclic loading which results in simultaneous development of transformation and plastic strains will be presented. The model is an extension of the one-dimensional model presented by Bo and Lagoudas, 1999a, Bo and Lagoudas, 1999b, Bo and Lagoudas, 1999c and Lagoudas and Bo (1999) for transformation induced plasticity due to thermally induced phase transformation under loading to three
Estimation of material parameters
As a final step to characterizing the thermomechanical behavior of SMAs undergoing cyclic loading, in this section the determination of the material parameters entering the model is discussed. Three groups of material parameters are identified. First, the parameters which are necessary to describe a stable transformation cycle are determined. In the context of the current model the stable transformation cycle is defined as a thermomechanical loading cycle during which no plastic strains are
Correlation with experimental data
The experimental results for NiTi undergoing cyclic loading will be simulated in this section. Two sets of experimental data will be used. First, in Section 4.1 the model will be calibrated using the results reported by Strnadel et al. (1995b) and presented in Section 2.1. Next, in Section 4.2 a set of experimental results obtained at Active Materials and Structures Laboratory at Texas A&M University will also be used to calibrate the model and the model simulations will be presented. Finally,
Conclusions
The derivation of a three-dimensional constitutive model for fully dense SMAs with simultaneous evolution of transformation and plastic strains has been presented in this Part I of a two-paper series. The model accounts for development of transformation and plastic strains during stress-induced martensitic phase transformation, as well as for the evolution of shape and size of the hysteresis with repeated transformation cycling. In the current formulation of the model the following quantities
Acknowledgements
The authors acknowledge the financial support of the Office of Naval Research Grant No. M00014-99-1-1069 monitored by Dr. Roshdy Barsoum and the support from the Texas Higher Education Coordinating Board TD&T Grant No. 000512-0278-1999. The authors also express their appreciation to Parikshith Kumar for his help in performing experiments on NiTi wires.
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