Rate and thermal sensitivities of unstable transformation behavior in a ...
International Journal of Plasticity 20 (2004) 577–605 www.elsevier.com/locate/ijplas
Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy Mark A. Iadicola, John A. Shaw* The University of Michigan, Department of Aerospace Engineering, Francois-Xavier Bagnoud Building, 1320 Beal Avenue, Ann Arbor, MI 48109-2140, USA Received in final revised form 3 February 2003
Abstract A special plasticity-based constitutive model with an up–down–up flow rule used within a finite element framework has previously been shown to simulate the inhomogeneous nature and the thermo-mechanical coupling of stress-induced transformation seen in a NiTi shape memory alloy. This paper continues this numerical study by investigating the trends of localized nucleation and propagation phenomena for a wider range of loading rates and ambient thermal conditions. Local self-heating (due to latent heat of transformation), the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), the size of the specimen’s nucleation barriers, the loading rate, and the nature of the ambient environment all interact to create a variety of mechanical responses and transformation kinetics. The number of transformation fronts is shown to increase dramatically from a few fronts under nearly isothermal conditions to numerous fronts under nearly adiabatic conditions. A non-dimensional film coefficient and non-dimensional conductivity are identified to be the major players in the range of responses observed. It is shown that the non-dimensional film coefficient generally determines the overall temperature response, and therefore force–displacement response, of a transforming specimen; whereas, the non-dimensional conductivity is the more important player in determining the number of nucleations, and therefore the number of transformation fronts, that may occur. # 2003 Elsevier Ltd. All rights reserved. Keywords: Shape memory alloy; Thermo–mechanical behavior; Rate effects; Localization; Phase transformation; Finite element analysis
* Corresponding author. Tel.: +1-734-764-3395; fax: +1-734-763-0578. E-mail address: [email protected] (J.A. Shaw). 0749-6419/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0749-6419(03)00040-8
578
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
Nomenclature X, S t u T, Tamb L, R p, Lc A C k h q:X, qR Q , N, P ", " Tad h0 f Nf x s r Bi h k
axial coordinate and transformation front position (m) time (s) displacement (m) axial temperature of wire, ambient temperature ( C) length and radius of wire (m) circumference and characteristic lateral length of wire (m) cross-section area of wire (m2) density (g/m3) specific heat (J/g–K) thermal conductivity (W/m–K) lateral heat transfer coefficient (W/m2–K) thermal diffusivity (m2/s) Dirac delta, or elongation (m1, m) axial and lateral heat flux (W/m2) volumetric heat source (W/m3) nominal axial, nucleation and propagation stresses (MPa) engineering axial strain and transformation strain homogeneous adiabatic temperature rise ( C) stress free specific enthalpy change (J/g) specific latent heat of transformation (J/g) internal heat source/inelastic work rate ratio number of transformation fronts non-dimensional axial coordinate non-dimensional transformation front position non-dimensional radial coordinate non-dimensional time non-dimensional temperature Biot number non-dimensional lateral heat transfer coefficient non-dimensional thermal conductivity
1. Introduction Shape memory alloys (SMAs) are distinguished by two remarkable behaviors, the shape memory effect and pseudoelasticity. The pseudoelastic response, in particular, refers to the fact that rather large strains (up to 8%) can be imposed by mechanical loading and then recovered via a hysteresis loop upon unloading. Pseudoelastic behavior arises from a reversible stress-induced martensitic transformation between different crystalline phases, Austenite (low strain phase) and detwinned Martensite (high strain phase). The behavior is being exploited in a variety of novel applications (Otsuka and Wayman, 1998).
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
579
It is now well known that under uniaxial loading in some SMAs, especially the NiTi-based alloys, that these transformations are accompanied by unstable mechanical behavior and localized transformation. This results in one or more propagating transformation fronts which produce evolving, distinctly nonuniform deformation and thermal fields. Under nearly isothermal conditions (slow loading rate in a thermally convective or conductive medium) distinct stress plateaus are seen in the mechanical response; whereas, under nearly adiabatic conditions (high loading rate in a thermally insulating medium) the mechanical response seemingly stabilizes, and a somewhat distorted load–displacement responses is observed (see experimental data in Fig. 1). This apparent rate effect is not the usual viscoelastic effect, but is due to an interaction between the inherent temperature sensitivity of the material’s transformation stresses (Clausius–Clapeyron relation, see experimental data in Fig. 2) and the self-heating or self-cooling that can occur due to latent heat changes within the transforming material (see, for example, calorimetry data in Fig. 3). Furthermore, the local nature of the transformation (in the neighborhood of transformation fronts) exacerbates the self-heating effect and causes significant rate effects to be observed at strain rates that one might normally consider quasi-static.
Fig. 1. Mechanical response (via miniature extensometer, E) and local temperature response (via small thermocouple, T) of pseudoelastic NiTi wire (experimental data from Shaw and Kyriakides, 1995): (a) nearly : : isothermal conditions (=L=1104 s1 in water), (b) nearly adiabatic conditions (=L=1102 s1 in air).
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M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
: Fig. 2. (a) Pseudoelastic force–displacement responses of NiTi strip at =L=1104 s1; (b) fit of nucleation and propagation stress for A!M (loading) transformation (Shaw, 2000).
Fig. 3. Differential scanning calorimetry of NiTi strip (Shaw, 2000).
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
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These phenomena have been studied experimentally in detail in Shaw and Kyriakides (1995, 1997). These rather complex interactions can present serious difficulties to the application engineer who wishes to design novel devices or new composite materials with SMA constituents, and the phenomena have important repercussions on the performance, reliability, and controllability of the material as passive or active constituents. The aforementioned rate sensitivity clearly affects the response time of the material and will be quite different for different ambient thermal environments (stagnant air, forced convention, or conductive contact). Therefore, laboratory or benchtop performance may, or may not, be representative of performance elsewhere during operation.1 The reliability of the device may be critically affected by the evolving cyclic response (hysteresis changes) and eventual fatigue of the material. Often, an SMA element is cyclically trained to shake out the initial unstable behavior or to induce a two-way shape memory effect. However, sites of nucleation and coalescence of transformation fronts tend to produce residual stresses and damage that can change the ensuing front motion (Iadicola and Shaw, 2002a), and we expect, ultimately lead to fatigue failure. Control strategies for devices usually rely on single point measurements of displacement and/or temperature, which may be inadequate to achieve predictable performance for material with strongly localized behavior. Thus, it is imperative that a clear understanding of the phenomena exists and that appropriate analytical tools are available for the engineer to achieve successful application designs. We conclude, therefore, that a successful constitutive model for this behavior must be able to predict the number and speed of propagating transformation fronts. One approach to model this behavior is the 1-D thermodynamic framework of (Abeyaratne and Knowles, 1993) where transformation fronts are modeled as strain discontinuities, across which jump conditions are enforced, and explicit nucleation criteria and kinetic relations are specified a priori. Levitas (2002) proposes a different formulation of the driving force for front propagation that includes the plastic deformation at the strain mismatch (front) in addition to the energy dissipated by the phase transformation. A somewhat different approach (applied to monotonic loading only) is the recent finite element simulation scheme of Shaw (2000). No kinetic relations are explicitly specified, but rather, the number of nucleation events and the kinetics of transformation fronts are solved as part of the coupled thermo-mechanical problem. It uses a 3-D continuum-level, plasticity-based model with a softening branch to simulate the unstable behavior seen in the stress-induced Austenite to Martensite (A!M) transformation (loading only). In this case, the transformation front is a local propagating neck with a finite extent. A temperature dependent ‘‘yield’’ surface is used to model the Clausius–Clapeyron behavior of the transformation stress, and the exothermic latent heat changes are modeled using a conversion of the plastic work rate into a volumetric heat source. The approach is successful at predicting details of the mechanical response, the number of nucleation events, and the evolution of 1
Nguyen et al. (2001) gives an interesting example where an SMA-actuated underwater control surface was designed, in which the operation and power consumption in air and water were quite different.
582
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transformation fronts for monotonic loading (that is, no unloading causing reverse transformation). The objective of the current work is to extend these simulations to a broader range of loading rates and ambient thermal conditions to investigate the trends of the material behavior, again for monotonic loading only. This parameter study is conducted with an eye towards better understanding of the phenomena, as well as producing quantitative results suitable for design purposes. Whereas the previous study Shaw (2000) simulated thin, flat strips, the current study will focus on thin wire, the most common and least expensive form of NiTi SMAs in use today. This also provides an axisymmetric setting that is less computationally intensive for finite element simulations than the fully 3-D calculations previously done for thin strips. In Section 2, an uncoupled thermal analysis is first provided to study the temperature evolution of a thin wire with traveling point heat sources or a uniform heat source. This is useful to understand the sensitivity of the thermal behavior to the transformation kinetics and to identify important non-dimensional parameters. Section 3 then presents finite element simulations of the coupled thermo-mechanical behavior. The kinetics of transformation, temperature profiles, and mechanical behavior are compared for a wide range of rate and ambient conditions. Some general relations are proposed that capture the trends of this parametric study.
2. Uncoupled thermal analysis In this section, the problem of interest is the thermal response of a thin SMA wire surrounded by a thermal bath during the exothermic (stress-induced) A!M transformation. Deformation of the wire during transformation is ignored. Three cases are considered in the following subsections, a single moving point heat source, two converging point sources, and uniform transformation, to show how the thermal problem is sensitive to different transformation kinetics. This section also identifies critical thermal parameters affecting the transient temperature response in an uncoupled setting before addressing the coupled behavior in Section 3. The temperature in the wire can be characterized by the time-dependent axial temperature field T(X,t) if the lateral Biot number, the ratio of the radial thermal conduction resistance to the lateral boundary thermal resistance (Incropera and DeWitt, 1996), is small (or BihLc/k 1 according to
ðx; Þ ¼
1 2X ð1 cosn Þ n e 1 en sinn x:
n¼1 nn
ð14Þ
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589
Fig.: 8. Temperature profiles for uniform transformation, corresponding to three elongation rates: : : (a) =L=4105 s1, (b) =L=4104 s1, (c) =L=4103 s1.
This case is an approximation to the limiting case where numerous nucleations occur along the length simultaneously, which occurs at relatively fast loading rates (see the next section). It also represents the transformation behavior in certain other SMAs (such as CuAlZn) where transformation does not localize.
3. Finite element thermo-mechanical simulations We now turn to numerical solutions of the coupled thermo-mechanical problem during the stress-induced A!M transformation. The specimen is assumed to be virgin polycrystalline fine grained NiTi wire. Our focus is on the macroscopic (smooth) deformation and temperature fields rather than details of the microstructure. The approach is similar to that used in Shaw (2000), but is applied to axisymmetric wire rather than thin strip SMA, and is applied over a much larger range of ambient conditions and loading rates. The problem is largely uniaxial in nature, and one-dimensional modeling is certainly possible (see, for example Ngan and Truskinovsky, 1999; Shaw, 2002), but for the single transformation of interest here the use of commercial finite element software ABAQUS (HKS, 1999) is convenient and certain details across transformation fronts are interesting. The purpose of these simulations is to evaluate the trends of deformation and temperature evolution, specifically the number of nucleation events, the temperature rise and their effect on the mechanical behavior, as a function of a few non-dimensional parameters. 3.1. Finite element model Fig. 9 shows a typical axisymmetric finite element mesh used, which consists of eight noded quadratic displacement and linear temperature elements (type CAX8HT). The simulation begins (at =0) with a zero (non-dimensional) displacement field ux(x,r)=0 (for 04x41 and 04r4R/L) and uniform temperature field at an ambient temperature, T(x,r)=Tamb, representing an SMA wire in a stress-free Austenite state. Just as for the previous linear thermal analysis an average value of the thermal conductivities of
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Fig. 9. Typical axisymmetric finite element mesh: 4200 elements (not to scale).
Austenite and Martensite is assumed. Radiation to the environment is neglected and radial heat transfer from the lateral surface follows the Fourier law qR=h(TTamb) with a constant film coefficient (h) characterizing the conduction/convection behavior of the ambient media. Accordingly, the following boundary conditions are imposed on the displacement field and temperature field, during the simulation ( > 0). Displacement boundary conditions are prescribed at the ends and along the axis according to ux ð0; rÞ
¼
0;
ux ð1; rÞ
¼
ð Þ;
ur ðx; 0Þ ¼
ð15Þ
0;
where ()="max is a monotonic (non-dimensional) displacement ramp, representing the prescribed average axial strain along the wire. The lateral surface, r=R/L, is traction free. Temperature boundary conditions are prescribed according to Tð0; rÞ ¼ Tð1; rÞ @Tðx; 0Þ=@r k@Tðx; R=LÞ@r
¼ Tamb ; ¼ 0;
ð16Þ
¼ h½Tðx; R=LÞ Tamb ;
where the ends are modeled with fixed temperature boundary conditions to represent the heat sink behavior of the grips, the midaxis has no radial temperature gradient due to axisymmetry, and the lateral surface radial heat flux follows the Fourier law with a convective film coefficient. Transformation fronts are the result of a material instability, which leads to an imperfection sensitive problem. The onset of transformation will first occur at the most favorable site, i.e. the location with the highest applied stress and the lowest temperature (lowest nucleation stress). In practice, this often occurs at the grips where stress concentrations are unavoidable, but the grips are not modeled here, so a small imperfection in the form of a dent (5% of the radius) is introduced at the left end (x,r)=(0,R/L) to control the location of initial nucleation. Subsequent nucleations occur naturally as determined by the analysis.
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
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The constitutive model is based on finitely deforming J2 flow theory of plasticity with isotropic hardening calibrated to trilinear (engineering) stress–strain curves each having a softening intermediate branch. The irreversibility of the constitutive model does not adversely affect the results provided unloading does not occur to a level below the stress for reverse transformation (and this was never an issue for the simulations that follow). The same temperature dependent fit that was used previously (Shaw, 2000) is used here (see Fig. 10). Each mechanical isotherm has the nucleation stress, N, and propagation stress, P (according to the Maxwell construction), as measured in a typical NiTi alloy (see Table 2). The coupled deformation–temperature analysis available in ABAQUS solves simultaneously for mechanical equilibrium and the heat equation. The time scales for inertial effects and martensitic transformation are assumed to be much shorter than the time scale associated with the global loading rate. The only other time scale is that for transient heat transfer, which may be of the same order as the duration of mechanical loading. This is assumed to govern the events seen at the macroscale, a view that is consistent with experimental observations at the macroscale for polycrystalline NiTi SMAs (Shaw and Kyriakides, 1997). Once the transformation has begun, a sequence of events leads to subsequent nucleations, briefly described as follows. Fig. 11 shows two instances in time just before and during a nucleation event (shown at the right end). In Fig. 11a, a single front is traveling from the left end, and the local temperature is somewhat higher than the rest of the wire. The transformation (propagation) stress is somewhat elevated locally, but equilibrium requires that the force be constant along the length, which leaves cooler regions in the wire in a metastable state (stress above the propagation stress, but below the nucleation stress). If the local self-heating becomes more severe, as in Fig. 11b, the propagation stress at the currently traveling front can exceed the nucleation stress at the untransformed site with the coolest temperature (shown at the opposite end), and a dynamic nucleation event will occur at this coolest site. This sequence can
Fig. 10. Temperature-dependent engineering stress–strain constitutive model (Shaw, 2000).
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Fig. 11. Thermo–mechanical sequence leading to subsequent nucleation event at the right end: (a) time t
Rate and thermal sensitivities of unstable transformation behavior in a shape memory alloy Mark A. Iadicola, John A. Shaw* The University of Michigan, Department of Aerospace Engineering, Francois-Xavier Bagnoud Building, 1320 Beal Avenue, Ann Arbor, MI 48109-2140, USA Received in final revised form 3 February 2003
Abstract A special plasticity-based constitutive model with an up–down–up flow rule used within a finite element framework has previously been shown to simulate the inhomogeneous nature and the thermo-mechanical coupling of stress-induced transformation seen in a NiTi shape memory alloy. This paper continues this numerical study by investigating the trends of localized nucleation and propagation phenomena for a wider range of loading rates and ambient thermal conditions. Local self-heating (due to latent heat of transformation), the inherent Clausius–Clapeyron relation (sensitivity of the material’s transformation stress with temperature), the size of the specimen’s nucleation barriers, the loading rate, and the nature of the ambient environment all interact to create a variety of mechanical responses and transformation kinetics. The number of transformation fronts is shown to increase dramatically from a few fronts under nearly isothermal conditions to numerous fronts under nearly adiabatic conditions. A non-dimensional film coefficient and non-dimensional conductivity are identified to be the major players in the range of responses observed. It is shown that the non-dimensional film coefficient generally determines the overall temperature response, and therefore force–displacement response, of a transforming specimen; whereas, the non-dimensional conductivity is the more important player in determining the number of nucleations, and therefore the number of transformation fronts, that may occur. # 2003 Elsevier Ltd. All rights reserved. Keywords: Shape memory alloy; Thermo–mechanical behavior; Rate effects; Localization; Phase transformation; Finite element analysis
* Corresponding author. Tel.: +1-734-764-3395; fax: +1-734-763-0578. E-mail address: [email protected] (J.A. Shaw). 0749-6419/03/$ - see front matter # 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0749-6419(03)00040-8
578
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
Nomenclature X, S t u T, Tamb L, R p, Lc A C k h q:X, qR Q , N, P ", " Tad h0 f Nf x s r Bi h k
axial coordinate and transformation front position (m) time (s) displacement (m) axial temperature of wire, ambient temperature ( C) length and radius of wire (m) circumference and characteristic lateral length of wire (m) cross-section area of wire (m2) density (g/m3) specific heat (J/g–K) thermal conductivity (W/m–K) lateral heat transfer coefficient (W/m2–K) thermal diffusivity (m2/s) Dirac delta, or elongation (m1, m) axial and lateral heat flux (W/m2) volumetric heat source (W/m3) nominal axial, nucleation and propagation stresses (MPa) engineering axial strain and transformation strain homogeneous adiabatic temperature rise ( C) stress free specific enthalpy change (J/g) specific latent heat of transformation (J/g) internal heat source/inelastic work rate ratio number of transformation fronts non-dimensional axial coordinate non-dimensional transformation front position non-dimensional radial coordinate non-dimensional time non-dimensional temperature Biot number non-dimensional lateral heat transfer coefficient non-dimensional thermal conductivity
1. Introduction Shape memory alloys (SMAs) are distinguished by two remarkable behaviors, the shape memory effect and pseudoelasticity. The pseudoelastic response, in particular, refers to the fact that rather large strains (up to 8%) can be imposed by mechanical loading and then recovered via a hysteresis loop upon unloading. Pseudoelastic behavior arises from a reversible stress-induced martensitic transformation between different crystalline phases, Austenite (low strain phase) and detwinned Martensite (high strain phase). The behavior is being exploited in a variety of novel applications (Otsuka and Wayman, 1998).
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
579
It is now well known that under uniaxial loading in some SMAs, especially the NiTi-based alloys, that these transformations are accompanied by unstable mechanical behavior and localized transformation. This results in one or more propagating transformation fronts which produce evolving, distinctly nonuniform deformation and thermal fields. Under nearly isothermal conditions (slow loading rate in a thermally convective or conductive medium) distinct stress plateaus are seen in the mechanical response; whereas, under nearly adiabatic conditions (high loading rate in a thermally insulating medium) the mechanical response seemingly stabilizes, and a somewhat distorted load–displacement responses is observed (see experimental data in Fig. 1). This apparent rate effect is not the usual viscoelastic effect, but is due to an interaction between the inherent temperature sensitivity of the material’s transformation stresses (Clausius–Clapeyron relation, see experimental data in Fig. 2) and the self-heating or self-cooling that can occur due to latent heat changes within the transforming material (see, for example, calorimetry data in Fig. 3). Furthermore, the local nature of the transformation (in the neighborhood of transformation fronts) exacerbates the self-heating effect and causes significant rate effects to be observed at strain rates that one might normally consider quasi-static.
Fig. 1. Mechanical response (via miniature extensometer, E) and local temperature response (via small thermocouple, T) of pseudoelastic NiTi wire (experimental data from Shaw and Kyriakides, 1995): (a) nearly : : isothermal conditions (=L=1104 s1 in water), (b) nearly adiabatic conditions (=L=1102 s1 in air).
580
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
: Fig. 2. (a) Pseudoelastic force–displacement responses of NiTi strip at =L=1104 s1; (b) fit of nucleation and propagation stress for A!M (loading) transformation (Shaw, 2000).
Fig. 3. Differential scanning calorimetry of NiTi strip (Shaw, 2000).
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
581
These phenomena have been studied experimentally in detail in Shaw and Kyriakides (1995, 1997). These rather complex interactions can present serious difficulties to the application engineer who wishes to design novel devices or new composite materials with SMA constituents, and the phenomena have important repercussions on the performance, reliability, and controllability of the material as passive or active constituents. The aforementioned rate sensitivity clearly affects the response time of the material and will be quite different for different ambient thermal environments (stagnant air, forced convention, or conductive contact). Therefore, laboratory or benchtop performance may, or may not, be representative of performance elsewhere during operation.1 The reliability of the device may be critically affected by the evolving cyclic response (hysteresis changes) and eventual fatigue of the material. Often, an SMA element is cyclically trained to shake out the initial unstable behavior or to induce a two-way shape memory effect. However, sites of nucleation and coalescence of transformation fronts tend to produce residual stresses and damage that can change the ensuing front motion (Iadicola and Shaw, 2002a), and we expect, ultimately lead to fatigue failure. Control strategies for devices usually rely on single point measurements of displacement and/or temperature, which may be inadequate to achieve predictable performance for material with strongly localized behavior. Thus, it is imperative that a clear understanding of the phenomena exists and that appropriate analytical tools are available for the engineer to achieve successful application designs. We conclude, therefore, that a successful constitutive model for this behavior must be able to predict the number and speed of propagating transformation fronts. One approach to model this behavior is the 1-D thermodynamic framework of (Abeyaratne and Knowles, 1993) where transformation fronts are modeled as strain discontinuities, across which jump conditions are enforced, and explicit nucleation criteria and kinetic relations are specified a priori. Levitas (2002) proposes a different formulation of the driving force for front propagation that includes the plastic deformation at the strain mismatch (front) in addition to the energy dissipated by the phase transformation. A somewhat different approach (applied to monotonic loading only) is the recent finite element simulation scheme of Shaw (2000). No kinetic relations are explicitly specified, but rather, the number of nucleation events and the kinetics of transformation fronts are solved as part of the coupled thermo-mechanical problem. It uses a 3-D continuum-level, plasticity-based model with a softening branch to simulate the unstable behavior seen in the stress-induced Austenite to Martensite (A!M) transformation (loading only). In this case, the transformation front is a local propagating neck with a finite extent. A temperature dependent ‘‘yield’’ surface is used to model the Clausius–Clapeyron behavior of the transformation stress, and the exothermic latent heat changes are modeled using a conversion of the plastic work rate into a volumetric heat source. The approach is successful at predicting details of the mechanical response, the number of nucleation events, and the evolution of 1
Nguyen et al. (2001) gives an interesting example where an SMA-actuated underwater control surface was designed, in which the operation and power consumption in air and water were quite different.
582
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transformation fronts for monotonic loading (that is, no unloading causing reverse transformation). The objective of the current work is to extend these simulations to a broader range of loading rates and ambient thermal conditions to investigate the trends of the material behavior, again for monotonic loading only. This parameter study is conducted with an eye towards better understanding of the phenomena, as well as producing quantitative results suitable for design purposes. Whereas the previous study Shaw (2000) simulated thin, flat strips, the current study will focus on thin wire, the most common and least expensive form of NiTi SMAs in use today. This also provides an axisymmetric setting that is less computationally intensive for finite element simulations than the fully 3-D calculations previously done for thin strips. In Section 2, an uncoupled thermal analysis is first provided to study the temperature evolution of a thin wire with traveling point heat sources or a uniform heat source. This is useful to understand the sensitivity of the thermal behavior to the transformation kinetics and to identify important non-dimensional parameters. Section 3 then presents finite element simulations of the coupled thermo-mechanical behavior. The kinetics of transformation, temperature profiles, and mechanical behavior are compared for a wide range of rate and ambient conditions. Some general relations are proposed that capture the trends of this parametric study.
2. Uncoupled thermal analysis In this section, the problem of interest is the thermal response of a thin SMA wire surrounded by a thermal bath during the exothermic (stress-induced) A!M transformation. Deformation of the wire during transformation is ignored. Three cases are considered in the following subsections, a single moving point heat source, two converging point sources, and uniform transformation, to show how the thermal problem is sensitive to different transformation kinetics. This section also identifies critical thermal parameters affecting the transient temperature response in an uncoupled setting before addressing the coupled behavior in Section 3. The temperature in the wire can be characterized by the time-dependent axial temperature field T(X,t) if the lateral Biot number, the ratio of the radial thermal conduction resistance to the lateral boundary thermal resistance (Incropera and DeWitt, 1996), is small (or BihLc/k 1 according to
ðx; Þ ¼
1 2X ð1 cosn Þ n e 1 en sinn x:
n¼1 nn
ð14Þ
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589
Fig.: 8. Temperature profiles for uniform transformation, corresponding to three elongation rates: : : (a) =L=4105 s1, (b) =L=4104 s1, (c) =L=4103 s1.
This case is an approximation to the limiting case where numerous nucleations occur along the length simultaneously, which occurs at relatively fast loading rates (see the next section). It also represents the transformation behavior in certain other SMAs (such as CuAlZn) where transformation does not localize.
3. Finite element thermo-mechanical simulations We now turn to numerical solutions of the coupled thermo-mechanical problem during the stress-induced A!M transformation. The specimen is assumed to be virgin polycrystalline fine grained NiTi wire. Our focus is on the macroscopic (smooth) deformation and temperature fields rather than details of the microstructure. The approach is similar to that used in Shaw (2000), but is applied to axisymmetric wire rather than thin strip SMA, and is applied over a much larger range of ambient conditions and loading rates. The problem is largely uniaxial in nature, and one-dimensional modeling is certainly possible (see, for example Ngan and Truskinovsky, 1999; Shaw, 2002), but for the single transformation of interest here the use of commercial finite element software ABAQUS (HKS, 1999) is convenient and certain details across transformation fronts are interesting. The purpose of these simulations is to evaluate the trends of deformation and temperature evolution, specifically the number of nucleation events, the temperature rise and their effect on the mechanical behavior, as a function of a few non-dimensional parameters. 3.1. Finite element model Fig. 9 shows a typical axisymmetric finite element mesh used, which consists of eight noded quadratic displacement and linear temperature elements (type CAX8HT). The simulation begins (at =0) with a zero (non-dimensional) displacement field ux(x,r)=0 (for 04x41 and 04r4R/L) and uniform temperature field at an ambient temperature, T(x,r)=Tamb, representing an SMA wire in a stress-free Austenite state. Just as for the previous linear thermal analysis an average value of the thermal conductivities of
590
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Fig. 9. Typical axisymmetric finite element mesh: 4200 elements (not to scale).
Austenite and Martensite is assumed. Radiation to the environment is neglected and radial heat transfer from the lateral surface follows the Fourier law qR=h(TTamb) with a constant film coefficient (h) characterizing the conduction/convection behavior of the ambient media. Accordingly, the following boundary conditions are imposed on the displacement field and temperature field, during the simulation ( > 0). Displacement boundary conditions are prescribed at the ends and along the axis according to ux ð0; rÞ
¼
0;
ux ð1; rÞ
¼
ð Þ;
ur ðx; 0Þ ¼
ð15Þ
0;
where ()="max is a monotonic (non-dimensional) displacement ramp, representing the prescribed average axial strain along the wire. The lateral surface, r=R/L, is traction free. Temperature boundary conditions are prescribed according to Tð0; rÞ ¼ Tð1; rÞ @Tðx; 0Þ=@r k@Tðx; R=LÞ@r
¼ Tamb ; ¼ 0;
ð16Þ
¼ h½Tðx; R=LÞ Tamb ;
where the ends are modeled with fixed temperature boundary conditions to represent the heat sink behavior of the grips, the midaxis has no radial temperature gradient due to axisymmetry, and the lateral surface radial heat flux follows the Fourier law with a convective film coefficient. Transformation fronts are the result of a material instability, which leads to an imperfection sensitive problem. The onset of transformation will first occur at the most favorable site, i.e. the location with the highest applied stress and the lowest temperature (lowest nucleation stress). In practice, this often occurs at the grips where stress concentrations are unavoidable, but the grips are not modeled here, so a small imperfection in the form of a dent (5% of the radius) is introduced at the left end (x,r)=(0,R/L) to control the location of initial nucleation. Subsequent nucleations occur naturally as determined by the analysis.
M.A. Iadicola, J.A. Shaw / International Journal of Plasticity 20 (2004) 577–605
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The constitutive model is based on finitely deforming J2 flow theory of plasticity with isotropic hardening calibrated to trilinear (engineering) stress–strain curves each having a softening intermediate branch. The irreversibility of the constitutive model does not adversely affect the results provided unloading does not occur to a level below the stress for reverse transformation (and this was never an issue for the simulations that follow). The same temperature dependent fit that was used previously (Shaw, 2000) is used here (see Fig. 10). Each mechanical isotherm has the nucleation stress, N, and propagation stress, P (according to the Maxwell construction), as measured in a typical NiTi alloy (see Table 2). The coupled deformation–temperature analysis available in ABAQUS solves simultaneously for mechanical equilibrium and the heat equation. The time scales for inertial effects and martensitic transformation are assumed to be much shorter than the time scale associated with the global loading rate. The only other time scale is that for transient heat transfer, which may be of the same order as the duration of mechanical loading. This is assumed to govern the events seen at the macroscale, a view that is consistent with experimental observations at the macroscale for polycrystalline NiTi SMAs (Shaw and Kyriakides, 1997). Once the transformation has begun, a sequence of events leads to subsequent nucleations, briefly described as follows. Fig. 11 shows two instances in time just before and during a nucleation event (shown at the right end). In Fig. 11a, a single front is traveling from the left end, and the local temperature is somewhat higher than the rest of the wire. The transformation (propagation) stress is somewhat elevated locally, but equilibrium requires that the force be constant along the length, which leaves cooler regions in the wire in a metastable state (stress above the propagation stress, but below the nucleation stress). If the local self-heating becomes more severe, as in Fig. 11b, the propagation stress at the currently traveling front can exceed the nucleation stress at the untransformed site with the coolest temperature (shown at the opposite end), and a dynamic nucleation event will occur at this coolest site. This sequence can
Fig. 10. Temperature-dependent engineering stress–strain constitutive model (Shaw, 2000).
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Fig. 11. Thermo–mechanical sequence leading to subsequent nucleation event at the right end: (a) time t
