Modeling Rate Dependent Response of Shape Memory ... - AIAA ARC

Modeling Rate Dependent Response of Shape Memory Alloys Using a Thermo-Mechanical Continuum Phase Field Approach Babatunde O. Agboola∗and Dimitris C. Lagoudas† Texas A and M University, College Station, TX, 77840, USA

A recently developed thermomechanically coupled, thermodynamically consistent, constitutive model based on phase field theory is used to describe the rate dependent response of polycrystalline shape memory alloys (SMAs). The model, which constitutively accounts for transformation rate, is used to study the evolution of phase transformation coupled with different mechanical and thermal response of SMA. The goal is to model the rate dependent behavior of SMAs. Phase transformation is simulated for an SMA axisymmetric rod. Simulation conditions are varied between isothermal and adiabatic conditions under quasi-static pseudoelastic loading. The effect of kinetic coefficient, which relates to the phase transformation kinetics, is studied. Stress relaxation and creep behavior reported in experiments is described. Heterogeneous rise in temperature, which occurs at the material interface as well as its influence on thermal hardening for different loading rates are also described. Results suggest that the rate dependency in SMA is a result of the latent heat of transformation, heat conduction and convection, strain rates, transformation kinetics and their couplings. Therefore, this work demonstrates that the reported model is able to capture the rate dependent response of SMA constitutively. The simulation results affirm the need for application specific testing conditions in order to account for the complex thermomechanical interaction, rate dependency and length scale effects in the design of SMAs. Based on these simulation results, the model provides new insight into the design of SMAs for thermomechanical engineering applications

Nomenclature d σ σd Λ b v q r  t G S SA SM σ η λ

Micro-traction Cauchy stress tensor, Pa Deviatoric stress tensor, Pa Transformation tensor body force velocity, m/s heat flux heat source Total strain Transformation strain Gibbs free energy, J/Kg Compliance tensor, Pa Austenite Compliance tensor, Pa Martensite Compliance tensor, Pa Axial stress, Pa Scalar order parameter. Gradient energy coefficient, N

∗ Graduate † Professor

Research Assistant, Aerospace Engineering, Student member. and Senior Associate Dean for Research, Aerospace Engineering, and Fellow

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T0 Reference temperature, K dx x-component of micro-traction vector p Scalar internal configurational(micro) force g Scalar external configurational (micro) force T Temperature field, K τ Kinetic Coefficient Pas H max Maximum transformation strain c Heat Capacity, J/Kg/K s Entropy, J/Kg/K s0 Reference entropy, J/Kg/K u internal energy, J/Kg u0 Reference internal energy, J/Kg ρ Density Kg/m3

I.

Introduction

Shape memory alloys (SMAs) are choice material when high work density is needed. SMAs are now used in a large number of structural and functional applications in the fields of aeronautical and aerospace engineering, biomedical and structural engineering etc.(see Jani et al1 for detailed review on recent application of SMAs). SMAs are metallic alloys that undergo reversible martensitic phase transformation between two phases, accompanied by hysteresis, which manifests in macroscopic deformation. One of the phases (austenite) is stable at high temperature and the other (martensite) is stable at lower temperature. The two phases have different crystal structures. SMAs can recover seeming permanent strains when subjected to particular thermo mechanical inputs. The shape recovery is driven by the reversible martensitic transformation between solid-state phases that take place in SMAs during thermal and/or mechanical loadings. The transformation results in two remarkable macroscopic phenomena (i.e. the shape memory effects (SME) and pseudoelasticity). These properties can be harnessed using different loading path in stress-temperature space. The SME is characterized by a recovery of permanent deformation, induced by loading at a low temperature, when the SMA is heated. Another way to harness the shape memory property is through thermally induced phase transition, whereby the SMA is cooled under stress to a temperature sufficient to transform the material into a martensite phase under constant load. Subsequent heating will recover the seeming permanent strain that was generated. Pseudoelasticity on the other hand describe the recovery of large inelastic strain, generated based on mechanical loading of SMA at a sufficiently high constant temperature where austenite is the stable phase, when the load is removed. As SMAs are designed for use in different applications, experiments have been performed both under stress and strain control. The goal of such experiments is to characterize the macroscopic response of SMAs. Under quasi-static loading rate, polycrystalline SMA wires, strips and tubes have been observed to exhibit stress peak (during loading) and stress valley (during unloading) which are followed by steady-state propagation stress plateaus.2–6 The inhomogeneous inelastic strain generated during pseudoelastic loading has been reported to be due to the nucleation and propagation of single or several fully transformed martensite bands, exhibiting Luders band like features as in plasticity, depending on the thermal condition. In general the features reported in experiments2–6, 9–12 are said to be due to complex interaction between mechanical work, heat production, the prevailing heat transfer conditions and the loading rate.7, 8 Rate dependency in SMA response includes observed strain and stress rate dependent stress-strain response, the pseudo-viscoelastic response (where SMAs behave like classical viscoelastic material as they exhibit transformation induced creep and stress relaxation at low temperature).13, 14 In order to aid in the design of SMA based device and components, models have been developed. Review of the available models can be found in Birman,15 Lagoudas et al.,16 Patoor et al17 and Paiva and Savi.18 Of all the different approaches, the continuum modeling is suitable for engineering use. One major reason is the ease of implementation in finite element packages, particularly for complex geometries. Among the continuum modeling that are in the literature, there are three categories (1) phenomenological24–38 (2) micromechanical19–23 and (3) phase front-based continuum models.39–43 The micromechanicsbased models utilize information about the microstructure of the SMA to predict the macroscopic response. Modeling of SMAs using this approach is still an active area of research. Their disadvantage is that they

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require substantial computational effort to analyze component response at the macroscopic scale. Phenomenological models, on the other hand, do not directly capture material behavior at the microscopic level, but consider energy potentials defined over homogenized material volumes and employ methods similar to those of classical plasticity to capture bulk behavior.43 These approach trade microscopic accuracy for global computational efficiency as they are calibrated by parameters measured at the macroscale through experimental observation. Since this modeling work is motivated by the afore mentioned engineering design relevant features of SMAs, SMA macroscopic response is modeled to be a result of nucleation and propagation of fully austenite and fully martensite transformation front or domain boundary. Earlier works have been carried out to model SMA behavior considering transformation based on evolving phase boundary. Abeyaratne and Knowles39–41 pioneered this approach, and it is based on the seminal work of Ericksen for a two phase material.44 Their approach however is limited since it is based on a sharp interface, which require exlicit tracking of the phase boundary. Other similar but static approach, looking at deformation from the lattice scale up thereby emphasizing microstructure of martensite, is that of Battacharya and his co-workers.45–48 In keeping with the microstructural features of SMA, phase field modeling based on the theory of Landau, Devonshire and Ginzburg- Landau has also enjoyed widespread research. A detailed review of the state of the art in phase field modeling is given in by Mamivand et al.49 Others have also made attempt to develop a phase frontlike models to capture the macroscopic localization and propagation of instability in SMAs3, 4, 50, 51 using constitutive approach. Some other 1-D purely mechanical approach and 3-D models exist.51, 52 A couple of non-isothermal models exist, which are not constitutive in their evolution equation development.53–55 Although models already exist for the simulating SMAs response, there is however a need for a thermodynamically consistent rate dependent constitutive continuum based SMA model. A model that overcomes the above stated challenges of the phenomenological and the phase front model is desired. In particular, for the design of SMA for aerospace application where stringent requirement (such as combination of sensing and actuation) and thermomechanical coupling are inevitable, a reliable and yet efficient model is needed. The subsequent sections will elaborate on the model and the simulated results. The model reported in this work is intended to comprise all the relevant features necessary to simulate SMAs for application design. The effect of latent heat during transformation on the pseudoelastic response of SMAs and the complex interaction between mechanical loading rate, kinetic of phase transition as well as the heat production or absorption during phase transition is discussed. A.

Modeling Approach

This work combines the strength of the classical phenomenological and the phase-front based modeling approaches through the introduction of a fundamentally different concept. The model is based on the Ginzburg-Landau type theory in its essence. It is phenomenological and constitutive. The free energy is calibrated (in the Landau-Devonshire and Ginzburg-Landau sense) to parameters measurable from experiments at the macroscale. Separation is made between balance laws and constitutive equation as classically done in continuum thermodynamics. Kinetic of phase transition results from the introduction of a balance law for configurational forces, which exist as conjugate to a scalar order parameter. The scalar order parameter is a new kinematic descriptor in addition to the displacement associated with the Newtonian forces. The purpose of the order parameter is to identify the two phases present in the SMA (in order to track their evolution) so that relevant macroscopic properties could be assigned accordingly. Consequently, in addition to the fundamental balance laws of continuum thermodynamics, a new balance law that results in the kinetic equation for phase transition is used. The new balance law is true for all material involving evolving interface whose phases are identified by a scalar order parameter. The approach is expected to be true for modeling plasticity as well. Constitutive equation specific to shape memory alloys is developed, not only to have a well posed problem, but mainly to accurately capture the response of SMAs. The constitutive field response functions and appropriate restriction by the second law of thermodynamics will be discussed. The model is thermomechanical and thermodynamically consistent, with rate dependency incorporated constitutively. The current model does not carry along with it the nucleation process in situ. The nucleation is assumed to occur at the ”grip” of the specimen. The overall goal of this model is to capture in one model, the shape memory effect, the pseudoelastic behavior, latent heat changes as well as possible unstable transformation behavior. However in the present form, the model is expected to be sufficient for modeling pseudoelastic as well as isobaric thermally induced phase transformation in SMAs. One remarkable property of this model is that, 3 of 19 American Institute of Aeronautics and Astronautics

unlike the traditional phase front models, the interface is diffuse (with an assigned thickness) and as such there is no need to explicitly track the propagating phase boundaries as discontinuous quantities, they arise naturally as smooth localized propagating fields in the boundary value problem. 1.

Balance Laws

The theory used in this work is based on the notion of configurational forces as explained by Gurtin (2000).57 The idea behind this approach is that every fundamental physical law involving energy should account for working due to each independent kinematic descriptor associated with disparate kinematic processes. The working or power expenditure is considered to be a product of a force density and the rate of change of the kinematic descriptor. In essence, to every kinematic descriptor a system of force is associated and to each density of forces a generalized velocity. The idea is that motions are a result of a cause, which is generally referred to as forces. Such a notion is consistent with a basic Newtonian idea of forces. However, configurational forces are non-Newtonian.57 A notion which unifies forces in the Newtonian and the ”Eshelbian” or configurational sense is to simply consider forces as conjugate to a kinematic descriptor. Therefore,for this work configurational forces are simply conjugate to a scalar order parameter. According to Gurtin (2000),57 these configurational forces are primitive objects consistent with their own force balance that is analogous to the balance of linear momentum. These forces expend power in response to the motion of the austenite-martensite interface. The local form of the configurational forces balance is given as ∇ · d + p + g = 0.

(1)

The balance of linear momentum in the local form results in ˙ ∇ · σ + b = ρv.

(2)

. The balance of angular momentum or moment of momentum results in the symmetry of the cauchy stress tensor σ = σT .

(3)

Considering the balance of energy with contribution of working from the configurational (micro) force results in the local form is given as ˙ − pη. ρu˙ = −∇ · q + ρr + σ · ˙ + d · ∇η ˙ 2.

(4)

Constitutive Equations

As noted earlier, the Gibbs free energy is chosen as the thermodynamic potential to describe the state of the SMA. Therefore the internal energy is transformed into the Gibbs potential using the legendre transform given as 1 G = u − σ ·  − T s. ρ

(5)

Combination of the legendre transform and the second law of thermodynamics in the Clasius-Duhem form equation 4 results in the dissipation rate, and it is writen as ˙ − −ρG˙ − ρT˙ s −  · σ˙ − pη˙ + d · ∇η

q · ∇T ≥ 0. T

(6)

Now employing the Coleman-Noll entropy principle58 on the response function of the constitutive fields, results in the following restriction on the constitutive equation of the cauchy stress σ,the micro-traction d and the entropy: 4 of 19 American Institute of Aeronautics and Astronautics

ˆ ∂G , (7) ∂σ ˆ ∂G d=ρ , (8) ∂∇η ˆ ∂G s=− . (9) ∂T A relationship is proposed between the primary order parameter and the inelastic strain resulting from the martensitic transformation. A relationship of such is necessary as the order parameter is the sole kinematic descriptor, introduced to capture phase transformation kinematics, and hence should be the primary contributor to dissipation. The relationship is given as  = −ρ

˙ t = Λη, ˙

(10)

where   σd 3 H max d , 2 σ ¯ 1 σ d = σ − tr(σ)I, r3 3 2 σ ¯d = kσ d k . 2

Λ=

(11) (12) (13)

Now, using the proposed relationship betwen the inelastic or transformation strain ˙ t and the order parameter, the restricted form of the dissipation rate becomes

−(p + ρ

ˆ ˆ ∂G ∂G + ρ t : Λ)η˙ ≥ 0. ∂η ∂

(14)

The following inequality holds for all [σ, η, ∇η, η, ˙ t , T ] −πdis (σ, η, ∇η, η, ˙ t , T )η˙ ≥ 0,

(15)

and πdis = p + ρ

ˆ ˆ ∂G ∂G + ρ t : Λ. ∂η ∂

(16)

So far, the response function for the internal configurational force is unknown. In order to determine this response function, granted smoothness,the most general form of πdis consistent with Eqn (15) is πdis = −τ η, ˙

(17)

t

where τ (σ, η, η, ˙ ∇η,  , T ) ≥ 0. Therefore the function for the internal configurational force is given as

p = −τ η˙ − ρ

ˆ ˆ ∂G ∂G − ρ t : Λ. ∂η ∂

(18)

According to Eq. (16) and Eq. (18) there are two contribution to the internal configurational force: a ∂G contribution ρ ∂G ˙ ∂η + ρ ∂t · Λ arising from change in the free energy and a dissipative contribution πdis = −τ η. The configurational forces balance equation,consistent with the second law of thermodynamics, result in the kinetic law for interface evolution and it is given in Eq. (19) ( assuming that τ is constant, so that it is a linear kinetic law).

τ η˙ = ρ∇ ·

ˆ ∂G ∂∇η

! −ρ

ˆ ˆ ∂G ∂G − ρ t : Λ + g. ∂η ∂

(19)

The external configurational force g = 0 for the present implementation. However, g can be chosen in general to satisfy the balance of configurational forces. 5 of 19 American Institute of Aeronautics and Astronautics

B.

Gibbs Free Energy Formulation

A modified Gibbs form of Ginzburg Landau type potential is used. The model is derived in 3-D using Lagrangian strain, which allow for large rotation in the shape memory alloy but the form described and implemented herein is the infinitesimal strain formulation. The 3-D Gibbs potential is taken to be seperable in the sense of decomposition into strain energy, a thermal energy and a gradient energy. For the Lagrangian formulation consistent with the assumption of infinitesimal deformation, the free energy is taken to be quadratic in σ. The free energy of the overall SMA can be determined assuming the strain due to thermal expansion is negligible and that transformation strain is only present in the martensite phase. Therefore, the total strain is taken as  = e + t $(η). Consequently, the free energy is given as   1 T 1 1 2 G = − σ : S(η)σ − σ : t $(η) + c (T − To ) − T ln( ) − so (η)T + uo (η) + λ |∇η| , 2ρ ρ To 2 (21) where S = S A + (S M − S A )m(η),

(22)

M A so = sA o + (so − so )f (η),

(23)

uo =

uA o

+

(uM o

−

uA o )κ(η),

(24)

and m(η) = mo η 2 + m1 η 3 + m2 η 4 + m3 η 5 , 2

(25)

$(η) = γo η + γ1 η ,

(26)

γ(η) = η$(η)

(27)

κ(η) = κo η 2 + κ1 η 3 + κ2 η 4 ,

(28)

2

3

4

f (η) = fo η + f1 η + f2 η .

(29)

Fourier’s law is assumed for the heat flux C.

3-D Balance Laws

The constitutive response functions are inserted into the balance of linear momentum, energy balance as well as the configurational forces balance; having restricted the constitutive equations by the second law of thermodynamics. The balance of linear momentum result in the mechanical equilibrium equation assuming no body forces and negligible inertia (hence quasi-static), and it is given as    ∇ · σ = ∇ · C(η)  − t $(η) = 0.

(30)

Similarly, the order parameter evolution equation results from the configurational forces balance as

τ η˙ =

dm d$ df dκ 1 σ : ∆Sσ + σ : t + ρ∆so T − ρ∆uo + σ : Λ$ + ρλ∇2 η, 2 dη dη dη dη

(31)

and the balance of energy becomes   d$ 1 dm dκ ρcT˙ = −∇ · q + σ : Λ$(η) + σ : t + σ : ∆Sσ − ρ∆uo − p η˙ + ρr. dη 2 dη dη

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(32)

1.

1-D Balance laws

Upon reduction of the constitutive model into 1-D, the balance of configurational forces becomes   0 0 0 1 ∂2η 1 σ2 max 0 − m + σH γ + ρ∆s T f − ∆u k + ρλ . τ η˙ = 0 0 2 EM EA ∂x2 The balance of energy simplifies to  2  0 0 0 ∂2T σ 1 4h(T (x, t) − T∞ ) 1 ˙ ρcT = k 2 + ( M − A )m + σHγ − ρ∆uo κ − p η˙ − , ∂x 2 E E d and the balance of linear momentum is    ∂u ∂ max E(η) −H γ = 0. ∂x ∂x

(32)

(32)

(32)

Looking at the above set of equations, it can be concluded that two additonal parameters (τ ) and (λ) have been incorporated because of the introduction of the configurational forces balance. These parameters allows for the rate dependency and the length scale effect in the constitutive model. D.

Model Parameters Calibration

The condition for austenite and martensite stability is used to calibrate the model to experiment. A LandauDevonshire type free energy is obtained such that at different temperature and stress the condition of stability and instability of austenite and martensite are met at the boundary corresponding to the Clausius-Clapeyron parameters CA and CM . Coefficients of the free energy polynomials, the reference entropy and reference internal energy are determined as a function of five material experimental parameters CA , CM , As , AM and H max . For vanishing stress, the boundary of martensite stability (upon heating or unloading) is the austenite start temperature (As )and the boundary for austenite stability upon cooling or loading is the martensite start temperature (Ms ). The idea behind the calibration of the model is the requirement that at high temperature above the austenite start temperature for zero stress (which is a phase stability boundary) the free energy function should have a single minimum only at vanishing order parameter (i.e. an order parameter value of zero which correspond to austenite phase). The condition is requisite for the stability of austenite at T ≥As . While at temperature lower than Ms , the only minimum should correspond to non-vanishing order parameter- which in this case is taken to have a value of 1 for the martensite phase. At intermediate temperature, the free energy is expected to have minima corresponding to both phases (i.e. Austenite and Martensite). A single free energy applies to both phases and based on the choice of free energy, it is expected that at T > T0 the martensite phase is metastable and T < T0 , the high temperature is metastable. Essentially, the martensite phase becomes unstable at T = As for a given stress which implies that Austenite is the only stable phase at T ≥ As and at T ≤ Ms Martensite is the stable phase . In the present formulation, the calibration is taken to be such that the only a single variant of martensite is favored as the load is high enough to bias the variant.

II.

Pseudoelastic Results and Discussions

An SMA wire is simulated as an axisymmetric rod under different strain rates and thermal boundary conditions. Heat transfer is assumed to be negligible radially across the rod such that temperature is constant at each cross section but only vary axially. The model is used to simulate the kinetic of phase transition, the resulting temperature profile due to heat generation and the stress-strain response under different mechanical loading conditions as well as thermal ambient conditions. COMSOL Multiphysics (a commercial finite element code) is used to solve the fully thermomechanically coupled balance laws. Results for an SMA wire with properties given in Table 1 are reported in figure 1 for the evolution of phase transformation with local increase in temperature due to latent heat of transformation. The result of figure 1 is for a water ambient medium which is representative of an isothermal loading condition. Notice that the temperature profile (white rods) is localized at the interface. Accumulation of strain due to the motion of transformation front is observed. Notice that the rod elongates as transformation progresses. The

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Table 1. Properties for simulation except otherwise stated

Properties Martensite Modulus Austenite Modulus Martensite Start Temperature Austensite Start Temperature Kinetic Coefficient Gradient energy coefficient Clasius Clapeyron Parameter (austenite) Clasius Clapeyron parameter (martensite) Heat Transfer coefficient (in air) Heat Transfer coefficient (in water)

Notation EM EA Ms As τ ρλ CA CM h h

Value (all) 15GPa 33 GPa 242 K 261 K 4.08 ×106 P as 1.5 N 4.5 Mpa K−1 4.5 Mpa K−1 6.5 W/m−2 K −1 890 W/m−2 K −1

Value figure (6) 30GPa 60GPa 272.5 K 318 K 1.5×107 5N 8.8 Mpa K−1 6.125 Mpa K−1

Figure 1. Interface evolution and accompanying temperature profile due to localized heat release and transfer

result shows that the transformation occur at the interface, since the peak of the heat released due to the exothermic nature of the austenite to martensite transformation at the interface. The heat released results in the increase in stress required to continue transformation. A.

Forward Transformation Stress-strain Response for Pseudoelastic NiTi SMA

The behavior of NiTi SMA is simulated for different cases, with properties as indicated in the plots and Table 1, under different loading rates and thermal ambient conditions. Case 1: Quasi-static loading rate (10−4 s−1 ) and water ambient condition (h = 890 W/(m2 K)) The first case simulates SMA in water thermal ambient condition under a very slow loading rate. The value of the heat transfer coefficient is taken from the work of Leo et al.56 According to figure 2a and b, martensite nucleates in austenite at a stress level corresponding to the isothermal critical stress on the stresstemperature phase diagram. After the nucleation, an initial strain softening is observed, which is followed by martensite domain front propagation and strain accumulation due to phase transition at a lower stress (rest stress) plateau. The transition results in a local rise in temperature at the interface due to the latent heat. However, because the ambient medium is considered to be water, the heat is quickly convected out of the specimen at a relatively faster rate compared to conduction through the material. Therefore only very small peak, 8 of 19 American Institute of Aeronautics and Astronautics

Figure 2. (a) Nominal stress and strain (b) Order parameter profile (c) Local strain vs nominal strian profile (d) Temperature change profile.

localized at the interface, is observed in the temperature profile. As a consequence, no thermal hardening is observed as the stress needed to sustain phase transformation remains the same as depicted by figure 2b. The absence of hardening can be explained by the fact that the temperature of the specimen only rose slightly at the interface and the peak remains at the same level through the transformation. According to figure 2d, the change in temperature from the initial temperature of the specimen is less than 1 K, and it is consistent with experimentally observation2 Case 2: Quasi-static loading rate (10−4 s−1 ) and air ambient condition (h = 6.5 W/(m2 K)) Unlike the water ambient medium, loading SMA in air ambient medium (captured by a lower heat transfer coefficient) results in the same initial behavior as the water ambient medium. However as phase transition continues, thermal hardening is observed as more mechanical work is needed to propagate the phase front due to the rise in temperature. Essentially, a higher stress is needed to keep martensite stable as the temperature rises locally and heat is conducted through the SMA. As shown by the temperature profile (see figure 2d), the rate at which heat is being removed by convection is relatively slower in this case compared to case 1, therefore heat gets conducted away from the interface to other part of the material due to the thermal gradient created by the local rise in temperature at the interface. The temperature increases as a result of latent heat generated locally at the interface. Notice that the front profile in figure 2b is slower than that of water. The difference between the stress-strain responses of the simulated NiTi SMA for the same quasi-static loading rate shows that the ambient medium in which an SMA is tested is very crucial for their design. It is important to note that the hardening of SMA or lack thereof depending on ambient media is reflected in the comparison between the stress-strain profile during inelastic deformation and the temperature profile. Notice the dashed red lines of figure 3, the profile is similar to that of the inelastic portion of the stress-strain plots for air and water ambient medium. It therefore suggests that the thermomechanical coupling occur

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Figure 3. Temperature profile for constant loading rate in (a) Air ambient medium and (b) Water ambient medium.

locally through the interface. Case 3: Quasi-static loading rate (10−4 s−1 ) and adiabatic thermal condition(h = 0 W/(m2 K)) In order to quantify the amount of heat released during the transformation from austenite to martensite, the loading of the SMA in adiabatic condition is simulated. In this case, no heat flux is allowed across the boundary of the material. Therefore, the latent heat generated in the specimen stays within the specimen. Heat conduction, however takes place due to the temperature gradient between the interface and the rest of the material. Under this condition, the temperature profile as shown in figure 2d is quite different as the temperature of the entire specimen rises considerable and the profile shows a switch in behavior as time increases due to thermal gradient changes. Case 4: Nominal strain-local strain profile. The local strain and the nominal strain is expected to be different because of strain localization as SMA transform by the propagation of a boundary between fully austenite and fully martensite phases. The observation is reported in experiments (see Shaw and Kyriakides (1995)2 ). In order to verify that this model actually capture the appropriate phase boundary motion relation, local strain is plotted against the nominal strain at the end of the specimen as depicted in figure 2c. The profile generated is consistent with that of experiment (see Shaw and Kyriakides (1995)2 ). Initially the local and nominal strain is the same during homogeneous deformation and elastic regime of the material. At the point of nucleation of martensite in austenite, a strain localization begins due to transformation resulting in discrepancy between the local and nominal strain and it varies with the thermal ambient condition like the stress-strain profile as well (compare figure 2 a and c). Once the transformation reaches the specimen end, the local and nominal strain becomes the same, which manifest in a jump in the local strain at the end of the specimen. Subsequently loading result in homogeneous elastic deformation of the fully martensite SMA. Case 5: Effect of loading rate in air ambient condition Simulation in air ambient medium under loading rate of 2 × 10−4 also show thermal hardening, similar to 1 × 10−4 for case 2. The reason is the same as that of case 2 but the profile of the stress strain plot is different from case 2 as the loading rate is much faster, The faster loading rate does not allows for much time for the heat to be taken out of the material through convection to the ambient medium. Therefore the stress increases much higher than that of case 2 (see figure 2a) since the degree of rise in temperature at the interface is higher. Higher stress value is, therefore, needed to keep martensite as the stable phase. The stress-strain profile of figure 2a and figure 4 are different as the loading rate has changed from 1 × 10−4 to 2 × 10−4 and then to 5 × 10−4 . Although the rate at which heat is being exchanged with the ambient medium (simulated by the heat transfer coefficient of air) is the same ,the loading rate increase result in different critical stress, and the value of the stress during inelastic strain accumulation is increased with the loading rate. For full pseudoelastic plot, this is expected to result in changes in the hysteresis. Another feature which is consistent with experimental observation50 is the vanishing of the strain softening as the

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Figure 4. Stress-strain plot under different strain rates in air ambient media.

loading rate increases.

Figure 5. Stress relaxation and associated stress temperature profile

So far, our simulation shows that the macroscopic stress strain response of given polycrystalline NiTi SMA, at the minimum depends on complex interaction between the phase transition kinetics, the external loading rate ( mechanical work rate) , the speed of heat convection and that of heat conduction away from the interface. Experimentalists have reported similar conclusion.3, 8, 50

III.

Pseudo-viscoelastic Results and Discussions

Pseudo-viscoelastic behavior has been reported for shape memory alloy,13, 14 which include strain creep at fixed stress and stress relaxation under fixed strain. Simulation and analysis of these two phenomena is the focus of this section.

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Figure 6. Transformation Induced Stress Relaxation

A.

Transformation Induced Stress Relaxation

A couple of researchers have reported a drop in stress (relaxation) due to strain arrest( fixed strain) and accumulation of strain (creep) under stress arrest( constant stress) during pseudoelastic loading of SMAs.13, 14 The ability of model to capture this feature will serve to validate the model as appropriate for SMA design. To simulate these behaviors, as described in figure 5, an SMA wire is loaded at a constant stress rate of about 3.3M P a/s through stage 1 (blue) and 2 (gray) up till a strain of about 0.018. The total strain is then arrested at this value of strain for about 600s. The stress is observed to drop to a rest stress as indicated by the line labeled 3. To further transform the SMA after the rest stress is reached, loading is required. It should be noted that the final rest stress under strain arrest is the value of the plateau stress value needed to propagate the front for an equivalent strain rate loading of the same SMA in water ambient condition (isothermal) as seen in figure 6a (see dashed line showing softening). Therefore, the result indicated that the model is able to predict stress relaxation in the SMA. A plot of the accompanying temperature profile suggests that during relaxation, the temperature of the interface drops as the system move towards thermal equilibrium (see figure 5b). In other to confirm that the current model is actually predicting stress relaxation in SMA, and to provide explanation for the stress relaxation phenomena, another SMA with different properties is simulated. The properties of the SMA is given in Table 1 (value for figure 6) and it is based on the experimental result by Matsui et al.13 To simulate the response of the SMA, the gradient energy coefficient (λ) as well as the kinetic coefficient (τ ) (a dynamic viscosity-like coefficient) of the model is calibrated to predict the response of the stress based loading of the SMA, which is reported in figure 6a (solid black line and subsequent loading and relaxation). The value of the gradient energy coefficient (ρλ) and the kinetic coefficient (τ ) are 5N and 5 × 107 P as respectively. It should be noted however that for the partial transformation simulation to match with experiment, the value of the heat transfer coefficient is 640W/m2 K. Once the simulation for the partial transformation (solid black line) and the experiment is the same, the total strain is then arrested. The heat transfer coefficient with value h = 4W/m2 K is used to simulate the stress relaxation. The final value of 12 of 19 American Institute of Aeronautics and Astronautics

Figure 7. Isothermal strain arrest during forward transformation under stress rate of 1 MPa/s

the rest stress as predicted by the model matches with that of the experiment for the set of parameters used (see figure 6b.). The value of τ , ρλ and heat transfer coefficient of 4W/m2 K are now used to simulate the experiment for strain rate loading. The result is also in good agreement with the experiment within acceptable discrepancies. According to the simulation result, this relaxation occur due to the further motion of the transformation as the SMA relaxes towards equilibrium (see figure 6d). The model shows that there is a time lag between when the strain was arrested and when the front stops. These observations may seem to suggest that the stress relaxation in SMA is solely due to the drop in temperature, whereby the thermal state favors the formation of martensite as lower stress value is required to ensure further phase transformation due to the thermomechanical coupling. However, this conclusion may not be true unless it can be shown that such further front propagation and consequently stress relaxation cannot be observed under isothermal condition. So, is the further front motion mainly due to drop in temperature? To arrive at a solid conclusion, strain arrest under isothermal condition using this model is simulated. Since there is no thermal effect during relaxation, it is expected that if the essential reason for the relaxation resulting from front motion is the drop in temperature then this phenomena should not occur under isothermal condition. The result of our simulation, under isothermal condition, does show that transformation front still moved during strain arrest, which leads to stress relaxation. (See figures 7) Therefore, apart from thermomechanical coupling, there has to be an intrinsic explanation for the drop in stress. Since thermal effect is eliminated, the plausible explanation is that as the system move towards a state of equilibrium by minimization of the available energy, the strain energy stored in the SMA is released to perform work in moving the transformation front. Since the total strain is kept fixed and further inelastic strain accumulate, the stress drops so as to maintain the constant strain imposed at the boundary. Notice that after a given time the stress in the SMA reaches an asymptotic value where the interface stops moving, such that for transformation to continue, more energy has to be supplied to perform work in moving the front by loading the SMA. 13 of 19 American Institute of Aeronautics and Astronautics

Figure 8. (a) Stress strain response during partial transformation and creep in SMA (b) Temperature profile during partial transformation (c) Change in stress with time (d) Change in strain with time

B.

Transformation Induced Creep

As reported by experiment,13, 14 accumulation of strain when stress is arrested (fixed stress) is characteristic of pseudoelastic loading of SMAs. In order to validate the versatility and accuracy of this model and its predictions, the transformation induced creep behavior of NiTi SMAs is also simulated. To do this, the SMA is loaded (using properties in Table 2) under a stress rate of 3.3M P a/s until a strain of about 0.018. At this point the stress is arrested. Figures 8 and 9 shows that strain continues to accumulate until the SMA is fully transformed. The strain accumulation is due to the fact that the transformation front continues to move (see figures 9a and b) as the stress state is above the rest stress associated with mechanical state that allows the two phase to exist in equilibrium. It should however be noted that this simulation was carried out under thermal condition that simulates air ambient medium. Unlike the stress relaxation, since the total strain is not fixed, there is no drop in stress. Inelastic strain continues to accumulate due to the motion of the transformation front until the entire material is fully transformed. It is worth noting that in this case as well, the profile of the stress-strain response follows the temperature profile (see figure 8b) . Another interesting observation is that the strain-time and stress-time profile matches very well with experimentally observed trend.(see figure 8c and d). Further comment on the creep result is revered for future work and publications. C.

Parameteric Study of Kinetic Coefficient and Gradient Energy Coefficient

Parametric study for the kinetic coefficient (τ ) and the gradient energy coefficient (λ) is carried out in order to understand their effect and to calibrate the parameter for different SMAs. Under a stress rate of 30M P a/s, the stress-strain response of the SMA is observed to change for different values of τ and λ. As λ increases, the stress corresponding to each strain becomes smaller for the same kinetic coefficient. The relationship between λ and the stress-strain profile is a result of length scale associated with the interface width (implicit in the

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Figure 9. Order parameter evolution during (a) partial transformation (b) creep during stress arrest and accompanying temp profile (c) and (d) respectively

gradient energy coefficient). As the gradient energy coefficient decreases, which correspond to a thinner interface, a higher stress is needed to generate the same inelastic strain (see figure 10). Therefore, this model may be improved or calibrated to include the effect of length scales on the response of the SMA. The length scale may be characteristic of inclusion size, inter-particle distance, grain size or the actual geometry dimension of the structural material being studied. The kinetic coefficient on the other hand has an opposite effect on the stress strain response as compared to the gradient energy coefficient. As the kinetic coefficient increases, the stress corresponding to the same amount of strain increases. The kinetic coefficient carries with it an intrinsic time scale and is expected to relate to a characteristic time of the SMA phase transformation. Since these parameters change the stress strain response, they should be calibrated for specific SMA. To achieve the calibration, maps of these two parameters that simulate experimental result for different SMAs may be generated or non-dimensionalization may be carried out. The relationship between these two parameters and the intrinsic length and time scale related quantities of specific shape memory alloys will be reported in future work as well.

IV.

Conclusions.

A recently developed thermodynamically consistent, continuum constitutive phase field model using the notion of configurational force balance and order parameter is reported. The model is used to simulate the rate dependent response of SMAs. Two different compositions of NiTi SMA are simulated. The model does couple different time and length scales, which correspond to the different phenomena responsible for the macroscopic response of SMA. Under pseudoelastic loading, our simulation result captures several experimentally observed features of the stress-strain response for forward transformation. The features include localized heating at the transformation front or interface, which results in thermal hardening; loading rate 15 of 19 American Institute of Aeronautics and Astronautics

Figure 10. Parametric study of effect of kinetic coefficient (τ )

dependence of critical stress for phase transformation; transformation induced stress relaxation; and creep in SMA. It can be concluded that the complex interaction between the loading rate, rate of interface evolution (transformation kinetics), latent heat of transformation, convection and conduction of heat (manifested in temperature change) of the SMA specimen is the reason for the rate dependent response of SMA. The gradient energy coefficient and the kinetic coefficient relates to a length and time scale in the SMA respectively, and different values result in different stress-strain response, which give insight to the fact they will be different for each SMA. To the best of this authors knowledge, the model reported in this paper is the first to present the modeling of SMA in a 3-D continuum, constitutive, thermodynamically consistent manner with full thermomechanical coupling based on the propagation of transformation front (phase boundary motion between austenite and martensite without the need to explicitly keep track of that boundary). It is theorized that the stress relaxation behavior of SMA is essentially due to the fact that the macroscopic straining of SMA result from the evolution of a phase boundary between fully austenite and fully martensite phases, apart from the thermomechanical coupling. All the different complex interaction resulting in the rate dependent SMA response under pseudoelastic loading converges to the evolution of a transformation front. Since, this model captures rate dependent phenomena and their thermomechanical couplings in SMA, it should be expected that many design problems that involve complex interaction can now be solved. This model is expected to benefit greatly applications that involve varying loading rates as well as those that involve SMA interacting with the solid or fluid media in which they are embedded. Of particular benefit are applications involving thermal interactions in high temperatures aerospace applications. Examples include: • SMA embedded in polymer for high temperature applications • Functionally graded ceramic metal (SMA) composites (GCMeC)

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• Aircraft aero-structural morphing members • Flutter mitigation through the vibration isolation and dampening properties of SMA Simulation of thermally induced phase transformation using this current model, and its subsequent improvement, is expected to give insight into the localized thermomechanical interaction of SMA and its aero-structural interaction as required for SMA design in aircraft applications. The applications may include the design of SMA torque tube, improved design of the variable geometry chevron (VGC), application design of other aircraft morphing structural members and other applications that utilizes SMA as reviewed by Jani et al and the references therein.1 Furthermore, the similarity between our simulations in comparison to experimental observation gives credibility to our model as advancing the state of the art in constitutive modeling of SMA. As an extension, this work suggests among others, that configurational forces balance may be a fundamental concept missing in continuum modeling of material with evolving inhomogeneity, surfaces and interfaces. Combination of this notion with phase field theory (diffuse interface), may lead the scientific community towards modeling several challenging engineering problems involving motion of defects and boundary between distinct phases. Hence, in the light of existing models in relation to mechanism reported in experiments for shape memory alloy response, this model is a significant contribution to the state of the art in SMA modeling and by extension a worthwhile contribution to continuum physics in general.

Acknowledgments This work acknowledges the sponsorship of the AFOSR under MURI award number FA9550-09-1-0686. The authors will also like to appreciate Dr. Theocharis Baxevanis (post doctoral research associate) for his comments and discussion.

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