Shape Memory Alloys (SMAs) are a unique class of metal alloys which can be deformed severely and afterwards recover their original shape after a thermomechanical cycle (shape memory effect), or a stress cycle within some appropriate temperature regimes (pseudoelasticity, also sometimes called in the literature superelasticity,
The basic idea of the phase field theory is that out of all complexities of statistical mechanics one can reduce the behavior of a system undergoing a phase transformation to that of a few order parameters (i.e., parameters that give a measure of the transformation development), governed by a free energy function, which depends on stress (or deformation), temperature and those parameters. A characteristic example of modeling phase transformations by Landau – Ginsburg theory is provided by Levitas and Preston, 2005.
Also, in the realm of the so – called non equilibrium (or irreversible) thermodynamics several models have been proposed which are based on the use of a set of thermomechanical equations describing the kinetics of the martensitic transformations. The constitutive equations are developed in a non – linear manner on the basis of the laws of thermodynamics. Depending on whether they utilize the full microscopic deformation or the phenomenological one, the thermodynamical models may be classified further as microscopic (e.g., Levitas and Ozsoy, 2009) or macroscopic (e.g., Müller and Bruhns, 2006).
Another approach, which besides being thermodynamically consistent may also furnish a concrete micromechanical justification, is through the employment of plastic flow theories. Recall that the martensite transformation is a diffusionless one during which there is no interchange on the position of neighboring atoms but atom movements resulting in changes in the crystal structure (e.g., see Smallman and Bishop, 2000, pp. 278 – 279). Based on this observation the martensite formation has been explained by a shear mechanism or by a sequence of two shear mechanisms. The shear mechanism can take place either by twinning or by sliding, depending on the composition and on the thermodynamical conditions (Smallman and Bishop, 2000, p. 280). Although in the book of Smallman and Bishop mainly martensitic transformation in steel is described, the authors discuss efforts for the development of a general theory of the crystallography of martensitic transformations. The crystallographic mechanisms of martensite in nickel titanium (NiTi, also known as Nitinol) are similar, i.e., slip or twinning, as in the alloys described in the book of Smallman and Bishop. As a result it can be considered that the role played by the different transformation systems in the martensitic transformations may be suitably parallelized by the role played by the slip systems in crystal plasticity. Models based on this idea have been proposed among others by Diani and Parks (1998), Thamburaja and Anand (2000) and Anand and Gurtin (2003). It should be emphasized that these models are also computationally attractive because a lot of work has been put recently in the algorithms of crystal plasticity, both in their purely algorithmic as well as in their mathematical aspects, resulting in the development of robust algorithms well suited for finite element applications. Accordingly, complex constitutive representations may be considered, since their numerical implementation is no longer intractable, no matter how complex they may be.
An alternative approach, within the context of plastic flow theories, has been proposed by Lubliner and Auricchio (1996) and Panoskaltsis et al. (2004), who developed three – dimensional thermomechanical constitutive models based on generalized plasticity theory in the small deformation regime, and by Panoskaltsis et al. (2011a, 2011b) within finite strains and rotations.
Generalized plasticity is a general theory of rate – independent inelastic behavior which is physically motivated by loading – unloading irreversibility and it may be mathematically founded on set theory and topology (Lubliner 1974, 1984, 1987). Its particular structure provides the theory with the ability to address “non – standard” cases such as non – connected elastic domains.
The objective of this work is twofold: First, to extend the previous works of SMAs modeling based on generalized plasticity, providing a general geometrical framework. This general framework will in turn constitute a basis for the derivation of constitutive models for materials undergoing phase transformations and for
This chapter is organized as follows: In section 2, a general multi – surface formulation of non – isothermal generalized plasticity, capable of describing the multiple and interacting loading mechanisms which occur during phase transformations (see Panoskaltsis et al., 2011a, 2011b)) is presented within the context of tensor analysis in Euclidean spaces. The derivation of the thermomechanical state equations on the basis of
2. Generalized plasticity for phase transformations
2.1. Formulation of the governing equations in the reference configuration
Generalized plasticity is a local internal variable theory of rate – independent behavior, which is based primarily on loading – unloading irreversibility. As in all internal – variable type theories, it is assumed that the local thermomechanical state in a body is determined uniquely by the couple (
The central concept of generalized plasticity is that of the
where < > stands for the Macauley bracket which is defined as:
and stands for a scalar function of the state variables. Accordingly, the value of must be positive at any inelastic state and zero at any elastic one. Finally,
It is emphasized that Eq. (1) has been derived under the assumption of a smooth loading surface at the current strain – temperature point, which implies that only one loading mechanism can be considered. On the other hand, phase transformations include multiple and sometimes interacting loading mechanisms, which may result in the appearance of a vertex or a corner at the current strain – temperature point. This fact calls for an appropriate modification of the rate Equation (1).
In order to accomplish this goal we assume that the loading surfaces are defined in the state space by a number – say n – of smooth surfaces, which are defined by expressions of the form
These surfaces can be either disjoint, or intersect in a possibly non – smooth fashion. Each of these surfaces is associated with a particular transformation mechanism which may be active at the current strain – temperature point. Then, by assuming that each equation defines an independent (non – redundant) active surface at the current stress temperature point, and in view of Eq. (1), we can state the rate equations for the evolution of the internal variables in the following general form
where and are functions of the state variables defined as in Eq. (1) and each set of them – defined by the index i – refers to the specific transformation associated with the part of the loading surface defined by From Eq. (3) one can deduce directly the
Then Eq. (3) implies the following loading – unloading conditions:
Hence, if we denote further by the number of parts for which (ii) holds, and we set:
the loading criteria in terms of the sets and may be stated as:
2.2. Equivalent spatial formulation
The equivalent assessment of the governing equations in the spatial configuration can be done on the basis of a push – forward operation (e.g., see Marsden and Hughes, 1994, pp. 67 – 68; Stumpf and Hoppe, 1997) to the basic equations. For instance, by performing a push – forward operation onto Eq. (3) the latter is written in the form
where F stands for the deformation gradient and g stands for the vector of the controllable variables in the spatial configuration and is composed by the Almansi strain tensor e – defined as the push – forward of the Green – St. Venant strain tensor – and the (scalar invariant) temperature T. Moreover in Eq. (5), q stands for the push forward of the internal variable vector, and stands for the Lie derivative (e.g., see Marsden and Hughes, 1994, pp. 93 – 104; Schutz, 1999, pp. 73-79; Stumpf and Hoppe, 1997), defined as the convected derivative relative to the spatial configuration. Finally, stands for the expression of the scalar invariant functions in terms of the spatial variables (e, T, q) and the deformation gradient F, stands for the push – forward of the tensorial functions and denotes the (scalar invariant) loading rates which are written in the form
where is the expression for the loading surface associated with the index i, in terms of the spatial variables. The (spatial) loading – unloading criteria flow naturally from Eq. (5) as:
where the sets and are now defined in terms of the spatial variables as,and
2.3. Description of rate effects
Rather recent experimental results (see, Nemat – Nasser et al., 2005a, 2005b) on a NiTi shape memory alloy, show that some of the phase transformations depend on the rate of loading. Such a behavior can be accommodated by the (geometrical) framework developed here, by noting that generalized plasticity can be combined consistently with a rate – dependent (viscoplastic) theory. In this case the rate equations for the internal variables may be written in the form
where the stand for additional functions of the state variables enforcing the rate – dependent properties of the transformation defined by the part of the loading surface associated with the index i. The crucial advantage of this approach lies on the compatibility of the two theories, in the sense that neither viscoplasticity, nor generalized plasticity employs the concept of the yield surface as its basic ingredient.
2.4. Transformation induced plasticity
From a further study of the experimental results of Nemat – Nasser et al. (2005a, 2005b) (see also Delville et al., 2011) it is observed that after a stress cycle within the appropriate limits for pseudo-elastic behavior permanent deformations appear, a fact which implies that a yielding behavior appears within the martensitic transformations.
Such a response can be described within our framework by introducing m additional (plastic) loading surfaces, which control the yielding characteristics of the material. These are assumed to be given by expressions of the form
where the functions and have an identical meaning with the functions and which appear in Eq. (3). The constitutive modeling of plasticity phenomena within the martensitic transformations is nowadays a very active area of research. Recent contributions include the phenomenological models by Hallberg et al. (2007, 2010) and Christ and Reese (2009).
A further observation of Eqs. (9) and (10) and their comparison with the basic Eqs. (2) and (3) reveal that both sets of equations show exactly the same qualitative characteristics. Accordingly, it is concluded that from a geometrical standpoint the phase transformation loading surfaces are indistinguishable from the plastic loading surfaces, which means that the internal variable vector
3. The invariant energy balance equation and the thermomechanical state equations
The concept of invariance plays a fundamental role in several branches of mechanics and physics. In particular, within the context of continuum mechanics the invariance properties of the balance of energy equation, under some groups of transformations, may be systematically used in order to derive the conservation laws, the balance laws and/or to determine some restrictions imposed on the equations describing the material constitutive response (e.g., Ericksen, 1961; Green and Rivlin, 1964; Marsden and Hughes, 1994, pp. 163 – 167, 200 – 203; Yavari et al., 2006; Panoskaltsis et al., 2011c). For instance, Marsden and Hughes (1994, pp. 202 – 203) by studying the invariant properties of the local form of the material balance of energy equation, under the action of arbitrary spatial diffeomorphisms, determined the thermomechanical state equations for a non – linear elastic material. The basic objective of this section is to revisit the approach given in Marsden and Hughes (1994, pp. 202 – 203), within the context of the Euclidean space used herein and to show how this can used as a basic constitutive hypothesis
3.1. Revisiting Marsden and Hughes’ theorem
Unlike the original approach of Marsden and Hughes where manifold spaces are used and the invariance of the local form of the material balance of energy equation is examined under the action of arbitrary spatial diffeomorphisms, which include also a temperature rescaling, we examine the invariance properties of the local form of the
where is the energy density, is the spatial mass density, is the Cauchy stress tensor, h is the heat flux vector per unit of surface of the spatial configuration, is the heat supply per unit mass and a superimposed dot indicates material time derivative. By introducing the Helmholtz free energy function obtained by the following Legendre transformation
where is the specific entropy, the local form of the energy balance can be written in the form
where denotes the push – forward operation.
Then the basic theorem of Marsden and Hughes (Theorem 3.6 p. 203), takes in our case the form:
since (see Marsden and Hughes, 1994, p. 98)
from which and by noting that and u can be arbitrarily specified, the thermomechanical state Equations (14) follow. By performing a pull – back operation to Eqs. (14) the following relations are derived
where S stands for the second Piola – Kirchhoff stress tensor, for the material mass density and for the expression of the Helmholtz free energy in the material configuration. It is concluded that Eqs. (20) are identical to the thermomechanical state equations of Marsden and Hughes (1994, p. 203). Thus, we can state the following proposition:
3.2. Thermomechanical state equations for a SMA material
Building on the previous developments we will derive the thermomechanical state equations for a
Axiom 2 is modified as follows:
In this case, as in the previous one, the derivation procedure is the following:
We evaluate Eq. (13) at time when (identity) and with w and u being the velocities ofand at Then
The critical step is the evaluation of the loading rates at which yields
Accordingly, the rate equation for the internal variables evaluated at timeyields
which in view of the rate Equations (5), reads
from which by subtracting the balance of energy Eq. (12) we can derive the identity
from which and by noting thatand u can be specified arbitrarily, we arrive at the expressions:
Therefore, unlike the classical elastic case, for the SMA material considered, the invariance of the local form of the energy balance under superposed spatial diffeomorphisms does not yield the standard thermomechanical state equations
vanish. If this is the case, the classical thermomechanical state equations (Eqs. (14)) can be derived, as in the classical elastic case, directly from Eqs. (29). Thus, we can state the following theorem:
It is interesting to note that in the classical theory of thermodynamics with internal variables Lubliner (1974, 1987) has arrived at a similar result by working entirely in the reference configuration and
4. A constitutive model
Up to now, the proposed formulation was presented largely in an abstract manner by leaving the kinematics of the problem and the number and the nature of the internal variables unspecified. The basic objective of this section is the introduction of a material model that will help make the application of the generalized plasticity concept in modeling phase transformations clearer. The model is based on a geometrically linear model proposed earlier within a stress space formulation by Panoskaltsis and co-workers (Panoskaltsis et al., 2004, Ramanathan et al., 2002) and which has been extensively used in several applications of engineering interesting (e.g., see Freed et al., 2008; Videnic et al., 2008; Freed and Aboudi, 2008; Freed and Banks – Sills, 2007).
There are two fundamental assumptions underlying the new model which is developed here. The first consists of the additive decomposition of the material strain tensor into elastic and inelastic (transformation induced) parts, i.e.,
Such a decomposition has its origins in the work of Green and Naghdi (1965). The second fundamental assumption is that the response of the material is isotropic. Accordingly, it is assumed that it can be described in terms of a
By noting that the martensitic transformations to be considered are accompanied by variations of the elastic properties of the SMA material and in view of the additive decomposition of strain (Eq. (31)), the Helmholtz free energy can be additively decomposed in elastic and inelastic (transformation) parts, as follows
It is emphasized that this is
where the terms
where and are type of parameters and
where , are the type of parameters when the material is fully austenite,, are these when the material is fully martensite and n and m are two additional model parameters. For the particular case n = m = 1 the rule of mixtures, which has been used extensively within the literature (e.g., Anand and Gurtin, 2003; Hallberg et al., 2007) is derived.
For the thermal part of the stored energy function, that is for the functions and we consider the following expressions:
where is the reference temperature, is the specific heat and the linear expansion coefficient, which may be assumed varying within the phase transformations according to expressions analogous to those given in Eq. (35).
Finally, the transformation part of the Helmholtz free energy is given as
where and stand for the configurational entropy and the configurational internal energy and for which we assume two expressions justified in the work of Müller and Bruhns (2006) (see also the thermomechanical theory of Raniecki et al., 1992; Raniecki and Lexcellent, 1998), namely
where , , , , and are the model thermal parameters.
Then in light of the first of Eqs. (20) the second Piola – Kirchhoff stress tensor, after extensive calculations, is found to be
where the dependence of the involved quantities on Z has been dropped for convenience.
The loading surfaces are assumed to be given in the
where denotes the Euclidean norm, stands for the deviatoric part of the stress tensor in the reference configuration and and are parameters. On substituting from Eq. (39) into Eq. (40) the equivalent expression for the loading surfaces in the
For the evolution of the transformation strain we assume a normality rule in the strain – space which is given as
where is a material constant, which is defined as the maximum inelastic strain (e.g., Boyd and Lagoudas, 1996; Lubliner and Auricchio, 1996; Panoskaltsis et al., 2004; Ramanathan et al., 2002), which is attained in the case of one – dimensional unloading in simple tension when the material is fully martensite.
The rate equation for the evolution of the martensite fraction Z, is determined on the basis of the geometrical framework described in section 2 as follows:
For the austenite to martensite transformation we consider the loading surfaces as:
where is a material parameter which can be determined by means of the well – known (e.g., see Lubliner and Auricchio, 1996; Panoskaltsis et al., 2004; Ramanathan et al., 2002; Christ and Reese, 2009) critical stress – temperature phase diagram for the SMAs transformation. Moreover we consider
where the parameters and stand for the martensite finish and martensite start temperatures respectively, and andare two additional parameters which may be determined from experimental results. Since is related to the finish values and to the starting values of the transformation, the loading surfaces and may be considered as the boundaries of the set of all states for which thetransformation can be active. Then the constant (see Eq. (3)) may be defined as
For the function several choices are possible (see Panoskaltsis et al., 2004). In this work, we use a linear expression (see Lickachev and Koval, 1992), which within the present strain – space formulation may be written in the form
where stands for the loading rate in the material description, that is
Similarly, for the inverse transformation we define the loading surfaces as follows
and the parameters ,,,and are material parameters, all related to the transformation. By applying analogous to the transformation case arguments, we derive the rate equation for the evolution of Z for the transformation as
As a result, the final form for the rate equation for the evolution of the internal variable Z (see Eq. (3)) takes the form
The thermomechanical coupling phenomena, which occur during the martensitic transformations may be studied on the basis of the energy balance equation. It should be mentioned here that with the aid of the fundamental concept of energy it is possible to relate different physical phenomena to one another, as well as to evaluate their relative significance in a given process in mechanics and more generally in physics (Lubliner, 2008, p. 44). This will be accomplished as follows:
The energy balance Eq. (12) can be written in a material setting as
where is the divergence operator,
This equation in turn, upon substitution of the thermomechanical state Eqs. (20), yields
The time derivative of the entropy density is determined by the second of Eqs. (20) as
Upon definition of the specific heat as
If we now define the
where is the
This expression has the obvious advantage of decoupling the elastic and inelastic contributions to material heating and is well suited for computational use.
It is noted that in an adiabatic process, that is in a process with
Eq. (62) takes the form
from which and by assuming that the temperature evolution due to
5. Computational aspects and numerical simulations
As a final step we examine the ability of our model in simulating qualitatively several patterns of the extremely complex behavior of SMAs under simple states of straining. Isothermal and non – isothermal problems are considered.
5.1. Isothermal problems
Focusing our attention first in the isothermal case we note that when the total strain tensor
As it has been mentioned the transformation is active when while the inverse transformation is active when Since we always have it is clear that only one phase transformation can be active at a given time of interest. Then we can treat the two phase transformations as two different inelastic processes and replace the general loading – unloading criteria by the following decoupled ones:
Then the governing equations, along with the aforementioned loading – unloading criteria, can be solved by a time discretization scheme based on backward Euler. The resulting system of the discretized equations is solved by means of a
To this end it is emphasized that predictor – corrector algorithms work well in case of domains which are connected. The commonly used predictor – corrector algorithms for elastoplasticity employ an elastic predictor and an inelastic corrector. The most important assumption is that the solution is unique for a particular set of values of the state variables. The predictor step
This is the case of SMAs, which have a transformation (inelastic) zone separating the fully martensite and fully austenite zones (being treated as elastic zones). During forward or reverse transformation, the predictor strain step is very important as we near the elastic – plastic (i.e. transformation) boundary. If the predicted solution lies within the transformation zone (i.e., outside the elastic range) the corrector step is activated and the resulting set of non –linear equations are solved. However,
The first problem we study is a standard problem within the context of finite inelasticity and is that of
where are the material coordinates and
All numerical tests that performed start with the specimen in the parent (austenite) phase, (Z=0).
The first simulation demonstrates the pseudoelastic phenomena within the SMA material. In this case the temperature is held constant at some value above. The purpose is to study a complete stress – induced transformation cycle. The results for this finite shear problem are shown for constant material stiffness as well for a linear (n = m = 1) and a power (n = m = 5) type of stiffness variation in Figures 1, 2 and 3. On loading, the material initially remains austenite (elastic region and straight shear stress – strain curve). As loading is continuing and the shear strain attains the value at which the material point crosses the initial loading surface for the transformation , the transformation starts (inelasticity and curvilinear shear stress – strain curve; coexistence of the two phases). If the loading continues and the strain crosses the final loading surface for the transformation the material is completely transformed into martensite and on further loading since the state of the material is elastic the shear stress – strain diagram is straight. Then, during unloading, the material is fully martensite (elastic region and straight shear stress – strain curve) until the strain crosses the initial loading surface of thetransformation, which is subsequently activated (phase coexistence, inelasticity and curvilinear shear stress – strain curve). On further unloading and when the strain meets the last boundary surface for the transformation , the material becomes fully austenite and on further unloading the stress – strain curve is straight going back to zero, which means that no permanent deformation exists and the austenite is completely recovered. This is expected as the martensite phase is not stable at a temperature above at zero stress level.
Next, the model is tested under
The ability of the model to simulate phase transformations and the corresponding stiffness variations under cyclic loading is demonstrated further by three additional tests. The first one illustrates the case of
5.2. Non – isothermal problems
In this section we examine the ability of the model in predicting pseudoelastic phenomena under
Nevertheless, since our objective is to discuss the proposed framework in its simplest setting, we consider two rather simple problems, namely a simple shear and a plane strain problem, where the equations of motion and the (mechanical) boundary conditions are trivially satisfied. Accordingly, within our simulations,
First, an adiabatic test in finite simple shear is considered. We assume that due to the dynamic rates resulting in adiabatic response, heat exchanges due to conduction, convection and radiation can be neglected in comparison to the temperature changes induced by inelastic (transformation) dissipation, which leads to thermomechanical processes that can be considered as homogeneous. The elastic constants, the mass density and the thermal parameters used in this simulation are those considered in the work of Müller and Bruhns (2006), that is:
while the other parameters are set equal to those used in the isothermal problems studied before. The shear stress – strain curves predicted by the model, for both adiabatic and isothermal cases, are shown in Figure 8. It is observed that the stress – strain curves have similar qualitative characteristics with the adiabatic and the isothermal curves of a perfect gas in a pressure – volume diagram, with the adiabatic stress curve being above the corresponding isothermal one. This fact has to be attributed to material heating due to inelastic dissipation during the transformation, which shifts the stress – strain curve upwards. Moreover, due to the higher stress attained during the transformation, the initial loading surface for the inverse transformationis triggered at a higher stress level, a fact which results in a corresponding higher stress – strain unloading curve. The corresponding temperature – shear strain curve for the adiabatic specimen is shown in Figure 9 (for constant stiffness). Consistently with the experimentally observed adiabatic response of a SMA material, the model predicts heating of the material during the forward transformation and cooling during the inverse transformation.
Next, we study a plane strain model, that of the biaxial extension of a material block. The straining occurs along axes while the block is assumed to be fixed along the direction. This problem is defined as
where λ and ω are the straining parameters.
The isothermal tests for and are conducted in order to show the ability of the model in predicting the shape memory effect. In the first of them, upon loading the transformation is activated, but since the temperature is less than the temperature required for the complete reverse transformation at zero stress, upon unloading the two phases coexist and permanent deformations appear. However, these deformations are recovered after increasing the temperature. In the second test the temperature initially is kept constant at a value less than the austenite start temperature at zero stress. As a result, at the end of the stress cycle the material is completely in the martensite phase and large permanent deformation appears. Nevertheless, like in the previous test, this deformation may be eliminated upon heating. For these new non – isothermal (i.e. heating) problems we assume thermal boundary conditions corresponding to convective heat exchange between the specimen and the surrounding medium on the free faces (with area ) of the specimen. In this case the normal heat flux is given by Newton’s law of cooling (e.g., see Simo and Miehe, 1992) as: with being the constant convection coefficient, which is chosen as , and is the surrounding medium temperature. By assuming that the size of the tested material is small, the contribution to the material heating due to heat conduction can be neglected, so that the temperature evolution equation (see Eq. (62)) can be written in the form
The results of these tests are illustrated in Figures 14 and 15, where the elongation along axis is plotted versus the surrounding medium temperature. The slight increase of the elongation of the SMA material due to the (elastic) thermal expansion occurring prior to the activation of the transformation, for initial temperature , is noteworthy (Figure 15).
6. Concluding remarks
In this chapter we developed a geometrical framework for the establishment of constitutive models for materials undergoing phase transformations and in particular for shape memory alloys. The proposed framework has the following characteristics:
It is quite general for the derivation of the kinetic equations governing the transformation behavior and it can describe multiple and interacting loading mechanisms.
It formulates general loading – unloading criteria, in both their material and spatial settings, that can be systematically employed for the numerical implementation of the derived constitutive models.
It can describe rate effects.
It can model non-isothermal conditions.
It can model transformation induced plasticity by considering it as an additional phase transformation.
It employs the invariance of the spatial balance of energy equation under the superposition of arbitrary spatial diffeomorphisms – that is spatial transformations which can change the Euclidean metric – as a basic constitutive hypothesis, in place of the second law of thermodynamics.
As an application a specific three – dimensional thermomechanical constitutive model for SMA materials is derived. The model can simulate several patterns – under isothermal and
The pseudoelastic behavior observed under monotonic loading.
The pseudoelastic behavior observed under several cyclic loadings.
The stiffness variations occurring during phase transformations.
The shape memory effect.
Additionally, the basic differences between the classical return mapping algorithms and the one used here for the case of not connected regions, have been outlined.