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Variational principle for shape memory alloys

The quasistatic problem of shape memory alloys is reviewed within the phenomenological mechanics of solids without microphysics analysis. The assumption is that the temperature variation rate is small. Reissner's type of generalized variational principle is presented, and its mathematical justification is given for three-dimensional bodies made of shape memory materials.


Introduction
Alloys like nitinol, the mechanics of which has been intensively studied for decades (Funakubo (Ed.), 1987;Brinson et al., 1996;Savi et al., 2002;Auricchio et al., 2007;Lagoudas, (Ed.), 2008), differ from many other metals because they can restore their initial shape after plastic deformation (force action) and heating. The stress-strain state of solids consisting of shape memory alloys is analyzed based on physical models, where martensitic and austenitic phase transformations occur under external forces and temperatures (Niezgodka et al., 1988;Bénilan et al., 1990;Auricchio and Sacco, 1997;Bonetti et al., 2016). Relevant mathematical problems are being studied with the spaces of generalized functions using variational inequalities. Many interesting problems have been solved toward this goal.
It should be noted that shape memory materials have one distinctive characteristic: the shape of the irreversibly changed specimens completely restores after loading and unloading (at constant temperature) if the specimens are heated at certain temperature (for each alloy). In other words, phenomenologically "initial and reverse stresses" occur the same way, up to a sign.
Because force loading causes plastic deformation, this brings up the question of whether it is possible to explain "reverse stress" based on the theory of plasticity without martensitic and austenitic transformations. The proposed paper serves this purpose. The variational principle of Reissner's type is presented and proved for the existence of the Lagrangian function saddle point defined for generalized velocity of strain and stress set in four-dimensional space and time. The developed variation principle by Reissner (Reissner, 1950(Reissner, , 1965 combines two main local minimum principles within the theory of perfect plasticity: minimum principle for strain rates and minimum principle for field of stress velocities (Koiter, 1960). 2 The statement of variational principles covers heuristic considerations on the definition of variational principles for shape memory materials without specific identification of input functional spaces. The next section contains mathematical definitions and proof of the principles. The main difficulties in proving the theorem of existence, based on the variational principle, are clearly seen on the idealized model when there is no hardening and the smooth loading surface is replaced with the von Mises surface. This simplified ideal plastic model is studied later, and consideration of translational hardening does not cause any further complications.

Statement of variational principles
In the case of monoaxial tension of ideal shape memory plastic materials at a constant temperature, the stress-strain curves depend on the temperature and look similar to those illustrated in Figure 1, T is called the shape recovery temperature.
The curves in Figure 1 allow for another interpretation. Let us consider two stages of medium deformation. At stage one, we would apply stress 0    at temperature 0 T . Then, we would relieve the load and measure residual strain 0  . At stage two, we would raise the temperature to 2 T without extra stress. The specimen should return to its initial state. Let us mathematically demonstrate setting of the deformation problem at stage one.
We would study the problem in relation to a solid three-dimensional statement with the von Mises yield criterion.
Here, k is the shear yield strength of the material, ij s is the stress deviator, ij  is the stress tensor, and ij  is Kronecker's delta.
Following (Koiter, 1960) For the parameters defining the continuum behavior, we choose stress tensor ij  and scalar  correlating with the plastic strain rate tensor p ij  as follows (Koiter, 1960;Kachanov, 1971): The point designates time differentiation, ij  is the strain rate tensor, and ijkl E is the elastic modulus tensor. The tensor ij 2  is taken per Prager's recommendation (Prager, 1958): The internal energy density variation rate is calculated per formula based on the following requirement: the process of elastic-plastic straining of solids at constant temperature and with no heat inflow should be described by Prandtl-Reuss law.
The formal contracting of this expression with tensor ij  leads to equality, The same result would be obtained if we formally required satisfaction of equations (2) -(4) and removed the rule of separating elastic deformation from plastic deformation within the laws of thermodynamics. Indeed, at q  = 0, the first law of thermodynamics means which is equivalent to the division of deformation into elastic and plastic parts: Finally, we use Drucker's postulate on normality of the tensor p ij  to the loading surface (1) and find Substituting expression (7) to condition (1), we define parameter p  and deviator The formula for the internal energy density variation rate transforms as follows: Thus, stating equations (3) and (4), we find division of deformation into reversible and irreversible parts and automatic application of the first law of thermodynamics under isothermal deformation of solids without heat inflow. The second law of thermodynamics also applies automatically because equation f can be identified with the change rate of the Helmholtz free energy, while no heat inflow q  =0 and formula (9) give the following expression: where T is the temperature, and s  is the entropy change rate that combines in one formula two main laws of thermodynamics under isothermal loading.
As a result, the set problem of isothermal deformation of elastic-plastic solids includes neither temperature nor laws of thermodynamics in its final form. It is still unclear how "latent heat of plastic melting" h transforms. This portion of energy is assumed to be scattered into the environment. The last process runs so fast (or deformation is so slow) that the main flow parameters do not change. These are the processes that are studied further. Born and Furth (Furth, 1940) noticed similarities between plastic yielding at constant temperature and melting. Both phenomena are followed by scattering of internal (latent) energy into the environment. Formula (4), like the Prandtl-Reuss equations, ensures a certain mechanism of scattering: in the beginning, external force energy at a constant temperature converts into internal energy of the specimen, then the latter one scatters into the atmosphere with a heat conductivity factor equal to infinity.
Let the system of volume forces i X act on the elastic-plastic body that occupies the area D with the boundary . On surface u D  , the velocities are equal to zero, and on p D  , the surface forces are equal to zero. The time interval when straining occurs is designated The assumption is that at any time there is a safe, statistically acceptable distribution of 5 stresses 0 ij  , when the following equality is valid at any velocities The problem of elastic-plastic behavior of solids is to find such tensor ij  and vector i u , that formula (5) and equations (1) and (7) are satisfied for the function of phase ) ( , The inside parameter  is defined from equalities (2) and (3), while the loading vector i X meets equation (10).
Relying on the results (Mosolov and Myasnikov, 1981;Panagiotopoulos, 1985), it was demonstrated (Neustadt, 1993) that the set problem is equivalent to finding the Lagrangian func- on the field of arbitrary velocities i u and deviators If equation (10) holds true, then there is a generalized solution to the "minimax" problem (11). This fact follows satisfaction of equation (6) and the first law of thermodynamics as (5) because when there is no external heat inflow ( q  = 0), the rate of internal energy variation can be presented as s T is the energy dissipation rate.
The mechanical meaning of saddle point existence in solids is as follows: there is a solution as long as the loading allows specifying an acceptable tensor 0 ij  at any point of the body.
Let us make sure that in case of "reverse" stress (if temperature is constant), the principle of the fixed point value can be used for a functional similar to (11).
We relieve stress 0 ij  to   is a stress tensor at point A (Fig. 3). In other words, "reverse" flow on the surface  S requires as much energy as flow on surface S. On the surface  S we have equation (9) by replacement of where parameter   is replaced with 2 h in equation (4), and in equation (8) Equations (10) and (12)  The variational principle in this form is proved in Neustadt, 2008. The next section covers generalizations for the problems when temperature and external stress change during loading. The equation for the internal energy variation rate can be used as (9) (with the factor ) (T k depending on temperature Bertram, 1982). Only equations of the first and second laws of thermodynamics (10), (12) are adjusted. It is also important that the temperature changes quite slowly, and sup-8 plied heat is used selectively (differentially), either for "shape restoration" or for temperature change. Shape memory alloys must absorb heat, i.e., release incoming energy into the environment at those points where the stress state corresponds to plastic flow.

Validation of the variational principle under changing temperature and stress
Let the solid occupy the three-dimensional area D with the boundary Then, we introduce a Hilbert space of functions H  as completion of tensors ij  differentiated with respect to time in the norm determined by the scalar product  (16). This space is not reflexive. It was studied in relation to viscoplasticity problems (Panagiotopoulos, 1985). Particularly, it was proved that the functions from The convex sets ij  and i u are chosen within the spaces H  and ) (M BD so that conditions (17) and (18) and following equations are satisfied (hereinafter, unless otherwise stated, integration is done on area ) M : Condition (19) means that only acceptable stress states (Figure 3) are considered, equations (20) and (21) are the first and second laws of thermodynamics, and the function ) ( 2 T k depends on the temperature. Let us highlight that the laws of thermodynamics "control" plastic deformations while the temperature is assumed to be a known function of time. In formulas (20) and (21), ) (t q  represents the rate of heat input, and ) (t q  is the heat absorption rate. Herewith, Finally, let us apply a load X i to the solid that could be balanced with a safe stress tensor 0 ij  : Now, we will write the Lagrangian function and prove the existence of and the following equation is satisfied 10 and a sequence could be chosen among The condition (25) (24), we have Here, c and 1 c are some invariables. The inequality (28) results in the satisfaction of equation (26), and the mixed variational principle is proved.
It should be noted again that the physical meaning of the constraints (20)-(22) is that the input heat is used either for the temperature change within the body or for irreversible plastic deformations on the surface  S . In this case, the energy that corresponds to plastic flow on the surfaces this is the Prandtl-Reuss law, which follows that for the differentiable functions satisfying the flow on the surface (30), equation (20) becomes The integral in the right part of the formula is non-negative; therefore, the heat input is directed, and the second law of thermodynamics is an effect of the first one.  (Josida, 1965), a subsequence can be chosen from the sequence  i u that weakly* tends to a limit within ) (M BD . It could be taken as a solution to the problem of the "reverse flow" of the shape memory alloys at temperature 2 T . It is clear that the variational principle does not hold true in the limiting state because two first members of the subintegral equation "disappear" in the right part of formula (20)  It should also be noted that it is required to prove regularity of the tensor ij  and the vector i u in order to convert the extremality condition of the Lagrangian function into equation (29). This problem has been studied by many authors, but is still far from being solved (Lewy and Stampacchia, 1970;Capogna and Garofalo, 2003;Focardi et al., 2017).

Conclusions
The key outcome of this research is that the mixed variational principle of Reissner's type can be applied to solids made of shape memory materials if the temperature variation rate is quite small: the process must be both quasi-static and "quasi-isometric".
Phenomenologically, the process can be described by the classical theory of plasticity without physics that takes into account austenitic and martensitic transformations. The further progress of the theory requires a number of mathematical problems to be solved.
In reality, the temperature and heat input rate are not independent. The last constraint can be dropped if the first law of thermodynamics (20)  or its solution that is obtained using formulas for the exponent and E ν -spectral decomposition of the Laplace distribution unit (Josida, 1965)