Preprint of conference paper "Modeling the Hysteresis of the Thermomechanical Response of Superelastic β-Titanium Alloys"

Modeling the Hysteresis of the Thermomechanical Response of Superelastic β-Titanium Alloys M. Frosta, A. Moskovkaa,b, M. Knapekc aInstitute of Thermomechanics, Czech Academy of Sciences, Dolejˇ skova 5, 18200 Praha, Czechia bDepartment of Mathematics, Faculty of Applied Sciences, University of West Bohemia, Technick´ a 8, 30100 Plzeˇ n, Czechia cDepartment of Physics of Materials, Faculty of Mathematics and Physics, Charles University, Ke Karlovu 5, 12116 Praha, Czechia 1. Introduction Thanks to the reversible martensitic transformation, shape memory alloys (SMA) exhibit remarkable deformation phenomena, including superelasticity or the shape memory effect, which are utilized in medical, aerospace, or other industries. Finite element simulations provide a cost-effective solution for meeting functional and shape design requirements of final products. The key input for the simulations is a constitutive model that can reliably capture the complex behavior of SMA exhibited in the stress-strain-temperature phase space. Hysteresis of the thermomechanical response is intrinsically related to the dissipation of energy during microstructural processes, such as phase transformations, martensite (de)twinning, or plastic deformation. Depending on the chemical composition, manufacturing, and processing of the material, various morphologies of hysteresis loops can be observed. In this work, we investigate a flexible approach for incorporating various morphologies of closed hysteresis loops into constitutive models. Specifically, we focus on models formulated within the framework of the Generalized Standard Materials (GSM). We demonstrate how a particular parametric function can control the shape of hysteresis loops, we analyze the requirements imposed on its parameters by irreversible thermodynamics, and illustrate the capabilities of the approach using the model of the thermomechanical response of selected Ti-Zr-Nb-Sn metastable β-titanium alloys, which are prospective Ni-free SMA suitable for hypoallergenic medical devices. 2. Parametrized Constitutive Functions for Hysteresis Loops in GSM The GSM theory provides a systematic and powerful formalism for the development of constitutive models, which are thermodynamically consistent [3]. Such models are defined by two potentials: the energy potential, W, and the dissipation potential, D. Both are expected to be functions of (suitable) control variables and descriptors of the state of the matter; the latter is also supposed to depend on fluxes of some of them. The constitutive behavior is then resolved through a (Coleman-Noll procedure-compatible) relation between (sub)gradients of dissipation potential and the generalized forces derived from the energy potential. Under some relatively mild conditions on the mathematical properties of the dissipation potential – non-negativity, convexity in flux(es), and vanishing for zero flux – satisfaction of the second law of thermodynamics can be formally proven. Let us consider a material undergoing a reversible martensitic transformation between the parent phase, called austenite, and the daughter phase, called martensite; the transformation is supposed to be thermodynamically irreversible, i.e., non-negligible energy is continuously dissipated during the process. To simplify the description, let us further suppose that the transformation is ideally athermal, resulting in purely rate-independent behavior. Let us consider a vector of control variables, κ, and one descriptor/internal scalar variable, ξ, indicating the volume fraction of martensite within a representative volume material; due to the characteristic lengths in the system, a mixture of phases within the volume is allowed, hence 0≤ξ≤1.(1) Both κ, ξ evolve in time; however, this dependence is not explicitly expressed in the notation for brevity. Let us assume the (stored) energy potential in the form (typical for SMA modeling) W(κ, ξ) = f(κ)ξ+g(κ),(2) where f, g are scalar functions. Let us suppose fis monotonic and Wis convex. For dissipative rate-independent transformation behavior, the potential of dissipation is a non-trivial positive homogeneous function of degree one. In its simplest form D(˙ ξ) = p0|˙ ξ|,(3) with a real parameter p0>0. The over-dot denotes the time derivative. The constitutive relation, i.e., the time-evolution of the system, is then resolved via the formula (cf. [3]) −∂W ∂ξ ∈∂˙ ξD, (4) with the term on the left being the driving force for the transformation. The symbol ∈denotes set inclusion and ∂˙ ξDdenotes the set of subderivatives (generally subgradients) of Dwith respect to the flux ˙ ξ. The initial state of ξis assumed to satisfy (1). Let us note that the relation (1) can be incorporated into the model, e.g., via enriching Weither by Lagrange multiplier terms [3] or a proper indicator function; such an extension does not significantly alter the treatment pursued below and is omitted for brevity. Then, (2)–(4) results in −f(κ) = (p0if ˙ ξ > 0, −p0if ˙ ξ < 0.(5) and ˙ ξ= 0 if |f(κ)| =p0(cf. [3]). This leads to a hysteretic response, see the blue line in Fig. 1. Employing the following notation of the positive and negative parts of the identity function x+:= max{0, x}, x−:= max{0,−x},(6) we can easily rewrite (3) as D(˙ ξ) = p0˙ ξ++p0˙ ξ−,(7) p0>0.(8) The abrupt phase change does not capture the typical phase kinetics of polycrystalline materials, which is usually gradual owing to significant material stress heterogeneities. To facilitate a smoother transition, Lagoudas et al. [1] proposed a “hardening-type” contribution to the stored energy within the context of SMA, formulated as a sum of specific power functions. Inspired by their work, let us consider a parametric function in a general form h(ξ) = psξ+pt nt ξnt+pi ni (1 −ξ)ni,(9) 1 0 0.4 0.2 0.6 0.8 -p0+p  -p0-p  p0-pri p0+prt  [1] -p00p0 f() [a.u.] Fig. 1. An example of hysteresis loops generated with the presented approach. A model determined via (2)–(4) is marked with the blue line, two models according to (4), (14) and (15) with two parametrizations differing just in exponents are shown in red and green. The black dashed line corresponds to these models without dissipation, i.e., D≡0. Initiation and termination points marked by arrows and related to the parameters of (15) with pf=pr=p0. The purple arrows indicate the direction of change in f(κ). with exponents 1< ni≤2,1< nt≤2.(10) This leads to the following extension of the previous model W(κ, ξ) = f(κ)ξ+g(κ) + h(ξ), D(˙ ξ) = p0˙ ξ++p0˙ ξ−,(11) together with (4) and with (1), (8) and (10) assumed to hold. To capture the differences in forward and reverse transitions, Lagoudas and co-workers considered a different parametrization of hfor the respective processes. Consequently, since the stored energy should return to its initial state after a full transformation cycle is accomplished (under the reversibility assumption), the authors derived some constraints on these parameters. Since the analogous analysis for minor loading cycles has not been addressed and since hysteresis can originate from dissipative processes, we investigate an alternative approach for incorporating diverse hysteresis loop morphologies below. The formula (4) allows shifting the contribution from stored to the dissipated energy. Indeed, let us first observe that function h extends the left-hand side of (5) with ∂h ∂ξ =ps+ptξnt−1−pi(1 −ξ)ni−1.(12) Now h(ξ)is strictly convex for ni, nt∈(1,2] and pi, pt≥0, pi+pt>0since ∂2h ∂ξ2= (nt−1)ptξnt−2+ (ni−1)pi(1 −ξ)ni−2>0∀ξ∈(0,1) (13) and the extension to the endpoints of the corresponding closed interval is accomplished via the limiting process. Moreover, this also means that the function ∂h/∂ξ attains its extrema for 0≤ξ≤1at the interval’s endpoints. As can be verified, we obtain a model formally equivalent to (11) via defining W(ξ) = f(κ)ξ+g(κ), D(˙ ξ) = p0˙ ξ++p0˙ ξ−+∂h ∂ξ ˙ ξ+−∂h ∂ξ ˙ ξ−,(14) 0.005 0.01 0.015 0.02 0.025 0.03 True Strain [1] 0 100 200 300 400 500 600 True Stress [MPa] S1 experiment S1 model S2 experiment S2 model Fig. 2. Hysteresis loops measured on two Ti-Zr-Nb-Sn samples with different chemical composition (dotted lines) fitted with the model defined via (4), (14) and (15) (full lines). together with (1), (4), (10). Following [1], one can expand the flexibility of such a model if different parameters of hare considered for the forward and reverse processes, respectively. After adjusting the constant terms, the explicit definition of the dissipation takes the form D(˙ ξ)=[pf+pftξnft−1−pfi(1 −ξ)nfi−1]˙ ξ++ [pr−priξnrt−1+prt(1 −ξ)nri−1]˙ ξ−.(15) For such a model, formal analysis provides the following sufficient conditions to assure nonnegativity of D(as a function of ˙ ξ) together with no abrupt phase change in the model: pf, pr, pfi, pft, pri, prt ≥0, pfi+pft, pri +prt >0, pf−pfi, pr−pri ≥0.(16) Fig. 1 shows two examples of hysteresis loops with different parametrization according to (15). 3. Application to Nickel-free Superelastic β-Titanium Alloys To illustrate the capabilities of the approach, superelastic hysteresis loops obtained from experiments on two superelastic Ti-Zr-Nb-Sn polycrystalline samples with different manufacturing processes (S1, S2) were fitted, see Fig. 2. Details on the material, experimental characterization, and parametrization of the model can be found in [2]. Acknowledgements This work has been financially supported by the CSF [project No. 24-10366S] and by the Ministry of Education, Youth and Sports of the Czech Republic via the project OP JAK Ferrmion [project No. CZ.02.01.01/00/22 008/0004591], co-funded by the European Union. References [1] Lagoudas, D., Hartl, D., Chemisky, Y., Machado L. and Popov, P., Constitutive model for the numerical analysis of phase transformation in polycrystalline shape memory alloys, Int J Plast 32-33 (2012) 155. [2] Moskovka, A., Hor´ ak, M., Valdman, J., Knapek, M., Janeˇ cek, M., Sedl´ ak, P. and Frost, M., Finite-Strain Constitutive Model for Shape Memory Alloys Formulated in the Logarithmic Strain Space, Shape Mem Superelasticity 11 (2025), in print. [3] Moumni, Z. and Nguyen, Q.S., A Model of Material with Phase Change and Applications, J de Phys IV Proc 6 (C1) (1996), C1-335.