I MPACT-I NDUCED P HASE T RANSITION F RONT P ROPAGATION IN A BAR A. Berezovski Institute of Cybernetics at Tallinn University of Technology, Tallinn 12618, Estonia
[email protected] G.A. Maugin Universit´e Pierre et Marie Curie, 75252, Paris C´edex 05, France
[email protected] Abstract Phase-transsition front propagation in a shape memory alloy bar is considered in one-dimensional setting. Both martensitic and austenitic phases are considered as isotropic materials. The change in cross-sectional area of the bar is neglected. Since thermal expansion coefficient of NiTi is small, the thermal strain in the material is negligible. Therefore, the isothermal case is considered. The behavior of the material is modeled by a phenomenological approach. The main focus is on the propagation of phase-transition fronts from the impact end. The attention is paid to stress and displacement distributions in the transformation regions.
1 Introduction Shape memory alloy components capable of absorbing dynamic loads may lead to a strong reduction in building’s responses under earthquake ground motion (Saadat et al.(2002), Van Humbeeck(2003), Desroches and Smith(2004)). The underlying mechanism is a reversible martensitic transformation between solid-state phases. A number of constitutive models have been proposed for materials capable of undergoing phase transformations, including purely phenomenological approaches, plasticity analogues, thermodynamically based continuum models, and detailed micromechanical models. Some of these models are reviewed in (Birman(1997), Bernardini and Pence(2002)). A polycrystalline shape memory alloy body subjected to external impact loading will experience deformations that will propagate along the SMA body as stress waves. Propagating phase transformation phenomena have been observed in uniaxial pseudoelastic NiTi (see, for example, Shaw and Kyriakydes(1997), Sun, Li and Tse(2000)). Extensive study of 1-D dynamic phase-transition front propagation in materials with transformation softening behavior has been conducted by Chen and Lagoudas(2000), Bekker et al.(2002), Shaw(2002), Stoilov and Bhattacharyya(2002), Lagoudas et al.(2003), Dai et al.(2004). Despite using different constitutive models, all of them have demonstrated the ability to reproduce the observed behavior of SMA materials. However, the used constitutive models simply prescribe the corresponding SMA behavior. In order to have the possibility to predict at least some of the properties of the SMAs at the macroscopic level of description, we need to turn to the non-equilibrium description of the phase-transition front propagation presented in (Berezovski and Maugin(2005)). In what follows we consider a simplest possible 1-D setting of the problem of impact-induced phase transformation front propagation in a SMA bar. Both martensitic and austenitic phases are considered as isotropic materials. The change in cross-sectional area of the bar is neglected. Since thermal expansion coefficient of NiTi is around 6 − 11 × 10−6 K −1 , the thermal strain in the material is negligible under the variation up to 100 K. Therefore, the isothermal case is considered. The main focus of the paper is on the propagation of phase-transition fronts from the impact end. The attention is paid to stress and displacement distributions in the transformation regions.
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2 Formulation of the problem We consider the boundary value problem of the tensile impact loading of a 1-D, SMA bar that is initially in an austenitic phase and that has uniform cross-sectional area A0 and temperature θ0 . The bar occupies the interval 0 < x < L in a reference configuration and the boundary x = 0 is subjected to the tensile shock loading σ(0, t) = σ ˆ (t) for t > 0. (1) The bar is assumed to be long compared to its diameter so it is under uniaxial stress state and the stress σ(x, t) depends only on the axial position and time. Supposing the temperature is constant during the process, it is characterized by the displacement field u(x, t), where x denotes the location of a particle in the reference configuration and t is time. Linearized strain is further assumed so the axial component of the strain ε(x, t) and the particle velocity v(x, t) are related to the displacement by ∂u ∂u , v= . (2) ∂x ∂t The density of the material ρ is assumed constant. All field variables are averaged over the cross-section of the bar. ε=
At each instant t during a process, the strain ε(x, t) varies smoothly within the bar except at phase boundaries; across a phase boundary, it suffers jump discontinuity. The displacement field is assumed to remain continuous throughout the bar. Away from a phase boundary, balance of linear momentum and kinematic compatibility require that ∂σ ∂ε ∂v ∂v = , = . (3) ρ ∂t ∂x ∂t ∂x Suppose that at time t there is a moving discontinuity in strain or particle velocity at x = S(t). Then one also has the corresponding jump relations (cf. Berezovski and Maugin(2005)) ρVS[v] + [σ] = 0,
VS[ε] + [v] = 0,
VSθ[S] = fSVS,
(4)
where VS is the material velocity of the discontinuity, square brackets denote jumps, S is the entropy per unit volume, and the driving traction fS(t) at the discontinuity is defined by (cf. Berezovski and Maugin(2005)) fS = −[W ]+ < σ > [ε],
(5)
where W is the free energy per unit volume. The second law of thermodynamics requires that fSVS ≥ 0
(6)
at strain discontinuities. If fS is not zero, the sign of VS, and hence the direction of motion of discontinuity, is determined by the sign of fS. Assuming that Hooke’s law holds for each phase σ = (λa + 2µa )ε,
σ = (λm + 2µm )(ε − εtr ),
(7)
where subscripts ”a” and ”m” denote austenite and martensite, respectively, and εtr is the transformation stress, we can then rewrite the relevant bulk equations of inhomogeneous linear isotropic elasticity as follows: ∂ε ∂v ∂v ∂ε = , ρ = (λ(x) + 2µ(x)) . (8) ∂t ∂x ∂t ∂x Here λ and µ are the Lame coefficients, values of which are constant but different depending on the martensitic or austenitic state. It is easy to see that the cross-differentiation of equations (8) leads to the conventional wave equation, solution of which is well-known if corresponding fields are smooth. The difficulties relate to an unknown motion of the phase boundary and to the jump relations across it. That is why we need to develop a numerical scheme which is compatible with the non-equilibrium jump relations at the moving phase boundary. Berezovski and Maugin
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3 Conservative wave propagation algorithm Following the main ideas of finite volume numerical methods (LeVeque(2002)), we divide the bar in a finite number of identical elements of elementary length ∆x. Integration over the finite volume element of equations (8) yields the following set of integral forms: Z ∂ εd x = v + − v − , (9) ∂t ∆x Z ∂ σd x = ((λ + 2µ)v)+ − ((λ + 2µ)v)− , (10) ∂t ∆x where superscripts ± denote the values of either ends of a discrete element. Introducing averaged quantities at each time step Z Z 1 1 vd x, ε¯ = εd x, v¯ = ∆x ∆x ∆x ∆x
1 σ ¯= ∆x
Z
σd x,
(11)
∆x
and numerical fluxes at the boundaries of each element Z tk+1 Z tk+1 1 1 F± ≈ σ ± dt, G± ≈ (2µ + λ)v ± dt, ∆t tk ∆t tk
(12)
we are able to write a finite-volume numerical scheme for equations (8) for a uniform grid (n) in one dimension in the form (k denotes time steps) ∆t + k k − k (¯ v )k+1 − (¯ v ) = (F ) − (F ) (13) n n n n , ρn ∆x ∆t + k (¯ σ )k+1 − (¯ σ )kn = (G )n − (G− )kn , (14) n ∆x where superscripts ”+” and ”-” denote inflow and outflow parts in the flux decomposition. The main difficulty in the construction of a numerical scheme is the proper determination of the numerical fluxes (LeVeque(2002)). Here we should remember that our discrete elements are not in equilibrium, especially in the presence of phase transformation. Even if we can associate the averaged quantities with local equilibrium parameters, we still need to have a description of the non-equilibrium states of discrete elements. The most convenient non-equilibrium thermodynamic theory is the thermodynamics of discrete systems (Muschik(1993)), where the thermodynamic state space of non-equilibrium discrete systems is extended by means of so-called contact quantities (which correspond to the numerical fluxes) in addition to the local equilibrium fields (which correspond to the averaged quantities). Following the ideas of the thermodynamics of discrete systems, we decompose the free energy density in each finite volume element into two terms (Muschik and Berezovski(2004)) ¯ + Wex , W =W
(15)
¯ is the averaged (local equilibrium) free energy, and Wex is the excess free energy. Continuing where W the consequent application of the thermodynamics of discrete systems, we introduce then a contact dynamic stress and an excess entropy ∂Wex ∂Wex Σ= , Sex = − , (16) ∂ε ∂θ which are determined similar to averaged (local equilibrium) stress and entropy σ ¯= Berezovski and Maugin
¯ ∂W , ∂ε
¯ ∂W S¯ = − . ∂θ
(17) 3
Now we can rewrite a finite-volume numerical scheme (13) - (14) by means of corresponding contact quantities instead of numerical fluxes: ∆t + k k − k (Σ (¯ v )k+1 − (¯ v ) = ) − (Σ ) (18) n n n n , ρn ∆x ∆t k k (2µ + λ)V+ n − (2µ + λ)V− n , ∆x where V denotes, by duality, the contact deformation velocity at a boundary of a discrete element. (¯ σ )k+1 − (¯ σ )kn = n
(19)
What we need now is to determine the values of contact quantities.
4 Contact quantities in the bulk In the bulk we apply the non-equilibrium jump relation (Berezovski and Maugin(2005)) [¯ σ + Σ] = 0,
(20)
which can be rewritten at the interface between elements (n) and (n − 1) as (Σ+ )n−1 − (Σ− )n = (¯ σ )n − (¯ σ )n−1 ,
(21)
It should be noted that the non-equilibrium jump relation (20) corresponds to the jump relation for linear momentum (3)1 rewritten in terms of averaged and contact quantities if we take into account that the velocity of stationary boundary between computational cells in bulk is zero. Under the same conditions the next jump relation (3)2 reduces to [¯ v + V] = 0. (22) or, more explicitly, (V+ )n−1 − (V− )n = (¯ v )n − (¯ v )n−1 .
(23)
The contact stresses and contact velocities at the boundary between computational cells are not independent from each other. The connection between contact stresses and contact velocities has the form V− where
n
=−
(Σ− )n , ρn cn
c=
V+
s
n−1
=−
(Σ+ )n−1 , ρn−1 cn−1
(24)
λ + 2µ . ρ
The relation (24) provides a closure of the system of equations for the determination of contact quantities (21) and (23)
5 Contact quantities at the phase boundary At the phase boundary we keep the continuity of contact stresses at the phase boundary (Berezovski and Maugin(2005)) [Σ] = 0, (25) which yields − Σ+ p−1 − Σp = 0.
(26)
Here subscripts p and p − 1 denote elements from both sides of the phase boundary. Berezovski and Maugin
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To determine the contact stresses at the phase boundary completely, the relation (26) should be complemented by the coherency condition (Maugin(1998)) which can be expressed in the small-strain approximation as follows [V] = 0. (27) We still keep the relations between contact stresses and contact velocities (24). This means that in terms of contact stresses equation (27) yields (Σ− )p (Σ+ )p−1 + = 0. ρp−1 cp−1 ρp cp
(28)
It follows from the conditions (26) and (28) that the values of contact stresses and velocities vanish at the phase boundary (Σ+ )p−1 = (Σ− )p = 0, (V+ )p−1 = (V− )p = 0. (29) Now all the contact quantities at the phase boundary are determined, and we can update the state of the elements adjacent to the phase boundary by means of the numerical scheme (18).
6 Velocity of the phase boundary After having the solution of a particular initial-boundary value problem, the material velocity at a moving discontinuity can be determined by means of the jump relation for linear momentum (4) VS[ρ¯ v ] + [¯ σ ] = 0,
(30)
where v¯ is the averaged velocity, ρ is the density. The application of the jump relation (4)2 gives [¯ v ] = −[¯ ε]VS,
(31)
and the jump relation for linear momentum (30) can be rewritten in the form that is more convenient for the calculation of the velocity at singularity ρVS2 [¯ ε] = [¯ σ ]. (32) The direction of the front propagation is determined by the positivity of the entropy production fSVS ≥ 0.
(33)
7 Application to earthquake loads As follows from the analysis of several earthquakes in (Marko et al.(2004)), typical duration of strong motion is in the range of 1.5 - 12.5 s and dominant frequencies are in the range 0.19 - 6.39 Hz. Comparing the duration of the strong motion induced by earthquakes with the duration of the stress pulse propagation through the SMA bar, which is approximately 200 µs, we should conclude that earthquake loads are quasistatic ones. Earthquake loads are displacement-controlled. Therefore, the damping capacity of a SMA component can be characterized by the decrease in the displacement due the martensitic transformation under quasi-static loading.
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7.1
Numerical procedure
Numerical solution of the system of equations (8) is obtained by means of the numerical scheme (18),(19), where the contact quantities are determined by (21),(23) and (24) in the bulk and by (29) at the phase boundary. A procedure similar to a cellular automaton is applied to the phase-transition front tracking. At any time step, the values of the driving force are calculated in cells adjacent to the phase boundary. If the value of the driving force at the phase boundary exceeds the critical one, the velocities of the phase front are computed by means of (32). Virtual displacements of the phase-transition front are calculated then for all possible phase boundaries adjacent to the cell. We keep the cell in the old phase state if the (algebraic) sum of the virtual displacements is less than the size of space step, and change it to another phase otherwise. All the calculations are performed with the Courant-Friedrichs-Levy number equal to 1. 7.2
Numerical simulations
The goal of the simulation is to analyze the phase-transition front propagation under the dynamic loading. Material properties for Ni-Ti SMA were extracted from the paper by McKelvey and Ritchie(2000), where all the details of experimental procedure are well explained. Simulations were performed for a bar of the length 25 cm at the temperature 37 o C. The Young’s moduli are 62 GPa and 22 GPa for austenite and martensite phases, respectively, Poisson’s ratio is the same for both phases and is equal to 0.33, the density for both phases is 6450 kg/m3 . In order to determine the corresponding decrease in the displacement, we consider a constant velocity stepwise loading at the left boundary of the bar. At the right boundary we apply a non-reflective boundary condition. In an elastic material, the step-wise loading with constant velocity leads to a step-wise constant stress distribution. However, due to the martensitic transformation, we obtain a two-step distribution of the stress inside the bar. Approximately after 200 µs, the whole bar will be transformed into the martensitic state, and further it will be deformed elastically. Concerning the displacement decrease, we made a comparison of the displacement distribution in the cases of the presence and of the absence of the martensitic phase transformation in the bar. This comparison is shown in the Figure 1. For pure austenite, one can see the linear increase of the displacement along the bar in the full correspondence with the elasticity theory. Another results appear in the case of the presence of the martensitic phase transformation. As one can see, in the considered time range the displacement at the right end of the bar is smaller than in the case of pure austenite. The full decrease in the displacement is the sum of contributions of the difference in properties of martensite and austenite, and of the transformation stress applied to the full region of the transformed martensite. The first contribution in the displacement decrease is connected only with time delay and will be eliminated as time goes on, whereas the second part will be held all the time. Just this displacement decrease, calculated as the product of the SMA component length and of the transformation stress, can be used as a characteristic damping property for application of SMA components to the damping of earthquake loads.
8 Conclusions Numerical simulations of the phase transition front propagation in a SMA bar under impact loading show that even in the simple isothermal one-dimensional case the developed model demonstrates a good agreement with experimental measurements. The numerical simulations allow us also to explain the decrease in displacement due to the martensitic phase transformation in the case of quasi-static loading. This displacement decrease is proposed to be considered as a characteristic damping property in the design of seismic
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austenite-200 microseconds 200 microseconds 150 microseconds 100 microseconds 50 microseconds 25 microseconds
1.6
Displacement (cm)
1.4 1.2 1 0.8 0.6 0.4 0.2 0 0
200
400
600
800
1000
Dimensionless length
Figure 1: Displacement distribution along a bar at different instants of time. NiTi SMA with phase transformation resistant SMA devices. Acknowledgements Support of the Estonian Science Foundation under contract No. 5765 and of FP6 contract WIND-CHIME (A.B.) is gratefully acknowledged. G.A.M. benefits from a Max Planck Award for International Cooperation (2001-2005). References Bekker, A., J.C. Jimenez-Victory, P. Popov and D.C. Lagoudas (2002), ”Impact Induced Propagation of Phase Transformation in a Shape Memory Alloy Rod”, International Journal of Plasticity, 18, 1447-1479. Berezovski, A. and G.A. Maugin (2005), ”Stress-Induced Phase-Transition Front Propagation in Thermoelastic Solids”, European Journal of Mechanics - A/Solids, 24, 1-21. Bernardini, D. and T.J. Pence (2002) , ”Shape Memory Materials: Modeling”, Encyclopedia of Smart Materials. Schwartz M. (ed.) Wiley, New York, 964-980. Birman, V. (1997), ”Review of Mechanics of Shape Memory Alloy Structures”, Applied Mechanics Review, 50, 629-645. Chen, Y.C. and D.C. Lagoudas (2000), ”Impact Induced Phase Transformation in Shape Memory Alloys”, Journal of Mechanics and Physics of Solids, 48, 275-300. Dai, X., Z.P. Tang, S. Xu, Y Guo and W. Wang (2004), Propagation of Macroscopic Phase Boundaries under Impact Loading. International Journal of Impact Engineering, 30, 385-401. Desroches, R. and B. Smith (2004), ”Shape Memory Alloys in Seismic Resistant Design and Retrofit: A Critical Review of their Potential and Limitations”, Journal of Earthquake Engineering, 8, 415-429. Lagoudas, D.C., K. Ravi-Chandar, K. Sarh and P. Popov (2003), ”Dynamic Loading of Polycrystalline Shape Memory Alloy Rods”, Mechanics of Materials, 35, 689-716. LeVeque, R.J. (2002), Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge, UK. Marko, J., D. Thambiratnam and N. Perera (2004), ”Influence of Damping Systems on Building Structures Subject to Seismic Effects”, Engineering Structures, 26, 1939-1956.
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Maugin, G.A. (1998), ”On Shock Waves and Phase-Transition Fronts in Continua”, ARI-An International Journal for Physical and Engineering Sciences, 50, 141-150. McKelvey, A.L. and R.O. Ritchie (2000), ”On the Temperature Dependence of the Superelastic Strength and the Prediction of the Theoretical Uniaxial Transformation Strain in Nitinol,” Philosophical Magazine A, 80, 1759-1768. Muschik, W. (1993), ”Fundamentals of Non-Equilibrium Thermodynamics”, Muschik, W., (Ed.), Non-Equilibrium Thermodynamics with Application to Solids. Springer, Wien, 1-63. Muschik, W. and A. Berezovski (2004), Thermodynamic Interaction between two Discrete Systems in Non-Equilibrium. Journal of Non-Equilibrium Thermodynamics, 29, 237-255. Saadat, S., J. Salichs, M. Noori, Z. Hou, H. Davoodi, I. Bar-on, Y. Suzuki and A. Masuda (2002), ”An Overview of Vibration and Seismic Applications of NiTi Shape Memory Alloy”, Smart Materials and Structures, 11, 218-229. Shaw J.A. (2002), ”A Thermomechanical Model for a 1-D Shape Memory Alloy Wire with Propagating Instabilities,” International Journal of Solids and Structures, 39, 1275-1305. Shaw, J.A. and S. Kyriakydes (1997), ”On the Nucleation and Propagation of Phase Transformation Fronts in a Niti Alloy,” Acta Materialia, 45, 673-700. Stoilov, V. and A. Bhattacharyya (2002), ”A Theoretical Framework of One-Dimensional Sharp Phase Fronts in Shape Memory Alloys,” Acta Materialia, 50, 4939-4952. Sun, Q.P., Z.Q. Li and K.K. Tse (2000), ”On Superelastic Deformation of Niti Shape Memory Alloy Micro-Tubes and WiresBand Nucleation and Propagation,” Proceedings of IUTAM Symposium on Smart Structures and Structronic Systems. Gabbert U, Tzou HS (Eds.), Kluwer, Dordrecht, 113-120. Van Humbeeck, J. (2003), ”Damping Capacity Of Thermoelastic Martensite In Shape Memory Alloys”, Journal of Alloys and Compounds, 355, 58-64.
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