Numerical simulation of phase-transition front propagation in thermoelastic solids A. Berezovski1 and G.A. Maugin2 1
2
Centre for Nonlinear Studies, Institute of Cybernetics at Tallinn University of Technology, Akadeemia tee 21, Tallinn 12618, Estonia
[email protected] Laboratoire de Mod´elisation en M´ecanique, Universit´e Pierre et Marie Curie, 4 place Jussieu, case 162, 75252, Paris Cedex 05, France
[email protected]
Summary. A thermodynamically consistent finite-volume numerical algorithm for martensitic phase-transition front propagation is described in the paper. The proposed numerical method generalizes the wave-propagation algorithm to the case of moving discontinuities in thermoelastic solids.
1 Introduction It is well-known that initial-boundary-value problems, formulated according to the usual principles of continuum mechanics, can suffer from a lack of uniqueness of the solution when the body is composed of a multiphase material (e.g. [1]). The solution in this case involves a propagating phase boundary which separates the austenite from the martensite; the speed of this interface remains undetermined by the usual continuum theory. A nucleation criterion and a kinetic relation for the velocity of the phase boundary are needed as well as the construction of a proper numerical algorithm. From a thermodynamic point of view, a phase transition is a non-equilibrium process; entropy is produced at the moving phase boundary. To perform simulations of practical examples, we need to move to a numerical approximation. In this case, we face a non-equilibrium behavior of finite-size discrete elements or computational cells. It is clear that the local equilibrium approximation is not sufficient to describe such a behavior. We have proposed to determine all the needed fluxes by means of non-equilibrium jump relations at the phase boundary [2]. These jump relations are connected with the contact quantities following from the thermodynamics of discrete systems [3]. In what follows we consider the simplest possible one-dimensional setting of the problem of impact-induced phase transformation front propagation in a shape-memory alloy (SMA) bar. Both martensitic and austenitic phases are considered as isotropic materials. The change in cross-sectional area of
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the bar is neglected. Since thermal expansion coefficient of SMA’s is around 10−5 K −1 , the thermal strain in the material is negligible under the variation up to 100 K. Therefore, the isothermal case is considered. The phase-transition front is viewed as an ideal mathematical discontinuity surface. However, the problem remains nonlinear even in this simplified description that requires a numerical solution. Extensive study of 1-D dynamic phase-transition front propagation in materials with transformation softening behavior has been conducted [4] -[9]. In spite of using different constitutive models, all of them have demonstrated the ability to reproduce the observed behavior of shape memory alloys. However, the used constitutive models are not sufficient to describe the phase-transition front propagation. Therefore, we need to turn to the non-equilibrium description of the phase-transition front propagation [10]. The main focus of the paper is the construction of a numerical scheme for the propagation of phase-transition fronts.
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 ε=
∂u , ∂x
v=
∂u . ∂t
(2)
The density of the material ρ is assumed constant. All field variables are averaged over the cross-section of the bar. 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 ρ
∂v ∂σ = , ∂t ∂x
∂ε ∂v = . ∂t ∂x
(3)
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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. [10]) ρVS [v] + [σ] = 0, VS [ε] + [v] = 0, VS θ[S] = fS VS , (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. [10]) fS = −[W ]+ < σ > [ε],
(5)
where W is the free energy per unit volume. The second law of thermodynamics requires that f S VS ≥ 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 = , ∂t ∂x
ρ
∂v ∂ε = (λ(x) + 2µ(x)) . ∂t ∂x
(8)
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.
3 Conservative wave propagation algorithm The system of equations (8) can be expressed in the form of conservation law ∂ ∂ q(x, t) + f (q(x, t)) = 0, ∂t ∂x where
µ q(x, t) =
¶ ε , ρv
(9)
µ and f (x, t) =
¶ −v , −ρc2 ε
(10)
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A. Berezovski and G.A. Maugin
p and c = (λ + 2µ)/ρ is the sound velocity. In the linear homogeneous case, equation (9) can be rewritten in the form ¶ µ ∂ ∂ 0 −1/ρ q(x, t) + A q(x, t) = 0, A = . (11) −ρc2 0 ∂t ∂x In finite volume numerical methods [11], the solution of the conservation law (9) is obtained in terms of averaged quantities at each time step Z 1 Q= q(x, t)d x, (12) ∆x ∆x and numerical fluxes at the boundaries of each element Z tk+1 1 F± ≈ f ± (q(x, t)) dt. ∆t tk
(13)
The corresponding finite-volume numerical scheme for a uniform grid (n) can be presented as follows (k denotes time steps) ¢ ∆t ¡ + k Qk+1 − Qkn = − (F )n + (F − )kn , (14) n ∆x where superscripts ”+” and ”-” denote inflow and outflow parts in the flux decomposition. Numerical fluxes are determined by means of the solution of the Riemann problem at interfaces between cells [11]. In the considered case, the solution of the Riemann problem at the interface between cells n − 1 and I n consists of two waves, which we denote LIn and LII n . The left-going wave Ln II moves into cell n − 1, the right-going wave Ln moves into cell n. In the linear case, these waves are proportional to eigenvectors rI and rII of the matrix A: I LIn = βnI rn−1 ,
II II LII n = βn rn .
(15)
In the conservative wave-propagation algorithm [12], the solution of the generalized Riemann problem is obtained by using the decomposition of flux difference fn (Qn ) − fn−1 (Qn−1 ) LIn + LII n = fn (Qn ) − fn−1 (Qn−1 ),
(16)
and the corresponding numerical scheme has the form ¢ ∆t ¡ II Qk+1 − Qkn = − Ln + LIn+1 . (17) n ∆x Coefficients β I and β II are determined from the solution of the system of linear equations µ ¶µ I ¶ µ ¶ 1 1 βn −(vn − vn−1 ) = . (18) ρn−1 cn−1 −ρn cn −(ρc2 εn − ρc2 εn−1 ) βnII However, our main goal is the phase-transition front propagation, where it is difficult even to formulate a Riemann problem at the moving phase boundary. Fortunately, we have a tool for the determination of numerical fluxes at the phase boundary. This is nothing else but the non-equilibrium jump relations [2], which should be fulfilled for each pair of adjacent discrete elements.
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4 Contact quantities and numerical fluxes In the non-equilibrium case, we decompose the free energy density into two terms [13] ¯ + Wex . W =W (19) Then contact stress Σ and an excess entropy Sex can be introduced [10] Σ=
∂Wex , ∂ε
Sex = −
∂Wex , ∂θ
(20)
similarly to conventional definition of averaged (local equilibrium) stress and entropy ¯ ¯ ∂W ∂W σ ¯= , S¯ = − . (21) ∂ε ∂θ Here overbars denote averaged quantities. In considered one-dimensional case, the non-equilibrium jump relations [2] take on the following form [¯ σ + Σ] = 0, in the bulk · µ ¶ ¸ ∂S ¯ θ +σ ¯ + Σ = 0, at the phase boundary. ∂ε σ
(22) (23)
What we need now is to determine the values of contact quantities. 4.1 Contact quantities in the bulk In the bulk we apply the non-equilibrium jump relation (22), which can be rewritten at the interface between elements (n) and (n − 1) as (Σ + )n−1 − (Σ − )n = (¯ σ )n − (¯ σ )n−1 ,
(24)
This jump relation should be complemented by the kinematic condition between material and physical velocity [14] which can be rewritten in the onedimensional case as follows [¯ v + V ] = 0. (25) Assuming that the jump of contact velocity is determined by the second term of the last relation [V] = [V ], (26) we obtain in the one-dimensional case (V + )n−1 − (V − )n = (¯ v )n − (¯ v )n−1 .
(27)
Using relations between contact stresses and contact velocities Σn+ = ρn cn Vn+ ,
Σn− = −ρn cn Vn− ,
(28)
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A. Berezovski and G.A. Maugin
we obtain then a system of linear equations for contact velocities µ ¶µ + ¶ µ ¶ 1 1 −(¯ vn − v¯n−1 ) −Vn−1 = . ρn−1 cn−1 −ρn cn −(ρc2 ε¯n − ρc2 ε¯n−1 ) Vn−
(29)
Comparing the obtained equation with (18), we conclude that + βnI = −Vn−1 ,
βnII = Vn− .
(30)
This means that the contact quantities correspond to numerical fluxes. Therefore, the conservative wave propagation numerical scheme (17) can be rewritten in terms of contact quantities ε¯k+1 − ε¯kn = n
¢ ∆t ¡ + V − Vn− , ∆x n
(ρ¯ v )k+1 − (ρ¯ v )kn = n
¢ ∆t ¡ + Σn − Σn− . ∆x
(31)
This means that the introduced non-equilibrium jump relations are consistent with conservation laws. From another point of view, this means that the wavepropagation algorithm is thermodynamically consistent. 4.2 Contact quantities at the phase boundary At the phase boundary we keep the continuity of contact stresses at the phase boundary [10] [Σ] = 0, (32) which yields
+ Σp−1 − Σp− = 0.
(33)
To determine the contact stresses at the phase boundary completely, the relation (33) should be complemented by the coherency condition [15] which can be expressed in the small-strain approximation as follows [V] = 0.
(34)
We still keep the relations between contact stresses and contact velocities (28). This means that in terms of contact stresses equation (34) yields (Σ + )p−1 (Σ − )p + = 0. ρp−1 cp−1 ρp cp
(35)
It follows from the conditions (33) and (35) 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.
(36)
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 (31).
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4.3 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,
(37)
where v¯ is the averaged velocity, ρ is the density. The application of the Maxwell-Hadamard lemma gives [16] [¯ v ] = −[¯ ε]VS ,
(38)
and the jump relation for linear momentum (37) can be rewritten in the form that is more convenient for the calculation of the velocity at singularity ρVS2 [¯ ε] = [¯ σ ].
(39)
The direction of the front propagation is determined by the positivity of the entropy production (6). We also apply the initiation criterion for the stressinduced martensitic phase transformation established in [10].
5 Conclusions Success in numerical simulations of moving discontinuities in solids depends crucially on the jump relations at the discontinuities. These jump relations should be specified before the construction of a numerical scheme. Since conventional continuum theory does not provide the corresponding jump relations, a non-equilibrium description of the phase-transition front propagation is adopted in the paper. It appears that the non-equilibrium description can serve as a basis in the construction of a numerical scheme [17, 18], which is very close to the conservative wave-propagation algorithm [12] based on the solution of a generalized Riemann problem at interfaces between computational cells. Moreover, the non-equilibrium jump relations at the phase boundary can be successfully implemented in the developed numerical scheme. Examples of the phase-transition front propagation simulations in thermoelastic media by means of the formulated algorithm can be found in [10, 19].
Acknowledgment Support of the Estonian Science Foundation under contract No.5756 is gratefully acknowledged.
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