Cyclic martensitic phase transformation of monocrystals at finite strains

Gautam Sagar∗ and Erwin Stein

Leibniz Universitat Hannover, Institute of Mechanics and Computational Mechanics, Appelstr. 9A, D-30167 Hannover,Germany.

Based on Bain’s principle, a C1-continuous thermo-mechanical micro-macro constitutive model for martensitic phase trans-formation (PT) of monocrystals at finite strains and hyperelastic free energy function is used. It is represented by a unifiednon-convex Lagrangian variational functional. The convexification problem is solved here by generalizing the explicit form ofthe lower Reuss bound for small strains given in [1] to finite strains. Abaqus is used for implementation of 3D finite elementsin space, via UMAT-interface which requires Jaumann rate of Kirchhoff stress tensor. Deterministic validation of the modelis presented by comparing verified numerical results with experimental data for Cu82Al14Ni4 [6] for quasiplastic PT.

1 Introduction

Martensitic phase transformation is a diffusionless first-order PT between ’high’temperature austenite and ’low’ temperature martensitic phases of shape memoryalloys (SMAs) and has many challenging applications in technical actuators andin medicine. The specific feature of martensitic phase transformations (PTs) is theability of SMAs to ’remember’ their initial state. We investigate the following PTcycle: 1.) Micro deformation from austenite to twinned and further to detwinnedmartensite, A → Mdetw, by cooling (which is not visible in computation), fol-lowed by elastic deformation due to mechanical loading at equilibrium tempera-ture θ0, θ0 = (As+Ms)/2, (As: austenite start temperature, Ms: martensite starttemperature, θ0 < Af , Af : austenite finish temperature); 2.) fast quasi-plastic PTat a critical stress at Mf , Mf ≈ θ0 (Mf : martensite finish temperature); 3.) elas-tic unloading at θ0; and finally 4.) back-PT to austenite, M → A, by heating upto second critical temperature θcrit, θcrit > Af (Af > As), Fig. 1. Fig. 1 PT cycle for SME

2 A unified model for PT of monocrystals

The macro and micro mechanical C1-continuous mathematical models are based on the multiplicative decomposition of the to-tal deformation gradient F into elastic F

e and transformation part Ft. Neo-Hookean hyperelastic material model is used. Theconcept of quasi-convexified free energy is applied for monocrystalline shape memory alloys in order to get the macroscopicconstitutive model at large strains. The extension of the total free energy by a Lagrangian functional L, see eq. (2), accountsfor kinematic constraints and mass conservation of the material with martensitic PT cycles under cooling, mechanical loadingand heating. Stress tensor, phase transformation vector and vector of thermodynamical forces are obtained in a canonical wayas partial derivatives of L, with respect to the conjugate field variables. The PT-criteria together with the evolution law forinternal variables completes the PT model.

3 Free energy

Based on the assumption by Ball and James [2] the macroscopic free strain energy of a crystal is given by the minimum of theenergies of all possible i.e. compatible n phase variants, Ψ = mini=1..n

[ψel

i (bei , J

ei ) + ψch

i

], where b

ei = F

(U

ti2)−1

FT

is elastic left Cauchy-Green deformation tensor, with PT stretch tensor Uti for phase i. ψch

i is the so-called chemical energyof the i-th phase variant. The extension of the vector of quasi-convexified free energies for the phase fractions from small tofinite strains, Ψ(ε, ξ) −→ Ψ(b, ξ) [4, 5], reads, Ψ(b, ξ) = ξ · ψ(b) + ΨM

FS(ξ).The approximation of free energy of mixing, ΨM

FS(bt, ξ), for a 2-phase system (1 means austenite, 2 means martensite)was given in [5] as

ΨMFS(ξ) ≈ −ξ2 ψt

2(bt

2, J t

2) + ξ2 ξ2 ψt

2(bt

2, J t

2). (1)

∗ Corresponding author E-mail: sagar@ibnm.uni-hannover.de, Phone: +49 511 762 9089, Fax: +49 511 762 19053

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

PAMM · Proc. Appl. Math. Mech. 7, 4060049–4060050 (2007) / DOI 10.1002/pamm.200700797

In PT process of Cu82Al14Ni4 treated here, only martensitic phase number 6 is active beside the austenitic parent phase i.e. intotal two phases have to be considered.

The presence of kinematic constraints of the material suggests the enhance-ment of the free energy function, by a Lagrangian functional presented forlinear kinematics in [3, 4] and for nonlinear theory in [5], as

L(b, ξ, γ, δ) = Ψ(b, ξ) − γ · ξ + δ(e∗ · ξ − 1), (2)

with the vector of phase fractions ξ =∑n

i=1ξiei, ei · ej = δij (1 austenite

phase and n-1 martensitic phases), the mass conservation condition e∗ · ξ−1 = 0, e∗ the normal vector of the (n-1)-dimensional PT-polytope, e∗ =∑n−1

i=1e∗i , |e

∗

i| = 1, ξi ≥ 0 and b

t is PT left Cauchy-Green tensor. ψ ∈ Rn

is the vector of phase energies with ψ(b) =∑n

i=1ψi(bi, ξ)ei, ei ∈ R

n .The vector γ and the scalar δ as Lagrangian multipliers for n phases haveto fulfill the Kuhn-Tucker conditions of the saddle point problem γi ≥0 ,−ξi ≤ 0γ · ξ = 0. The energy dissipation condition for the driving force(thermodynamical conjugate force), f = ∂L/∂ξ, is D = f · ξ ≥ 0, and alocal maximum dissipation principle reads f · ξ → Max. With analogies tothe theory of stable inelastic deformations in elastoplasticity it is deducedthat the ’transformation function’ φ = ‖f‖ − fc ≤ 0, – here in a simplesingle surface form – has to be convex (similar to a convex yield function)with the critical driving force fc as an energy barrier for initiating PT.

calculated dataexperimental data

Fig. 2 Engineering stress-strain curve of SMEspecimen; experimental and computed data withlinearized strains [4].

4 Verification and validation of PT-model

The numerical results of the finite strain PT-model are capturing more de-tailed information for stress-strain behavior during quasiplastic PT. Thezigzag type experimental stress-strain curve within phase transformation atloading, called ’yield tooth’, Fig. 2, is approximated within the finite el-ement analysis by a smoothly decreasing and then increasing axial stress- axial strain curve which could not be achieved with linearized kinemat-ics yielding a constant axial stress during PT and also presented here forcomparison, Fig. 2. Figs. 2, 3 show that PK1-transformation-stress withnon-linear kinematics is 1.6% less than with linear kinematics, and approx-imately 2% less than the experimentally measured stress [6]. The residualPT strains after elastic unloading for non-linear kinematics are smaller by3% than those with linear strain.The presented model can also be used forsimulation of superelastic PT-effect.

Acknowledgement: Support for this work is provided by the German Sci-ence Foundation (DFG) under project No. Ste 238/51-1&2, which is grate-fully acknowledged.

Fig. 3 Engineering stress-strain curve of SMEspecimen; computed data with linearized strain(linear) and finite strains (non-linear).

References

[1] S. Govindjee, A. Mielke, and G. J. Hall, The free energy of mixing for n-variant martensitic phase transformations using quasi-convexanalysis, J. Mech. Phys. Solids, 51(4), I–XXVI (2003).

[2] J. M. Ball and R. D. James, Fine phase mixtures and minimizers of energy, Arch. Rational Mech. Anal., 100(1), 13–52 (1987).[3] S. Govindjee and C. Miehe, A multi-variant martensitic phase transformation model: formulation and numerical implementation,

Comput. Methods Appl. Mech. Eng., 191(3–5), 215–238 (2001).[4] E. Stein and O. Zwickert, Theory and finite element computations of a unified cyclic phase transformation model for monocrystalline

materials at small strain, Comp. Mech., 6, 429-445 (2007).[5] E. Stein and G. Sagar, Theory and finite element computation of cyclic martensitic phase transformation at finite strain, Int. J. Num.

Meth. Engng., online printed first, 32 pages (2007).[6] Z. Xiangyang, S. Qingping and Y. Shouwen. A non-invariant plane model for the interface in CuAlNi single crystal shape memory

alloys, J. Mech. phys. solids, 48, 2163-2182 (2000).

© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

GAMM Sections 4060050