Algorithmic aspects and finite element solutions for advanced phase field approach to martensitic phase transformation under large strains

Abstract

A new problem formulation and numerical algorithm for an advanced phase-field approach (PFA) to martensitic phase transformation (PT) are presented. Finite elastic and transformational strains are considered using a fully geometrically-nonlinear formulation, which includes different anisotropic elastic properties of phases. The requirements for the thermodynamic potentials and transformation deformation gradient tensor are advanced to reproduce crystal lattice instability conditions under a general stress tensor obtained by molecular dynamics (MD) simulations. The PFA parameters are calibrated, in particular, based on the results of MD simulations for PTs between semiconducting Si I and metallic Si II phases under complex action of all six components of the stress tensor (Levitas et al. in Phys Rev Lett 118:025701, 2017a; Phys Rev B 96:054118, 2017b). The independence of the PFA instability conditions of the prescribed stress measure is demonstrated numerically for the initiation of the PT. However, it is observed that the PT cannot be completed unless the stress exceeds the stress peak points that depend on which stress measure is prescribed. Various 3D problems on lattice instability and following nanostructure evolution in single-crystal Si are solved. The effect of stress hysteresis on the nanostructure evolution is studied through analysis of the local driving force and stress fields. It is demonstrated that variation of internal stress fields due to differing boundary conditions may lead to completely different PT mechanisms.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14
Fig. 15
Fig. 16

References

  1. 1.

    Artemev A, Jin YM, Khachaturyan AG (2001) Three-dimensional phase field model of proper martensitic transformation. Acta Mater 49:1165–1177

    Article  Google Scholar 

  2. 2.

    Babaei H, Levitas VI (2018) Phase field approach for stress- and temperature-induced phase transformations that satisfies lattice instability conditions. Part 2: simulations for phase transformations Si I\(\leftrightarrow \)Si II. Int J Plast 107:223–245

    Article  Google Scholar 

  3. 3.

    Bangerth W, Hartmann R, Kanschat G (2007) Deal. II—a general purpose object oriented finite element library. ACM Trans Math Softw 33(4):1–27

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Barsch GR, Krumhansl JA (1984) Twin boundaries in ferroelastic media without interface dislocations. Phys Rev Lett 53:1069–1072

    Article  Google Scholar 

  5. 5.

    Blank VD, Estrin EI (2014) Phase transitions in solids under high pressure. CRC Press, Boca Raton

    Google Scholar 

  6. 6.

    Chen LQ (2002) Phase-field models for microstructure evolution. Annu Rev Mater Res 32:113–140

    Article  Google Scholar 

  7. 7.

    Falk F (1983) Ginzburg–Landau theory of static domain walls in shape-memory alloys. Z Physik B Condens Matter 51:177–185

    Article  Google Scholar 

  8. 8.

    Finel A, Le Bouar Y, Gaubert A, Salman U (2010) Phase field methods: microstructures, mechanical properties, and complexity. C R Phys 11:245–256

    Article  Google Scholar 

  9. 9.

    Domnich V, Gogotsi Y (2004) Indentation-induce phase transformations in ceramics. In: Gogotsi Y, Domnich V (eds) High pressure surface science and engineering. Institute of Physics, Bristol and Philadelphia, pp 443–466

  10. 10.

    He Y, Zhong L, Fan F, Wang C, Zhu T, Mao SX (2016) In situ observation of shear-driven amorphization in silicon crystals. Nat Nanotechnol 11(10):866

    Article  Google Scholar 

  11. 11.

    Hennig RG, Wadehra A, Driver KP, Parker WD, Umrigar CJ, Wilkins JW (2010) Phase transformation in Si from semiconducting diamond to metallic beta-Sn phase in QMC and DFT under hydrostatic and anisotropic stress. Phys Rev B 82:014101

    Article  Google Scholar 

  12. 12.

    Hill R, Milstein F (1977) Principles of stability analysis of ideal crystals. Phys Rev B 15:3087–3096

    Article  Google Scholar 

  13. 13.

    Hornbogen E (1998) Legierungen mit Formgedächtnis. Rheinisch–Westfälische Akademie der Wissenschaften, Vorträge 388

  14. 14.

    Jacobs AE (1992) Finite-strain solitons of a ferroelastic transformation in two dimensions. Phys Rev B 46:8080–8088

    Article  Google Scholar 

  15. 15.

    Javanbakht M, Levitas VI (2016) Phase field simulations of plastic strain-induced phase transformations under high pressure and large shear. Phys Rev B 94:214104

    Article  Google Scholar 

  16. 16.

    Javanbakht M, Levitas VI (2018) Nanoscale mechanisms for high-pressure mechanochemistry: a phase field study. J Mater Sci 53:13343–13363

    Article  Google Scholar 

  17. 17.

    Ji C, Levitas VI, Zhu H, Chaudhuri J, Marathe A, Ma Y (2012) Shear-induced phase transition of nanocrystalline hexagonal boron nitride to wurtzitic structure at room temperature and lower pressure. Proc Natl Acad Sci USA 109:19108–19112

    Article  Google Scholar 

  18. 18.

    Jin YM, Artemev A, Khachaturyan AG (2001a) Three-dimensional phase field model of low-symmetry martensitic transformation in polycrystal: simulation of \(\zeta _2\) martensite in AuCd alloys. Acta Mater 49:2309–2320

    Article  Google Scholar 

  19. 19.

    Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day Inc, Toronto

    Google Scholar 

  20. 20.

    Levitas VI (2004) Continuum mechanical fundamentals of mechanochemistry. In: Gogotsi Y, Domnich V (eds) High pressure surface science and engineering. Institute of Physics, Bristol, pp 159–292 (Section 3)

    Google Scholar 

  21. 21.

    Levitas VI (2004a) High-pressure mechanochemistry: conceptual multiscale theory and interpretation of experiments. Phys Rev B 70:184118

    Article  Google Scholar 

  22. 22.

    Levitas VI (2013) Phase-field theory for martensitic phase transformations at large strains. Int J Plast 49:85–118

    Article  Google Scholar 

  23. 23.

    Levitas VI (2013b) Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater 61:4305–4319

    Article  Google Scholar 

  24. 24.

    Levitas VI (2013c) Interface stress for nonequilibrium microstructures in the phase field approach: exact analytical results. Phys Rev B 87:054112

    Article  Google Scholar 

  25. 25.

    Levitas VI (2014) Unambiguous Gibbs dividing surface for nonequilibrium finite-width interface: static equivalence approach. Phys Rev B 89:094107

    Article  Google Scholar 

  26. 26.

    Levitas VI (2014a) Phase field approach to martensitic phase transformations with large strains and interface stresses. J Mech Phys Solids 70(2014):154–189

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Levitas VI (2018) Phase field approach for stress- and temperature-induced phase transformations that satisfies lattice instability conditions. Part I. General theory. Int J Plast 106:164–185

    Article  Google Scholar 

  28. 28.

    Levitas VI, Javanbakht M (2010) Surface tension and energy in multivariant martensitic transformations: phase-field theory, simulations, and model of coherent interface. Phys Rev Lett 105:165701

    Article  Google Scholar 

  29. 29.

    Levitas VI, Javanbakht M (2014) Phase transformations in nanograin materials under high pressure and plastic shear: nanoscale mechanisms. Nanoscale 6:162–166

    Article  Google Scholar 

  30. 30.

    Levitas VI, Preston DL (2002a) Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. I. Austenite \(\leftrightarrow \) Martensite. Phys Rev B 66:134206

    Article  Google Scholar 

  31. 31.

    Levitas VI, Preston DL (2002b) Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. II. Multivariant phase transformations and stress-space analysis. Phys Rev B 66:134207

    Article  Google Scholar 

  32. 32.

    Levitas VI, Shvedov LK (2002) Low pressure phase transformation from rhombohedral to cubic BN: experiemnt and theory. Phys Rev B 65(10):104109(1–6)

    Article  Google Scholar 

  33. 33.

    Levitas VI, Warren JA (2016) Phase field approach with anisotropic interface energy and interface stresses: large strain formulation. J Mech Phys Solids 91:94–125

    MathSciNet  Article  Google Scholar 

  34. 34.

    Levitas VI, Preston DL, Lee DW (2003) Three-dimensional Landau theory for multivariant stress-induced martensitic phase transformations. III. Alternative potentials, critical nuclei, kink solutions, and dislocation theory. Phys Rev B 68:134201

    Article  Google Scholar 

  35. 35.

    Levitas VI, Levin VA, Zingerman KM, Freiman EI (2009) Displacive phase transitions at large strains: phase-field theory and simulations. Phys Rev Lett 103:025702

    Article  Google Scholar 

  36. 36.

    Levitas VI, Chen H, Xiong L (2017a) Triaxial-stress-induced homogeneous hysteresis-free first-order phase transformations with stable intermediate phases. Phys Rev Lett 118:025701

    Article  Google Scholar 

  37. 37.

    Levitas VI, Chen H, Xiong L (2017b) Lattice instability during phase transformations under multiaxial stress: modified transformation work criterion. Phys Rev B 96:054118

    Article  Google Scholar 

  38. 38.

    Mamivand M, Zaeem MA, el Kadiri H (2013) A review on phase field modeling of martensitic phase transformation. Comput Mater Sci 77:304–311

    Article  Google Scholar 

  39. 39.

    Milstein F, Marschall J, Fang H (1995) Theoretical bcc–fcc transitions in metals via bifurcations under uniaxial load. Phys Rev Lett 74:2977–2980

    Article  Google Scholar 

  40. 40.

    Olson GB, Cohen M (1972) A mechanism for the strain-induced nucleation of martensitic transformation. J Less Common Met 28:107

    Article  Google Scholar 

  41. 41.

    Olson GB, Cohen M (1986) Dislocation theory of martensitic transformations. In: Nabarro FRN (ed) Dislocations in solids, vol 7. Elsevier Science Publishers B V, New York, pp 297–407

    Google Scholar 

  42. 42.

    Olson GB, Roytburd AL (1995) Martensitic nucleation. In: Olson GB, Owen WS (eds) Martensite, Ch 9. The Materials Information Society, Russell Township, pp 149–174

    Google Scholar 

  43. 43.

    Patten J (2004) Ductile regime machining of semiconductors and ceramics. In: Gogotsi Y, Domnich V (eds) High pressure surface science and engineering. Institute of Physics, Bristol and Philadelphia, p 543632

    Google Scholar 

  44. 44.

    Vedantam S, Abeyaratne R (2005) A Helmholtz free-energy function for a Cu–Al–Ni shape memory alloy. Int J Non-Linear Mech 40:177–193

    Article  MATH  Google Scholar 

  45. 45.

    Wang Y, Khachaturyan AG (2006) Multi-scale phase field approach to martensitic transformations. Mater Sci Eng A 438:55–63

    Article  Google Scholar 

  46. 46.

    Wang J, Yip S, Phillpot SR, Wolf D (1993) Crystal instabilities at finite strain. Phys Rev Lett 71:4182–4185

    Article  Google Scholar 

  47. 47.

    Wasmer K, Wermelinger T, Bidiville A, Spolenak R, Michler J (2008) In situ compression tests on micron-sized silicon pillars by Raman microscopy Stress measurements and deformation analysis. J Mater Res 23(11):3040–3047

    Article  Google Scholar 

  48. 48.

    Wriggers P (2008) Nonlinear finite element methods. Springer, Heidelberg

    Google Scholar 

  49. 49.

    Zarkevich NA, Chen H, Levitas VI, Johnson DD (2018) Lattice instability during solid-solid structural transformations under a general applied stress tensor: example of Si I Si II with metallization. Phys Rev Lett 121(16):165701

    Article  Google Scholar 

  50. 50.

    Zhu J, Wu H, Wang D, Gao Y, Wang H, Hao Y, Yang R, Zhang T, Wang Y (2017) Crystallographic analysis and phase field simulation of transformation plasticity in a multifunctional \( \beta \)-Ti alloy. Int J Plast 89:110–129

    Article  Google Scholar 

  51. 51.

    Zienkiewicz OC, Taylor RL (2000) The finite element method: volume 2- solid mechanics. Butterworth-Heinemann, Woburn

    Google Scholar 

Download references

Acknowledgements

The support of NSF (CMMI-1536925), ARO (W911NF-17-1-0225), ONR (N00014-16-1-2079), and Iowa State University (Vance Coffman Faculty Chair Professorship) are gratefully acknowledged. The simulations were performed at Extreme Science and Engineering Discovery Environment (XSEDE), allocations TG-MSS140033 and MSS170015.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Valery I. Levitas.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Appendix

Appendix

A: Derivation of the weak form of the mechanical equilibrium equation and linearization

In Sect. 3, we outline the weak forms for the mechanical equilibrium equations without details. We present the detailed derivation in this appendix. As described in Sect. 3, we have used a non-monolithic scheme to solve the governing equations, i.e. for solving for the displacements, we assume that the order parameter is constant in all iterations.

The weak form of the equilibrium equation (Eq. 13) is given by Eq. (39). Integrating Eq. (39) by parts and using the Gauss divergence theorem, the weak form is rewritten and can be presented as

$$\begin{aligned} R({\pmb {u}},\delta {\pmb {u}})=\int _{\Omega _0} {\pmb {P}}^T\varvec{:}{\varvec{\nabla }}_0\delta {\pmb {u}} dV_0 - \int _{S_{0T}} \bar{{\pmb {p}}}{{\cdot }}\delta {{\pmb {u}}}\,dS_0=0 \end{aligned}$$
(74)

where we have used the identity \({\varvec{\nabla }}_0\cdot ({\pmb {P}}^T\cdot \delta {\pmb {u}})= ({\varvec{\nabla }}_0\cdot {\pmb {P}})\cdot \delta {\pmb {u}}+{\pmb {P}}^T:{\varvec{\nabla }}_0\delta {\pmb {u}}\) and recall that \( \bar{{\pmb {p}}}\) is the specified traction on the traction boundary \(S_{0T}\). Noticing that

$$\begin{aligned}&{\pmb {E}}=\frac{1}{2}({\pmb {F}}^T{\pmb {.}}{\pmb {F}}-{\varvec{I}}),\quad \delta {\pmb {F}}={\varvec{{\nabla }}}_0\delta {\pmb {u}}\quad \text {and}\quad \nonumber \\&{\varvec{{\nabla }}}_0(.)={\varvec{\nabla }}(.){\pmb {.}}{\pmb {F}} \end{aligned}$$
(75)

the variation of the Lagrangian strain is expressed as

$$\begin{aligned} \delta {\pmb {E}}= & {} \frac{1}{2}({\varvec{{\nabla }}}_0\delta {\pmb {u}}^T{{\cdot }} {\pmb {F}}+{\pmb {F}}^T{{\cdot }}{\varvec{{\nabla }}}_0\delta {\pmb {u}})\nonumber \\= & {} \frac{1}{2}{\pmb {F}}^T{{\cdot }}({\varvec{\nabla }}\delta {\pmb {u}}^T+{\varvec{\nabla }}\delta {\pmb {u}}){{\cdot }}{\pmb {F}} ={\pmb {F}}^T{{\cdot }}\delta {\varvec{\varepsilon }}{{\cdot }}{\pmb {F}} \end{aligned}$$
(76)

with \(\delta {\varvec{\varepsilon }}\) given by Eq. (41). Utilizing the relations between the Piola–Kirchhoff and Cauchy stresses given by Eqs. (10) and (11), as well as using Eq. (76), the first integrand in Eq. (74) can be rewritten as

$$\begin{aligned} {\pmb {P}}\varvec{:}{\varvec{\nabla }}_0(\delta {\pmb {u}})= {\pmb {S}}\varvec{:}{\pmb {F}}^T{{\cdot }}{{\varvec{\nabla }}}(\delta {\pmb {u}}){{\cdot }}{\pmb {F}}= {\pmb {S}}\varvec{:}\delta {\pmb {E}}={\varvec{{\tau }}}\varvec{:}\delta {\varvec{\varepsilon }}, \end{aligned}$$
(77)

where \( {\varvec{{\tau }}}=J {\varvec{\sigma }}\) is the Kirchhoff stress. Therefore, the weak form of the equilibrium equation Eq. (74) can be written in the form given by Eq. (40).

Because we will use the Newton’s iteration for computing the displacements, we must linearize the weak form given by Eq. (40). In doing so we expand the weak form in a Taylor series about \({\pmb {u}}\)

$$\begin{aligned} R({\pmb {u}}+ \Delta {\pmb {u}}, \delta {\pmb {u}})= & {} R({\pmb {u}}, \delta {\pmb {u}})\nonumber \\&+\,\Delta R({\pmb {u}}, \Delta {\pmb {u}}, \delta {\pmb {u}})+o(\Delta {\pmb {u}}) = 0, \end{aligned}$$
(78)

where \(\Delta {\pmb {u}}\) is an increment of the displacement vector, \(\delta {\pmb {u}}\) has been kept fixed, \(o(\Delta {\pmb {u}})\) consists of the higher order terms in \( \Delta {\pmb {u}}\) such that \(\lim _{\Delta {\pmb {u}}\rightarrow {\pmb {0}}} o(\Delta {\pmb {u}})/|\Delta {\pmb {u}}|= 0\), and \(\Delta R({\pmb {u}}, \Delta {\pmb {u}}, \delta {\pmb {u}})\) is the directional derivative of R defined as [48]

$$\begin{aligned} \Delta { F}({\pmb {u}}, \Delta {\pmb {u}}, \delta {\pmb {u}})= & {} D F({\pmb {u}},\delta {\pmb {u}})\cdot \Delta {\pmb {u}} \nonumber \\= & {} \left. \frac{d}{d\epsilon } F({\pmb {u}}+\epsilon \Delta {\pmb {u}},\delta {\pmb {u}})\right| _{\epsilon =0} \end{aligned}$$
(79)

for any differentiable functional or function F.

The linearized form of residual R in Eq. (40) can be expressed as

$$\begin{aligned} D R\cdot \Delta {\pmb {u}}=\int _{\Omega _0}\Delta {\pmb {S}}\varvec{:}\delta {\pmb {E}} dV_0+\int _{\Omega _0}{\pmb {S}}\varvec{:}\Delta (\delta {\pmb {E}})dV_0. \end{aligned}$$
(80)

We will now derive an amenable form of the integrands in Eq. (80).

Noticing that \({\pmb {U}}_t\) is independent of \({\pmb {u}}\), we derive the expression for the increment of \({\pmb {S}}\) using Eq. (11):

$$\begin{aligned} \Delta {\pmb {S}} = J_t{\pmb {U}}_t^{-1} \cdot \Delta \hat{{\pmb {S}}}\cdot {\pmb {U}}_t^{-1} = J_t{\pmb {U}}_t^{-1} \cdot ({{\pmb {C}}}:\Delta {{\varvec{E}}}_e) \cdot {\pmb {U}}_t^{-1}, \nonumber \\ \end{aligned}$$
(81)

where \({{\pmb { C}}}\) is the fourth order elastic modulus tensor with respect to \(\Omega _t\) and is given by Eqs. (27) and (44). Using Eq. (3)\(_1\), we show that the increments \(\Delta {\pmb {E}}_e\) and \(\Delta {\pmb {E}}\) are related by

$$\begin{aligned} \Delta {\pmb {E}}_e = {\pmb {U}}_t^{-1}\cdot \Delta {\pmb {E}}\cdot {\pmb {U}}_t^{-1}, \end{aligned}$$
(82)

which we use to rewrite Eq. (81) as

$$\begin{aligned} \Delta {\pmb {S}}= & {} J_t {\pmb {U}}_t^{-1} \cdot ({{\pmb { C}}}:{\pmb {U}}_t^{-1}\cdot \Delta {{\varvec{E}}}\cdot {\pmb {U}}_t^{-1})\cdot {\pmb {U}}_t^{-1} \nonumber \\= & {} {\varvec{{{\mathcal {C}}}}}: \Delta {\pmb {E}}, \end{aligned}$$
(83)

where \({\varvec{{{\mathcal {C}}}}}\) is the fourth order elasticity tensor defined in the reference configuration \(\Omega _0\), which is related to \({{\pmb { C}}}\) by

$$\begin{aligned} {{\mathcal {C}}}^{IJKL}=J_t (F_t^{-1})^{Ii}(F_t^{-1})^{Jj}(F_t^{-1})^{Kk}(F_t^{-1})^{Ll}{ C}^{ijkl}. \end{aligned}$$
(84)

Note that the indices in upper case, i.e. IJ,  etc. are for \(\Omega _0\) and the indices in lower case, i.e. ij,  etc., are for \(\Omega _t\). Using Eqs. (76) and (83), we rewrite the first integrand of Eq. (80) as

$$\begin{aligned} \Delta {\pmb {S}}:\delta {\pmb {E}}= & {} \delta {\pmb {E}}: {\varvec{{{\mathcal {C}}}}}: \Delta {\pmb {E}} \nonumber \\= & {} {\pmb {F}}^T\cdot \delta {\varvec{\varepsilon }}\cdot {\pmb {F}}:({\varvec{{{\mathcal {C}}}}}: {\pmb {F}}^T\cdot \Delta {\varvec{\varepsilon }}\cdot {\pmb {F}} )\nonumber \\= & {} \delta {\varvec{\varepsilon }}:J{{\pmb {\textsf {C}}}}:\Delta {\varvec{\varepsilon }}, \end{aligned}$$
(85)

where we have used (see Chapter 10 of [51])

$$\begin{aligned} \Delta {\pmb {E}} ={\pmb {F}}^T\cdot \Delta {\varvec{\varepsilon }}\cdot {\pmb {F}}, \end{aligned}$$
(86)

with \(\Delta {\varvec{\varepsilon }}\) given by Eq. (45) and \({{\pmb {\textsf {C}}}}\) as the fourth order elasticity tensor defined in \(\Omega \), which is given by Eq. (43).

Next, let us simplify the second integrand in Eq. (80). It can be obtained from Eq. (76) that

$$\begin{aligned} \Delta (\delta {\pmb {E}})= & {} \frac{1}{2}({\varvec{\nabla }}_0 \Delta {\pmb {u}}^T{{\cdot }}{\varvec{\nabla }}_0\delta {\pmb {u}}+{\varvec{\nabla }}_0 \delta {\pmb {u}}^T{{\cdot }}{\varvec{\nabla }}_0 \Delta {\pmb {u}})\nonumber \\= & {} \frac{1}{2}{\pmb {F}}^T{{\cdot }}({\varvec{\nabla }} \Delta {\pmb {u}}^T{{\cdot }}{\varvec{\nabla }}\delta {\pmb {u}}+{\varvec{\nabla }} \delta {\pmb {u}}^T{{\cdot }}{\varvec{\nabla }} \Delta {\pmb {u}}){{\cdot }}{\pmb {F}}. \end{aligned}$$
(87)

Thus, noticing that \( {\pmb {S}}={\pmb {F}}^{-1}{{\cdot }}{\varvec{{\tau }}}{{\cdot }}{\pmb {F}}^{-T} \), the second integrand of Eq. (80) is expressed as

$$\begin{aligned} {\pmb {S}}\varvec{:}\Delta (\delta {\pmb {E}})={\varvec{\nabla }}\delta {\pmb {u}}\varvec{:}{\varvec{{\tau }}}{{\cdot }}{\varvec{\nabla }}\Delta {\pmb {u}}^T. \end{aligned}$$
(88)

Therefore, Eq. (80) simplifies to the form given by Eq. (42).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Babaei, H., Basak, A. & Levitas, V.I. Algorithmic aspects and finite element solutions for advanced phase field approach to martensitic phase transformation under large strains. Comput Mech 64, 1177–1197 (2019). https://doi.org/10.1007/s00466-019-01699-y

Download citation

Keywords

  • Phase-field approach
  • Martensitic phase transformation
  • Lattice instability condition
  • Nanostructure