In the present work, the effect of a thermodynamically consistent inelastic interface stress on nanovoid evolution in NiAl is studied. Such interface stress is introduced for the solid–gas interface of nanovoids within the concept of the phase field approach. The Cahn–Hilliard (CH) equation using the Helmholtz free energy describes the evolution of nanovoid concentration. The interface stress changes the total stress distribution and affects the elastic stress field. Thus, due to the significant effect of the elastic energy on nanovoid dynamics, it can indirectly affect nanovoid nucleation and growth. The highly nonlinear coupled CH and elasticity equations are solved using the finite element method and the COMSOL code. The coupling appears due to the presence of the nonlinear nanovoid inelastic strain in the total strain, the presence of the nonlinear inelastic interface stress in the stress tensor and the presence of elastic energy in the Helmholtz free energy. Several examples of thermal-induced nanovoid evolutions are presented to investigate the effect of the solid–gas interface stress. The obtained results show the significant effect of the interface stress on the total stress distribution, and consequently a different distribution of thermodynamic driving force which can affect the nanostructure evolution and the deformation. Mainly, the interface stress represents a promotive effect on nanovoid growth which results in a faster nanovoid growth and a larger nanovoid concentration and region.
AspKÅgrenJ.Phase-field simulation of sintering and related phenomena – A vacancy diffusion approach. Acta Mater2006; 54: 1241–1248.
2.
BoettingerWJWarrenJABeckermannCKarmaA.Phase-field simulation of solidification. Annu Rev Mater Res2002; 32: 163–194.
3.
EmmerichH.Advances of and by phase-field modelling in condensed-matter physics. Adv Phys2008; 57: 1–87.
4.
HenryHLevineH.Dynamic instabilities of fracture under biaxial strain using a phase field model. Phys Rev Lett2004; 93: 105504.
5.
MoelansNBlanpainBWollantsP.Quantitative analysis of grain boundary properties in a generalized phase field model for grain growth in anisotropic systems. Phys Rev B2008; 78: 024113.
6.
SteinbachI.Phase-field models in materials science. Model Simul Mater Sci Eng2009; 17: 073001.
7.
YangMZhangJWeiHZhaoYGuiWSuHJinTLiuL.Study of γ′ rafting under different stress states – A phase-field simulation considering viscoplasticity. J Alloys Compd2018; 769: 453–462.
8.
Kubendran AmosPGSchoofESchneiderDNestlerBChemo-elastic phase-field simulation of the cooperative growth of mutually-accommodating Widmanstätten plates. J Alloys Compd2018; 767: 1141–1154.
9.
TschukinOSchneiderDNestlerB.An elasto-chemical phase-field model for isotropic solids. Eur J Mech - A/Solids2019; 73: 181–191.
10.
JacobsAECurnoeSHDesaiRC.Simulations of cubic-tetragonal ferroelastics. Phys Rev B2003; 68: 224104.
11.
ArtemevAJinYKhachaturyanAG.Three-dimensional phase field model of proper martensitic transformation. Acta Mater2001; 49: 1165–1177.
12.
LevitasVILeeD-WPrestonDL.Interface propagation and microstructure evolution in phase field models of stress-induced martensitic phase transformations. Int J Plast2010; 26: 395–422.
13.
LevitasVIJavanbakhtM.Phase-field approach to martensitic phase transformations: Effect of martensite–martensite interface energy. Int J Mater Res2011; 102: 652–665.
14.
SeolDJHuSYLiYLChenLQOhKH.Cubic to tetragonal martensitic transformation in a thin film elastically constrained by a substrate. Met Mater Int2003; 9: 221–226.
15.
RasmussenKØLookmanTSaxenaABishopARAlbersRCShenoySR.Three-dimensional elastic compatibility and varieties of twins in martensites. Phys Rev Lett2001; 87: 055704.
16.
ChenL-Q.Phase-field models for microstructure evolution. Annu Rev Mater Res2002; 32: 113–140.
17.
YedduHKBorgenstamAÅgrenJ.Effect of martensite embryo potency on the martensitic transformations in steels—A 3D phase-field study. J Alloys Compd2013; 577: S141–S146.
18.
HuSYLiYLZhengYXChenLQ.Effect of solutes on dislocation motion —a phase-field simulation. Int J Plast2004; 20: 403–425.
19.
WangYUJinYMKhachaturyanAG.Phase field microelasticity modeling of dislocation dynamics near free surface and in heteroepitaxial thin films. Acta Mater2003; 51: 4209–4223.
20.
RodneyDLe BouarYFinelA.Phase field methods and dislocations. Acta Mater2003; 51: 17–30.
21.
WangYLiJ.Phase field modeling of defects and deformation. Acta Mater2010; 58: 1212–1235.
FarrahiGHJavanbakhtMJafarzadehH.On the phase field modeling of crack growth and analytical treatment on the parameters. Contin Mech Thermodyn2018.
24.
LevitasVIJafarzadehHFarrahiGHJavanbakhtM.Thermodynamically consistent and scale-dependent phase field approach for crack propagation allowing for surface stresses. Int J Plast2018; 111: 1–35.
25.
JafarzadehHFarrahiGHJavanbakhtM.Phase field modeling of crack growth with double-well potential including surface effects. Contin Mech Thermodyn2019.
26.
NguyenTTRéthoréJBaiettoM-C.Phase field modelling of anisotropic crack propagation. Eur J Mech - A/Solids2017; 65: 279–288.
27.
MaySVignolletJde BorstR.A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations. Eur J Mech - A/Solids2015; 52: 72–84.
28.
MozaffariNVoyiadjisGZ.Phase field based nonlocal anisotropic damage mechanics model. Phys D Nonlinear Phenom2015; 308: 11–25.
29.
MillettPCTonksM.Application of phase-field modeling to irradiation effects in materials. Curr Opin Solid State Mater Sci2011; 15: 125–133.
30.
LiYMaDWangB.Influence of bulk free energy density on single void evolution based on the phase-field method. Comput Mater Sci2019; 163: 100–107.
31.
BasakALevitasVI.Phase field study of surface-induced melting and solidification from a nanovoid: Effect of dimensionless width of void surface and void size. Appl Phys Lett2018; 112: 201602.
32.
BerezovskiAGiorgioIDella CorteA.Interfaces in micromorphic materials: Wave transmission and reflection with numerical simulations. Math Mech Solids2016; 21(1): 37–51.
33.
TurcoEGiorgioIMisraAdell’IsolaF. King post truss as a motif for internal structure of (meta)material with controlled elastic properties. R Soc Open Sci2017; 4(10): 171153, 1–20.
34.
MisraALekszyckiTGiorgioIGanzoschGMüllerWHdell’IsolaF. Pantographic metamaterials show atypical Poynting effect reversal. Mech Res Commun2018; 89: 6–10.
35.
dell’IsolaFSeppecherPSpagnuoloM,et al. Advances in pantographic structures: Design, manufacturing, models, experiments and image analyses. Contin Mech Thermodyn2019; 31(4): 1231–1282.
36.
PlacidiLMisraABarchiesiE.Simulation results for damage with evolving microstructure and growing strain gradient moduli. Contin Mech Thermodyn2019; 31: 1143–1163.
37.
dell’IsolaFDella CorteAGiorgioI.Higher-gradient continua: The legacy of Piola, Mindlin, Sedov and Toupin and some future research perspectives. Math Mech Solids2017; 22: 852–872.
38.
YuH-CLuW.Dynamics of the self-assembly of nanovoids and nanobubbles in solids. Acta Mater2005; 53: 1799–1807.
39.
HuSHenagerCH.Phase-field modeling of void lattice formation under irradiation. J Nucl Mater2009; 394: 155–159.
40.
HuSYHenagerCH.Phase-field simulation of void migration in a temperature gradient. Acta Mater2010; 58: 3230–3237.
41.
RokkamSEl-AzabAMillettPWolfD.Phase field modeling of void nucleation and growth in irradiated metals. Model Simul Mater Sci Eng2009; 17: 064002.
42.
MillettPCEl-AzabARokkamSTonksMWolfD.Phase-field simulation of irradiated metals i. Comput Mater Sci2011; 50: 949–959.
43.
MillettPCEl-AzabAWolfD.Phase-field simulation of irradiated metals ii. Comput Mater Sci2011; 50: 960–970.
44.
LiYHuSSunXGaoFHenagerCHKhaleelM.Phase-field modeling of void migration and growth kinetics in materials under irradiation and temperature field. J Nucl Mater2010; 407: 119–125.
45.
SemenovAAWooCH.Interfacial energy in phase-field emulation of void nucleation and growth. J Nucl Mater2011; 411: 144–149.
46.
SemenovAAWooCH.Phase-field modeling of void formation and growth under irradiation. Acta Mater2012; 60: 6112–6119.
47.
LiYHuSMontgomeryRGaoFSunX.Phase-field simulations of intragranular fission gas bubble evolution in UO2 under post-irradiation thermal annealing. Nucl Instruments Methods Phys Res Sect B Beam Interact Mater Atoms2013; 303: 62–67.
48.
VanceIWMillettPC.Phase-field simulations of pore migration and morphology change in thermal gradients. J Nucl Mater2017; 490: 299–304.
49.
JavanbakhtMSadegh GhaediM.Thermal induced nanovoid evolution in the vicinity of an immobile austenite-martensite interface. Comput Mater Sci2020; 172: 109339.
50.
JavanbakhtMGhaediMSBarchiesiECiallellaA.The effect of a pre-existing nanovoid on martensite formation and interface propagation: A phase field study. Math Mech Solids2020; 1081286520948118.
51.
JavanbakhtMGhaediMS.Nanovoid induced martensitic growth under uniaxial stress: effect of misfit strain, temperature and nanovoid size on PT threshold stress and nanostructure in NiAl. Comput Mater Sci2020; 184: 109928.
52.
JavanbakhtMGhaediMS.Nanovoid induced multivariant martensitic growth under negative pressure: Effect of misfit strain and temperature on PT threshold stress and phase evolution. Mech Mater2020; 103627.
53.
PorterDAEasterlingKEE. Phase Transformations in Metals and Alloys, Third Edition (Revised Reprint); 3rd Edition; CRC Press, 2009; ISBN 9781439883570.
54.
LevitasVI.Thermodynamically consistent phase field approach to phase transformations with interface stresses. Acta Mater2013; 61: 4305–4319.
55.
JavanbakhtMSadegh GhaediM.Phase field approach for nanovoid dynamics with interface stresses at the nanoscale. Int J Eng Sci2020.
56.
LowengrubJTruskinovskyL.Quasi-incompressible Cahn–Hilliard fluids and topological transitions. P Roy Soc A-Math Phy1998; 454.1978: 2617–2654.
57.
PengtaoYFengJLiuCShenJ.A diffuse-interface method for simulating two-phase flows of complex fluids. J Fluid Mech2004; 515: 293.
58.
GaoYZhangYSchwenDJiangCSunCGanJ.Formation and self-organization of void superlattices under irradiation: A phase field study. Materialia2018; 1: 78–88.
59.
ChenG-XZhangJ-MXuK-W.Self-diffusion of Ni in B2 type intermetallic compound NiAl. J Alloys Compd2007; 430: 102–106.
60.
WuXLiYHuangMLiuWHouZ.Precipitation kinetics of ordered γ′ phase and microstructure evolution in a Ni Al alloy. Mater Chem Phys2016; 182: 125–132.
61.
YangMWeiHZhangJZhaoYJinTLiuLSunXF.Phase-field study on effects of antiphase domain and elastic energy on evolution of γ′ precipitates in nickel-based superalloys. Comput Mater Sci2017; 129: 211–219.
62.
PrikhodkoSCarnesJIsaakDArdellA. Elastic Constants of a Ni-12.69 AT. % Al Alloy From 295 to 1300 K. Scr Mater1997; 38: 67–72.
63.
SchwenDAagesenLKPetersonJWTonksMR.Rapid multiphase-field model development using a modular free energy based approach with automatic differentiation in MOOSE/MARMOT. Comput Mater Sci2017; 132: 36–45.
