The nanoscale multiphase phase-field model for stress and temperature-induced multivariant martensitic transformation under large strains developed by the authors in Basak and Levitas (J Mech Phys Solids 113:162–196, 2018) is revisited, the issues related to the gradient energy and coupled kinetic equations for the order parameters are resolved, and a thermodynamically consistent non-contradictory model for the same purpose is developed in this paper. The model considers N+1 order parameters to describe austenite and N martensitic variants. One of the order parameters describes austenite↔martensite transformations, and the remaining N order parameters, whose summation is constrained to the unity, describe the transformations between the variants. A non-contradictory gradient energy is used within the free energy of the system to account for the energies of the interfaces. In addition, a kinetic relationship for the rate of the order parameters versus thermodynamic driving forces is suggested, which leads to a system of consistent coupled Ginzburg–Landau equations for the order parameters. An approximate general crystallographic solution for twins within twins is presented, and the explicit solution for the cubic to tetragonal transformations is derived. A large strain-based finite element method is developed for solving the coupled Ginzburg–Landau and elasticity equations, and it is used to simulate a 3D complex twins within twins microstructure. A comparative study between the crystallographic solution and the simulation results is presented.
more »
« less
An exact formulation for exponential-logarithmic transformation stretches in a multiphase phase field approach to martensitic transformations
A general theoretical and computational procedure for dealing with an exponential-logarithmic kinematic model for transformation stretch tensor in a multiphase phase field approach to stress- and temperature-induced martensitic transformations with N martensitic variants is developed for transformations between all possible crystal lattices. This kinematic model, where the natural logarithm of transformation stretch tensor is a linear combination of natural logarithm of the Bain tensors, yields isochoric variant–variant transformations for the entire transformation path. Such a condition is plausible and cannot be satisfied by the widely used kinematic model where the transformation stretch tensor is linear in Bain tensors. Earlier general multiphase phase field studies can handle commutative Bain tensors only. In the present treatment, the exact expressions for the first and second derivatives of the transformation stretch tensor with respect to the order parameters are obtained. Using these relations, the transformation work for austenite ↔ martensite and variant ↔ variant transformations is analyzed and the thermodynamic instability criteria for all homogeneous phases are expressed explicitly. The finite element procedure with an emphasis on the derivation of the tangent matrix for the phase field equations, which involves second derivatives of the transformation deformation gradients with respect to the order parameters, is developed. Change in anisotropic elastic properties during austenite–martensitic variants and variant–variant transformations is taken into account. The numerical results exhibiting twinned microstructures for cubic to orthorhombic and cubic to monoclinic-I transformations are presented.
more » « less- Award ID(s):
- 1904830
- PAR ID:
- 10546868
- Publisher / Repository:
- SAGE Publications
- Date Published:
- Journal Name:
- Mathematics and Mechanics of Solids
- Volume:
- 25
- Issue:
- 6
- ISSN:
- 1081-2865
- Format(s):
- Medium: X Size: p. 1219-1246
- Size(s):
- p. 1219-1246
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
A thermodynamically consistent multiphase phase-field approach for stress and temperature-induced martensitic phase transformation at the nanoscale and under large strains is developed. A total of N independent order parameters are considered for materials with N variants, where one of the order parameters describes A ↔ M transformations and the remaining N − 1 independent order parameters describe the transformations between the variants. A non-contradictory gradient energy is used within the free energy of the system to account for the energies of the interfaces. In addition, a non-contradictory kinetic relationships for the rate of the order parameters versus thermodynamic driving forces is suggested. As a result, a system of consistent coupled Ginzburg-Landau equations for the order parameters are derived. The crystallographic solution for twins within twins is presented for the cubic to tetragonal transformations. A 3D complex twins within twins microstructure is simulated using the developed phase-field approach and a large-strain-based nonlinear finite element method. A comparative study between the crystallographic solution and the simulation result is presented.more » « less
-
Scale-free phase-field approach and corresponding finite element method simulations for multivariant martensitic phase transformation from cubic Si I to tetragonal Si II in a polycrystalline aggregate are presented. Important features of the model are large and very anisotropic transformation strain tensor 𝜺𝑡 = {0.1753; 0.1753; −0.447} and stress-tensor dependent athermal dissipative threshold for transformation, which produce essential challenges for computations. 3D polycrystals with stochastically oriented grains are subjected to uniaxial strain- and stress-controlled loadings under periodic boundary conditions and zero averaged lateral strains. Coupled evolution of discrete martensitic microstructure, volume fractions of martensitic variants and Si II, stress and transformation strain tensors, and texture are presented and analyzed. Macroscopic variables effectively representing multivariant transformational behavior are introduced. Macroscopic stress–strain and transformational behavior for 55 and 910 grains are close. Large transformation strains and grain boundaries lead to huge internal stresses of tens GPa, which affect microstructure evolution and macroscopic behavior. In contrast to a single crystal, the local mechanical instabilities due to phase transformation and negative local tangent modulus are stabilized at the macroscale by arresting/slowing the growth of Si II regions by the grain boundaries. This leads to increasing stress during transformation. The developed methodology can be used for studying similar phase transformations with large transformation strains and for further development by including plastic strain and strain-induced transformations.more » « less
-
A phase-field model for thermomechanically-induced fracture in NiTi at the single crystal level, i.e., fracture under loading paths that may take advantage of either of the functional properties of NiTi–superelasticity or shape memory effect–, is presented, formulated within the kinematically linear regime. The model accounts for reversible phase transformation from austenite to martensite habit plane variants and plastic deformation in the austenite phase. Transformation-induced plastic deformation is viewed as a mechanism for accommodation of the local deformation incompatibility at the austenite–martensite interfaces and is accounted for by introducing an interaction term in the free energy derived based on the Mori–Tanaka and Kröner micromechanical assumptions and the hypothesis of martensite instantaneous growth within austenite. Based on experimental observations suggesting that NiTi fractures in a stress-controlled manner, damage is assumed to be driven by the elastic energy, i.e., phase transformation and plastic deformation are assumed to contribute in crack formation and growth indirectly through stress redistribution. The model is restricted to quasistatic mechanical loading (no latent heat effects), thermal loading sufficiently slow with respect to the time rate of heat transfer by conduction (no thermal gradients), and a temperature range below 𝑀𝑑, which is the temperature above which the austenite phase is stable, i.e., stress-induced martensitic transformation is suppressed. The numerical implementation of the model is based on an efficient scheme of viscous regularization in both phase transformation and plastic deformation, an explicit numerical integration via a tangent modulus method, and a staggered scheme for the coupling of the unknown fields. The model is shown able to capture transformation-induced toughening, i.e., stable crack advance attributed to the shielding effect of inelastic deformation left in the wake of the growing crack under nominal isothermal loading, actuation-induced fracture under a constant bias load, and crystallographic dependence on crack pattern.more » « less
-
Quenching and partitioning (Q&P) processing of third-generation advanced high strength steels generates multiphase microstructures containing metastable retained austenite. Deformation-induced martensitic transformation of retained austenite improves strength and ductility by increasing instantaneous strain hardening rates. This paper explores the influence of martensitic transformation and strain hardening on tensile performance. Tensile tests were performed on steels with nominally similar compositions and microstructures (11.3 to 12.6 vol. pct retained austenite and 16.7 to 23.4 vol. pct ferrite) at 980 and 1180 MPa ultimate tensile strength levels. For each steel, tensile performance was generally consistent along different orientations in the sheet relative to the rolling direction, but a greater amount of austenite transformation occurred during uniform elongation along the rolling direction. Neither the amount of retained austenite prior to straining nor the total amount of retained austenite transformed during straining could be directly correlated to tensile performance. It is proposed that stability of retained austenite, rather than austenite volume fraction, greatly influences strain hardening rate, and thus controls strength and ductility. If true, this suggests that tailoring austenite stability is critical for optimizing the forming response and crash performance of quenched and partitioned grades.more » « less
