Micromechanical modeling of bainitic phase transformation

We develop a micromechanical material model for phase transformation from austenite to bainite for a polycrystalline low alloys steel. In this material (e.g. 51CrV4) the phase changes from austenite to perlite-ferrite, bainite or martensite, respectively. This work is concerned with phase transformation between austenite and n-bainite variants in differently orientated grains. The characteristic features of bainite formation are the combination of time-dependent transformation kinetics and lattice shearing in the microstructure. These effects are considered on the microscale and transferred to the polycrystalline macroscale by means of homogenisation of stochastically orientated grains. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A coupled phase transformation and solute diffusion model for bainitic transformation

In materials science one distinguishes between upper and lower bainite, where both microstructures develop due to different diffusion processes. In this work we describe these different mechanisms with a new diffusion model coupled to a multiphase-field equation. Numerical examples demonstrate the expected behavior. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling of stress relaxation associated with an electrically-induced phase transformation

Stress relaxation due to an electrically-induced phase transformation in a ferroelectric crystal bar stretched by a hard-loading device is studied in the one-dimensional setting of electromechanics. According to the proposed model, the stress relaxation rate is governed by a nonlinear ordinary differential equation which resulted from the kinetic relation that controls the evolution of the phase transformation. A numerical analysis basing on simplifying approximation indicates that the stress starts to decrease when the intensity of the applied electric field reaches a critical value and that a stronger electric field results in a quicker stress reduction.     

Modeling of deformation-induced phase transformation during very high cycle fatigue (VHCF) using the boundary element method

A microstructural simulation model is proposed which accounts for damage accumulation in shear bands and deformation-induced martensite formation in a metastable austenitic stainless steel (AISI304). The model is numerically solved using the two-dimensional (2-D) boundary element method. By using this method, sliding displacements can be directly evaluated in shear bands and austenite grains as well as generated martensite domains with their individual mechanical properties and shape deformation can be considered. (© 2014 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Discrete transformation hypergroups and transformation hypergroups with phase tolerance space

Tolerance spaces and algebraic structures with compatible tolerances play an important role in contemporary algebra and its applications. In this contribution we present transformation hyperstructures, namely semihypergroups and hypergroups, acting on tolerance spaces. Some basic concepts concerning the mentioned structures are introduced and their fundamental properties are examined on suitable constructions.     

Cyclic martensitic phase transformation of monocrystals at finite strains

Based on Bain's principle, a C¹-continuous thermo-mechanical micro-macro constitutive model for martensitic phase transformation (PT) of monocrystals at finite strains and hyperelastic free energy function is used. It is represented by a unified non-convex Lagrangian variational functional. The convexification problem is solved here by generalizing the explicit form of the lower Reuss bound for small strains given in [1] to finite strains. Abaqus is used for implementation of 3D finite elements in space, via UMAT-interface which requires Jaumann rate of Kirchhoff stress tensor. Deterministic validation of the model is presented by comparing verified numerical results with experimental data for Cu₈₂Al₁₄Ni₄ [6] for quasiplastic PT. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

On the multiple hyperbolic systems modelling phase transformation kinetics

We discuss Cahn’s time cone method modelling phase transformation kinetics. The model equation by the time cone method is an integral equation in the space-time region. First, we reduce it to a system of hyperbolic equations, and in the case of odd spatial dimensions, the reduced system is a multiple hyperbolic equation. Next, we propose a numerical method for such a hyperbolic system. By means of alternating direction implicit methods, numerical simulations for practical forward problems are implemented with satisfactory accuracy and efficiency. In particular, in the three dimensional case, our numerical method on the basis of reduced multiple hyperbolic equation is fast.     

On oscillatory integral Hilbert transformation with trigonometric polynomial phase

Let $ \mathcal{P}_n $ denote the set of algebraic polynomials of degree n with the real coefficients. Stein and Wpainger [1] proved that $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \leqslant C_n , $$ where C _n depends only on n. Later A. Carbery, S. Wainger and J. Wright (according to a communication obtained from I. R. Parissis), and Parissis [3] obtained the following sharp order estimate $$ \mathop {\sup }\limits_{p( \cdot ) \in \mathcal{P}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{ip(x)} }} {x}dx} } \right| \sim \ln n. $$ . Now let $ \mathcal{T}_n $ denote the set of trigonometric polynomials $$ t(x) = \frac{{a_0 }} {2} + \sum\limits_{k = 1}^n {(a_k coskx + b_k sinkx)} $$ with real coefficients a _k , b _k . The main result of the paper is that $$ \mathop {\sup }\limits_{t( \cdot ) \in \mathcal{T}_n } \left| {p.v.\int_\mathbb{R} {\frac{{e^{it(x)} }} {x}dx} } \right| \leqslant C_n , $$ with an effective bound on C _n . Besides, an analog of a lemma, due to I. M. Vinogradov, is established, concerning the estimate of the measure of the set, where a polynomial is small, via the coefficients of the polynomial.     

Transformation strains for bainitic variant evolution in steel

In our work, we supplement a thermodynamically consistent multi-scale model for the bainitic phase transformation in a low alloy steel, which takes into account the mechanisms of elasto-viscoplasticity, phase transformations and heat conduction as well as the poly-crystalline structure of steel. In order to obtain realistic simulation results, the microscopic conversion procedures for the austenite-to-bainite transformation have to be described in an appropriate way. To this end, the transformation strains for the crystallographic variants have been adjusted. For calculation of transformation strains a theory for the formation of dislocated martensite, more precisely a hierarchical packet-block structure, is used. The calculated transformation strains are used in a simulation of bainite phase transformation in a polycrystalline RVE. (© 2017 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Effects of heat treatment on the phase transformation in powder metallurgical multi-functional coating

Heat treatment is an important part of the manufacturing of metallic products, especially in powder coating processes. It provides an efficient way to improve the properties of the metal as e.g. hardness by controlling the rate of diffusion and the rate of cooling within the microstructure. The process-integrated powder coating by radial axial rolling of rings is a new hybrid production technique which is introduced in [1]. The applied temperatures in hot rolling are within the range of austenitizing temperatures for the investigated steels [2]. Therefore, reasonably controlling the temperature is an important task [3]. The paper is concerned with the integration of heat treatment of the rolled ring into the subsequent cooling process. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Modeling phase diagrams as stochastic processes with application in vehicular traffic flow

Motivated by the similarity between the fundamental diagram of vehicular traffic and the Maxwell–Boltzmann distribution of ideal gases, this paper proposed a methodology to model the fundamental diagram as a stochastic process which also applies to other real-world systems with similar nature. A concrete example is provided to illustrate the application of the methodology where the fundamental diagram of vehicular traffic is modeled as a stochastic process to capture the scattering effect in flow–density relationship. A verification study was conducted on the model using empirical data and the statistical analysis shows that the overall quality of the fitted stochastic process is acceptable. Related existing efforts are referenced to the proposed stochastic fundamental diagram where their similarities and differences are elaborated. Further discussion is carried out on the significance of the stochastic fundamental diagram as well as the proposed methodology with an additional real-world example to illustrate its applications.     

Prediction of temperature distribution and phase transformation on the run-out table in the process of hot strip rolling

In this paper a mathematical model based on the finite element method and the Scheil additivity rule is presented for predicting the temperature distribution and phase transformation behavior on the run-out table during the hot strip rolling of a low carbon steel. The model considers the austenite to ferrite and pearlite transformations, the temperature-dependent material properties of the cooling austenite as well as the austenite work hardening effect on the kinetics of austenite transformation. To determine the validity of the model predictions, the time-temperature histories of a low carbon steel rod in different cooling media were measured and also hot rolling experiments were performed. Good agreement between the predictions and the experimental results indicates the reliability of the model.     

Visco‐elasto‐plastic model for martensitic phase transformation in shape‐memory alloys

Evolution of fine structure in martensite undergoing an isothermal process is modelled on a microscopic level by using a positive homogeneous dissipation potential which can reflect a specific energy needed for a phase transformation between different variants of martensite. The model thus naturally incorporates an activation phenomenon. Existence of a weak solution is proved together with convergence of finite‐element approximations. Numerical experiments showing the expected rate‐independent hysteresis response are also presented. Copyright © 2002 John Wiley & Sons, Ltd.