**Objectives**

The main goals of this project proposal are

(i) Contribute to the knowledge and development of an efficient numerical tool capable of accurately simulate the evolution of the thermomechanical and metallurgic state of multiphase steels parts submitted to heat treatments and subsequent forming behaviour;

(ii) Characterise the mechanical, metallurgical and thermal properties of multiphase steels from their microscale particularities and phases (austenite, ferrite, perlite, bainite and martensite);

(iii) Development of mathematical models that define the evolution of grain during heat treatments and subsequent forming. This includes phase kinetics, diffusion and thermomechanical grain characterization.

(iv) Integration and implementation of the constitutive models in a modular and interactive finite element code that would represent the microscale level. A grain will be numerically discretized in multiple finite elements. Phenomena, such as grain nucleation and growth, coarsening, etc. will be modelled.

(v) Development and implementation of a micro-macro homogenization algorithm in which the microscale is calculated by the previous developed FE code. Integrate the referred algorithm in efficient way using parallelization techniques;

(vi) Perform a set of experimental tests in order to validate the previously defined technological objectives.

While the application of the finite element method to model the non-uniform deformation of metal single crystals and polycrystals at the micro/ mesoscopic length scales through explicit discretization of the individuals crystals can be found very recently, the use of these techniques (and the use of hierarchical homogenization) in heat treatments processes is original.

**Research plan and methods**

The importance of fully controlling the heat treatments is not only related to the recent development of high strength steels, and their impressive mechanical properties. The knowledge of the residual stresses developed during the heat treatments undergone by most metal components/parts is also very important considering that, in many cases, unexpected failure occurs due to the presence of residual stresses which have combined with the service stresses to seriously shorten component life. Consequently, it is of most value the prediction of these stresses developed during the heat treatments. Reliable methods of determining the magnitude and distribution of residual stress fields are therefore required to quantify their effect and to avoid detrimental failure.

For all the importance of fully understand the thermomechanical and metallurgical phenomena in steels (during heat treatments and subsequent forming), several phenomenological theories and models have been developed. However, the majority of these models consider only the macroscopic scale level.

Prospects for manipulating microstructure to achieve enhanced performance demand models that distinguish between nucleation, migration, absorption/desorption, trapping, and bypass or annihilation of dislocations at various material length scales that manifest work hardening behavior. Most of these phenomena cannot be considered in isolation, which is a hallmark of plasticity – it is a highly coupled phenomenon, in general, with

important effects of both short and long range character attributed both to the physics of dislocation cores and long range interactions of dislocation arrays.

Considering that the mechanical behaviour of high performance steels (such as multi-phase steels) is attributable to their microstructure, the modelling of the mechanical behaviour of these steel materials has to be done based on microstructural levels. Nevertheless, macroscopic properties are also required for industry engineers. Therefore, a hierarchical multi-scale representation must be taken in account and, within this project, both scales will be taken as full FEM models.

Several options are available concerning the model of the microstructural phenomena. In this decade, crystal plasticity based models and the so-called self-consistent models have proven to be very useful and accurate reproducing valuable properties such as grain orientation and anisotropy. However, grain boundaries, phase distribution and occurrence of local heterogeneities cannot be represented in these models. The prediction of the creation, growth and role of grain boundaries is of utmost importance for the microstructural behaviour of the polycrystalline material in heat treatments and subsequent forming behaviour. It should be highlighted that grain boundaries may act either as barriers for dislocation glide, or conversely as sources for bulk dislocations.

Therefore, and in order to solve the previous problems, in this project proposal, a hierarchical multi-scale homogenization method will be used considering that all scales are taken as FEM models. This approach for modelling heat treatments is innovative.

During the manufacture and/or heat treatment process, phase transformation phenomena take place. In this project, within the microstructural representation as representative volume element (RVE), the material grains (particles) are explicitly discretized by finite elements. Therefore, each finite element will be considered as a fraction of a grain. Thermomechanical, metallurgical (including transformation kinetics, grain size, etc.) and diffusion equations will be developed and implemented at the microlevel in order to reproduce these phenomena.

A finite element representing an elementary part of the material has also an energetic (thermodynamics) evolution equation attached. This energetic evaluation (including temperature and thermal gradients, Gibbs free energy, activation energy, etc.) will define the metallurgical phase-change during the heat treatment (not only considering phase starting temperatures). Finite element surfaces will be used, when defined, as grain boundaries allowing grain regrowth. Therefore, thermodiffusion equations must be also developed and implemented in order to reproduce grain dynamics (including recovery, recristalization and grain growth).

The non-uniform deformation within the individual crystals of a polycrystal of steels will be also studied by the same methodology of discretizing each grain. For this case, even anisotropy levels can be analysed microstructurally and used macroscopically. Such approach will be also used to study grain-scale heterogeneous deformations that lead to the formation of macroscopic shear bands in plane strain compression.

The implementation of hierarchical homogenization models (particularly in plasticity) will certainly lead to CPU times of several hours or days. Thus, it is most important to implement adequate numerical programming techniques in order to reduce CPU times. The use of parallel/distributed programming is by itself a definite step to achieve that. Although parallel computation is a highly developed computer science discipline, it is not widely used by the scientific/engineering community. In this project, the parallelisation of the finite element code will be performed by a substructuring technique, where the mesh is partitioned into subdomains, each of which is assigned to a specific processor.

The main goals of this project proposal are

(i) development of an efficient numerical tool capable of reproduce with accuracy the evolution of the thermomechanical and metallurgic state of multiphase steels parts submitted to heat treatments and subsequent forming behaviour; This tool will contribute to the knowledge of the phenomena observed and can enlarge the field of application of MP steels.

(ii) Characterise the mechanical, metallurgical and thermal properties of multiphase steels from their microscale particularities and phases (austenite, ferrite, perlite, bainite and martensite).

(iii) Development of mathematical models that define the evolution of grain during heat treatments and subsequent forming. This includes phase kinetics, diffusion and thermomechanical grain characterization. The grain boundaries and interactions among the different crystals will be also taken in account.

(iv) Integration and implementation of the constitutive models in a modular and interactive finite element code that would represent the microscale level. A grain will be numerically discretized in multiple finite elements. Phenomena, such as grain nucleation and growth, coarsening, etc. will be modelled.

(v) Development and implementation of a hierarchical micro-macro homogenization algorithm in which the microscale is calculated by the previous developed FE code. Integrate the referred algorithm in efficient way using parallelization techniques;

(vi) Perform a set of experimental tests in order to validate the previously defined technological objectives and to validate the theories formulated within the project.

These goals will be divided in two main tasks with the correspondent milestones:

1) The first task is dedicated to the study and implementation of all microstructural phenomena in a finite element model. In this task, experimental test will be also executed in order to verify some micro-observations and validate the developed model.

It is expected by the end of this task to obtain a reliable model, already implemented in a FE code, that can reproduce the phenomena observed in both heat treatment and subsequent deformation processes such as (i) phase change (including the austenite, ferrite, perlite, bainite and martensite phases); (ii) phase transformation plasticity phenomena; (iii) grain regrowth; (iv) grain recovery and recrystallization; (v) grain boundaries and anisotropy.

2) The second task corresponds to the implementation of the hierarchical homogeneization algorithm. At first, mean field methods will be applied and simple tests will be performed. These methods have the advantage of a straightforward implementation. Then asymptotic expansion homogenisation (AEH) methods, considering thermoviscoplasticity, will be implemented. In the AEH methodology, overall material properties can be derived from the mechanical behaviour of selected periodic microscale representative volumes (also known as representative unit-cells, RUC). Notice that the project team has been working with these methods in the last few years (see [PT2]). This task must also include the implementation of CPU parallelization techniques due to the large computation time required by the constant change in scale-computations and scale-update.

The use of the techniques mentioned in this project (I. application of the finite element method to model the polycrystals at the microscopic length scales through explicit discretization of the individual crystals and II. hierarchical homogenization) in the simulation of heat treatments processes is original.

**Project team**

A. Gil Andrade-Campos (PI)

João A. Oliveira

Frédéric Barlat

Patrícia Vasconcelos

Bruno Barroqueiro

**Previous projects**

This project has as predecessor the project PTDC/EME-TME/68975/2006, untitled Numerical analysis and modelling of heat treatments on metallic geometrically complex parts.

The aim of heat treatments is to achieve material properties that could not be obtained otherwise. In general terms, the heat treatment of aluminium alloys, for instance, consists of solution heat treatment, quenching and age hardening. The alloy is heated to the maximum practicable temperature (723-823 K) and kept at that temperature during the time necessary to achieve the maximum solution of the constituents. The alloy is then exposed to quenching at room temperature water in order to ensure that the dissolved constituents remain in the solution. Finally, the aluminium alloy is age hardened.

The heat treatment processes, and particularly the quenching process, induce severe temperature gradients during the rapid cooling of the material from the complete solution temperature to room temperature. During the cooling process the material contracts in a non-uniform way inducing complex residual stress fields. These stresses frequently exceed the material’s yield stress. Such occurrence can induce permanent deformations that can unacceptably change the geometry of the part. Residual stresses and strains will also influence the final characteristics and mechanical properties of the part and may compromise its industrial application.

Strain control during the heat treatments processes resorts, almost exclusively, to empiric experience. Reducing the cooling rate during quenching will reduce residual stresses. However it can also significantly worsen mechanical properties and corrosion resistance.

At present, except for simple geometries, the only economic and prompt way to assess residual stresses is to measure them with destructive testing techniques. These techniques bring to an end the subsequent utilization of the part. This fact stresses out the need for numerical analyses of heat treatment processes that predict these thermal residual stresses.

The main objectives of this project proposal are (i) contribute to the knowledge and development of efficient numerical tools capable of simulation with accuracy the evolution of the thermomechanical and metallurgic state of metal (steel and aluminium alloy) parts with complex geometry submitted to heat treatments; (ii) development and implementation of non-linear constitutive models and algorithms of thermomechanical coupling for metals; (iii) integrate the referred models in efficient way using parallelization techniques; (iv) development and construction of the pivot equipment that allow executing and monitoring experimental quenching tests; (v) experimentally determine the numerical process input data as, for example, the heat transfer coefficients associated in the process by the use of inverse methodology; (vi) development of an automatic numerical tool to calculate the temperature-dependent surface heat transfer coefficient during quenching process. The inverse methodology tool will be based on evolutionary algorithms and gradient-based optimization methods; (vii) execute a set of experimental tests in order to validate the previously defined technological objectives and (viii) divulgation of the results and numerical tool developed.

The project will be performed by a multidisciplinary research team composed by experts from mechanical technology, heat transfer, applied and computational mechanics, guaranteeing that there will be very useful work produced in industrial and scientific terms.