Advances in Materials Science and Engineering

Volume 2016, Article ID 2357534, 20 pages

http://dx.doi.org/10.1155/2016/2357534

## Application of Metamodels to Identification of Metallic Materials Models

AGH University of Science and Technology, Aleja Adama Mickiewicza 30, 30-059 Kraków, Poland

Received 30 September 2015; Accepted 6 December 2015

Academic Editor: Antonio Riveiro

Copyright © 2016 Maciej Pietrzyk et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Improvement of the efficiency of the inverse analysis (IA) for various material tests was the objective of the paper. Flow stress models and microstructure evolution models of various complexity of mathematical formulation were considered. Different types of experiments were performed and the results were used for the identification of models. Sensitivity analysis was performed for all the models and the importance of parameters in these models was evaluated. Metamodels based on artificial neural network were proposed to simulate experiments in the inverse solution. Performed analysis has shown that significant decrease of the computing times could be achieved when metamodels substitute finite element model in the inverse analysis, which is the case in the identification of flow stress models. Application of metamodels gave good results for flow stress models based on closed form equations accounting for an influence of temperature, strain, and strain rate (4 coefficients) and additionally for softening due to recrystallization (5 coefficients) and for softening and saturation (7 coefficients). Good accuracy and high efficiency of the IA were confirmed. On the contrary, identification of microstructure evolution models, including phase transformation models, did not give noticeable reduction of the computing time.

#### 1. Introduction

Continuous progress in numerical modelling of metals processing has been observed during more than half of the century. It became evident that the accuracy of simulations depends mainly on the correctness of the description of boundary conditions and properties of processed materials. The latter problem was the subject of the present work. A number of material models can be found in the scientific literature. Potential extensive predictive capabilities of these models are useful only when proper identification was performed on the basis of experiments. Interpretation of the results of various experiments is the main part of the identification, which usually uses inverse analysis (IA) with finite element (FE) simulation of the test [1–4]. That approach allows elimination of the influence of various disturbances, such as friction or heating due to deformation or due to phase transformations. The published results show that IA improves the accuracy of interpretation of experimental tests significantly.

Various material models used in simulations of thermomechanical processing and various experimental tests performed to identify these models were investigated in the present work. These models describe flow stress, microstructure evolution, and phase transformations. Plastometric tests [4], stress relaxation tests [5], and dilatometric tests [6] are used, respectively, to identify these models. Direct problem model is the model, which is used in the IA to simulate experiment. Thus, finite element (FE) method is used to simulate plastometric tests while JMAK (Johnson-Mehl-Avrami-Kolmogorov) equation describes microstructure evolution and phase transformations in stress relaxation and dilatometric tests. The direct problem model has to be executed at least once for each calculation of the objective function in the inverse analysis. However, the tests are often performed in various conditions (temperatures, strain rates, and cooling rates) and each calculation of the objective function in the inverse analysis requires several runs of the direct problem model. It means that identification of the models would require long computing times. Thus, making the inverse analysis more efficient was the main objective of this work. Application of the metamodel in the inverse analysis was proposed in [7] and the possibility of application of this technique to various experiments was further explored in the present work.

The idea of numerical models substitution with metamodels in the inverse solution has been explored for some time now and solutions for structural mechanics [9], dynamic systems [10], or damage [11] were published. To the authors’ knowledge, there are no such solutions for materials models used in processing. Therefore, exploring capabilities of metamodels as substitutions for direct problem models in the inverse analyses of material tests was the particular objective of this work. To reach this goal, various models were analysed from the point of view of their mathematical formulation and physical phenomena, which can be accounted for by the models. Selection of the appropriate metamodel for a considered application was one of the objectives of the work. Various metamodels were tested with respect to accuracy and number of training data and capability of implementation to the optimization procedure in the inverse analysis. The focus in the paper is on flow stress models, but identification of the phase transformation models is addressed briefly, as well, to make the picture of the identification complete.

#### 2. Models and Metamodels

Modelling of materials processing requires knowledge of material properties, which depend on many factors like grain size, grain boundaries, dislocation density, stacking fault energy, and so forth. Due to their complexity and scale, accounting for all these factors is difficult. To overcome this problem polycrystals are described by homogenized models, which represent statistically all mentioned microstructural phenomena.

##### 2.1. Flow Stress Models

A large number of flow stress models for metal forming were published in the second half of the XXth century. These models are characterized by various complexity of mathematical formulation and various predictive capabilities. There were several attempts to classify these models; see, for example, [12, 13]; but there is still a lack of convincing hints for a selection of the most appropriate model for a particular application. Analysis of various models inspired the authors to divide the models into three groups: (i) conventional models, (ii) internal variable method (IVM), and (iii) multiscale models.

By conventional model, we understand closed form equations, which describe flow stress as a function of temperature, strain, and strain rate. Introduction of the internal variables instead of strain as independent variables allowed accounting for the inertia of microstructural phenomena. Dislocation density is the most commonly used internal variable and a variety of dislocation density based models were developed following fundamental works of Estrin, Kocks, and Mecking [14, 15] and Sandström and Lagneborg [16]. Multiscale models have developed rapidly in the first decade of this century. Since substitution of these models with metamodels does not seem possible, they are not considered in the present work.

###### 2.1.1. Conventional Models

Conventional models give good results when conditions of deformation are reasonably monotonous and these models are commonly used in simulations of industrial metal forming processes. The first attempt to describe the flow stress as a function of process parameters is attributed to Hollomon, who proposed the power equation describing flow stress relation on strain. To account for the influence of temperature and strain rate, Hollomon equation was extended to the following form: where is the flow stress, is the effective strain, is the effective strain rate, is the absolute temperature, is the Universal Gas Constant, and , , , and are coefficients, and the last one is interpreted as activation energy for plastic deformation.

Equation (1) gives good description of the flow stress in the range of temperatures and strain rates corresponding to the high values of the Zener-Hollomon parameter: where is the activation energy for plastic deformation.

At lower material softening due to dynamic recrystallization (DRX) becomes important. Beyond , the stacking fault energy (SFE) is the material parameter, which determines the tendency of this material to dynamic recrystallization. The lower is SFE, the larger strain is needed to launch DRX. Among a number of amendments of (1) to account for softening, correction term proposed by Hensel and Spittel [17] should be mentioned:where – are coefficients and is the temperature in °C.

For larger strains, flows stress calculated from (3) drops to zero or even to negative values, which is not physical and is the main drawback of this model. In real materials, after rapid drop of the flow stress due to DRX, state of saturation is reached and the flow stress remains constant [18]; see curve with round symbols in Figure 1. Auxiliary arrow in this figure shows that the higher is the Zener-Hollomon parameter and the higher is the stacking fault energy (SFE), the lower is the tendency of the material to dynamic recrystallization.