Experimental estimation of the Inelastic Heat Fraction from thermomechanical observations and inverse analysis

Abstract

A new method to estimate the inelastic heat fraction (Taylor and Quinney Beta coefficient) during the deformation of a titanium material is proposed. It is based on (1) thermomechanical full field measurements during the loading and on (2) an inverse analysis. First, two cameras (a visible and an infrared) are used to measure the kinematic and the thermal fields on each face of a notched flat sample loaded in tension. Second, two coupled finite element simulations (a mechanical, then a thermal one) of the same tests are conducted. Associated with a Levenberg-Marquardt optimization algorithm, they are able to give, in a first step, optimized values of the anisotropic elastoplastic model parameters. Then, in a second step, parameters of four strain dependent Beta models are identified. Finally, the thermal responses of these models are compared to the experimental values.

Introduction

Heat generation during the deformation can be estimated either by an experimental approach or by a modeling one [1,2]. In these two approaches, the intrinsic dissipation is often computed through In-elastic Heat Fraction (IHF also denoted ft). As mentioned by [3], the ft factor, introduced and calculated in the pioneer work of Taylor and Quinney [4], was an integrated factor defined by the ratio of the dissipated energy to the plastic work.

In many Finite Element codes, the ft factor was assumed to be constant and typically about 0.9 for metals (see [5] for example). However, several experimental studies have shown that this assumption does not hold in many material types ([2, 4, 6-9]). In these studies, ft was different than 0.9 and was strain (and sometimes strain rate) dependent. A summary of such variations in ft values can be found in [8].


Several experimental techniques have been proposed to measure, at a macroscopic scale, the ft or the stored energy ratio. In this work, a new method to estimate the inelastic heat fraction (ft) during the deformation of a titanium material is proposed.

It is based on (1) thermomechanical full field measurements during the loading and on (2) an inverse analysis.

After a presentation of the proposed procedure, various models of ft evolution with the strain are chosen a priori and identified through an inverse method. The chosen method is the Finite Element Updating (FEU), broadly used in several applications.

Overview of the proposed procedure

In thermomechanical analysis, the material self-heating is often simulated in the FE calculations as a conversion rate of the plastic power per unit volume wL into heat:

tmpFC-41_thumb

where w’d is the dissipated power per unit volume generated by irreversibilities (plasticity, damage…), and ft is the so called Taylor and Quinney factor. In the case of titanium, the thermomechanical coupling power per unit volume w^mc is only due to the thermoelastic power per unit volume w!hel. The heat sources, noted w’ch, can then be computed using the following equation:

tmpFC-42_thumb

In the present work, two FEU inverse methods are successively run (Fig. 1):

(i) The first one denoted FEU-M is a mechanical inverse identification and aims to determine the parameters of the mechanical model. Global measurement (reaction force of the sample Fexp(t)) and local displacement data obtained by DIC are used to identify both Ludwick’s hardening parameters and Hill’s anisotropic parameters. Mechanical parameters are assumed not to depend on the sample self-heating in the observed range of temperature [101.

Mechanical parameters and inelastic heat fraction identification flowchart

Fig. 1: Mechanical parameters and inelastic heat fraction identification flowchart

(ii) The second one denoted FEU-T, identifies the parameters of four different p evolution models:

tmpFC-44_thumb

where E p is the axial component of the Green Lagrange plastic strain tensor. The first model gives a constant value for p  and models 2 to 4 are strain dependent. The three first models exhibit an increasing number of parameters (a to f) and the fourth one is the model proposed by Zenhder [11] with two parameters.

This identification takes into account the plastic power w^, the thermo-mechanical coupling power (wt’!e;) both computed in the previous FEU-M analysis and the experimental thermal field Texp measured by the infrared camera.

Experimental Setup

The tested samples are cut from titanium rolled sheets (thickness: 1.6 mm). The rolling and the transverse directions are respectively denoted 0° and 90°. A notched geometry (Fig. 2) is chosen, according to the work of [12], so that the obtained strain fields exhibit a shear band in the central zone of the sample.

Digital Image Correlation (DIC) is used for kinematic measurement purpose on one side of the sample. DIC is processed using 7D software [13]. The dimension of the grid is set to 16 x 16 pixels and the dimension of the pattern used to compare sub images is 16 x 16 pixels. The displacement field has a bi-linear form and the gray level interpolation is bi-cubic.

The other side of the sample is used for thermal field measurement. A fixed infrared camera (Cedip Jade III MW) captured thermal field at a 10 Hz frequency with a resolution of 320×240 pixels.

Experimental setup with kinematic and thermal full-field measurement devices facing different sides of the sample.

Fig. 2: Experimental setup with kinematic and thermal full-field measurement devices facing different sides of the sample.

Numerical modelling

A 3D Finite Element model is built to duplicate the experimental geometry. The only visible part of the sample is modeled. In other words, the free ends, hidden in the grips, are not considered. The part is meshed using 7056 solids elements (3 layers). Due to the unavailability of through-thickness measurements (whether from a kinematic or a thermal point of view), the surface data are repeated over the 3 layers of elements. Two distinct problems need to be solved successively: the mechanical one and the thermal one. Hence, two cost-functions are defined. The first is needed in the mechanical problem and is then built using strictly mechanical data: the two components of the displacement and the global force. The second, defining the thermal problem, is built from sample temperature.

The cost functions are minimized using a Levenberg Marquardt algorithm and sensitivities are computed through a forward finite difference scheme. Mechanical boundary conditions are applied in each node of the sample top and bottom borders according to the corresponding DIC measurement. Thermal boundary conditions are modeled as thermal fluxes on the front/back face of the sample (convection with the ambient air) and on the top/bottom borders (through grip thermal conduction). The parameters of those heat fluxes are experimentally estimated from independent tests.

Results

• Mechanical identification :

The six parameters of the model (two from the hardening law and four from the in-plane Hill’s criterion) are identified from two tests led with samples cut at 0°and 90° from the rolling direction. Results show a good agreement between measured and predicted displacement fields (Fig. 3).

Evolution, daring loading of the relative error (in %) on longitudinal displacement, after identification, for experiment at 0°.

Fig. 3: Evolution, daring loading of the relative error (in %) on longitudinal displacement, after identification, for experiment at 0°.

Hence, the displacement and the stress fields computed with the optimized parameters are used to assess the plastic power wL per unit volume and the thermoelastic coupling power per unit volume w’thel. Thus, given a value of ft, allows the estimations of the heat sources. Fig. 4 shows the time evolution of the volume plastic power in the sample. Higher value of wL (and so the heat sources) are located in the reduced section of the sample with higher intensities near the edges.

Evolution, during loading, of the calculated plastic power fields (in W.m-3).

Fig. 4: Evolution, during loading, of the calculated plastic power fields (in W.m-3).

Thermal identification:

The thermal problem is solved for a single test led at 0°. Identified parameters of the four models are gathered in Table 1. The nodels no.3 and no.4 exhibit the lowest values of the cost function and are thus the most able to predict the measured emperature. Conversely, model no.1 is certainly not sufficient to model the observed phenomenon.

Table 1: Identified thermal parameters of the four models of P.

Model no.1

Model no.2

Model no.3

Model no.4

Parameter

a

b c

d

e

f

h

Initial value Final value

1

0.65

0.3 0.3 0.79 0.43

0.3 0.99

1

0.21

0.3 0.01

1

4.08

Final cost function value

2.199

1.910

1.704

1.810

Moreover, the evolution of the p ratio versus strain is plotted in Fig. 5.a. Since the ratio of plastic power converted into heat is not defined fortmpFC-48_thumbthe plot has been chosen to start attmpFC-49_thumb. Fig. 5.b shows the temperature evolution predicted by the four models and measured at the sample surface for 3 points (P1, P2 and P3) defined in Fig. 4. Fig. 5.b confirms the above considerations among which model 3 is the most able to predict the observed temperature field. Thus, considering the presented results, the evolutionary value of p along strain seems obvious. However, the chosen model to represent this evolution is of the utmost importance. For instance, the model no.2 does not bring major improvements compared to the constant model assumed in many former studies.

: (a) Predicted evolutions of P for the 4 identified models. (b) Comparison between the experimental response and the predicted temperature for the 4 identified models, in the three points P1, P2 and P3 given in Fig. 4.

Fig. 5: (a) Predicted evolutions of P for the 4 identified models. (b) Comparison between the experimental response and the predicted temperature for the 4 identified models, in the three points P1, P2 and P3 given in Fig. 4.

Conclusions

In the present paper, a new method to identify the inelastic heat fraction (p) during a quasi static tensile test is proposed. The method is based on an inverse Finite Element Update approach, coupling a Finite Element code (Abaqus) and a Levenberg-Marquardt algorithm, and using simultaneous DIC and IR thermal full field measurements. This method is first used to identify the mechanical parameters of material constitutive equations by the means of DIC measurements. Then, the obtained results are used to assess the inelastic heat fraction for the same test. Four strain evolution models for p are tested. Up to the maximum load, results show a good agreement between measured and predicted temperature fields. The identified evolution models of p, on a titanium material, confirm a strain dependency but remain probably too simple to retrieve the whole energetic process. Furthermore, the obtained values of p are very different from the broadly used value (0.9). The best prediction is obtained with a non linear evolution of p with the strain, governed by three parameters (model no.3). The strain evolution is close to the one given by Zehnder model but leads to better temperature estimations at the beginning of the loading. Improvements of p models, in order to obtain non monotonous evolution with the strain, may also be proposed with the same approach.

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