A Coupled Electrical–Thermal–Mechanical Modeling of Gleeble Tensile Tests for Ultra-High-Strength (UHS) Steel at a High Temperature

Abstract

A coupled electrical–thermal–mechanical model is proposed aimed at the numerical modeling of Gleeble tension tests at a high temperature. A multidomain, multifield coupling resolution strategy is used for the solution of electrical, energy, and momentum conservation equations by means of the finite element method. Its application to ultra-high-strength steel is considered. After calibration with instrumented experiments, numerical results reveal that significant thermal gradients prevail in Gleeble tensile steel specimen in both axial and radial directions. Such gradients lead to the heterogeneous deformation of the specimen, which is a major difficulty for simple identification techniques of constitutive parameters, based on direct estimations of strain, strain rate, and stress. The proposed direct finite element coupled model can be viewed as an important achievement for subsequent inverse identification methods, which should be used to identify constitutive parameters for steel at a high temperature in the solid state and in the mushy state.

Appendix: Material Properties of The Uhs Steel

To calculate the solidification path of the considered UHS steel, the microsegregation model presented by Won et al.[23] is used here, which takes into account the steel composition and cooling rate. The model is listed briefly as follows:

$$ f_{\text{s}} = \left( {{\frac{1}{1 - 2\beta k}}} \right)\left[ {1 - \left( {{\frac{{T_{\text{f}} - T}}{{T_{\text{f}} - T_{\text{L}} }}}} \right)^{{\left( {1 - 2\beta k} \right)/(k - 1)}} } \right] $$

$$ \beta = a\left[ {1 - \exp \left( {{\frac{ - 1}{a}}} \right)} \right] - \frac{1}{2}\exp \left[ {{\frac{ - 1}{2a}}} \right] $$

$$ a = 33.7\dot{T}^{ - 0.244} $$

where k is the equilibrium partition coefficient of carbon (taken as 0.265) and T _f is the melting temperature of pure Fe (taken as 1808 K (1535 °C)). The liquidus temperature can be determined by the following:

$$ T_{\text{L}} = 1808 - 78\left( {{\text{pct}}\,{\text{C}}} \right) - 7.6\left( {{\text{pct}}\,{\text{Si}}} \right) - 4.9\left( {{\text{pct}}\,{\text{Mn}}} \right) - 34.4\left( {{\text{pct}}\,{\text{P}}} \right) -38\left( {{\text{pct}}\,{\text{S}}} \right)$$

where temperature is K. The cooling rate $\dot{T}$ is taken as 0.17 K/s. The calculated relation between temperature and solid fraction is shown in Figure 16. The latent heat L is 272 kJ/kg. Thermophysical and electrical properties are given in Table AI.

Fig. 16

The calculated relation between solid fraction and temperature for the steel

Table AI Thermal and Electrical Properties of the Considered UHS Steel

In regard to the mechanical behavior, the UHS steel is supposed to behave as an elastic-viscoplastic material, obeying the classical J ₂ theory with isotropic linear hardening. The Young’s modulus is taken from Mizukami et al.[24] and is as follows:

$$ E\left[ {GPa} \right] = 968 - 2.33\left( {T - 273} \right) + 1.90 \times 10^{ - 3} \left( {T - 273} \right)^{2} - 5.18 \times 10^{ - 7} \left( {T - 273} \right)^{3} $$

Poisson’s ratio is arbitrarily taken as 0.3. Two different constitutive viscoplastic models have been considered in the present study.

The additive model from Kozlowski et al. (model III)[25] corresponds to Eq. [17] and is as follows:

$$ \bar{\sigma } = \sigma_{\rm y} + H_{\text{evp}} \bar{\varepsilon }^{n} + K_{\text{evp}} \dot{\bar{\varepsilon }}^{m} $$

The material exhibits a plastic yield stress under which its behavior is purely elastic. This plastic yield stress is the sum of the initial yield stress σ _y and the strain-hardening contribution $H_{\text{evp}} \bar{\varepsilon }^{n} .$ The following constitutive parameters used here are taken from Reference 25, and they are supposed to cover a wide range of austenitic plain carbon steels:

$$ \sigma_{\text{y}} = 0 $$

$$ m = {1 \mathord{\left/ {\vphantom {1 {\left( {8.132 - 1.540 \times 10^{ - 3} T} \right)}}} \right. \kern-\nulldelimiterspace} {\left( {8.132 - 1.540 \times 10^{ - 3} T} \right)}} $$

$$ n = - 0.6289 + 1.114 \times 10^{ - 3} T $$

$$ H_{\text{evp}} = 130.5 - 5.128 \times 10^{ - 3} T $$

$$ K_{\text{evp}} = \left( {\frac{1}{C}\exp \left( {\frac{Q}{RT}} \right)} \right)^{m} $$

where C = 46550 + 71400c + 12000c ², Q = 371.2 kJ/mol. The carbon content c is in wt pct, and the stress H _evp and K _evp is in MPa.

The model proposed by Han et al.[22] corresponds to Eq. [30] and is as follows:

$$ \bar{\sigma } = {\frac{{\bar{\varepsilon }^{n} }}{\alpha }}\text{arsinh}\left[ {\left( {\frac{1}{A}\exp \left( {\frac{Q}{RT}} \right)} \right)^{m} \dot{\bar{\varepsilon }}^{m} } \right] $$

The corresponding material parameters used here are taken from Seol et al.,[8] who characterized a steel grade with a composition approaching the composition of the grade considered here—0.14 wt pct C, 0.40 wt pct Si, and 1.28 wt pct Mn. Their parameters are listed in Table AII.

Table AII Parameters Value of Eq. [30], Taken from Seol et al.[8]

To illustrate the differences between these two constitutive models, stress–strain curves have been plotted for a constant strain rate 0.001 s⁻¹ and three different temperatures (Figure 17). Significant differences are observed between the calculated curves.

Fig. 17

Plotted stress–strain curves of Eq. [17] with parameters taken from Ref. 25 and of Eq. [30] with parameters taken from Ref. 8