An Inverse Piecewise Flow Curve Determination Method for Torsion Tests at Elevated Temperature

Abstract

This paper presents an extended method for determining flow curves under shear loading using torsion tests, a technique often used to characterize plastic behavior in metal forming. Torsion tests are advantageous due to their ability to achieve flow curves up to large strains (~3) while maintaining stable specimen geometry during deformation. However, the strain and strain rate distribution across the specimen are non-uniform, increasing radially from the rotation axis. Traditional analytical methods, such as the Fields and Backofen approach, address this non-uniformity by considering average strain and strain rates. Conversely, inverse approaches, which rely on fitting constitutive equations through iterative procedures, are more sensitive to the choice of empirical equations and can be computationally expensive. To address these issues, this study adapts an inverse piecewise flow curve determination method from compression tests for use in torsion tests. A stepwise methodology is proposed to calculate constant strain rates and isothermal flow curves, where flow curves for the lowest strain rates are first determined and subsequently used to derive flow curves at higher strain rates. The proposed approach was applied to the case-hardened steel 16MnCrS5, with tests conducted at temperatures ranging from 900 °C to 1100 °C and strain rates from 0.01 s⁻¹ to 1 s⁻¹. The experimental data obtained were successfully replicated by the flow curves with a maximum deviation of only 1%. The results demonstrate the efficiency and accuracy of the stepwise inverse approach for determining flow curves in torsion tests, making it appropriate for characterizing material behavior for metal-forming applications.

1. Introduction

One essential feature that influences material flow in forming simulations is the flow curve of the material [1]. Its determination is crucial in predicting forming forces and material response. This information is obtained using laboratory-scale material testing techniques such as tensile, compression, and torsion tests [2]. Among these testing methods, torsion tests are utilized to determine flow curves at large strains, up to approximately 3. Torsion tests maintain consistent sample geometry throughout the testing process, unlike tensile and compression tests, where specimens exhibit necking and bulging, respectively [3]. As a result, the same procedure for determining flow curves can be applied throughout the deformation history. However, due to the nature of torsion tests, there is an inhomogeneous distribution of strain and strain rate in the sample gauge section [2]. During the tests, these values are zero at the axis of symmetry and maximum at the full radius. This gradient illustrates the strain or strain distribution along the measurement length, highlighting the maximum deformation experienced at the outer edges, while the center remains unaffected [4].

This gradient, particularly the strain rate gradient, complicates the calculation of constant strain flow curves using analytical methods, and also data corresponding to several experiments at various rotation speeds would be required in order to determine a single flow curve at a constant strain rate. In addition, the deformation heating caused by the conversion of plastic work into heat makes it difficult to maintain similar temperature distribution along the gauge length of the specimen, and, therefore, the temperature in the specimen deviates from its initial temperature. This results in non-isothermal flow curves if the data from the experiments are directly converted to flow curves without pre-processing [2]. Inverse methods overcome these challenges, where a Finite Element (FE) simulation of the torsion test experiments is set up, and the unknown parameters in the chosen empirical equation are iteratively determined such that the simulated and experimental conditions match. Since simulations can explicitly reflect the inhomogeneities that occur in the torsion experiments, no additional efforts are required to determine constant strain rate flow curves through this method. However, these are computationally expensive, and the prediction accuracy highly depends on the chosen analytical equation [5]. The established analytical and inverse methods to determine flow curves through torsion tests are reviewed below.

2. State of the Art

2.1. Analytical Methods for Torsion Test Flow Curve Determination

Consider a cylindrical gauge section of an axisymmetric torsion specimen with length L and radius a. Torque M is applied at one end of the specimen, which creates shear stress τ and shear strain $\overline{\gamma }$.

The shear stress in the specimen is calculated using Equation (1) as follows [2]:

(1)$\overline{\gamma }=\mathrm{}{\displaystyle \frac{r\alpha }{L}}$

where $\alpha$ is the twist angle in radians. When the specimen is twisted, the incremental torque component dM at a partial radius r is given by

(2)$dM=2\pi r^2\tau dr$

where $\tau$ is the current shear stress in the element tubular section. The total torque in the gauge section is given by [2]

(3)$M=2\mathrm{}\pi {\int }_0^a\tau r^{2}dr$

Nadai [6] developed a graphical approach to convert the experimental torque–twist data to flow stress and obtain a solution to Equation (3), which can be re-written as

(4)$M\mathrm{}=\mathrm{}{\displaystyle \frac{2\pi }{{\alpha \prime }^3}}\underset{0}{\overset{\gamma }{\int }}\tau \mathrm{}{\gamma }^{2}d\gamma$

where $\alpha \prime ={\displaystyle \frac{\alpha }{L}}$. Nadai assumes that the shear stress is only dependent on the local values of shear strain, i.e., $\tau =\tau \left(\gamma \right)$, and, therefore, the integral in Equation (4) only depends on the upper limit. Differentiation of Equation (4) with respect to $\gamma \prime$ yields

(5)$\tau \mathrm{}=\mathrm{}{\displaystyle \frac{1}{2\pi a^3}}(3M\mathrm{}+\alpha \prime {\displaystyle \frac{dM}{d\alpha \prime }})$

where the term ${\displaystyle \frac{dM}{d\alpha \prime }}$ is the slope of the M vs. $\alpha \prime$ at various points.

Although this method is straightforward, Nadai’s graphical technique does not take into account the dependence of shear stress on the twist rate, which causes significant errors in hot torsion test experiments [7]. Additionally, the accuracy with which the slope of the torque–twist data is gathered affects the estimation of shear stress. Nadai’s method was extended by Fields and Backofen [8] for rate-sensitivity materials. The equation to determine the shear stress from the torque–twist data is similar to Nadai’s method and is given as

(6)$\tau \mathrm{}=\mathrm{}{\displaystyle \frac{1}{2\pi a^3}}(3M_c\mathrm{}+\alpha \prime {\displaystyle \frac{d{M}_c}{d\alpha \prime }})$

Here, however, the term $M_c$ in Equation (6) is referred to as “constructed torque” and is different from the torque $M$ in Equation (3). The constructed torque–twist curve must satisfy the following condition:

(7)${\displaystyle \frac{{\dot{\alpha }}_1}{{\alpha }_1}}={\displaystyle \frac{{\dot{\alpha }}_2}{{\alpha }_2}}=\mathrm{}......{\displaystyle \frac{{\dot{\alpha }}_a}{{\alpha }_a}}$

Fields and Backofen have shown that the Equation (6) can be re-written as follows:

(8)$\tau \mathrm{}=\mathrm{}{\displaystyle \frac{M}{2\pi a^3}}(3\mathrm{}+m\mathrm{}+n)$

where m and n are the strain hardening and strain rate sensitivity parameters, respectively. To convert experimental torque–twist data into shear stress, the Fields and Backofen approach is used in the literature, particularly for hot tensile testing where the influence of the strain rate cannot be disregarded. However, a linear relationship must be assumed between $\ln \mathrm{M}$ and $\ln \alpha$ as well as between $\ln \mathrm{M}$ and $\ln \dot{\alpha }$. This presumption is inaccurate at elevated temperatures when the flow curve has multiple regimes of hardening and softening [4].

Koddham et al. [4] proposed a method to obtain instantaneous strain hardening and strain rate sensitivity parameters for flow curve determination at elevated temperatures. This method requires at least three experiments at each twist rate. Khoddam et al. fit the experimental torque–twist curve to rational models as they show excellent goodness at various strain rates and temperatures. This rational model is described by

(9)$M\left(\alpha \right)={\displaystyle \frac{\sum _{i=1}^{q+1}{b}_i{\alpha }^{q+1-i}}{{\alpha }^s+\sum _{j=1}^s{b}_{q+j}{\alpha }^{s-j}}}$

where q and s are polynomial terms of the highest orders in the numerator and denominator, respectively, and ${\mathrm{b}}_{\mathrm{i}}\prime \mathrm{s}$ are the coefficients. Using this representation, Khoddam et al. have derived the strain rate sensitivity parameters for experiments at three twist rates (${\dot{\alpha }}_1$, ${\dot{\alpha }}_2$, ${\dot{\alpha }}_3$). The parameters are as follows:

(10)$m_A={\displaystyle \frac{\left(M_A-M_B\right) {\dot{\alpha }}_1}{\left(2M_A-M_B\right) {\dot{\alpha }}_1-M_A{\dot{\alpha }}_2}}$

(11)$m_B=0.5{\dot{\theta }}_2\left({\displaystyle \frac{M_A-M_B}{M_A{\dot{\alpha }}_1-M_B{\dot{\alpha }}_2}}+{\displaystyle \frac{M_B-M_c}{2M_A{\dot{\alpha }}_1-M_C{\dot{\alpha }}_2-M_B{\dot{\alpha }}_3 }}\right)$

(12)$m_c={\displaystyle \frac{{(M}_C-M_B) {\dot{\alpha }}_3}{M_C ({\dot{\alpha }}_3-{\dot{\alpha }}_2)}}$

The strain hardening parameter (n) can be calculated using the following equation:

(13)$n={\displaystyle \frac{\alpha }{M}} {\displaystyle \frac{\partial M}{\partial \alpha }}$

where ${\displaystyle \frac{\partial \mathrm{M}}{\partial \alpha }}$ is the gradient of the torque–twist curve. This method’s drawback is that a minimum of three experiments are required for each twist rate.

Finally, the shear stress–shear strain curve must be converted into an equivalent stress–strain curve. For pure shear condition and assuming the material to be isotropic, the von Mises relation shown below is most used for this conversion [9]

(14)${\sigma }_f=\sqrt{3} \tau \phi ={\displaystyle \frac{\gamma }{\sqrt{3}}}$

In most scientific studies, flow curves obtained from torsion tests are determined under the assumption of an isothermal condition, referring to a constant deformation temperature. This approach often disregards deformation heating and other heat transfer mechanisms, where temperature compensation is not performed or considered. Also, the analytical methods for temperature changes due to conduction and radiation must consider the specimen geometry as well and, therefore, are specific to a considered geometry and testing machine. In addition, these analytical methods are straightforward and do not explicitly account for temperature, strain, or strain rate inhomogeneity. Instead, they incorporate these factors indirectly only through sensitivity parameters. In [10], it is claimed that even a small error in the strain rate sensitivity and strain hardening parameters can result in large deviations in the final flow curve.

2.2. Inverse Methods for Torsion Test Flow Curve Determination

Flow curve determination methods based on inverse modeling, where the flow curve is represented by empirical constitutive functions, are developed to address the difficulties caused by the specimen’s inhomogeneity [11]. In this, a simulation model replicating the experimental conditions is developed, and the flow curve is approximated by a constitutive equation. The coefficients of the constitutive equation are then optimized by minimizing the error between the experimental and simulated results. In [12], the parameters of a power law type constitutive equation using inverse modeling of torsion tests are determined, and strain hardening and strain rate sensitivity parameters using a newly proposed cost function are estimated, respectively. In [13], an inverse approach based on a temperature-dependent material model to predict isothermal flow curves of hot torsion tests is employed. Moreover, an inverse flow curve determination method based on a unified material model is proposed in [14]. A unified material model takes into account several constitutive models, each for a particular temperature range. However, the constitutive model that is selected a priori has a significant impact on how accurate these methods are [11]. Furthermore, these models usually fail to describe flow curves for materials displaying complex flow behavior, such as several dynamic recrystallization cycles.

To overcome the limitations corresponding to analytical and inverse methods, an inverse piecewise method named “flow curve determination through explicit piecewise inverse modelling (FepiM)” is developed for compression tests by the current authors [15,16]. This approach is used to consider the strain inhomogeneity in compression tests. Additionally, only the temperature changes caused by deformation heating are considered in the case of compression tests, and these effects were taken into account via a stepwise flow curve determination approach. However, due to the nature of torsion tests, the strain and strain rate inhomogeneities exist together with the effect of deformation heating, and, therefore, additional changes have been considered in the method. This universally enables the determination of constant strain rate flow curves for hot working simulations using the FepiM approach, obviating the need for a constitutive model.

3. Materials and Methods

3.1. FE Model and Consecutive Flow Curve Point Determination

To determine flow curves using FepiM, an FE model of the torsion test specimen that constitutes the geometry and boundary conditions during testing is required. The torsion experiment is conducted on an axisymmetric cylindrical specimen, which has a total length of L. However, the gauge section where most of the plastic deformation is concentrated has a length of l and a radius a. The FE model is developed in the commercial FE software Abaqus/Standard 2020. During the deformation, both the top and the bottom ends of the axisymmetric cylindrical specimen are clamped, and a rotation/twist is applied at one of the ends. The specimen is modeled using axisymmetric twist elements CGAX4T (4-node axisymmetric, thermally coupled quadrilateral, twist element). The twist element type has two advantages with respect to FepiM. Firstly, due to the axisymmetry in the specimen, the rotational symmetry reduces computational costs. Secondly, twist elements are capable of handling high deformation without element distortion. The FE model before deformation is illustrated in Figure 1a.

The basic procedure of consecutive flow curve point determination for torsion tests is illustrated in Figure 1b. The experimental torque vs. twist angle (TT) curves and the material’s initial flow stress are inputs for the FepiM flow curve determination. The initial flow stress value is identified by considering the beginning of plastic deformation in the TT curve and can be calculated from analytical methods [8]. Initially, the experimental TT is divided into different twist angle steps, and the goal is to establish a flow curve point for each of these steps. The flow curve point is obtained by minimizing the error between the experimental torque and simulative torque. Once convergence is obtained, the flow curve corresponding to the next twist angle step is determined. Also, once a flow curve point is obtained, it remains fixed for the consecutive twist angle steps. However, due to the inhomogeneities in the strain and strain rate distribution, the flow curve determination is not straightforward, and the approach for determining constant strain rates and isothermal flow curves will be discussed in detail in the sections below. It may be computationally costly to determine a flow curve point at each location on the experimental MD curve. Instead, only a few significant twist angle steps are resampled and used, depending on how much the slope of the MD curve changes. These resampled points are called evaluation points (EPs), and the procedure to determine these points is described in [15].

3.2. Implementation of an Iterative Approach in Abaqus/Standard and Flow Curve Determination at Room Temperature

An explicit iterative approach is followed to determine a piecewise flow curve using experimental twist angle steps and the FE model described in the section above. The detailed approach for determining the flow curve points is described in this section via a flow diagram shown in Figure 2.

FepiM is built with the software package MATLAB^® 2018b, and MATLAB^® 2018b scripts are used to regulate and optimize the flow curve points. The Abaqus/Standard UHARD subroutine [17] is used to connect FepiM to Abaqus/Standard by assigning and modifying flow curve data. The implementation steps involved in the determination of a new flow curve point (${\phi }_{\mathrm{i}+1},{\sigma }_{\mathrm{i}+1}$) is discussed in detail below and illustrated in Figure 2. The experimental FD curve is resampled to N EPs as input data, and the goal is to obtain flow curve points (${\phi }_{\mathrm{j}},{\sigma }_{\mathrm{j}}$) (j = 1 to N) at each EP. The flow curve data until (${\phi }_{\mathrm{i}},{\sigma }_{\mathrm{i}}$) are assumed to be known and are fixed for the consecutive twist angle steps.

Step (a): Initial gradient estimation. Initially to estimate a new flow curve point (${\phi }_{\mathrm{i}+1},{\sigma }_{\mathrm{i}+1}$), the known flow curve until (${\phi }_i,{\sigma }_i$) must be extrapolated with an initial slope of $\theta$. This slope is calculated using the two preceding flow curve points (${\phi }_j,{\sigma }_j$) (j = i and i + 1).

Step (b): Flow curve extrapolation. In this step, the flow curve (${\phi }_i,{\sigma }_i$) is extrapolated with the slope $\theta$ calculated in step (a) until a very large strain, which cannot be encountered in the current displacement step. The flow curve data are represented in a tabular form containing the original curve, as well as the extrapolated data, which are saved in an ASCII file.

Step (c): The start of the FE displacement step. Using these flow curve data from step (b), the Abaqus FE torsion test corresponding to the twist angle step is started.

Step (d): Flow stress assignment. During the simulation, a flow stress value must be assigned to each element in the model. This assignment is performed using an additional subroutine UHARD integrated with Abaqus/Standard. The UHARD subroutine obtains the local equivalent plastic strain ${\phi }^{\mathrm{e}}$ of each element during the simulation from the FE solver. The subroutine takes the input flow curve data and assigns the flow stress ${\sigma }^{\mathrm{e}}$ to each element by linear interpolation. The requirement of UHARD, as well as the necessity of storing the flow curve data in an ASCII file, will be explained in detail later.

Step (e): Extract global simulate force. At the end of the twist angle step, the global simulated moment $M_{i+1,sim}$ is extracted from the simulation model.

Step (f): Compare local force and convergence check. The difference between the simulated and experimental moment ${\mathrm{M}}_{\mathrm{i}+1,\mathrm{e}\mathrm{x}\mathrm{p}}$ is determined at the EP. If the absolute error is under a user-defined tolerance limit (TOL), the iterations are terminated and the procedure in step (h) is performed; otherwise, more iterations for the same twist angle step are performed according to step (g).

Step (g): Modification of the slope $\theta$. Steps (b) to (f) are repeated with a new slope $\theta \to {\theta }^{\prime }$ if the tolerance criterion in step (f) is not met. The new slope is calculated based on the relative error between the experimental and simulated moment as follows:

(15)${\theta }^{\prime }=\theta +H (M_{i+1,exp}-M_{i+1,sim})$

H is a user-defined parameter that affects convergence. When a larger H is chosen, the anticipated slope exceeds the ideal value throughout iterations, and when H is too small, the number of convergence iterations is enormous.

Step (h): Progress to the new displacement step. Following convergence in step (g) or (h), the flow curve point corresponding to displacement step i + 1 is obtained by locating the element with the highest plastic strain and the associated von Mises stress. These two values represent the new flow curve point (${\phi }_{i+1},{\sigma }_{i+1}$). Then, FepiM moves to the next displacement step, and the steps from (a) to (h) are repeated until N displacement steps are reached.

The Abaqus restart functionality is employed here so that after evaluating a flow curve point, the simulation can move to the next twist angle step and avoid running the simulation from the beginning. However, Abaqus does not enable users to change the flow curve data during the simulations, which is necessary, for example, in step (g) during the modification of the slope of the extrapolated flow curve or at step (h) to add new flow curve data corresponding to the new displacement step. To address this issue, the UHARD subroutine is developed, which receives the flow curve tabular data from an external ASCII file and allows the flow curve data to be updated in the ASCII file after each EP is computed or also between iterations.

This method described above could be used to calculate individual flow curves when materials do not exhibit strain rate sensitivity and when temperature variations can be disregarded. This is because the flow stress corresponding to different locations in the torsion test sample will lie on the same flow curve, though they would have a different local strain rate. Torsion tests exhibit inhomogeneous strain rate distribution, and assigning flow curve data corresponding to a single flow curve using the approach described in step (d) in Figure 2 would not be appropriate, especially when determining flow curves at elevated temperatures. In the next section, a particular emphasis is placed on flow curve determination considering the strain rate inhomogeneity.

3.3. Torsion Flow Curve Determination at Elevated Temperatures

Due to the inhomogeneous distribution of the strain rate in torsion test specimens, constant strain rate flow curves cannot be described in a single step, as shown in Section 3.2. Therefore, a stepwise approach is proposed in this section, and this approach is described initially for flow curve determination at the lowest strain rate in the experimental matrix. In the further steps, the flow curves at the next strain rates are determined in ascending order of strain rates. This division is necessary due to the limited availability of test data corresponding to different strain rates, and an approximation method is proposed to assume test data at strain rates close to zero. As a simplification and to understand this approach, the procedure is demonstrated for a set of two flow curves at strain rates of 0.01 and 0.1 s⁻¹ and a single temperature. The dependency of temperature during the flow curve determination will be taken up later.

Flow curve determination at the lowest strain rate

At a given temperature and for an experimental matrix, the flow curve at the lowest strain rate is computed in the first step (for the example experimental matrix, 0.01 s⁻¹ is the lowest strain rate). However, for the experiment performed at 0.01 s⁻¹, the local strain rate in the specimen varied from zero at the axis of rotation to 0.01 s⁻¹ at the maximum radius. As a result, the inhomogeneity in the strain rate cannot be taken into account when computing the flow curve using the piecewise approach discussed in Section 3.2. This is because flow curve data corresponding to lower strain rates are required for assigning flow stress to elements with strain rates less than 0.01 s⁻¹. Generally, experimental data are limited, and conducting experiments at multiple strain rates can be expensive. Hence, an analytical method is proposed in this chapter to approximate the flow curve data below the strain rate of 0.01 s⁻¹ using existing experimental TT curve data (TT curves at 0.01 and 0.1 s⁻¹).

The proposed analytical method to approximate the flow strain rate flow curves takes into consideration the strain rate sensitivity in the experimental TT curves at 0.01 s⁻¹ and 0.1 s⁻¹. In this case, Equation (16) proposes the calculation of the strain rate sensitivity parameter m

(16)$m={\left({\displaystyle \frac{\mathit{ln}{M}^{0.01}-\mathit{ln}{M}^{0.1}}{\mathit{ln}\left(0.01\right)-\mathit{ln}\left(0.1\right)}}\right)}_{\alpha }$

where M^0.01 and M^0.1 are the moment values at twist angle $\alpha$ corresponding to the experiments at strain rates 0.01 s⁻¹ and 0.1 s⁻¹, respectively. Using this parameter m, the flow curve at a lower strain rate can be approximated. In this current example, the flow curve at 0.001 s⁻¹ is approximated using Equation (17) as follows:

(17)${\left({\sigma }_f^{0.001}\right)}_{\phi }=e^{-m\mathit{ln}{\displaystyle \frac{\left(0.01\right)}{\left(0.001\right)}} }{\left({\sigma }_f^{0.01}\right)}_{\phi }$

where ${\sigma }_{\mathrm{f}}^{0.001}$ and ${\sigma }_{\mathrm{f}}^{0.01}$ are the flow stress at strain rates of 0.001 s⁻¹ and 0.01 s⁻¹, respectively. ${\sigma }_{\mathrm{f}}^{0.01}$ is obtained using the iterative FepiM approach, as shown in Figure 2, where the detailed implementation of the flow curve determination at 0.01 s⁻¹ is discussed below.

The experimental TT curve at 0.01 s⁻¹ is the input to the iterative FepiM approach for determining the corresponding flow curve. As a first step, the gradient of the flow curve is extrapolated to a large strain and is given to the FE model (green curve in Figure 3). In addition, the flow curve at 0.001 s⁻¹ is approximated using Equations (16) and (17) (dotted red curve in Figure 3). In the simulation model, one end of the torsion specimen is rotated with a constant angular velocity such that the maximum strain rate simulation is 0.01 s⁻¹. As the simulation starts, the UHARD reads the input flow curves and assigns flow stress to individual elements (${\sigma }_{\mathrm{f}}^{\mathrm{e}}$) in the FE model by interpolation.

Figure 3 also shows an example of interpolation for two elements A and B with different local strain and strain rates. Assume two elements A and B, where element A corresponds to a point at the maximum radius and, therefore, has a local strain rate of 0.01 s⁻¹, and the flow stress of this element lies directly on the flow curve of 0.01 s⁻¹. However, element B corresponds to a region between the axis of rotation and the maximum radius and has a local strain rate of 0.005 s⁻¹. Hence, the flow stress of this element lies between the flow curves of 0.001 and 0.01 s⁻¹ and is obtained by interpolation. In this manner, the flow stress of every element in the simulation model is obtained considering the local plastic strain and strain rate.

At the end of the simulation, the simulated torque is extracted for the displacement step, and if the difference between the experimental and simulated torque is below a tolerance limit, the iterations are stopped or the procedure discussed above is repeated with a new gradient ${\theta }^{\prime }$. Upon convergence, the maximum plastic strain in the simulation and corresponding flow stress value on the extrapolated flow curve of 0.01 s⁻¹ is considered the new flow curve point. This procedure is repeated for every twist angle step in the TT curve to obtain a complete constant strain rate flow curve at 0.01 s⁻¹. This procedure is similar to that described in steps (a)–(h) in Section 3.2. However, the only difference here is that the flow curve at 0.001 s⁻¹ is approximated in parallel to the piecewise flow curve determination of 0.01 s⁻¹ using the strain rate sensitivity parameter m.

Flow curve determination at higher strain rates

In the second step of the flow curve determination, the flow curves at the higher strain rates in the chosen experimental matrix (flow curves above the strain rate of 0.01 s⁻¹) are determined sequentially in ascending order of strain rate. The general procedure is similar for all flow curves and is described below.

For the chosen example experimental matrix, the flow curve at 0.1 s⁻¹ is determined next. The iterative approach is similar to the procedure described earlier in Section 3.2. The flow curve at 0.1 s⁻¹ is extrapolated until a large strain is not encountered in the current displacement step. This flow curve is then the input to the simulation model, where the torsion specimen is rotated with an angular velocity such that the maximum strain rate in the specimen is 0.1 s⁻¹. This extrapolated flow curve is accompanied by the already determined flow curves at lower strain rates (0.001 s⁻¹ and 0.01 s⁻¹). The UHARD subroutine uses this complete flow curve data for interpolation and also for the flow stress assignment.

Figure 4 shows the illustration of flow curve interpolation for three elements, A, B, and C, in the model. Element A is at the axis of rotation and, therefore, has a lower strain and strain rate compared to the other elements. Similarly, element C is at the maximum radius and, therefore, has the highest strain and strain and strain rate. Based on the local strain and strain rate, the UHARD subroutine assigns the flow stress value to each element in the FE model as shown. Here, during the iterations, only the gradient for the curve at 0.1 s⁻¹ is modified; the other flow curves do not change. Also, in this case, when convergence between the simulated and experimental torque is obtained, the flow curve point is determined based on the maximum plastic strain in the simulation and is the corresponding von Mises stress on the flow curve at 0.1 s⁻¹. This process is repeated at each twist angle step to obtain a constant strain rate flow curve at 0.1 s⁻¹.

This stepwise approach is repeated to determine the constant strain rate flow curve at other strain rates, e.g., 1 s⁻¹. In addition, to obtain isothermal flow curves, the temperature changes in the specimen due to deformation heating and other heat transfer mechanisms must also be considered. In order to take the temperature changes into account, a combination of online-offline temperature compensation is used. The procedure is similar to that of temperature compensation described by the current authors for compression tests [17], where the flow curve determination starts from the highest temperatures. Here, the average temperature change in the gauge section due to deformation heating and other heat transfer mechanisms is obtained from the simulation, and the flow stress is compensated with respect to this temperature change to obtain isothermal flow curves. Thereafter, isothermal flow curves at the other lower temperatures are obtained directly during the FepiM flow curve determination itself. This procedure is detailed in the sub-section below by considering a sample experimental matrix of two temperatures and two strain rates.

3.4. Synopsis of the Approach

A stepwise approach of flow curve determination in the ascending order of strain rates is necessary for obtaining constant strain rate flow curves. Similarly, a combination of online-offline temperature compensation is used to determine isothermal flow curves for torsion tests. Therefore, a systemic stepwise approach must be followed to determine the isothermal as well as constant strain rate flow curves. The procedure is shown with an example experimental matrix of two temperatures (1000 °C and 1100 °C) and two strain rates (0.01 s⁻¹ and 0.1 s⁻¹) in Figure 5.

Step 1. The procedure starts from the determination of the non-isothermal flow curve at the highest temperature and lowest strain rate (1100 °C and 0.01 s⁻¹). Here, in addition to this, the non-isothermal flow curve at the same strain rate but the next highest temperature (1000 °C and 0.01 s⁻¹) is also determined. This is necessary to compensate for the temperature change due to deformation heating for the flow curve at 1100 °C and 0.01 s⁻¹. The required isothermal flow curve is then obtained by an offline compensation procedure by considering the average temperature change in the deformation zone and with the procedure described in Section 3.2 and Section 3.3. The output of this procedure is an isothermal and constant strain rate flow curve at 1100 °C and 0.01 s⁻¹.

Step 2. With the similar procedure in step 1, the flow curve at 1100 °C and 0.1 s⁻¹ is determined in this step. However, the already determined non-isothermal flow curves at 0.01 s⁻¹ are also provided as inputs here and are used for flow stress assignment in UHARD. Here, an offline temperature compensation is followed, where the average temperature change during the non-isothermal flow curve determination is considered for determining the required isothermal flow curve at 1100 °C and 0.1 s⁻¹.

Step 3. The flow curve at the next highest temperature and lowest strain rate (1000 °C and 0.01 s⁻¹) is then computed. Since the flow curves at 1100 °C are already known, an online temperature compensation considering the temperature inhomogeneity is possible.

Step 4. In the final step, the isothermal and constant strain rate flow curve at 1000 °C and 0.1 s⁻¹ is determined with the help of all the flow curves determined from steps 1 to 3. Also, the temperature changes in the simulation are compensated considering the inhomogeneities by online compensation.

It must be noted that the flow curves at 0.001 s⁻¹ (dotted boxes) are obtained by approximating them with the strain rate sensitivity parameter determined by Equations (16) and (17). These curves are necessary to consider the strain rate inhomogeneity when computing the flow curves at 0.01 s⁻¹ (the lowest strain rate in the experimental matrix).

3.5. Material and Experimental Procedure

The hot torsion test investigations are carried out using the Bähr STD815 torsion plastometer, which has a maximum torque capacity of 50Mn (see Figure 6). The temperature is controlled using a thermocouple of type S, and the frequency is automatically adjusted to ensure the correct testing temperature. The specimens are heated using induction coils, and the thermocouples are welded to the center of the specimen’s gauge length to regulate the temperature. An argon atmosphere is used to avoid scaling in the specimen. Also, the figure depicts the precise original specimen geometry proposed by the machine manufacturer. The gauge section, where the plastic deformation is concentrated, has a cylinder cross-section with dimensions of 10 mm in length and 6 mm in diameter. The experimental chamber consists of the top and bottom clamps and the specimen is clamped between them, and the required twist is applied to the bottom clamp.

In order to determine flow curves, the torsion tests are strain rate controlled, meaning the rate of twist is controlled such that the maximum strain rate at the maximum radius in the deformation zone remains constant during the experiment. The following linear relationship shown in Equation (18) between time and rotation angle is employed to maintain a constant strain rate:

(18)$t={\displaystyle \frac{R}{\sqrt{3}\times 60\times L\times \dot{\phi } }} \alpha$

where t is time in seconds, a and l are the specimen gauge radius and length, respectively, $\dot{\phi }$ is the desired strain rate, and $\alpha$ is the deformation twist angle in degrees.

The experimental investigation in this work is carried out on case-hardened steel 16MnCrS5, which is widely used in caliber rolling and forging processes and has numerous applications in the automotive industry, such as gear shafts and gear wheels [18]. The experimental torque–twist (TT) curves are recorded on the torsion plastometer at strain rates of 0.01 s⁻¹, 0.1 s⁻¹, and 1 s⁻¹ and nominal start temperatures of 900 °C, 1000 °C, and 1100 °C. The torsion specimens are precisely heated to 1200 °C at 10 °K/s and held for 3 min for homogenization. Then, they are cooled to the testing temperature at a rate of 3.33 °K/s. The specimens are held for 3 min before deformation for homogenization and to reduce the skin effect caused by induction heating. During deformation, the torque is measured with a load cell, and the temperature at the center of the specimen is measured and controlled using the thermocouple welded at the surface of the gauge section.

Also, at a later stage, for better validation of the FepiM flow curves and comparison with the analytical method, a torsion plastometer experiment is conducted with a different specimen geometry. The dimensions of the new specimen are shown in Figure 7. The gauge section has a length of 17.89 mm, the radius is 12 mm, and the overall specimen length is the same as earlier. The experiment is performed with a start temperature of 950 °C, such that the maximum strain rate of 0.5 s⁻¹ is maintained in the specimen. The experimental TT curve for this experiment is shown in Section 5.2.

4. Results

4.1. Experimental Results of Torsion Experiments

Using the experimental procedure discussed in Section 3.5, torsion experiments for 16MnCrS5 were conducted. Each experiment was repeated twice, and the average TT curves for each experimental condition are shown in Figure 8. The experiments were performed until a twist angle of 1800°, which corresponds to a plastic strain of ~3 in the flow curve. These TT curves serve as input to FepiM as well as the analytical method and are post-processed to flow curves. Due to dynamic recovery (DRX) and dynamic recrystallization (DRV), the TT curves exhibit complex non-monotonous behavior, particularly at high twist angles, and this complexity is expected to be replicated in the corresponding flow curves as well.

This section may be divided by subheadings. It should provide a concise and precise description of the experimental results and their interpretation, as well as the experimental conclusions that can be drawn.

4.2. Torsion Test Flow Curves Determined from FepiM

The FepiM torsion test flow curves are determined using the experimental TT curves and the stepwise approach described in Section 3.3. The FE model used for the inverse flow curve determination is shown in Section 3.2. As discussed earlier, an axisymmetric FE model is used in the determination of flow curves used in FepiM. For the discretization of the FE model, an element size of 0.5 mm is considered at the gauge section where most of the plastic deformation is concentrated. The other regions of the specimen are meshed with a coarser mesh, and the entire torsion specimen consists of 588 elements. The element size at the deformation zone did not show a major influence on the simulated TT curve, but simulation with an element size above 0.5 mm shows high element distortion above a plastic strain of 1, and reducing the element size further increases computational time. To capture the hot deformation, the FE simulations are thermo-mechanically coupled. The specimens are heated up using an induction coil, and modeling the deformation process coupled with magnetic induction would complicate the inverse modeling further. Instead, at the beginning of the simulation, the temperature of elements in the gauge section is set to the start temperature of the corresponding experiment. Since the induction coil is located at the center of the specimen covering the gauge length, similar to the torsion plastometer model of Wang et al. [18], a gradient of 100 °C is assumed from the end of the gauge length to the end of the specimen to account for the inhomogeneity in the start temperature. The initial start temperature profile for the experiments at 1000 °C is shown in Figure 9. The initial temperature profile along the radial direction due to skin effects is ignored because of sufficient homogenization before the deformation begins. Also, similar to the model by Wang et al., the heat radiation with an emissivity of 0.2 to the surrounding environment and a dissipation coefficient of 0.95 were considered. Since heating is controlled by induction heating, the emissivity is deliberately chosen lower to avoid large temperature drops due to radiation.

The FepiM flow curves for 16MnCrS5 at 0.01 1 s⁻¹, 0.1 s⁻¹, and 1 s⁻¹ and start temperatures of 900 °C, 1000 °C, and 1100 °C are determined using the sequential approach described in Section 3.3, where the flow curve at the lowest strain rate and highest temperature (1100 °C, 0.01 s⁻¹) is determined first and flow curves at the lowest temperature and highest strain rate (900 °C, 0.01 s⁻¹) are determined at the end.

Initially, the flow curves determined by different analytical methods, namely, Fields and Backofen [9] and Khoddam et al. [4], are compared, and the best of both methods is used later to compare the FepiM flow curves. For the analytical methods, it is assumed that the temperature in the gauge section is constant, and isothermal conditions are assumed.

Comparison of flow curves from the Fields and Backofen and Khoddam et al. methods

The analytical flow curves are determined using the TT curves for experiments at 900 °C. These flow curves are determined with the Fields and Backofen (FB) and Khoddam et al. methods discussed in Section 2.1. The results are shown in Figure 10.

In general, the hardening behavior in flow curves determined by both methods appears similar, but the steady-state flow stress is slightly higher for the FB flow curves compared with the flow curves of Khoddam et al. The difference between the two methods can be attributed to the calculation of the strain hardening and strain rate parameters. For instance, the relation between $\ln \mathrm{M}$ and $\ln \alpha$ for the experiment at 900 °C and 0.01 s⁻¹ is plotted in Figure 11. According to the FB method, this relation is assumed to be linear, and the slope of this plot is considered as the strain hardening parameter. But, this is only true until softening occurs. On the other hand, the strain hardening parameter in the Khoddam et al. methods is considered an instantaneous value, according to Equations (10)–(12). As a result, the method developed by Khoddam et al. is regarded as a more appropriate analytical approach, and the FepiM flow curves will be compared with the flow curves generated by this method in the following sections.

Comparison with FepiM flow curves with the analytical method (Khoddam et al. [4])

The FepiM flow curves determined using the sequential approach in comparison to the analytical method of Khoddam et al. for temperature ranges of 900 °C to 1100 °C and strain rate ranges of 0.01 s⁻¹ to 1 s⁻¹ are shown in Figure 12, Figure 13 and Figure 14. These flow curves exhibit complex non-monotonous flow curve behavior during hot deformation, with the flow stress rapidly increasing at first and reaching a peak stress, corresponding to a peak strain. The flow stress then gradually decreases until the value corresponds to a steady flow stress at which equilibrium in the strain hardening and strain softening process is reached. In general, both methods could capture this complex behavior.

Also, the flow stress is higher in the FepiM flow curves determined at 900 °C and 1000 °C and the strain rates of 0.01 s⁻¹ and 0.1 s⁻¹ than in the analytical flow curves. Furthermore, at lower temperatures, 900 °C, and 1000 °C, the initial hardening in the FepiM flow curves is greater than that in the analytical flow curves. Although the method of Khoddam et al. takes into account the instantaneous strain hardening and strain rate parameters, inhomogeneity in the strain and strain rate is not explicitly considered in the model. This could be one of the reasons for the discrepancy between the two methods. In Section 5, a detailed comparison of both methods is presented.

5. Discussion

A qualitative analysis will be performed in this section between the FepiM flow curves and the analytical Khoddam et al. method, in which these curves are used in FE simulations, and the results of the simulations are compared to experimental TT curves. For this aim, the FepiM and analytical flow curves will be used in the torsion test simulations that replicate the experiments described in Section 3.5, and the resultant simulated curves from both methods will be compared with the experimental TT curves.

5.1. Validation with Torsion Test Simulations

Flow curves in the temperature ranges of 900 °C to 1100 °C and strain rate ranges of 0.01 s⁻¹ to 1 s⁻¹ are determined sequentially using the FepiM stepwise approach. Therefore, the flow curves as a complete flow curve field are validated in this section with Abaqus/Standard, and the flow curve assignment is performed internally with FE software without the UHARD subroutine developed in this chapter. A quantitative comparison of FepiM and the analytical method of Khoddam et al. is performed. To accomplish this, the overall flow curve field determined by both methods is used in the FE simulation of torsion tests. The same FE model used to determine the inverse flow curves is used here. The simulated TT curves extracted from the simulations can be directly compared with the experimental TT curves using this analysis. The following equation is used to calculate the relative error between the experimental and simulated TT curves:

(19)$\left|∆\right|={\displaystyle \frac{M_{simulation}-M_{experiment}}{M_{experiment}}}\times 100$

The torque is extracted from the simulation models at every 20° twist angle and is compared with the experimental results. Figure 15 shows the relative error between the simulated TT curves (with analytical flow curves and FepiM flow curves) and the experimental TT curves at various testing conditions. The temperature of the specimen during deformation rises above the nominal starting temperature due to deformation heating, and because flow curves above 1100 °C are not available and flow stress extrapolation when simulating with Abaqus is not possible.

The relative error in FepiM flow curve simulations is consistently lower than in analytical flow curve simulations. The error in the analytical flow curves is higher at the start of the simulation, which is also reflected in the flow curves shown in Figure 12, Figure 13 and Figure 14, where the initial hardening analytical curves deviate from the FepiM flow curves. The average error ($\overline{∆}$) in the analytical flow curves is between 2.01 and 9.61%, whereas the average error in the FepiM flow curves is always below 0.5%. Figure 16 shows a summary of the average error for each testing condition.

Also, the maximum and average error at every 20° twist angle for each testing condition is obtained for the simulation models with FepiM and analytical flow curves. The results of this analysis are shown in Figure 17. The FepiM flow curve simulations have a lower maximum and average error compared to the simulations with the analytical flow curve simulations. This type of maximum and average analysis provides an overview of the accuracy of the flow curves when used in process simulations.

The lower error in the FepiM flow curves corresponds to the consideration of inhomogeneity with respect to the strain and strain rate during the inverse flow curve determination. This is because although the method of Khoddam et al. takes into account the instantaneous values of strain hardening and strain rate sensitivity parameters, these parameters are considered in an average sense for all the sets of flow curves via the Equations (10)–(12). However, in the FepiM method, the assignment of flow stress during the flow curve determination is based on the local strain; the strain rate is performed in FepiM inverse determination, and no approximations are considered for determining the constant strain rate flow curves.

Also, the difference in the flow curves can be due to the considered temperature boundary conditions in the FE model. In these models, the temperature changes due to deformation heating, conduction, and heat loss due to radiations are also considered, and isothermal flow curves are determined after considering these effects. For example, the temperature contour plots of the simulations at a starting temperature of 1000 °C and a twist angle of 720° are shown in Figure 18.

The maximum plastic strain corresponds to $\phi$ $\approx$1 for these plots. As seen, the temperature deviates from the starting temperature in the deformation zone. Especially, for the simulations at 0.01 s⁻¹, the temperature falls below the starting temperature, and for the simulation at 1 s⁻¹, the temperature is above the starting temperature. However, in the analytical method of Khoddam et al., the temperature changes during deformation are not considered, and isothermal conditions are assumed. This assumption is, to some extent, valid for lower flow stress materials, like aluminum, because the heat gained due to deformation heating is also lower. The analytical methods in the literature overlook the occurrence of deformation heating and other thermal boundary conditions [8,11]. Also, the temperature changes with analytical methods are difficult because the calculations are specific to the specimen geometry and testing equipment and they cannot be generalized.

5.2. Validation with a Different Specimen Geometry

As discussed earlier, to validate the FepiM torsion test flow curves, experiments with a different specimen geometry were tested (see Figure 7). To simulate the experiment, a FE model with twist elements similar to that used for inverse modeling is used. The simulations were performed with analytical as well as FepiM flow curve fields, as shown in Section 4.2. The TT curves are extracted from the simulated models and compared with the experimental TT curve. The results are shown in Figure 19.

The twist rotation is only considered until 150° of deformation because the maximum plastic strain exceeds >3 and flow curves beyond that value are not available. As seen, the simulation with analytical flow curves underestimates the experimental torque. Except at the beginning between a twist angle of 0 and 20°, the FepiM flow curves could well predict the experimental TT curve. More importantly, the peak value of torque is well predicted by the FepiM flow curves that generally represent the peak stress and strain in flow curves. However, the simulated TT curve obtained from the analytical flow curves is always lower than the experimental TT curve, and the peak torque could not be predicted by these flow curves. As shown in the relative error plot in Figure 19b, except for the 0–20° twist angle range, the error in the simulated FepiM TT curve is lower than that of the simulated analytical TT curve. Additionally, the average relative error is lower in the FepiM TT curve.

6. Summary and Conclusions

The present manuscript describes a method to derive isothermal and constant strain rate flow curves for inhomogeneous torsion test specimens under hot forming conditions using a piecewise inverse method. While torsion tests ensure stable sample geometry, they introduce inhomogeneous strain and strain rate distributions within the deformation zone. Traditional analytical methods cannot explicitly account for these inhomogeneities, relying instead on sensitivity parameters to approximate average values. Conversely, inverse methods require an a priori constitutive equation, which can struggle to represent complex material behavior under hot forming conditions. To overcome these challenges, the piecewise inverse method (FepiM) was extended for torsion tests using a stepwise approach, wherein flow curves at the lowest strain rates were determined first and subsequently used to obtain higher strain rate flow curves. An offline–online temperature compensation procedure was also integrated to ensure accurate isothermal flow curve determination.

Using FepiM and the stepwise approach, flow curves were obtained for case-hardened steel 16MnCrS5 across a temperature range of 900–1100 °C and strain rates from 0.01 to 1 s⁻¹. Comparisons with the analytical method proposed by Khoddam et al. demonstrated that FepiM achieved better accuracy, predicting experimental data with a relative error below 0.5%, whereas the analytical method exhibited higher errors, ranging from 2.1% to 9.61%. Validation using torsion experiments with different specimen geometries further confirmed the effectiveness of FepiM. While the analytical method consistently underestimated torque throughout the deformation, FepiM initially struggled to predict the torque–time (TT) curve but improved significantly in later stages, accurately capturing the maximum torque. This stepwise inverse approach method can, therefore, be implemented for characterizing material behavior under hot forming conditions, making it a promising method for determining flow curves until a large strain (~3).

Author Contributions

Conceptualization, A.V. and H.B.; methodology, A.V.; validation, A.V. and H.B.; formal analysis, A.V., H.B., D.B., and E.S.; investigation, A.V.; writing—original draft preparation, A.V.; writing—review and editing, A.V., H.B., D.B., and E.S.; supervision, H.B. and E.S.; project administration, D.B.; funding acquisition, D.B. and E.S. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

The raw data supporting the conclusions of this article will be made available by the authors on request.

Conflicts of Interest

The authors declare no conflict of interest.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures

Figure 1. (a) FE model of a torsion test specimen, (b) illustration of flow curve point determination for torsion tests.

Figure 2. Flow diagram depicting the implementation of the FepiM iterative approach with MATLAB® and Abaqus/Standard.

Figure 3. Flowchart illustrating the iterative approach at the lowest strain rate.

Figure 4. Flow stress assignment for flow curve determination at higher strain rates.

Figure 5. Example of the stepwise procedure to obtain isothermal and constant strain rate flow curves at two different temperatures and strain rates.

Figure 6. Torsion plastometer Bähr STD815.

Figure 7. Specimen with a different gauge geometry used for the torsion plastometer test.

Figure 8. Experimental TT curves of 16MnCrS5 at three different strain rates and temperatures.

Figure 9. Starting temperature profile of the simulation model at 1000 °C.

Figure 10. Comparison of flow curves determined by the Fields and Backofen method and the Khoddam method (adapted from ref. [4]).

Figure 11. Plot of [Forumla omitted. See PDF.] and [Forumla omitted. See PDF.] at 900 °C and 0.01 s−1.

Figure 12. Comparison of 16MnCrS5 flow curves at 1100 °C determined with FepiM and with the analytical Khoddam et al. method.

Figure 13. Comparison of 16MnCrS5 flow curves at 1000 °C determined with FepiM and with the analytical Khoddam et al. method.

Figure 14. Comparison of 16MnCrS5 flow curves at 900 °C determined with FepiM and with the analytical Khoddam et al. method.

Figure 15. Relative error between simulated and experimental TT curves at different testing conditions.

Figure 16. Summary of average relative error in simulations for all the testing conditions.

Figure 17. Average and maximum error for all the torsion simulations.

Figure 18. Temperature contour plots of torsion simulations for a twist angle of 720° in the deformation zone for experiments at 1000 °C.

Figure 19. (a) Comparison of simulated and experimental TT curves for a new specimen geometry, (b) relative error between simulated and experimental TT curves for a new specimen geometry.

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