Material Characterisation, Modelling, and Validation of a UHSS Warm-Forming Process for a Heavy-Duty Vehicle Chassis Component

Abstract

The lightweighting of heavy-duty vehicles (HDVs) is an effective strategy to reduce fuel consumption and lower CO₂ emissions in the transport sector. The widespread application of ultra-high-strength steels (UHSSs) in HDV construction offers a viable solution, particularly for thick-walled chassis components. This study aimed to support the lightweighting of heavy vehicles by developing a methodology capturing the entire warm-forming process in the range of 430–580 °C for thick-walled UHSSs—from material characterisation, including elastoplastic and fracture properties, to downstream forming process simulations. A novel 7 mm thick UHSS grade, WARMLIGHT-980 (ultimate tensile strength (UTS) of 980 MPa), intended for warm forming was investigated at 430, 505, and 580 °C using samples of reduced thickness. The results showed that thickness reduction had minimal influence on mechanical response at elevated temperatures, enabling flexible specimen design. The thermal uniformity improved in thinner samples, enhancing testing reliability. The calibrated hardening and fracture models demonstrated strong agreement with experimental data. Validated simulations of thick-walled components confirmed the accuracy of the modelling approach. The findings support the development of reliable, temperature-dependent models for warm-forming applications and contribute to the design of lighter, more sustainable HDV components without compromising structural integrity.

1. Introduction

According to the 2011 European Commission Transport White Paper, the European Union (EU) must reduce greenhouse gas (GHG) emissions from the transport sector by 70% by 2050 compared to 2008 levels to meet its environmental goals [1]. Heavy-duty vehicles (HDVs) contribute approximately 26% of total CO₂ emissions in EU road transport, with 85% from trucks and the rest from buses and coaches [2]. Most truck emissions originate from long-haul and regional distribution vehicles, highlighting the urgent need for energy-efficient HDVs. Lightweighting is identified as a key strategy in this transition [3,4], with the latter emphasising it as a near-term opportunity to reduce GHG.

During the past two decades, the European Union has funded more than 50 projects dedicated to advancing lightweight design in vehicle manufacturing. These initiatives have primarily focused on materials such as steels, particularly the Advanced High-Strength Steels (AHSSs) family, as well as aluminium alloys, magnesium alloys, and polymer-based composites. The results have demonstrated significant potential for weight reduction in various vehicle components, with a strong emphasis on body-in-white (BiW) structures for passenger cars. Recently, there has been a surge in projects related to electric vehicles (EVs), driving further innovation in materials and forming processes. Despite broad efforts, AHSSs, and especially press-hardened steels (PHSs), have emerged as the most effective solution to combine lightweight design, mechanical performance, and cost efficiency [5]. For more than a decade, press hardening of boron steels, particularly the 22MnB5 grade, has been widely applied to structural components in the automotive sector.

Press hardening could be extended beyond BiW applications to include chassis components, thereby offering additional potential for decreasing weight. However, this depends on acquiring a comprehensive understanding of the fatigue resistance. The broader adoption of AHSSs and PHSs in the lightweight construction of HDVs offers a promising strategy to reduce fuel consumption and CO₂ emissions across Europe. Integrating ultra-high-strength steels (UHSSs), with ultimate tensile strengths around or exceeding 1000 MPa, can enable even greater weight savings. However, such materials require forming processes capable of handling complex, thick-walled geometries. In sectors such as trucks and heavy machinery, where mechanical loads are substantial, fatigue resistance remains a critical design constraint.

In this study, warm forming of UHSS components for HDVs is investigated. Warm forming refers to metal forming of complex-shaped components in the 200–600 °C range [6], differing from hot forming (e.g., PHS), which involves heating above the austenitisation temperature, followed by simultaneous forming and quenching to achieve martensitic microstructures [7]. Although fatigue behaviour is not addressed in this study, the successful implementation of UHSS and warm forming must consider fatigue resistance in thick sheets. In particular, fatigue performance at welded joints is a known limitation for AHSS applications in HDVs. Although an EU-funded project, FATWELDHSS, has shown improvements in weld fatigue resistance [8], its practical application to truck components remains limited due to poor reproducibility and additional process steps. As a result, the use of welds or joints should be avoided in HDV designs employing AHSS. The lightweight design of HDV chassis components is highly dependent on the stiffness and fatigue performance. Therefore, a transition from conventional hot-rolled, cold-formed high-strength steels (with yield stresses around 500 MPa and ultimate tensile strengths (UTSs) between 550 and 700 MPa) to complex-shaped hot-rolled, warm-formed UHSSs with UTSs around 1000 MPa might be a viable approach while maintaining or improving fatigue properties. Parareda et al. [9] have shown that a warm-formed UHSS, WARMLIGHT-980 (WL980), can offer improved fatigue resistance compared to a hot-formed 22MnB5, highlighting the possibilities for broader implementation of the warm-forming process for HDVs.

Recent research on warm forming has focused mainly on aluminium alloys. Prakash et al. [10] modelled warm deep drawing of AA5083-O up to 300 °C, observing improved formability. Shrivastava et al. [11] highlighted the widespread application of warm forming to improve the formability of aluminium alloy sheets, and studied the effect of process variables on frictional behaviour in warm strip drawing of AA5182 using finite element modelling. Tokita et al. [6] investigated the stretch formability of galvanised and uncoated high-strength steels. Saito et al. [12] explored the V- and U-bend performance of 1.6 mm 980 MPa high-strength steel (HSS) at warm-forming temperatures, reporting reduced springback with increased temperature and holding time. Sun et al. [13] analysed a novel warm-forming process for 1.6 mm MS1180 steel, identifying optimal conditions at 400–450 °C with heating rates > 50 K/s. Pandre et al. [14] characterised 1 mm DP590 steel to predict the forming limit under warm-forming conditions. However, recent studies have not addressed the mechanical characterisation and modelling of warm-forming processes of thick-walled steel plates specifically targeting HDVs.

Previous studies by Bao and Wierzbicki [15] demonstrated the computational modelling of aluminium tensile tests under various triaxialities. Jonsson and Kajberg [16] characterised 1.4–1.55 mm steel sheets at room temperature using digital image correlation (DIC) and implemented a plane-stress fracture criterion. Sjöberg et al. [17] conducted similar studies on 1.6 mm thick Alloy 718 at elevated temperatures using DIC to evaluate the fracture strain. These validated characterisation methods are applied here to support computer-aided engineering (CAE) of a warm-forming process under plane-stress assumptions.

To enable accurate simulation of warm-forming processes using CAE, the elastoplastic and failure behaviour must be characterised at elevated temperatures. This work adapts established characterisation methods developed for thin steel sheets and applies them to 7 mm sheets under warm-forming conditions. Due to practical limitations in testing 7 mm material under plane-stress conditions, a thickness reduction was necessary. However, the impact of the reduced thickness on the mechanical properties must be carefully evaluated. In industrial applications, forming simulations typically use shell elements for tooling and blank geometry. However, because of the weakness of the shell elements in providing an accurate response after the onset of necking as a 3D stress state inevitably forms [18], the blank in this study is modelled with solid elements.

This research aims to support the lightweighting of HDVs by developing a methodology that encompasses the full warm-forming process of thick-walled UHSS, from material characterisation, including elastoplastic and fracture properties, to downstream forming simulations. The elastoplastic and fracture behaviour of WARMLIGHT-980 is therefore characterised at three temperatures: 430 °C, 505 °C, and 580 °C. A temperature-dependent work-hardening model and a fracture model are calibrated for this specific temperature interval. The models are validated by a three-point bending test and the outcome from a simulation of the warm-forming process for an HDV crossmember. The results contribute to a better understanding of the behaviour of the material under warm-forming conditions and support the development of accurate, simulation-based forming strategies for thick-walled UHSS components in HDVs.

2. Materials and Methods

2.1. Ultra-High Strength Steel for Warm Forming

This study uses a 7 mm thick prototype hot-rolled martensitic ultra-high-strength steel, WARMLIGHT-980 (WL980), from voestalpine Stahl GmbH (Linz, Austria). With a specifically designed thermal process, this steel achieves a target UTS of at least 980 MPa at room temperature after warm forming. As received, WL980 is direct quenched to a martensitic microstructure. The martensite strength instability at elevated temperatures was handled by the manufacturer by adding microalloying elements to counteract this behaviour. The measurable effect of several alloying elements on the resistance of martensite to softening at elevated temperatures is described by Grange et al. [19] The chemical composition of the WL980 steel is presented in

Table 1. Chemical composition in mass percentage of the WARMLIGHT-980 steel.

C	Si	Mn	Al	Cr	Ni + Mo	V + Nb	Ti	B
0.09	0.10	1.6	0.05	0.9	0.7	0.1	0.02	0.0025

. Upon focussing only on the material properties, the martensitic steel undergoes an annealing process in tandem with the forming process, and the martensite becomes tempered in comparison with the as-received condition. Microstructure images depicting the WL980 steel in the as-received and tempered condition were produced. The samples in their as-received and tempered states were prepared using the standard metallographic procedure for microstructure analysis. The samples were first mechanically ground using P240-, P600-, P1200-, and P4000-grit SiC paper, and then polished with 1-micrometer diamond paste and 0.05-micrometer colloidal silica (OPS) solution. The samples were subsequently etched with 3% nital. The secondary electron (SE) images of the etched samples were acquired with the help of an Everhart–Thornley detector (ETD) in an FEI Magellan 400 field emission scanning electron microscope (SEM, Thermo Fisher Scientific Inc., Waltham, MA, USA) with extreme high resolution (XHR) at an accelerating voltage of 3 kV and beam current of 50 pA. The SEM images depicting the microstructure of WL980 steel in its as-received state are illustrated in Figure 1a,b and those after annealing are illustrated in Figure 1c,d.

Thermal History

To achieve the final intended properties, the material is annealed to a certain level with the aid of a tempering parameter, namely, the Hollomon–Jaffe parameter [20]. The tempering parameter ($HP$) is presented in Equation (1), where T is the temperature in kelvin, t the hold time in hours, and C is a constant that depends on the chemical composition of the steel, mainly the carbon fraction. For WL980 steel, $C=20$ was used according to the manufacturer’s recommendation.

$$HP=T(\log t+C)$$

(1)

The thermal process was tailored to reach a Holloman–Jaffe parameter value of 16,000. To facilitate the translation of heat treatments across various testing setups with diverse ovens and induction heaters, the calculation method for the Holloman–Jaffe parameter employed an accumulative approach. This approach considered the entire thermal history rather than focussing solely on the duration at the target holding temperature. This modified Hollomon–Jaffe parameter approach was proposed by Tsuchiyama [21], and Revilla et al. [22] used it and showed its importance for relatively short holding times. The calculation scheme consists of dividing the heat treatment cycle into small isothermal steps where the tempering from the previous time step is translated to a fictitious holding time at the new temperature and added to the actual holding time at that isothermal step. The modified tempering parameter is presented in Equation (2), where $T_n$ is the isothermal holding temperature (in kelvin) of the n-th step, $t_n$ the equivalent holding time of the n-th step, $\alpha$ is the heating/cooling rate at $T_{n-1}$, and $\Delta t$ is the size of the time step (in hours). $T_1$ is defined as the starting temperature (in kelvin) of the heat treatment.

$$\begin{array}{ccc}\hfill P_n& =T_n(\log t_n+20),\hfill & \hfill \mathrm{where}\\ \hfill t_n& ={10}^{[(T_{n-1}/T_n)·(\log t_{n-1}+20)-20]}+\Delta t,t_1=\Delta t\hfill & \hfill \mathrm{and}\\ \hfill T_n& =T_{n-1}+\alpha \Delta t\hfill \end{array}$$

(2)

In these experiments, carried out under elevated-temperature conditions, the annealing process was conducted in situ. The designed thermal history used for the material tests is shown in Figure 2. Starting at room temperature, the material was heated by 2.7 K/s up to a holding temperature of 580 °C, then kept there for 240 s, then the test was started or the samples were cooled to the specific testing temperature at 5 K/s.

2.2. Experiments

To mechanically characterise the novel hot-rolled steel in terms of elastoplastic properties and failure behaviour at enhanced temperatures, several different specimen geometries were used. The testing methodology can be divided into three distinct parts:

• Thickness reduction investigation;
• Plasticity/fracture property investigation;
• Young’s modulus investigation.

To achieve the representative temperatures necessary for warm forming, inductive heating was employed. Due the enhanced temperatures, all deformation measurements during the tensile tests were conducted by applying digital image correlation (DIC). To obtain failure strains at various stress triaxialities under the plane-stress condition, different specimen geometries are typically applied such as notched, hole, and shear geometries. However, due to the thickness of 7 mm of prototype steel, relatively large specimens must be produced for these geometries to fulfil plane stress. Large specimens are in turn associated with practical limitations such as grip dimensions, loading capacity, etc., and in this case, especially, the size of the inductive coil and its capacity. A coil that is too large will demand carefully designed heat insulation to protect the testing equipment. The originally 7 mm thick plate was therefore machined using electrical discharge machining (EDM) to a thickness of 1.2 mm, followed by machining to the final geometries shown in Figure 3a–d. It is assumed that any thickness in the range 1–1.5 mm is able to achieve a plane-stress condition. A pre-study was conducted to check whether the thickness reduction approach produces representative mechanical properties or not. The work-hardening behaviour up to the onset of necking in uniaxial tensile tests, considering both the original and reduced thicknesses, was examined. The tests were carried out at temperatures of 430 °C, 505 °C, and 580 °C, with three repetitions at each. Independent of the testing temperature, all specimens were heated up to 580 °C to reproduce the conditions of the warm-forming process, and thereby achieve the tempered microstructure depicted in Figure 1a–d. Two dogbone-type specimens were used with cross-sections of 7 × 7 mm² and 12.5 × 1.2 mm², respectively. The parallel lengths were 60 mm and 70 mm (gauge length 50 mm in both cases), respectively. The dimensions were chosen for some practical reasons. The parallel length should remain within the induction coil as much as possible during the test. This is most easily met by a short parallel length when testing very ductile material. The cross-sectional area of the thickness-reduced specimens was chosen to meet the ISO 6892-1:2019-standard [23]. However, the width of thicker specimens was limited to 7 mm to keep the heating power low, and to match the needed parallel length according to the above-mentioned standard. To determine the Young’s modulus by loading and unloading sequences (described in the following paragraph) at warm-forming temperatures, a cross-section of 12.5 × 7 mm² and a parallel length of 100 mm were chosen. The wider cross-section and longer parallel length for these thicker specimens were chosen to enable strain field measurement using DIC over a larger area, resulting in a larger data set compared to the former mentioned thick specimens.

All tests were performed on a servo-hydraulic testing machine, Dartec M1000/RK (Dartec Ltd., Bournemouth, UK), complemented by an induction heater, EFD Sinac 5 SH (EFD Induction, Skien, Norway) and a compressed air system for forced cooling. The induction heater and compressed air system were controlled by a Eurotherm 3504 PID controller (Eurotherm, Worthing, UK). To assess the deformation fields and subsequently determine the failure strains, a digital image correlation system (DIC), GOM ARAMIS v6.3 (Carl Zeiss GOM Metrology GmbH., Braunschweig, Germany), was utilised. The camera sensor used was the GOM Aramis 5M with the lenses Tokina AT-X Pro D Macro 100 mm (Tokina Co., Ltd., Tokyo, Japan) for the shear specimen, and Sigma 50 mm f/2.8 EX DG Macro (Sigma Co., Kawasaki, Japan) for all other specimens. The DIC system was used to create a virtual extensometer of measurement length 50 mm for the dogbone-type specimen and 14 mm for the fracture-characterisation specimen. Custom induction coils were developed for all specimen geometries to achieve a homogeneous heat distribution over the gauge areas of the specimens. To study the degree of uniformity of the temperature, a thermal camera, FLIR SC4000 (FLIR Systems, Wilsonville, OR, USA), was used. The software used to analyse the thermography images was FLIR ThermaCam Resarcher PRO 2.8, where a factory calibration profile was used for the specific camera. In order to guarantee a uniform emissivity factor across the evaluated surface, primarily because of the non-uniform accumulation of oxidation, the specimens were coated with a boron nitride spray coating possessing a known and consistent emissivity factor. The emissivity factor was then further tuned so that the temperature reading from the thermal camera (at the temperature of interest) matched the reading from a thermocouple situated on the opposite side of the specimen. A representative sample for each geometry was filmed with the thermal camera to assess the heat distribution. Subsequent samples in the test series were monitored with a single thermocouple in the middle of the gauge area. The experimental setup is shown in Figure 4.

To examine the reduction in material thickness, and assess the plastic properties and failure strain, a crosshead speed of 0.1 mm/s was used. For the uniaxial tests this speed resulted in a strain rate of 0.002 s⁻¹, which is within the ISO 6892-1:2019-standard [23]. The DIC system served as a virtual extensometer of 50 mm measurement length. For the Young’s modulus investigation, dogbone-type tensile specimens were used. The universal testing machine was switched to load control and programmed to perform 10 loading and unloading cycles at 0.1 Hz in the elastic region. DIC was used to capture displacements; the surface was then evaluated for average strain in the axial direction with the help of the GOM Aramis software V6.3. The stress vs. strain data were sorted by loading and unloading phases, and a simple linear regression (SLR) analysis was performed with the intention of finding the corresponding slopes of the elastic region. The SLR model is explained by Zou et al. [24], and the SLR was performed in Matlab^® 2019 using the polyfit function. The average Young’s modulus was calculated by taking the average of the loading and unloading phases for the corresponding temperature. The failure strains for the different triaxialities tested were identified by visual inspection of the strain fields from the DIC system of the three repetitions of each geometry and temperature, with the exception of the shear specimens, where failure was classified as soon as the first visual sign of a surface crack showed itself. This occurred long before any indication of fracture appeared in the load versus displacement data. In the case of shear samples, the deformation was too large for the DIC system to be able to track it to final failure. Hence, the failure strain was determined as the last value that could be read in the analysis. The virtual measurement length, $l_0$, was determined to be 0.2 mm using the equation provided in the work by Ilg et al. [25]:

$$l_0\approx 2·\left(\mathrm{facet}\mathrm{point}\mathrm{distance}\right).$$

(3)

As failure strains were investigated at 1.2 mm thickness and needed to be applied at 7 mm thickness, the specimens were treated as miniatures with a scale factor of 1:5.833 to the original thickness of 7 mm. The fundamental miniaturisation technique used was formulated by Gorji et al. [26], where they employed miniature specimens scaled down by a factor of 1:20 to evaluate the plasticity and fracture characteristics. Then, these results were compared with analyses conducted on a full-scale specimen.

The stress–strain curves obtained in the pre-study are depicted in Figure 5a–c, demonstrating similar behaviour across all three temperatures. The reduction in thickness resulted in a percentage change in the maximum true stress ranging from −1.45% at 430 °C to −0.16% at 580 °C, as detailed in

Table 2. A presentation of average yield stress of the 7 mm samples and a comparison of the true stress at the onset of necking for the 7 mm and the 1.2 mm specimens, based on three repetitions. Values are given with the standard deviation.

Temperature (°C)	7 mm—Average Yield Stress (MPa)	7 mm—Average Peak Stress (MPa)	1.2 mm—Average Peak Stress (MPa)	Percentage Change (%)
430	824 ± 3	895 ± 1	882 ± 4	−1.45
505	755 ± 1	809 ± 3	801 ± 5	−0.97
580	607 ± 5	652 ± 6	651 ± 7	−0.16

. This suggests that, at least until necking occurs, the material properties remain largely unaffected by the thickness reduction. It can be observed that necking occurs already at an effective plastic strain of approximately 0.01.

2.3. Calibration of Temperature Dependent Work-Hardening Model

El-Magd et al. [27] suggested a work-hardening model based on a model presented by Mecking and Kocks [28]. Although the El-Magd model was originally deduced for fcc metals, it was successfully demonstrated to describe flow curves of bcc metals as well. To include temperature influence, it is here complemented with a temperature-dependent factor, analogously with the well-known rate- and temperature-dependent Johnson–Cook model [29]. It should be mentioned that strain-rate dependence is not included in this study. The temperature-dependent work-hardening model is thus given by

$$\sigma \left({\overline{\epsilon }}^p,T\right)=\left[C_1+C_2{\overline{\epsilon }}^p+C_3\left(1-e^{-C_4{\overline{\epsilon }}^p}\right)\right]\left(1-D{T}^{\ast m}\right)$$

(4)

where the original El-Magd material constants $C_1$, $C_2$, $C_3$, and $C_4$ require calibration. The linear term $C_2{\overline{\epsilon }}^p$ was introduced to obtain a good representation in the range of large strains [27]. Furthermore, calibration of the added temperature-sensitive parameters D and m is necessary. $T^{\ast }$ is the homologous temperature defined by

$$T^{\ast }={\displaystyle \frac{T-T_0}{T_m-T_0}}$$

(5)

where T is the current material temperature, $T_0$ is the reference temperature, and $T_m$ is the melting temperature of the material. In this study, $T_0=430$ °C and $T_m=1500$ °C were used.

By setting ${\overline{\epsilon }}^p=0$ and $T=T_0=430$ °C in Equation (4), it is obvious that $C_1$ equals the yield stress $R_{p0.2}$ at 430 °C. The two temperature-sensitive parameters D and m can thereafter be directly determined by considering the two yield stresses at 505 °C and 580 °C. With ${\overline{\epsilon }}^p=0$, Equation (4) can be reformulated to give m as

$$m={\displaystyle \frac{ln\left(1-\frac{R_{p0.2}^{580}}{R_{p0.2}^{430}}\right)-ln\left(1-\frac{R_{p0.2}^{505}}{R_{p0.2}^{430}}\right)}{ln{T}^{\ast ,580}-ln{T}^{\ast ,505}}}$$

(6)

The parameter D is then derived by using any of the yield stresses at 505 °C or 580 °C. At 580 °C, Equation (4) gives

$$D={\displaystyle \frac{1-\frac{R_{p0.2}^{580}}{R_{p0.2}^{430}}}{{\left(T^{\ast ,580}\right)}^m}}$$

(7)

Since necking occurred already at an effective plastic strain of approximately 0.01, as reported in the pre-study (see Figure 5a–c), it was decided that $C_3$ and $C_4$ should be calibrated to represent the response in the small-strain region up to necking. The two parameters were calibrated in the least-squares sense by minimising the residual between the experimental and model stress responses. The employed experimental true stress–true strain data originated from the 7 × 7 mm² thick dogbone samples in the pre-study.

The parameter $C_2$, corresponding to the linear term in Equation (4), representing the response from necking to large strains, was calibrated using an inverse modelling scheme. For the inverse modelling, a simulation model of the R30 tensile specimen with a mesh size of 0.1 mm was used. Subsequently, $C_2$ was varied in such a way that the force elongation behaviour up to just before fracture for the model closely aligned with the experimental data at all three test temperatures. The optimisation process was conducted in LS-Opt using the partial-curve-mapping algorithm [30]. The inverse modelling flowchart from LS-Opt is shown in Figure 6.

2.4. Calibration of Fracture Criterion

Based on the evaluated fracture strains and the derived stress triaxialities at the fracture initiation points, the modified Mohr–Coulomb (MMC) criterion [31] for plane stress was calibrated. However, the criterion was modified to account for the temperature dependence of the fracture strain in the forming temperature range. In order to model the temperature dependency of the equivalent strain at fracture, a factor was introduced. The factors were applied in the same sense as the temperature-dependent work-hardening model. The MMC model used in conjunction with the von Mises yielding function is defined as

$${\overline{\epsilon }}_f^p\left(\eta ,\overline{\theta },T\right)={\left\{c_2\left[\sqrt{{\displaystyle \frac{1+c_1^2}{3}}}cos\left({\displaystyle \frac{\overline{\theta }\pi }{6}}\right)+c_1\left(\eta +{\displaystyle \frac{1}{3}}sin\left({\displaystyle \frac{\overline{\theta }\pi }{6}}\right)\right)\right]\right\}}^{-{\displaystyle \frac{1}{n}}}\left(1+d{T}^{\ast }\right)$$

(8)

and for plane stress, the Lode parameter is given by

$$\overline{\theta }=1-{\displaystyle \frac{2}{\pi }}arccos\left(-{\displaystyle \frac{27}{2}}\eta \left({\eta }^2-{\displaystyle \frac{1}{3}}\right)\right).$$

(9)

The MMC contains the three original parameters, $c_1$, $c_2$, and n, plus the additional parameter d, describing the temperature dependence. The four parameters were calibrated in the least-squares sense by minimising the residual, i.e., the root mean squared error. Failure strains were taken from DIC measurements, and the test was modelled in LS-Dyna to extract the evolution of the triaxiality throughout the test. Since triaxiality is not constant throughout the tests, a weighted average triaxiality was used in the calibration. The weighted average triaxiality is given by

$${\eta }_{int}={\displaystyle \frac{1}{{\overline{\epsilon }}_f^p}}{\int }_0^{{\overline{\epsilon }}_f^p}\eta \left({\overline{\epsilon }}^p\right)d{\overline{\epsilon }}^p$$

(10)

where ${\overline{\epsilon }}_f^p$ is the measured effective plastic strain at fracture. The MMC curves were later extrapolated to 3D surfaces for all three temperatures to accommodate the simulation models.

The damage model used in the subsequent warm-forming simulations was the generalised incremental stress-state-dependent damage model (GISSMO) in LS-Dyna [32]. The thermal option of the model was activated by incorporating the calibrated fracture surfaces into a 4D table. As all DIC fracture strains were very conservative, the damage parameters in GISSMO were set at $ecrit=0.01$, $dmgexp=1.0$, and $fadexp=100$. This was so that the damage model would not contribute to the softening of the elements. The model was then regularised to account for the dispersion in the mesh size of the blank. The miniaturisation scale factor was then applied to the corresponding mesh sizes in the regularisation curve to account for the 7 mm thickness of the blank.

2.5. Validation by Three-Point Bending

To validate the calibrated plasticity model, a thermomechanical simulation of a three-point bending experiment was carried out. The results, in terms of roller displacement versus force, were compared with experimental data from one repetition at three different temperatures: 430 °C, 505 °C, and 550 °C. The geometry of the experiments was a blank with dimensions of 150 × 50 mm² and a thickness of 7 mm. The rollers had a radius of 10 mm and a length of 70 mm. The roller span was set at 70 mm. Hence, the simulation model had the same dimensions. The middle roller was set at a displacement of 10 mm, as any additional displacement would result in friction becoming a significant contributing factor to the observed force response. The simulation was carried out on Ansys LS-Dyna R13.0 where the blank and the rollers were discretised with a solid 8-point hexahedron element type 1, which is an element formulation that employs an assumed strain approach to avoid shear locking that can be seen in ordinary fully integrated solid elements [33]. The thermal material model used for the specimen and the rollers was type 1, which is used for its isotropic thermal properties. The blank used a 2.5 mm mesh with 5 elements through the thickness. The rollers (half-cylinders) featured a mesh comprising 36 elements distributed over an arc of 180 degrees, which led to an element length of roughly 0.87 mm. Furthermore, the rollers contained 35 elements along their length. The contacts used were the “forming one way surface to surface” contact with the thermal option activated. The frictional coefficients were calibrated by a trail-and-error approach by matching the load response from the experiment with the simulation results at 550 °C. These calibrated static and dynamic friction values were subsequently applied across the entire temperature range in the simulations. The friction coefficients used in the LS-Dyna contact were 0.3 for static and 0.2 for dynamic friction. The simulation model and the corresponding meshes are shown in Figure 7. The rollers used a rigid material model. The developed temperature-dependent work-hardening model was implemented using material type 106, an elastic–viscoplastic material card with thermal effects [32].

2.6. Validation by Forming of an HDV Chassis Crossmember

The thermomechanical forming process of an HDV crossmember was simulated and evaluated against an actual prototype crossmember manufactured using WL980 steel. In the absence of force data from the prototype’s forming process, the comparison between the modelled and prototype crossmember focused on examining the thinning at specific critical locations on the crossmember. The thermomechanical simulation model used the same basic approach as the three-point bending model. The blank with a thickness of 7 mm was discretised with a 2.5 mm mesh, mainly consisting of solid 8-point hexahedron elements of type 1. The tooling surfaces were discretised as rigid bodies with shell element formulation 2, Belytschko–Tsay. The type-1 thermal material model, designed for isotropic thermal characteristics, was applied to both the blank and tool components. The contacts used were the same as for the three-point bending model. A computer-aided design depiction of the HDV crossmember to be formed and the discretised simulation model are shown in Figure 8a,b. In order to assess the thinning of the modelled crossmember, the process involved exporting the formed mesh surfaces as polygon geometries. Subsequently, these geometries were imported into the CAE software (Siemens NX 12.0), where they were converted to a solid body, allowing a thickness analysis to be conducted. For comparative analysis, a 3D-scanning technique was utilised to study the reduction in thickness of the prototype.

3. Results

3.1. Heat Distribution

When testing at enhanced temperatures, it is of importance to achieve a temperature distribution close to uniform in the region of interest. The heat distribution maps obtained in the pre-study concerning the influence of reduced sample thickness are shown in Figure 9a–c for 1.2 mm thickness, and Figure 9d–f for 7 mm thickness, with some statistics presented in

Table 3. Results from the heat distribution analysis of 1.2 mm and 7 mm tensile tests.

Thickness (mm)	Target Temperature (°C)	Average Temperature (°C)	Temperature Span (°C)	Standard Deviation
1.2	430	431	19	2.8
1.2	505	511	23	4.4
1.2	580	587	24	3.9
7	430	422	30	7.3
7	505	498	31	7.1
7	580	574	35	7.4

. The heat distribution analysis of the dogbone specimens shows a temperature standard deviation from 2.8 °C for the thinner 1.2 mm specimen at 430 °C to 7.4 °C for the 7 mm thick sample at 580 °C. The analysed region of interest is representative of the measurement length used by the virtual extensometer of the DIC system.

A similar analysis of the temperature distribution was conducted for the non-uniform specimens utilised to obtain failure strains at various stress triaxialities. Representative heat maps are shown for 580 °C in Figure 10a–d, where the region of interest has been selected to include the localisation region. On the hole samples, the region of interest is chosen on the side of the hole where the controlling thermocouple is located. The results of the heat analysis for 580 °C are presented in

Table 4. Heat distribution analysis results from the fracture characterisation tests.

Specimen Type	Target Temperature (°C)	Average Temperature (°C)	Temperature Span (°C)	Standard Deviation
Shear (a)	580	579	11	1.7
Hole (b)	580	578	11	1.8
R30 (c)	580	579	12	2.4
R7.5 (d)	580	577	12	3.0

. The standard deviation of the tests ranged from 1.7 °C for the shear specimen to 3.0 °C for the R7.5 specimen. The tests carried out at 580 °C presented the greatest challenge in achieving uniform heating and exhibited the most significant heat gradient, thus this temperature was chosen for analysis.

3.2. Young’s Modulus

The stress–strain data points and the fitted linear regression line for the Young’s modulus investigation at 430 °C, 505 °C, and 580 °C are shown in Figure 11a–f. The resulting Young’s modulus and goodness of fit are presented in

Table 5. Results from the Young’s modulus investigation.

Temperature (°C)	Loading		Unloading		Average Young’s Modulus (GPa)
	Young’s Modulus (GPa)	Goodness of Fit, ${\mathit{R}}^2$	Young’s Modulus (GPa)	Goodness of Fit, ${\mathit{R}}^2$
430	185	0.984	186	0.989	185
505	177	0.988	178	0.985	177
580	156	0.978	161	0.975	159

for the loading and unloading phases at all three temperatures. An average Young’s modulus is also given for each temperature. The average Young’s modulus ranged from 185 GPa at 430 °C down to 159 GPa at 580 °C. The quality of the fit, along with visually inspecting the residual dispersion of the regression line shown in Figure 11a–f, provides a solid foundation for the testing approach. Additionally, by presuming a Young’s modulus of 200 GPa at room temperature, the findings are consistent within 2% of those detailed by Chen et al. [34] when applying the proposed reduction factors for the Young’s modulus of high-strength steel, linearly interpolated to correspond with the specific temperatures evaluated in this research. The presumed Young’s modulus at room temperature was approximated using the reduction factors through a trial-and-error strategy. This involved iteratively adjusting the values to minimise discrepancies between the estimated and test-based results. The Young’s modulus results were then used as material input for subsequent thermomechanical simulations.

3.3. Calibration of Temperature-Dependent Work-Hardening Model

In

Table 6. Calibrated parameters of the modified El-Magd work-hardening model.

C1 [MPa]	C2 [MPa]	C3 [MPa]	C4	D	m
824	205	59	363	6.82	1.65

, the calibrated parameters in the El-Magd work-hardening model given by Equation (4) are presented. As mentioned, $C_1$ was given by the yield stress at 430 °C. The temperature-sensitive parameters m and D were analytically calculated according to Equations (6) and (7). Parameters $C_3$ and $C_4$ were calibrated in the least-squares sense by considering the flow stress response in the small-strain region up to necking detected at an effective plastic strain of approximately 0.01. The final parameter $C_2$ was calibrated by considering the load vs. extension responses of the notched R30 specimen at the three investigated temperatures. The calibration was performed by inverse modelling using LS-Opt [30]. The measured and computed (with calibrated parameters) load vs. extension responses are depicted in Figure 12a; they agree well at all three tested temperatures. The flow stress curves at the three investigated temperatures obtained by the calibrated El-Magd model are shown in Figure 12b as dashed lines. Excellent fits up to necking at an effective plastic strain 0.01 were found. After necking, when a uniform and uniaxial stress state is no longer present, the experimentally obtained flow-stress curves obviously do not reflect the work-hardening and consequently deviate massively from the model responses. The strain rate in the calibration experiments was characterised by considering the local strain rate in the most deformed region of the notched specimens. For the R30 specimen a strain rate of approximately 0.3 s⁻¹ was achieved.

3.4. Calibration of Fracture Criterion

The calibrated parameters for the MMC fracture criterion are shown in

Table 7. Calibrated parameters of the temperature-dependent modified Mohr–Coulomb (MMC) fracture criterion.

${\mathit{c}}_1$	${\mathit{c}}_2$	n	d
0.352	1.58	0.358	1.30

, where $c_1$, $c_2$ and n were from the original expression of MMC and d is the added parameter for the temperature dependence. The MMC curves for plane stress are presented in Figure 13a and the 3D MMC surface in Figure 13b.

3.5. Three-Point Bending Validation

The calibrated models were validated by three-point bending tests. Figure 14 shows the measured and computed force vs. displacement histories at 430 °C, 505 °C, and 550 °C. The highest selected temperature was 550 °C due to cooling during the handling time when moving the specimen from the oven to the testing apparatus. The computed force responses follow the experimental data closely but start to diverge around 10 mm displacement when friction starts becoming a major contributing factor to the force response.

3.6. Thinning Comparison: Model vs. Prototype Crossmember

The thermomechanical simulation of the warm-forming process for the HDV chassis crossmember suggests that forming geometrically complex structural components is feasible, as the damage model showed no signs of failure. Furthermore, the prototype crossmembers did not show evidence of fractures. To assess the accuracy of the simulation model relative to the final outcomes of the prototypes, a detailed thinning analysis was performed. This involved comparing the computed thickness reduction with 3D scans of one prototype crossmember. As depicted in Figure 15, ten specific points were marked on the thickness distribution plot of the modelled crossmember. The thickness at these points was compared to corresponding locations on the prototype crossmember. Only points on the right side were selected, as they were also representative of the corresponding points on the left side of the crossmember. The findings of this comparison are presented in

Table 8. Thickness comparison: model and 3D-scanned prototype. The initial thickness of the blank is 7 mm.

Point	1	2	3	4	5	6	7	8	9	10
3D Scanning [mm]	5.88	6.56	6.55	6.34	7.40	7.33	6.53	6.44	6.51	5.94
Model [mm]	6.41	6.52	6.50	6.36	7.37	7.32	6.50	6.41	6.50	6.37
Percentage difference [%]	8.63	0.61	0.77	0.32	0.41	0.14	0.46	0.47	0.15	6.99

.

The most significant disparities were observed at point one, which exhibited a percentage difference of 8.63%, and at point ten, with a difference of 6.99%. In contrast, the percentage differences at all other points were substantially below 1%. The larger difference of points one and ten, might be remedied either by or a combination of tuning the frictional constant or using a smaller mesh size, as the thinning in that area appeared as a quite narrow band in the 3D scanning. Furthermore, the GISSMO damage model did not indicate fracture in any element of the model. The maximum strain rate of approximately 1 s⁻¹ was detected at the bending radius around point 10 in Figure 15. This rate is higher than the rate of 0.3 s⁻¹ obtained for the R30 specimens in the calibration experiments. It is concluded that this difference do not have a significant effect on the thinning in the simulations.

4. Discussion and Conclusions

The influence of the thickness of the specimen on the mechanical response at elevated temperatures showed a consistent behaviour at all temperatures tested. As demonstrated in the thickness comparison, reducing the specimen thickness from 7 mm to 1.2 mm resulted in small differences in true stress at the onset of necking, with a maximum deviation of only 1.45% at 430 °C. This indicates that the mechanical properties, particularly as long as uniform stress state is present, remain largely unaffected by reductions in thickness. The practical implication is that thinner geometries can be used in experimental setups without introducing an appreciable bias in mechanical response, thus allowing for greater flexibility in specimen design and testing configurations.

The thermal imaging analysis further supports the observed mechanical consistency. The thinner specimens exhibited better temperature homogeneity, as indicated by lower standard deviations in thermal distribution. This is of course an important factor to ensure accurate mechanical testing at elevated temperatures. At 580 °C, the 7 mm thick samples presented a standard deviation of 7.4 °C compared to only 3.9 °C for the 1.2 mm specimens, highlighting the benefit of reduced thermal gradients in thinner geometries. The uniformity not only enhances the test reliability, but also reduces uncertainty in parameter calibration for the modelling.

The heat distribution study conducted at 580 °C on specimens used for fracture characterisation served not only as a temperature homogeneity check, but also as a validation step for the induction heating methodology. The results demonstrated that a high degree of thermal uniformity could be achieved even in specimens of varying geometries, with standard deviations ranging from 1.7 °C to 3.0 °C. These values are sufficiently low to indicate that the induction heating approach produces consistent and well-controlled thermal conditions, supporting its use in subsequent mechanical testing and fracture model calibration. Ensuring spatially stable and repeatable temperature fields helps isolate the mechanical response from thermal artefacts, thus strengthening the reliability of the fracture calibration procedure and enhancing confidence in the fracture model. With this in mind, the thermal analysis is a key factor in verifying the robustness of the experimental methodology as a whole.

Young’s modulus was, of course, found to decrease with increasing temperature, these results also matched closely with the proposed reduction factors reported by Chen et al. [34], suggesting that the testing and analysis methodology employed here is robust and reliable. This temperature-dependent reduction in stiffness was an important input for the simulation models.

The calibration of the temperature-dependent El-Magd work-hardening model showed good agreement with the experimental tensile data. The work-hardening response aligned closely with the experimental flow curves up to the UTS, with a slight increase in hardening thereafter. Together with the validation from the three-point bending test, this consistent correlation demonstrates the appropriateness of the selected model for thermomechanical simulations in warm-forming processes.

Similarly, the modified Mohr–Coulomb fracture criterion, which incorporates temperature dependence, was successfully calibrated and implemented in the LS-Dyna GISSMO failure model. The addition of a temperature-sensitive parameter (d in Equation (8)) allowed the model to reflect the evolving failure behaviour of the material under thermal conditions. This establishes a more precise basis for forecasting the onset of fractures in warm-forming processes, where the relationship between stress triaxiality and temperature is dynamic.

Validation through three-point bending simulations further reinforced the applicability of the model. The predicted force–displacement responses closely mirrored the experimental measurements, with divergence observed only at larger displacements, most likely due to frictional effects. This minor discrepancy does not detract from the general validity of the model but suggests that future iterations could benefit from an enhanced friction calibration for more accurate post-yield predictions.

Despite the robustness of the experimental and modelling approach, some limitations must be acknowledged. Firstly, the inherent noise floor of the camera system introduces a baseline uncertainty in strain measurements, particularly for low strain levels. Secondly, at elevated temperatures, refractive index fluctuations in the warm air surrounding the tensile specimen can distort the optical path, leading to apparent local displacements in the DIC data. These air-induced refraction effects can manifest as low-frequency spatial noise. While these effects were minimised through careful system setup and environmental control, they still contribute to measurement scatter, especially in the pre-necking regime. Future improvements could involve implementing optical shielding, maybe in conjunction with testing in a vacuum environment, or dual-camera systems with correction algorithms to further mitigate these thermally induced artefacts. Secondly, the study did not explore the minute springback behaviour observed from the production of the prototypes; this would be of interest for further studies. Thirdly, this study did not investigate the response of the material or optimise the process concerning the strain rate dependency. Investigating this aspect in future studies could contribute to the optimisation of the warm-forming process.

In conclusion, the combined experimental and modelling efforts in this study have provided a robust characterisation of high-strength steel under warm-forming conditions. The minimal influence of thickness reduction on mechanical properties, the high fidelity of thermal uniformity in thinner specimens, and the successful calibration of both work-hardening and fracture criteria support the development of accurate, temperature-dependent simulation models. In addition, a comprehensive validation of the methodology was conducted by effectively modelling the chassis crossmember. The results indicated minimal differences in thinning compared to the prototype crossmember. These insights form a strong foundation for accurate process simulations and optimisation in industrial warm-forming applications, enabling the design of lighter and more environmentally efficient components for HDVs, without compromising structural integrity. This will help to reduce the environmental impact of the HDV transport sector.