1 Introduction
Deep drawing is a well-established and frequently used sheet metal forming process for manufacturing structural components. Prior to the large-scale production of a new component, the drawing process usually is optimized regarding sheet thinning [
1] and reduced distortion after spring back [
2,
3]. Significant residual stresses, which are frequently neglected during the assessment of sheet forming processes, develop during the forming process [
4], that may affect the components behaviour in service [
5]. Considering or utilizing these residual stresses, for example via targeted induced compressive residual stresses at predominantly tensile stressed areas, may reduce the superimposed acting stresses and can counteract crack initiation. By this means, the components lifetime can be significantly increased. Hence, regarding the components production process, this enables the use of reduced metal sheet thickness to fulfill the ever increasing light-weight requirements and reduces material costs at the same time.
Numerical methods, such as finite element method (FEM), nowadays allow for fast and precise simulation of the forming process and thereby the prediction of the resulting residual stress distribution [
6]. Simulative approaches provide powerful tools for efficient and targeted optimization of production processes without elaborate experimental tests, provided that the material behaviour is modeled precisely. In sheet metal forming, multiphase materials, such as dual phase and duplex stainless steels are frequently used since they perfectly meet the requirements for deep drawn components, that is, these materials combine high strength and good ductility and in case of the duplex stainless steels they exhibit a very good corrosion resistance. However, it must be taken into account that multiphase materials develop phase-specific micro residual stresses in addition to the macro residual stresses, which also can have an impact on the components fatigue behaviour [
7]. This means that appropriate knowledge of the phase-specific residual stress, i.e., in particular the phase-specific micro residual stress, is essential for the meaningful evaluation of sheet metal forming processes. However, this requires a greater effort on the part of the experimental residual stress analysis and of the numerical process simulation.
In general, phase-specific residual stress can only be experimentally determined by means of diffraction methods, e.g. by using X‑ray or neutron radiation. The numerical simulation of micro residual stress for entire components, require elaborate approaches. For an overview of multiscale computational techniques it is referred to Kanoutè et al. [
8]. The computational cost of a full-field simulation of an underlying microstructure is often too high for complex structures with a non-linear material behavior.
This work focuses on the residual stress evolution in sheet metal forming of duplex stainless steel X2CrNiN23‑4, consisting of ferrite and austenite phases in equal volume fraction. Here, the mean field homogenization is used as a numerically efficient method for the simulation of residual stress on the phase level during the simulation of structural parts. As a result of the sheet rolling process, both phases possess significant crystallographic textures, which must be taken into account and bring additional challenges for precise experimental analysis and numerical simulation of residual stress. To meet these challenges, elaborate approaches from the fields of technical mechanics, metal sheet processing and materials testing are combined within an interdisciplinary cooperation. In [
9], a numerically efficient two-step simulation approach for the prediction of phase-specific residual stresses is developed. In a first step finite-element simulations are performed to predict the macroscopic stress and strain states. Based on the finite-element results a mean-field homogenization scheme is used in a second step to estimate the phase-specific residual stresses. In [
10] an incremental two-scale simulation approach is presented, which allows for the consideration of the history of phase-specific micro residual stress evolution for complex forming processes. With the phase-specific and anisotropic strain hardening of the duplex stainless steel X2CrNiN23‑4 investigated in [
11], the mean field approach can now be applied for sheet metal forming simulation considering the anisotropic and phase-specific material behaviour. In this work, the influence of the drawing depth of circular deep drawn cups made of duplex stainless steel X2CrNiN23‑4 on the residual stress distribution is investigated. The mean field approach is applied for prediction of the global macro and phase-specific micro residual stress distribution for three different drawing depths. For validation of the numerical results, a deep drawn cup made of duplex stainless steel X2CrNiN23‑4 was manufactured with dimensions that correspond with the simulation. The numerical results are validated by experimentally determined residual stress distributions that were analyzed by means of X‑ray diffraction for multiple measuring points along the cup’s wall. With this work, it is aimed to contribute to the understanding of residual stress generation during deep drawing processes, especially for duplex stainless steels.
3 Numerical simulation
The numerical simulation of the phase-specific residual stresses is performed in a two step approach introduced in Simon et al. [
9]. In the first step, the macroscopic residual stress distribution is calculated by a finite-element modeling using Abaqus Standard by Dassault Systèmes. The resulting effective stress and strain states before and after unloading are used as an input for the micromechanical mean-field modeling. Hereby, steady phase-specific plastic strains are assumed during the unloading step.
The macroscopic elasto-plastic material response is characterized experimentally by tensile tests in rolling direction (
\(0{\mathrm{{}^{\circ}}}\)), in
\(45{\mathrm{{}^{\circ}}}\) to the rolling direction and in transverse direction (
\(90{\mathrm{{}^{\circ}}}\)) (see Fig.
3). For simplicity, the macroscopic elastic behavior is modeled isotropic with Young’s modulus
\(E=195\,{\mathrm{GPa}}\) and Poisson ratio
\(\nu=0.3\). The Ludwik-Hollomon approach is chosen for the evolution of the flow stress
\(\sigma_{\mathrm{F}}(\varepsilon_{\mathrm{p}})=a\varepsilon_{\mathrm{p}}^{n}\) with the equivalent plastic strain
\(\varepsilon_{\mathrm{p}}(t)=\int_{0}^{t}||\dot{\boldsymbol{\varepsilon}}_{\mathrm{p}}(\tilde{t})||\mathrm{d}\tilde{t}\) and material parameters
\(a\) and
\(n\). Based on the results of the tensile tests the parameters of the Ludwik-Hollomon approach are chosen as
\(a=840\,{\mathrm{MPa}}\) and
\(n=0.076\). The approximated strain hardening is visualized in Fig.
3.
The anisotropic behavior of the material is taken into account according to the Hill yield criterion based on the determined
\(r\)-values listed in Table
2.
The simulation setup of the deep drawing process up to drawing depths of \(20\,{\mathrm{mm}}\), \(30\,{\mathrm{mm}}\) and \(40\,{\mathrm{mm}}\) is as follows. One quarter of the sheet metal blank is modeled with 34310 eight-node brick elements (C3D8), where an element number of 10 over the blank thickness of \(s_{0}=1.5\,{\mathrm{mm}}\) is used. The coefficient of friction between the blank and a tool surfaces is chosen as \(\mu=0.1\). In correspondence with the experiment a blank holder force of \(F_{\mathrm{B}}=180\,{\mathrm{kN}}\) is applied.
The macroscopic simulation gives the effective stress
\(\bar{\boldsymbol{\sigma}}\), the effective strain
\(\bar{\boldsymbol{\varepsilon}}\) and the effective plastic strain
\(\bar{\boldsymbol{\varepsilon}}_{\mathrm{p}}\) for the loaded state as well as after the unloading step. The corresponding quantities on the microscale of the material are given point-wise by localization relations with the 4th-order strain localization tensor
\(\mathbb{A}(\boldsymbol{x})\), the 4th-order stress localization tensor
\(\mathbb{B}(\boldsymbol{x})\) and the corresponding strain and stress fluctuation fields
\(\boldsymbol{a}(\boldsymbol{x})\) and
\(\boldsymbol{b}(\boldsymbol{x})\). Averaging over a phase
\(\xi\) it follows for the strain and stress within the phase
\(\boldsymbol{\varepsilon}_{\xi}=\mathbb{A}_{\xi}[\bar{\boldsymbol{\varepsilon}}]-\boldsymbol{a}_{\xi}\) and
\(\boldsymbol{\sigma}_{\xi}=\mathbb{B}_{\xi}[\bar{\boldsymbol{\sigma}}]-\boldsymbol{b}_{\xi}\). As shown in [
9,
10], for a two phase material, i.e.,
\(\xi=1,2\), explicit expressions are given for the localization tensors and fluctuation fields based on the phase volume fractions
\(c_{\xi}\), the effective stiffness tensor
\(\bar{\mathbb{C}}\), the phase-specific stiffness tensors
\(\mathbb{C}_{\xi}\) and the plastic strains within the phases
\(\boldsymbol{\varepsilon}_{\mathrm{p}\xi}\). Backstress effects are not considered in the present work due to the lack of data for non-proportional loading histories.
For simplicity, the microstructure of the material is assumed to be statistically isotropic, which allows the characterization of the phase-specific stiffness tensors by the Young’s modulus
\(E_{\xi}\) and the Poisson ratio
\(\nu_{\xi}\). Furthermore, it is assumed that both for the effective material behavior as well as for the phase averages the incremental form Hooke’s law is valid, i.e.
\(\Delta\bar{\boldsymbol{\sigma}}=\bar{\mathbb{C}}[\Delta\bar{\boldsymbol{\varepsilon}}-\Delta\bar{\boldsymbol{\varepsilon}}_{\mathrm{p}}]\) and
\(\Delta{\boldsymbol{\sigma}}_{\xi}={\mathbb{C}}_{\xi}[\Delta{\boldsymbol{\varepsilon}}_{\xi}-\Delta{\boldsymbol{\varepsilon}}_{\mathrm{p}\xi}]\). This allows the usage of the small strain homogenization scheme within an objective incremental setting of large deformations. For details on the incremental two-scale material modeling it is referred to Hofinger et al. [
10]. In this work the phase-specific plastic strains are given by the generalized Ramberg-Osgood relation
$$\boldsymbol{\varepsilon}_{\mathrm{p}\xi}=\varepsilon^{0}_{\xi}\left(\frac{\sqrt{\frac{2}{3}}||\boldsymbol{\sigma}^{\prime}_{\xi}||}{\sigma^{\mathrm{F}}_{\xi}}\right)^{m_{\xi}}\frac{\boldsymbol{\sigma}^{\prime}_{\xi}}{||\boldsymbol{\sigma}^{\prime}_{\xi}||},$$
(3)
which accounts for multiaxial stress states. Hereby, the reference strain
\(\varepsilon^{0}_{\xi}\), the strain exponent
\(m_{\xi}\) and the flow stress
\(\sigma^{\mathrm{F}}_{\xi}\) are introduced as material parameters. From this, the effective plastic strain is given by the localization relation
\(\bar{\boldsymbol{\varepsilon}}_{\mathrm{p}}=\sum_{\xi}c_{\xi}\mathbb{B}^{\sf T_{H}}_{\xi}[\boldsymbol{\varepsilon}_{\mathrm{p}\xi}]\). The material parameters for the mean-field modeling of the duplex stainless steel X2CrNiN23‑4, listed in Table
3, are chosen based on the experimental results of Simon et al. [
11]. Note that for simplicity the phase-specific behavior is modeled statistically isotropic. The blank sheet’s initial residual stress state has been analysed using X‑ray diffraction and the strain gauge hole drilling method at several positions on the sheet surface. Only minor residual stresses
\(|\sigma|<30\,\text{MPa}\) were obtained. Therefore, the initial residual stress state was not further considered in the simulation of the deep drawing process.
Table 3
Material parameters of austenite and ferrite for mean-field modeling based on experimental results of Simon et al. [
11]
Austenite | \(190\,{\mathrm{GPa}}\) | 0.31 | \(420\,{\mathrm{MPa}}\) | 6 | 0.002 |
Ferrite | \(202\,{\mathrm{GPa}}\) | 0.288 | \(380\,{\mathrm{MPa}}\) | 11 | 0.002 |
5 Conclusion
In this work, the evolution of the residual stress distributions in deep-drawn circular cups made of duplex stainless steel X2CrNiN23‑4 is investigated regarding different drawing depths. A two-scale numerical simulation approach was applied to calculate the macro and phase-specific micro residual stresses for ferrite and austenite in every integration point of the entire component. For this purpose, the phase-specific strain hardening of the particular material state is considered. The numerical results are compared to phase-specific residual stresses experimentally determined by means of X‑ray diffraction at the surface of an accordingly deep-drawn cup.
From the numerical and experimental findings the following conclusions can be drawn regarding the numerical prediction of macro and phase-specific micro residual stresses:
-
The calculated macro residual stress distributions are in very good agreement with near-surface XRD analysis at the inner and outer surfaces of the cup’s wall, indicating that the chosen numerical simulation approach is well-suited for the prediction of the effective residual stresses at the component level.
-
Compared to the macro residual stress level, only minor phase-specific micro residual stresses of maximum \(\pm 100\,\text{MPa}\) develop, being almost homogeneous in regions of comparable degree of plastic deformation. However, according to these experimental findings, the phase-specific micro residual stresses are slightly overestimated by the numerical simulations.
-
Apart of this slight overestimation, the first time applied phase-specific input data is well suited for the prediction of phase-specific residual stresses for the deep-drawn duplex stainless steel.
-
The rather low phase-specific residual stresses determined experimentally and numerically prove that the strain hardening behaviour of the two phases ferrite and austenite is very similar for the investigated state of the duplex stainless steel X2CrNiN23‑4.
Furthermore, the influence of the drawing depth on the macro residual stress distribution for the investigated deep drawing process can be summarized as follows:
-
Significant residual stresses develop in the near-surface area of the inner and outer walls as well as in the transition zone to the bottom and flange radius. In contrast, bottom and flange remain almost free of process induced residual stresses, as expected.
-
At the cup’s inner wall high compressive residual stresses were obtained, with significantly higher residual stresses in drawing direction compared to the circumferential direction.
-
At the outside wall predominantly tensile residual stresses occur for both, drawing and circumferential direction.
-
With increasing drawing depth, the tensile and compressive residual stresses, which are acting in drawing direction, increase significantly, whereas the circumferential residual stresses remain constant in magnitude.
-
Moreover, the regions with high residual stress at the cup’s frame are more extensive for larger drawing depths.
Planned experiments under cyclic loading conditions will provide the foundation for determining the necessary material parameters to account for backstress effects in order to further improve the numerical approach.