Residual stresses in hot bulk formed parts: two-scale approach for austenite-to-martensite phase transformation

For the numerical analysis, a cross section of the cylindrical specimen applying plane strain conditions is considered. Its dimensions are given in Fig. 4. As simplification, only two parts of the axial section are modeled, which represent the thin and thick side and are defined with an opening angle \(2\,\alpha =10\,^\circ \). Due to the eccentricity of the hole, the inner lateral surface and the outer one are not coaxial.

Dirichlet boundary conditions are used to apply the cooling, as shown in Fig. 3, on \(\partial {{{\mathcal {B}}}}_{{\bar{\theta }}}\) of the macroscale BVPs representing thin and thick side of the cylinder. By cooling over the lateral surface, it is assumed that the heat flux in circumferential direction only has minor impact, and thus, \({\bar{q}}_0=0\) on boundary \({{{\mathcal {B}}}}_{{\bar{\varvec{q}}}}\) is applied. For simplification, rotational symmetry is considered, although it is not given, so that all nodal displacements on \(\partial {{{\mathcal {B}}}}_{{\bar{\varvec{u}}}}\) are fixed orthogonal to the radial cuts, see Fig. 5.

The symmetry of both boundary value problems is not further exploited to obtain more precise results regarding the stress analysis, which will be done along the x-axis for \(y=0\). Using the given configuration, integration points, for which the stresses are assessed, are positioned along this axis. Exploitation of the symmetry would change that insofar, that not integration points but nodes would lie on the x-axis with \(y=0\). Thus, an additional extrapolation would be required to compute the stresses at the nodes, which would result in an additional interpolation.

The phase transformation from austenite to martensite is considered through a temperature-dependent evolution of the volume fractions \(c^{\text {A}}\) and \(c^{\text {M}}\). In order to avoid fluctuations in this measure, the macroscopic temperature \(\bar{\theta }_n\) from the previous time step \(t_n\) is directly transferred to the microscopic level as state variable \(\theta _n=\bar{\theta }_n\,\forall \varvec{x}\in \text {RVE}\) in the current time step \(n+1\). Based thereon, the volume fractions of austenite \(c^{\text {A}}(\theta _n)\) and martensite \(c^\mathrm{M}(\theta _n)\) as well as the material parameters are determined. Initially, there is \(100\%\) austenite, i.e., \(c^\mathrm{A} = 1\). After cooling below the martensitic start temperature \(\theta _{\text {Ms}}\), martensite starts to form and the phase fraction of austenite decreases. To represent this process, an element-wise switch of the related material is carried out. Based on the martensitic volume fraction \(c^\mathrm{M}=1-c^\mathrm{A}\), a section of the finite elements is randomly identified to be martensite. In the final state, there is \(87\%\) martensite and \(13\%\) austenite, which results in the exemplary distribution shown in red and white, respectively, in Fig. 5. It has to be noticed that there is no difference in the final volume fractions, whether a material point at the bulk material or near the surface is considered. The applied cooling route results in an almost uniform phase distribution as obtained in simulation at the IFUM. When switching from austenite to martensite, a volume expansion of \(2\%\) of the martensitic phase is applied to represent the change from fcc to bct lattice, as described in Sect. 2. It is incorporated as \(\varepsilon _{\text {vol}}^*=0.02\) as shown in Table 1. Afterwards, a classical radial return algorithm for the von Mises yield criterion is applied to calculate the microscopic material response, see Simo and Hughes [40]. The solution steps of the microscopic BVP are summarized in Table 1.

The numerical modeling of the cooling process includes the setup of an appropriate material model and the calibration of material parameters. Therefore, the Institute of Forming Technology and Machines, Leibniz University Hannover, (IFUM) provides data, which is obtained using the thermodynamic simulation software JMatPro [10]. Therewith, mechanical and thermal material parameters are calculated based on the chemical composition of the considered Cr-alloyed steel 1.3505 (100Cr6), which is given in Table 2. As stated in Sect. 3.2, the macroscopic temperature \(\theta _n\) is passed onto the microscale as state variable, what matches the assumption of an isothermal mechanical problem on the minor scale. The provided data are evaluated with respect to that state variable as describes in the following.

The data are generated in a temperature range from 50 to \(950\,^\circ \text {C}\) with step size \(50\,^\circ \text {C}\) for each material phase. The Young’s modulus E, the Poisson’s ratio \(\nu \), the heat conduction coefficient k, the product of specific heat capacity and density \(c_\rho \) and the heat expansion coefficient \(\alpha _\mathrm{T}\) are provided as function of the temperature \(\theta \). Furthermore, the yield stress y is given depending on the accumulated plastic strains e for specific temperatures.

A first dataset provides information of the mechanical material parameters, namely E and \(\nu \), used to derive bulk modulus \(\kappa \) and shear modulus \(\mu \), and thermal material parameters k, \(c_\rho \) and \(\alpha _\mathrm{T}\). These parameters determine the material behavior for the elastic regime. The data are piecewise linear interpolated for the thermal and mechanical material parameters between the sampling points. The results are plotted in Fig. 6.

For the plastic regime, a second dataset provides the yield stress y dependent on the accumulated plastic strains \(e\in [0, 4]\) for certain temperatures \(\theta \in [50\,^\circ \text {C},1050\,^\circ \text {C}]\). It is obtained in dilatometer tests using a strain rate of \({\dot{{\varvec{\varepsilon }}}}=0.001\,\text {s}^{-1}\). The data for \(\theta \) are given in equidistant steps of \(50\,^\circ \text {C}\) but for e non-equidistant, i.e., the step sizes areNote that, JMatPro provides in general data for the accumulated plastic strains in a range of [0, 4] but for the considered cooling process of a thick-walled steel part, only small strains are expected. In order to obtain the model parameters y and h with respect to the current temperature \(\theta _n\), a piecewise two step linear interpolation is used within the numerical simulation. Therefore, the material parameters depend on current temperature \(\theta _n\) and accumulated plastic strains \(e_n\) at one microscopic material point. Both quantities are used to determine the rows and columns in Table 3 of the provided data that has to be interpolated, i.e., \(\theta _{i-1}\le \theta _n<\theta _{i}\) and \(e_{j-1}\le e_n<e_j\).

The first interpolation step is done row-wise along the temperature. Based on \((\theta _{i-1},y_{j-1,i-1})\) and \((\theta _{i},y_{j-1,i})\) an interpolated value \(y(e_{j-1},\theta _n)\) is computed using Eq. (11)\(_1\). Analogously, \(y(e_j,\theta _n)\) is computed between \((\theta _{i-1},y_{j,i-1})\) and \((\theta _{i},y_{j,i})\), cf. Eq. (11)\(_2\). Afterwards, the results \(y(e_{j-1},\theta _n)\) and \(y(e_{j},\theta _n)\) are used to interpolate the sought value \(y(e_n,\theta _n)\) in a second step, cf. Eq. (11)\(_3\). Here, the linear hardening parameter h is obtained as the slope of the linear relation in Eq. (11)\(_3\).This interpolation scheme is applied for the yield stress of austenite and martensite. Since martensite starts to form below a temperature of \(200\,^\circ \text {C}\), a temperature range from 20 to \(200\,^\circ \text {C}\) is considered here. In contrast, austenite is present at all temperatures of the cooling process, such that the full temperature range is evaluated. Values for y and h for different temperatures and accumulated plastic strains are exemplary given in Fig. 7.

First, a constant time step size of \(\Delta t = 0.1\,\text {s}\) is chosen for the complete cooling time of \(80\,\text {s}\), in order to determine a reasonable macroscopic discretization. In Fig. 8, the stresses \({\bar{\sigma }}_{yy}\) for the x-axis at \(y=0\) and computational time \(t=20\,\text {s}\) are depicted for the coarsest macroscopic mesh size \(\overline{\text {C}}\) and all three microscopic discretizations. Due to microstructural phenomena related to the ongoing phase transformation an unreasonable stress distribution can be observed independent of the microscopic mesh size. Hence, a refinement on the macroscale to a medium \(\overline{\text {M}}\) or fine \(\overline{\text {F}}\) mesh size needs to be analyzed. Figure 9 illustrates the stress distribution for those macroscopic meshes at time \(t=20\,\text {s}\). Obviously, a coarse microscopic discretization \({\text {C}}\) does not show a reasonable stress distribution as it is indicated by the yellow \(\overline{\text {M}}\) \({\text {C}}\) and green \(\overline{\text {F}}\) \({\text {C}}\) curve.

Stresses computed by the other four combination of medium and fine macroscopic and microscopic discretization seem to be more suitable. Computations with a microscopically medium discretization \(\overline{\text {M}}\) \({\text {M}}\) or \(\overline{\text {F}}\) \({\text {M}}\) do not provide sufficiently smooth stress results, especially regarding the thin side BVP. Thus, the microscopic mesh should be further refined to mesh \({\text {F}}\). In order to decide between the two combination, i.e., \(\overline{\text {M}}\) \({\text {F}}\) and \(\overline{\text {F}}\) \({\text {F}}\), it is inevitable to balance numerical costs with numerical accuracy. While the combination of both fine meshes \(\overline{\text {F}}\) \({\text {F}}\) show best results, a greater number of BVPs have to be solved in every time step. At the same time, the stress plots for \(\overline{\text {M}}\) \({\text {F}}\) do not show a strong deviation from \(\overline{\text {F}}\) \({\text {F}}\). Thus, in the following sections, these discretizations are considered on macroscale and microscale, respectively. As given in Fig. 10, a further refinement on the macroscale to discretizations \(\overline{\text {FF}}\) or \({\overline{\mathbb {F}}}\) confirm the results of mesh size \(\overline{\text {F}}\).

Mesh discretization \(\overline{\text {M}}\) \({\text {F}}\) is considered in the following analysis regarding the time step size. Therefore, the time step size is set to either \(\Delta t = 0.5\,\text {s}\) or \(\Delta t = 1\,\text {s}\) for the complete cooling time of \(80\,\text {s}\). Additionally, the influence of the time stepping scheme especially during phase transformation is examined. Thus, a refinement of the time step size to \(\Delta t = 0.05\,\text {s}\) or \(\Delta t = 0.01\,\text {s}\) is applied between 8 and \(40\,\text {s}\) of computational time and remains \(\Delta t =0.1\,\text {s}\) before and after this range.

One notices that larger time steps do not show convergence on the macroscale. A finer time stepping scheme does not significantly alter the results, cf. Fig. 11.

Note that for a discretization \(\overline{\text {M}}\) \({\text {M}}\)  a refinement of the time step size can improve the results. In the previous Sect. 4.1 it has been discussed that a discretization does not show sufficiently smooth results, especially with respect to the thin side BVP. Choosing a time step size of \(\Delta t =0.05\,\text {s}\) improves the smoothness of the stress distribution, whereas \(\Delta t =0.01\,\text {s}\) does not lead do significant further improvement, see Fig. 12.

Using a suitable discretization as well as an appropriate time step size as determined in Sects. 4.1 and 4.2, the evolution of tangential stresses during the cooling process is investigated. Thus, in Figs. 13, 14 and 15, the tangential stresses \({\bar{\sigma }}_{yy}\) along the x-axis at \(y=0\) are given for meaningful points in time. It becomes possible to relate the stress development to the phenomena occurring in phase transformation such as the volume expansion of the unit cells. Based on this knowledge, the numerical environment enables a prediction of residual stress distributions for altered process parameters, e.g., cooling rate or cooling time.

Initially, the material 1.3505 (100Cr6) is a purely austenitic due to heating over \(1000\,^\circ \text {C}\). The cooling of the lateral surface of both—thick and thin—parts of the cylindrical specimen progresses slowly to the inner part of both geometries. It evokes small tensile stresses in the outer regions of the bodies. Naturally, the inner part of the thin side cool faster compared to the thick side. The stress distribution for \(x\ge 15\,\text {mm}\) after \(5\,\text {s}\) of computation time show compressive stresses, see Fig. 13b.

After \(t=10\,\text {s}\), cf. Fig. 13c, the outer lateral surfaces are cooled below \(\theta _{\text {Ms}}=196.96\,^\circ \text {C}\) and the phase transformation is initiated. It influences the tangential stresses clearly, as high compressive stresses occur due to the martensitic volume expansion. Once the temperature at inner and outer lateral surface of the thick side falls below the martensitic start temperature, compressive stresses evolve in those regions, see Fig. 13e, while the inner part shows tensile stresses. With ongoing phase transformation, the compressive stresses increase further, see Figs. 13f and 14c.

On the thin side, a different stress evolution can be observed. The outer lateral surface of the BVP is cooled faster than the inner lateral surface. The related temperature progress into the inner parts of the body evokes the phase transformation. Thus, compressive stresses are found at the outer lateral surface and in the inner parts, while tensile stresses occur at the inner lateral surface, see Fig. 13d. With initiation of the phase transformation on the inner lateral surface, a superposition of the previously introduced tensile stresses takes place and compressive stresses are observed, cf. Fig. 13e.

Qualitatively, this stress distribution remains, until a computation time of \(17\,\text {s}\) is reached, cf. Fig. 14d. At that time, the inner part of the thick side is subjected to phase transformation. Thereby, compressive stresses are evoked which affect the outer parts of the geometry, where at first the compressive stresses are reduced and later on turn over to tensile stresses, cf. Figs. 14e and 15c. For the thin side, the stresses distribution changes its sign as well, such that compressive stresses occur on the inner lateral surface, while tensile stresses can be found at the outer lateral surface, see Fig. 15d.

For the use of targeted residual stress states to improve the components properties, compressive stresses in regions close to the surface seem reasonable. In the process simulation in Sect. 4.3, these only occur for the outer surface of the thin side boundary value problem. Thus, the cooling process has to be adapted, prospectively, such that the desired characteristics can be established. As a first attempt the total cooling time is altered in the following to analyze the influence of the final tangential stress distributions.

Note that with assuming a simple scaling to shorten or lengthen the cooling time, no modification of the material parameters and microscopic constituents depending on the cooling rate is considered. Hence, neither additional phases such as pearlite, ferrite or bainite nor a varied function for the phase fraction, cf. Eq. (1), are taken into account in the following study. Thereby, the results presented in the following show the principle usability of the proposed numerical model; however, the above-mentioned modifications are inevitable to consider the changed parameter setup.

By changing the cooling time from 80 s to 40 s or 160 s by scaling the evolution of the temperature applied to the lateral surfaces, the cooling is fastened or slowed down, respectively, see Fig. 16a. The austenite-to-martensite phase transformation, which is directly related to temperature \(\theta \), is influenced and thereby, different stress distributions can be found. Shortening or extending the cooling time leads qualitatively to similar tangential stress curves. Quantitatively, a faster cooling during \(40\,\text {s}\) shows higher stresses than previous computation, while a slower cooling during \(160\,\text {s}\) results in lower stress values at the end of computation time, see Fig. 16b.