In this study, we systematically investigate the influence of sand casting defects, particularly shrinkage porosity, on the mechanical behavior and damage evolution of low-alloy cast steel G20Mn5 joints. The Gurson-Tvergaard-Needleman (GTN) damage model is adopted to capture the ductile damage mechanism, and its parameters are calibrated through a combination of tensile tests on notched specimens and finite element simulations. We then conduct a full-scale casting simulation of a typical K-type cast steel joint using the ProCAST software to quantify the spatial distribution and volume fraction of shrinkage porosity. Finally, a coupled elastoplastic-damage finite element analysis is performed on the joint containing initial sand casting defects, revealing how the presence of porosity and subsequent damage accumulation affect the load-bearing capacity. Our results demonstrate that the GTN model effectively couples the equivalent plastic strain of the matrix material with damage variables, providing a more realistic assessment of structural performance in the presence of sand casting defects.
Introduction
Steel structures are widely employed in large-span spatial structures due to their excellent mechanical properties, flexible aesthetic forms, and good weldability. Among them, cast steel joints are often the critical load-transferring components. However, during the sand casting process, various defects such as shrinkage porosity, gas porosity, and inclusions inevitably arise due to complex solidification behavior, alloy composition, and mold design. Among these, sand casting defect of shrinkage porosity is the most common and detrimental to the structural integrity. The presence of these defects significantly reduces the effective cross-sectional area, creates stress concentrations, and accelerates damage initiation and propagation. Traditional design approaches often rely on large safety factors to compensate for the uncertain influence of casting flaws, leading to oversized and uneconomical structures. Therefore, a fundamental understanding of how sand casting defects affect the damage evolution and failure of cast steel joints is urgently needed.
Over the past decades, damage mechanics has provided a powerful framework to describe the gradual degradation of materials from initial deformation to final fracture. For cast steel, the ductile damage process can be well captured by the GTN model, which accounts for void nucleation, growth, and coalescence. The model incorporates microstructural parameters such as initial void volume fraction, nucleation strain, and critical void fraction at failure. In this work, we first calibrate the GTN model for G20Mn5 low-alloy cast steel using an inverse method based on experimental data. Then, we use the ProCAST casting simulation software to predict the distribution and magnitude of sand casting defects in a K-type joint. Finally, we apply the calibrated GTN model in a finite element analysis of the joint, considering the degraded elastic properties due to porosity and the damage evolution under monotonic loading. Our aim is to establish a rigorous methodology for incorporating sand casting defects into the structural performance assessment of cast steel components.
GTN Damage Model and Parameter Calibration
Constitutive Framework
The GTN model defines a yield function that accounts for the softening effect of microvoids:
$$
\Phi(\sigma_{eq},\sigma_m,\sigma_h,f^*) = \left(\frac{\sigma_{eq}}{\sigma_m}\right)^2 + 2f^* q_1 \cosh\left(-\frac{3q_2\sigma_h}{2\sigma_m}\right) – 1 – q_3 (f^*)^2
$$
where \(\sigma_{eq}\) is the von Mises equivalent stress, \(\sigma_m\) the flow stress of the matrix material, \(\sigma_h\) the hydrostatic stress, and \(f^*\) the effective void volume fraction. The parameters \(q_1\), \(q_2\), and \(q_3\) are adjustment coefficients introduced by Tvergaard and Needleman. The matrix material follows an isotropic hardening law:
$$
\sigma_y = \sigma_y^0 + Q \left[1 – \exp(-b \bar{\varepsilon}^{pl})\right]
$$
where \(\sigma_y^0\) is the initial yield stress, \(\bar{\varepsilon}^{pl}\) the equivalent plastic strain, and \(Q\), \(b\) are hardening constants.
The evolution of the actual void volume fraction \(f\) is governed by two mechanisms: growth of existing voids and nucleation of new ones:
$$
df = df_g + df_n
$$
$$
df_g = (1-f) d\varepsilon_{ij}^p
$$
$$
df_n = \frac{f_N}{S_N \sqrt{2\pi}} \exp\left[-\frac{1}{2}\left(\frac{\bar{\varepsilon}^{pl} – \varepsilon_N}{S_N}\right)^2\right] d\bar{\varepsilon}^{pl}
$$
To account for the loss of load-carrying capacity upon void coalescence, the effective void volume fraction \(f^*\) is a piecewise function of \(f\):
$$
f^*(f) =
\begin{cases}
f, & f \leq f_c \\
f_c + \frac{f_u^*-f_c}{f_F – f_c}(f-f_c), & f_c < f \leq f_F \\
f_u^*, & f > f_F
\end{cases}
$$
where \(f_c\) and \(f_F\) are the critical void volume fractions at the onset of coalescence and final failure, respectively, and \(f_u^* = 1/q_1\).
Material Characterization and Parameter Identification
We adopted the low-alloy cast steel G20Mn5, whose chemical composition is given in Table 1. The elastic modulus \(E_0 = 209\) GPa, yield strength \(\sigma_y^0 = 290\) MPa, and initial yield strain \(1.29 \times 10^{-3}\) were obtained from smooth round bar tensile tests. The hardening parameters were fitted from the stress-strain curve: \(Q = 382.41\) MPa, \(b = 10.79\).
To calibrate the four unknown GTN parameters (\(f_0, f_N, f_c, f_F\)), we performed an inverse analysis on a notched round bar specimen (notch radius 2 mm, referred to as ZR2) tested in uniaxial tension. The experimental load-displacement curve showed a fracture load of 11.995 kN and fracture displacement of 0.952 mm. We built a 3D FE model of the ZR2 specimen using C3D8 elements with a fine mesh (0.3 mm) in the notch region. A series of simulations were conducted according to an orthogonal array L9(3^4). The parameters and results are summarized in Table 2.
| C | Si | Mn | Ni | Cr | Mo |
|---|---|---|---|---|---|
| 0.23 | 0.58 | 1.38 | 0.01 | 0.22 | 0.01 |
| Run | \(f_0\) | \(f_c\) | \(f_F\) | \(f_N\) | \(\Delta F\) (kN) | \(\Delta D\) (mm) |
|---|---|---|---|---|---|---|
| 1 | 0.002 | 0.02 | 0.4 | 0.01 | -0.397 | 0.025 |
| 2 | 0.002 | 0.03 | 0.3 | 0.02 | -0.596 | -0.042 |
| 3 | 0.002 | 0.04 | 0.2 | 0.03 | -0.348 | -0.101 |
| 4 | 0.005 | 0.02 | 0.3 | 0.03 | — | — |
| 5 | 0.005 | 0.03 | 0.2 | 0.01 | -0.348 | -0.067 |
| 6 | 0.005 | 0.04 | 0.4 | 0.02 | -0.621 | -0.067 |
| 7 | 0.0001 | 0.02 | 0.2 | 0.02 | — | — |
| 8 | 0.0001 | 0.03 | 0.4 | 0.03 | — | — |
| 9 | 0.0001 | 0.04 | 0.3 | 0.01 | -1.714 | 0.183 |
Note: Runs 4, 7, and 8 produced unacceptable deviations and are omitted.
The best agreement was achieved with Run 1: \(f_0 = 0.002\), \(f_c = 0.02\), \(f_F = 0.4\), \(f_N = 0.01\). These values were adopted for all subsequent analyses. The classical parameters were set as \(q_1 = 1.5\), \(q_2 = 1.0\), \(q_3 = 2.25\), \(\varepsilon_N = 0.3\), \(S_N = 0.1\).
Casting Simulation of K-Type Joint Using ProCAST
Model Setup and Gating System Design
To obtain realistic distributions of sand casting defects, we performed a sand casting simulation of a typical K-type cast steel joint using the ProCAST software. The joint material was assigned as Steel_38MnSiVS5 (composition similar to G20Mn5), and the mold material as silica sand. The initial casting temperature was set to 1560°C, the mold temperature to 25°C, and the pouring rate to 0.32 m/s. The heat transfer coefficients were: mold-air 10 W/(m²·K), casting-mold 500 W/(m²·K).
We designed a gating and riser system suitable for sand casting, as illustrated schematically in the figure below. The system includes a pouring cup, sprue, runner, ingates, and risers to ensure smooth filling and adequate feeding during solidification.
Results of Shrinkage Porosity Prediction
After filling (approximately 13.61 s) and complete solidification, the shrinkage porosity distribution was extracted using ProCAST’s postprocessor. The maximum shrinkage porosity occurred at the thick-walled junction between the main tube and branch tubes, where solidification was slowest. The overall shrinkage porosity ratio across the joint ranged from 1% to 2%, with an average value of 1.5%. This porosity was then used as input for subsequent mechanical analysis.
Mechanical Analysis of K-Type Cast Steel Joint with Sand Casting Defects
Modeling Approach
We constructed a 3D finite element model of the K-type joint including the main tube, two branch tubes (braces), and segments of the lower chord member. The geometry was meshed with C3D8 elements: the joint region was refined to 0.3 mm, while the chords and braces used 1 mm elements. The boundary conditions are shown in the loading scheme: the left end of the lower chord was fully fixed, the right end was supported only in the vertical direction, and axial displacements were applied to the ends of the two braces.
The sand casting defect distribution obtained from ProCAST was mapped onto the FE mesh. The elastic modulus and Poisson’s ratio of the cast steel were degraded as functions of local porosity fraction \(f’\) (shrinkage ratio) using empirical relations from literature:
$$
E(f’) = E_0 \left(1 – \frac{f’}{0.5}\right)^{2.5}
$$
$$
\nu(f’) = \nu_0 + \frac{f’}{f’_\infty} (\nu_\infty – \nu_0)
$$
with \(\nu_0 = 0.3\), \(\nu_\infty = 0.14\), \(f’_\infty = 0.472\). For the average porosity of 1.5%, the modulus reduction was modest but non-negligible.
The calibrated GTN model was implemented via a user-defined material subroutine (UMAT) in ABAQUS/Standard. The analysis was performed under monotonic displacement control until structural failure.
Damage Evolution Results
Figure 6 (conceptual) shows the contour plots of maximum principal stress, void volume fraction, and equivalent plastic strain at a late loading stage. The stress concentration was observed at the root of the branch-to-main tube connection, consistent with geometric discontinuity. Both the void volume fraction and plastic strain exhibited similar distribution patterns, confirming the coupling effect inherent in the GTN model. The evolution of the total void volume fraction and its components (growth and nucleation) is summarized in the following table for a representative element at the critical region.
| Equivalent plastic strain | \(f\) (total) | \(f_g\) (growth) | \(f_n\) (nucleation) |
|---|---|---|---|
| 0.00 | 0.0020 | 0.0020 | 0.0000 |
| 0.05 | 0.0025 | 0.0023 | 0.0002 |
| 0.10 | 0.0034 | 0.0028 | 0.0006 |
| 0.15 | 0.0050 | 0.0034 | 0.0016 |
| 0.17 | 0.0068 | 0.0037 | 0.0031 |
| 0.20 | 0.0120 | 0.0041 | 0.0079 |
At a strain of about 0.17, the nucleation component exceeded the growth component, indicating that void nucleation became the dominant damage mechanism at that stage. The final fracture occurred when the total void fraction reached the critical value \(f_F = 0.4\) in the most damaged element, leading to loss of load-carrying capacity.
Effect of Sand Casting Defects on Structural Response
We compared two cases: a defect-free joint with nominal material properties, and a joint with 1.5% average shrinkage porosity (homogeneously distributed). The presence of sand casting defects reduced the ultimate load by approximately 8.2% and the ultimate displacement by 12.5%. Moreover, the damaged zone (elements with elevated void volume fraction) was significantly larger in the defective joint, indicating that the initial porosity promotes earlier damage initiation and faster damage propagation. This underscores the importance of accounting for sand casting defects in the design of cast steel joints.
Conclusion
In this work, we have performed a comprehensive study on the damage evolution of cast steel joints containing typical sand casting defects. The key findings are summarized as follows:
- The GTN damage model with an isotropic hardening law has been calibrated for low-alloy cast steel G20Mn5 using an inverse method based on notched round bar tests. The optimal parameter set is \(f_0 = 0.002\), \(f_c = 0.02\), \(f_F = 0.4\), \(f_N = 0.01\).
- ProCAST casting simulation of a K-type joint reveals that shrinkage porosity predominantly forms at the thick-walled junction regions, with an average volume fraction of 1.5% under the designed gating system.
- By coupling the porosity-dependent elastic properties with the GTN damage model, the finite element analysis shows that sand casting defects accelerate damage evolution, reduce ultimate strength and ductility, and shift the damage mechanism from void growth-dominated to void nucleation-dominated at larger strains.
- The proposed methodology provides a rational basis for incorporating the effect of sand casting defects into the structural design and integrity assessment of cast steel components, potentially allowing for more optimized and economical designs without compromising safety.
Future work should extend this study to cyclic loading and consider the stochastic nature of sand casting defects for probabilistic fatigue life prediction.