In our research on cast steel joints used in large-span spatial structures, we have systematically investigated the influence of sand casting defects, particularly shrinkage porosity, on the mechanical performance and damage evolution of low-alloy cast steel G20Mn5. Sand casting is the predominant manufacturing process for these complex steel components, but unavoidable sand casting defects such as shrinkage cavities, gas porosity, and inclusions significantly degrade the material’s load-bearing capacity and ductility. To accurately predict the structural behavior under service loads, we integrated advanced constitutive modeling, numerical casting simulation, and finite element analysis. This article presents our methodology and findings, emphasizing how sand casting defects govern the damage initiation and propagation in cast steel joints.
1. Modeling of Ductile Damage Using the GTN Model
To capture the progressive deterioration of cast steel containing sand casting defects, we employed the Gurson-Tvergaard-Needleman (GTN) damage model. This micromechanical model couples the evolution of void volume fraction with plastic strain, making it ideal for materials with pre-existing porosity. The yield function is given by:
$$ \Phi = \left( \frac{\sigma_{eq}}{\sigma_m} \right)^2 + 2 f^* q_1 \cosh\left( -\frac{3 q_2 \sigma_h}{2 \sigma_m} \right) – 1 – q_3 (f^*)^2 = 0 $$
where $\sigma_{eq}$ is the von Mises equivalent stress, $\sigma_m$ is the flow stress of the matrix material, $\sigma_h$ is the hydrostatic stress, $f^*$ is the effective void volume fraction, and $q_1$, $q_2$, $q_3$ are Tvergaard constants. The matrix hardening is described by an exponential law:
$$ \sigma_y = \sigma_y^0 + Q \left[ 1 – \exp(-b \bar{\varepsilon}^{pl}) \right] $$
with $\sigma_y^0$ the initial yield stress, $Q$ the hardening coefficient, $b$ the hardening exponent, and $\bar{\varepsilon}^{pl}$ the equivalent plastic strain. The evolution of the actual void volume fraction $f$ accounts for void growth and nucleation:
$$ \dot{f} = \dot{f}_g + \dot{f}_n $$
$$ \dot{f}_g = (1 – f) \dot{\varepsilon}_{kk}^{pl} $$
$$ \dot{f}_n = \frac{f_N}{s_N \sqrt{2\pi}} \exp\left[ -\frac{1}{2} \left( \frac{\bar{\varepsilon}^{pl} – \varepsilon_N}{s_N} \right)^2 \right] \dot{\bar{\varepsilon}}^{pl} $$
where $f_N$ is the volume fraction of second-phase particles that can nucleate voids, $\varepsilon_N$ and $s_N$ are the mean and standard deviation of the nucleation strain.
2. Calibration of GTN Parameters for Cast Steel G20Mn5
We calibrated the GTN model parameters for low-alloy cast steel G20Mn5 using experimental data from tensile tests on smooth and notched round bars (Figure 1 in the original study). The material’s elastic modulus was measured as 209 GPa, yield strength 290 MPa, and initial yield strain $1.29\times10^{-3}$. The hardening parameters were fitted as $Q = 382.41$ MPa and $b = 10.79$. The Tvergaard constants were taken as $q_1 = 1.5$, $q_2 = 1$, $q_3 = 2.25$, and the nucleation parameters as $\varepsilon_N = 0.3$, $s_N = 0.1$. The remaining four parameters ($f_0$, $f_N$, $f_c$, $f_F$) were identified through an orthogonal experimental design combined with finite element simulations of the notch-bar test.
| Test | $f_0$ | $f_c$ | $f_F$ | $f_N$ | $\Delta F$ (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 |
*Tests 4, 7, and 8 produced large deviations and were discarded.
From the calibration, the optimal parameter set obtained was: $f_0 = 0.002$, $f_N = 0.01$, $f_c = 0.02$, $f_F = 0.4$. These values reflect the typical initial void content and damage characteristics of sand-cast G20Mn5 steel with sand casting defects.
3. Simulation of Sand Casting Process for K-Joints
To quantify the spatial distribution and volume fraction of sand casting defects in actual cast steel joints, we performed casting process simulations using ProCAST software. A K-type joint commonly used in spatial trusses was modeled with a sand mold and a carefully designed gating/riser system (Figure 2 in the original). The material was selected as Steel_38MnSiVS5 (composition similar to G20Mn5) and the mold as silica sand. The key process parameters are listed below.
| Parameter | Value |
|---|---|
| Pouring temperature | 1560 °C |
| Initial mold temperature | 25 °C |
| Inlet velocity | 0.32 m/s |
| Heat transfer coefficient (mold-air) | 10 W/(m²·K) |
| Heat transfer coefficient (casting-mold) | 500 W/(m²·K) |
The filling time was approximately 13.61 seconds, and the solidification sequence showed that the thickest region at the intersection of the main tube and branch tubes solidified last, leading to the highest risk of shrinkage porosity. The predicted shrinkage porosity distribution (Figure 4 in the original) indicated that the entire K-joint had an average porosity of approximately 1.5%, with local maxima near 2%. This porosity is directly attributed to sand casting defects formed during solidification contraction without sufficient liquid metal feed.
4. Effect of Sand Casting Defects on Mechanical Response
We incorporated the simulated sand casting defects into a finite element model of the K-joint using a homogenized approach. The elastic modulus and Poisson’s ratio were degraded as functions of local porosity $f’$ (volume fraction of shrinkage cavities):
$$ E(f’) = E_0 \left(1 – \frac{f’}{0.5}\right)^{2.5} $$
$$ \nu(f’) = \nu_0 + \frac{f’}{f’_\infty} (\nu_\infty – \nu_0) $$
where $E_0 = 209$ GPa, $\nu_0 = 0.3$, $f’_\infty = 0.472$, $\nu_\infty = 0.14$. The average porosity $f’=0.015$ was applied uniformly to the cast steel region for simplicity, though the methodology can accommodate field-dependent distributions.
The K-joint model (including braces and chord) was meshed with 8-node hexahedral elements. The boundary conditions and loading are shown in Figure 5 of the original study: the left end of the chord was fixed, the right end was a roller, and axial displacements were applied at the ends of the two braces. The GTN model with the calibrated parameters was used for the cast steel region.
4.1 Damage Evolution Results
The numerical results revealed that sand casting defects significantly affect the stress and damage distribution. The maximum principal stress concentrated at the weld toe region between the main tube and the brace, as expected. However, the void volume fraction ($f$) and equivalent plastic strain ($\bar{\varepsilon}^{pl}$) exhibited similar patterns, confirming the coupling described by the GTN model. The damage evolution is summarized in the following table.
| Load step | $\bar{\varepsilon}^{pl}$ (max) | $f$ (max) | $f_g$ (growth) | $f_n$ (nucleation) |
|---|---|---|---|---|
| 0.1 | 0.02 | 0.0023 | 0.0022 | 0.0001 |
| 0.3 | 0.08 | 0.0038 | 0.0035 | 0.0003 |
| 0.5 | 0.15 | 0.0065 | 0.0050 | 0.0015 |
| 0.7 | 0.22 | 0.0120 | 0.0070 | 0.0050 |
| 0.9 | 0.30 | 0.0250 | 0.0100 | 0.0150 |
At lower strains, void growth dominated the damage process. However, once the equivalent plastic strain exceeded approximately 0.17, void nucleation from second-phase particles became the primary contributor, surpassing the growth contribution. This transition is crucial for understanding the failure mechanism of cast steel with sand casting defects, as the initial porosity accelerates the nucleation phase, leading to earlier fracture.
5. Discussion and Implications
Our integrated approach demonstrates that sand casting defects cannot be treated as purely geometric imperfections; they actively participate in the damage process through void growth and nucleation. The GTN model captures the progressive deterioration of material stiffness and strength, which is essential for realistic structural assessment. The casting simulation provides a direct link between the manufacturing process and the mechanical properties, enabling manufacturers to optimize gating and riser design to minimize sand casting defects in critical regions.
Furthermore, the combination of casting simulation and damage mechanics offers a powerful tool for life prediction of cast steel joints subjected to monotonic or cyclic loading. By mapping the initial porosity field from ProCAST into the finite element model, we can evaluate the influence of localized sand casting defects on the overall load-bearing capacity. Our results show that even a moderate average porosity of 1.5% can reduce the ductility and fracture resistance of the joint, which must be accounted for in design codes.
6. Conclusion
Through experimental calibration of the GTN model, casting process simulation, and nonlinear finite element analysis, we have quantified the effect of sand casting defects on the damage evolution of low-alloy cast steel G20Mn5 K-joints. The key findings are:
- The optimal GTN parameters for G20Mn5 are $f_0 = 0.002$, $f_N = 0.01$, $f_c = 0.02$, $f_F = 0.4$.
- ProCAST simulation predicted an average shrinkage porosity of 1.5% in the K-joint, localized at thick sections.
- The GTN model coupled plastic strain and void growth, showing that void nucleation dominates after $\bar{\varepsilon}^{pl} \approx 0.17$.
- Initial sand casting defects amplify the damage rate and reduce the ultimate deformation capacity of the joint.
This methodology provides a rational basis for designing cast steel structures with controlled sand casting defects and for predicting their service life, contributing to safer and more economical engineering practice.