This paper focuses on the issue of deformation during the clamping of thin-walled parts in frozen sand molds. By establishing mathematical models, conducting experiments, and performing numerical simulations, it comprehensively analyzes the influencing factors and provides a basis for optimizing the selection of frozen sand molds and fixture design. The research results show that water content, sand mold material, and contact surface materials have significant impacts on the clamping deformation of frozen sand molds, which can guide the actual production process.
1. Introduction
1.1 Background of Frozen Sand Mold Technology
Frozen sand mold is a sand casting technology that uses pure water as a binder. It involves mixing an appropriate amount of water with fine sand and then freezing it at a low temperature to form a frozen sand blank. Subsequently, the frozen sand mold is machined based on the principle of subtractive manufacturing. This technology has numerous advantages. For example, during the mold – making process, there is less dust. After pouring, the sand mold can self – disperse, and it has no pungent smell. Also, the molding sand can be reused, enabling the rapid manufacturing of single – piece, small – batch, and complex metal parts. However, after the processing of the frozen sand mold in the forming machine tool, it needs to be transported. Since it is a brittle object at low temperatures, how to clamp the frozen sand mold stably without damaging it has become a key research point.
1.2 Research Status
In the field of frozen sand mold research, many scholars have conducted in – depth studies. Academician Shan Zhongde proposed research plans for digital mold – free composite casting equipment and intelligent green casting factories. SHAN et al. studied the influence laws of different freezing temperatures and water contents on the tensile strength and air permeability of 100 – mesh and 200 – mesh silica sand and zircon sand frozen sand molds. Yan Qinlao et al. improved the processing efficiency of complex casting sand molds by optimizing the numerical control processing path. SHI et al. deeply analyzed the evolution laws of the phase field and temperature field during the unidirectional freezing process of frozen sand molds with different pore structures through numerical simulation. YANG et al. analyzed the liquid – solid phase change mechanism of the binder during the additive manufacturing process of frozen sand molds and the changes in the normal temperature field and phase change field of the pre – cooled powder bed porous medium at different temperatures.
In addition, many scholars have studied materials with similar mechanical properties to frozen sand molds, such as frozen soil, frozen sandstone, and concrete. They have analyzed the effects of different factors on the mechanical properties of these materials, such as triaxial (uniaxial) compressive strength, stress – strain curves, and deformation parameters. However, there is relatively little research on the optimization of elastic deformation clamping parameters when stably clamping and not damaging the frozen sand mold, which requires further exploration.
2. Experimental Parameters and Clamping Scheme
2.1 Compressive Strength Measurement
At – 20°C, compressive strength tests were carried out on frozen sand molds made of different water contents and different sand mold materials. The sand mold materials included 100 – mesh silica sand, 100 – mesh zircon sand, and 100 – mesh chromite sand. As shown in Table 1, with the increase of water content, the compressive strength of the frozen sand mold gradually increases. For example, when the water content (mass fraction) of silica sand is 4%, 6%, and 8%, the compressive strengths are 2.187 MPa, 2.448 MPa, and 2.551 MPa respectively.
| Sand Mold Material | Water Content (Mass Fraction) | Compressive Strength (MPa) |
|---|---|---|
| 100 – mesh Silica Sand | 4% | 2.187 |
| 100 – mesh Silica Sand | 6% | 2.448 |
| 100 – mesh Silica Sand | 8% | 2.551 |
| 100 – mesh Zircon Sand | 4% | 2.429 |
| 100 – mesh Chromite Sand | 4% | 2.531 |
| Table 1: Compressive Strength of Different Frozen Sand Molds |
2.2 Friction Coefficient Measurement
When the handling device clamps the frozen sand mold, the outer surface of the frozen sand mold is in direct contact with the fixture. The friction coefficients between several materials with good performance at low temperatures and frozen sand molds made of different water contents and different sand mold materials were measured. As shown in Table 2, when the water content (mass fraction) of silica sand is 6%, the friction coefficients with rubber plates, steel plates, and polyurethane plates are 0.563, 0.322, and 0.397 respectively.
| Sand Mold Material | Water Content (Mass Fraction) | Contact Surface Material | Friction Coefficient |
|---|---|---|---|
| Silica Sand | 6% | Rubber Plate | 0.563 |
| Silica Sand | 6% | Steel Plate | 0.322 |
| Silica Sand | 6% | Polyurethane Plate | 0.397 |
| Table 2: Friction Coefficients of Different Combinations |
2.3 Typical Frozen Sand Mold Parts
The typical frozen sand mold part has a regular internal structure, with dimensions of 400mm×400mm×100mm. This study focused on frozen sand molds with wall thicknesses of 10mm, 20mm, 30mm, 40mm, and 50mm. The grid – division model of the typical frozen sand mold part has a grid size of 5mm, as shown in Figure 1. [Insert Figure 1: Typical components(a) and grid – division model(b) of frozen sand mold]
2.4 Clamping Experimental Scheme
The frozen sand mold is a special object made of water and 100 – mesh fine sand at – 20°C. During the clamping process, loose sand may fall off its surface, and the frozen sand mold will undergo a certain degree of elastic deformation when it is stably clamped. A special frozen sand mold fixture is required for the handling operation. When designing and using the fixture, the strength and stability of the frozen sand mold need to be considered to ensure the safety of the handling process. According to Hooke’s law, before brittle materials exceed their proportional limit and plastic materials reach their yield limit, they can be approximately regarded as completely elastic bodies. In this experiment, a safety factor of 1.2 was taken to ensure the reliability of clamping. The experiment was simulated based on the actual clamping scheme, with the load applied evenly on both sides of the frozen sand mold for clamping operations, as shown in Figure 2. [Insert Figure 2: Schematic of actual processing of frozen sand molds and load gripping direction]
3. Theoretical Analysis of the Forces on Frozen Sand Molds
3.1 Three – Dimensional Mathematical Model of Force Equilibrium of Frozen Sand Molds
At – 20°C, the frozen sand mold is a brittle object. In the case of stable clamping without damaging it, it can be described by the elastic mechanics theory of three – dimensional objects. During the clamping process, its surface bears uniform forces, and its interior is subjected to pressure or tension exerted by the fixture. To maintain the internal force balance of the frozen sand mold, the internal stress and external force need to be in equilibrium. The stress and deformation degree of the frozen sand mold are directly related to the external force. Only when the external force applied to the frozen sand mold does not exceed its compressive strength will the frozen sand mold undergo elastic deformation. When the external force disappears, the frozen sand mold will return to its original shape, and the corresponding internal stress will cause elastic deformation inside the frozen sand mold. The force – balance equations are as follows: \(\sum F_{x}=\frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+\frac{\partial \tau_{x x}}{\partial z}+x = 0\) \(\sum F_{y}=\frac{\partial \tau_{x y}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+\frac{\partial \tau_{z y}}{\partial z}+y = 0\) \(\sum F_{z}=\frac{\partial \tau_{x z}}{\partial x}+\frac{\partial \tau_{y z}}{\partial y}+\frac{\partial \sigma_{z}}{\partial z}+z = 0\)
These equations can describe the force – balance state when the frozen sand mold is clamped.
3.2 Two – Dimensional Mathematical Model of the Strain Plane of Frozen Sand Molds
When clamping the frozen sand mold, the axial direction is not stressed. The two – dimensional force diagram of the thin – walled part of the frozen sand mold is shown in Figure 3. The length direction is taken as the z – axis, and the two end – faces cannot move along the z – axis direction. Assuming that the frozen sand mold is cut into many thin slices along the z – axis direction, the stress conditions of these slices are the same. However, due to the shape of the casting part required, the thickness of the plane is different, so the strain on each plane is also different. The stress – strain and displacement components of these frozen sand mold slices can be regarded as functions of x and y in the xoy plane and are independent of the z – coordinate. Therefore, it can be approximately considered that the axial displacement \(w = 0\) at each point on any cross – section of the frozen sand mold. The three strain components \(\varepsilon_{x}\), \(\varepsilon_{y}\), and \(\gamma_{s}\) parallel to the xoy coordinate plane only deform within the cross – section. Thus, the three – dimensional strain problem of clamping the frozen sand mold can be transformed into a two – dimensional plane – strain problem, and then the situation with the largest strain of the frozen sand mold can be analyzed under different conditions. [Insert Figure 3: 2D force diagram of thin – walled components in frozen sand mold] The relevant equations are as follows: Balance equations: \(\frac{\partial \sigma_{x}}{\partial x}+\frac{\partial \tau_{y x}}{\partial y}+x = 0\) \(\frac{\partial \tau_{x y}}{\partial x}+\frac{\partial \sigma_{y}}{\partial y}+y = 0\) Geometric equations: \(\varepsilon_{x}=\frac{\partial u}{\partial x}\) \(\varepsilon_{y}=\frac{\partial v}{\partial y}\) \(\gamma_{x y}=\frac{\partial u}{\partial y}+\frac{\partial v}{\partial x}\) Physical model: \(\varepsilon_{x}=\frac{1 – \mu^{2}}{E}(\sigma_{x}-\frac{\mu}{1 – \mu}\sigma_{y})\) \(\varepsilon_{y}=\frac{1 – \mu^{2}}{E}(\sigma_{y}-\frac{\mu}{1 – \mu}\sigma_{x})\) \(\gamma_{x y}=\frac{2(1+\mu)}{E}\tau_{x y}\)
3.3 Compressive Constitutive Relationship of Frozen Sand Molds
In engineering, frozen sand molds and some materials (such as frozen soil, rock, and concrete) exhibit reversible deformation characteristics within the small – deformation range, that is, they can return to their original state after unloading. The microscopic mechanism of elastic deformation is that under the action of force, the distance between microscopic particles of the object changes, but its microscopic structure remains unchanged. When the force is unloaded, the interaction force between microscopic particles will make them return to the initial state. Therefore, elastic deformation is only related to the magnitude of the load, and there is a one – to – one correspondence between stress and strain. The following model is used to define the constitutive equation of the elastic stage of the frozen sand mold under compression at low temperatures: \(\begin{cases}\sigma<\sigma_{*}\\\sigma = E\varepsilon_{e}\\\sigma_{s}=\sigma_{0}-KV_{p}\\\varepsilon=\frac{V}{L}=\varepsilon_{e}+\frac{V_{p}}{L}\end{cases}\) where \(\sigma_{*}\) and \(V_{P}\) are the strength and plastic deformation of the sand – grain micro – element at the current state; \(\sigma_{0}\) is the strength limit at the initial state or when \(V_{P}=0\); \(\varepsilon\) and \(\varepsilon_{e}\) are the apparent strain and elastic strain respectively; V is the compression deformation; L is the length of the sand – grain micro – element. Based on the established constitutive model, at a low temperature of – 20°C, for frozen sand molds with water contents (mass fractions) of 4%, 6%, and 8%, made of silica sand, zircon sand, and chromite sand, and with rubber, polyurethane, and steel plates as the contact surfaces of the clamping plates, finite – element simulation analysis was carried out on the frozen sand molds by applying different compressive stresses. The parameter values of the compressive constitutive model of the frozen sand mold are shown in Table 3.
| Frozen Sand Mold | Temperature \(T(^{\circ}C)\) | Water Content (Mass Fraction) | Compressive Strength \(\sigma_{bc}(MPa)\) | Elastic Modulus \(E(MPa)\) | Poisson’s Ratio \(\mu\) | Sand – Grain Diameter (Mesh) | Density \((g\cdot cm^{-3})\) |
|---|---|---|---|---|---|---|---|
| Silica Sand | – 20 | 4% | 2.187 | 122.15 | 0.211 | 100 | 2.44 |
| Silica Sand | – 20 | 6% | 2.448 | 123.23 | 0.232 | 100 | 2.34 |
| Silica Sand | – 20 | 8% | 2.551 | 124.13 | 0.241 | 100 | 2.30 |
| Zircon Sand | – 20 | 4% | 2.429 | 126.21 | 0.252 | 100 | 4.09 |
| Chromite Sand | – 20 | 4% | 2.531 | 129.17 | 0.273 | 100 | 4.15 |
| Table 3: Parameters of the Constitutive Model for Frozen Sand Molds under Compression |
4. Numerical Simulation Research on the Clamping Deformation of Frozen Sand Mold Thin – Walled Parts
4.1 Analysis of the Clamping Deformation of Frozen Sand Molds with Different Wall Thicknesses at Different Water Contents
Simulation experiments were carried out on frozen sand molds made with different water contents. The water contents (mass fractions) of the frozen sand molds were 4%, 6%, and 8% respectively, the material was 100 – mesh silica sand, and the contact – surface material of the clamping plate was a steel plate. Using the above experimental parameters, numerical simulation and analysis were carried out on frozen sand molds with different wall thicknesses.
As shown in Figure 4, when clamping the silica sand frozen sand mold with a water content (mass fraction) of 4%, at a wall thickness of 40mm, the compressive stress is 0.073699MPa, which is less than the compressive strength. The frozen sand mold is stably clamped without damage but undergoes a certain elastic deformation. At this time, the strain at the easily deformed part of the frozen sand mold is smaller than that of other wall – thickness cases, and the maximum elastic strain at the easily deformed thin – walled part is 0.00060409. [Insert Figure 4: Silica sand frozen sand mold elastic strain with water content (mass fraction) of 4%]
For the silica sand frozen sand mold with a water content (mass fraction) of 6%, when the wall thickness is 40mm, the maximum elastic strain at the easily deformed part is 0.00063091, as shown in Figure 5(a). For the silica sand frozen sand mold with a water content (mass fraction) of 8%, when the wall thickness is 40mm, the minimum elastic strain at the easily deformed part is 0.00064367, as shown in Figure 5(b). [Insert Figure 5: Silica sand frozen sand mold elastic strain with different water contents (mass fractions)]
In summary, when the wall thickness is the same, the deformation of the frozen sand mold with a water content (mass fraction) of 4% is the smallest among the three water – content cases.
4.2 Analysis of the Clamping Deformation of Frozen Sand Molds with Different Wall Thicknesses Made of Different Materials
Simulation experiments were carried out on frozen sand molds made of different materials. It was known from the previous section that the frozen sand mold with a water content (mass fraction) of 4% has the smallest elastic deformation at the most easily deformed part. By using the control – variable method, the water content (mass fraction) of the sand mold was selected as 4%, the sand – mold materials were 100 – mesh zircon sand and 100 – mesh chromite sand respectively, and the contact – surface material of the clamping plate was a steel plate. Numerical simulation analysis was carried out on frozen sand molds with different wall thicknesses.
4.3 Analysis of the Clamping Deformation of Frozen Sand Molds with Different Wall Thicknesses under Different Contact Surfaces
Simulation experiments were carried out by clamping frozen sand molds with different contact surfaces of clamping plates. It was known from the previous two sections that when the contact surface is a steel plate, the frozen sand mold made of silica sand with a water content (mass fraction) of 4% has the smallest elastic deformation at the most easily deformed part. By using the control – variable method, the water content (mass fraction) was selected as 4%, the sand – mold material was 100 – mesh silica sand, and the contact – surface materials of the clamping plates were rubber plates and polyurethane plates respectively. Numerical simulation and analysis were carried out on frozen sand molds with different wall thicknesses.
During the clamping process, the compressive stresses when the rubber plate and the polyurethane plate stably clamp the frozen sand mold are 0.0449832MPa and 0.060581MPa respectively, which are less than the compressive strengths and no damage occurs. As shown in Figure 7(a), when the silica sand frozen sand mold with a water content (mass fraction) of 4% is stably clamped by the rubber plate surface with a wall thickness of 40mm, the maximum elastic strain at the easily deformed part is 0.00036748. As shown in Figure 7(b), when the silica sand frozen sand mold with a water content (mass fraction) of 4% is stably clamped by the polyurethane plate surface with a wall thickness of 40mm, the maximum elastic strain at the easily deformed part is 0.00049657.
It can be clearly seen that compared with polyurethane plates and steel plates as contact surfaces for clamping frozen sand molds, the rubber plate surface causes the smallest deformation when clamping frozen sand molds.
[Insert Figure 7: Elastic strain when rubber plate surface and polyurethane plate surface clamp silica sand frozen sand mold with water content (mass fraction) of 4%]
5. Comprehensive Discussion
5.1 Influence Mechanism of Water Content
The water content in the frozen sand mold plays a crucial role in its mechanical properties during clamping. When the water content is relatively low, there is not enough water – ice bonding to effectively transfer the clamping force, resulting in a relatively large deformation. As the water content increases, the ice formed in the sand mold acts as a stronger binder, enhancing the overall strength of the frozen sand mold. However, when the water content is too high, excessive ice may cause internal stress concentration during the freezing process, which also affects the stability during clamping. From the experimental and simulation results, it can be seen that a water content (mass fraction) of 4% provides an optimal balance, resulting in the smallest deformation under the same wall – thickness condition.
5.2 Influence of Sand Mold Materials
Different sand mold materials have different physical and mechanical properties. Silica sand, zircon sand, and chromite sand have different particle shapes, sizes, and chemical compositions, which lead to differences in their strength, elasticity, and friction characteristics. Silica sand has relatively good elastic – deformation resistance during clamping compared to zircon sand and chromite sand. This may be related to its more uniform particle distribution and relatively stable chemical properties, which enable it to better withstand the clamping force and reduce deformation.
5.3 Role of Contact Surface Materials
The contact surface material between the fixture and the frozen sand mold significantly affects the clamping deformation. Rubber plates have a relatively soft texture and good elasticity. When in contact with the frozen sand mold, they can better adapt to the surface of the frozen sand mold, distribute the clamping force more evenly, and reduce the local stress concentration, thus resulting in the smallest deformation. Polyurethane plates and steel plates are relatively harder. The contact with the frozen sand mold may cause relatively large local stresses, leading to larger deformations.
6. Engineering Applications and Future Research Directions
6.1 Engineering Applications
The research results have important guiding significance for the actual production process of frozen sand mold casting. In the selection of frozen sand molds, considering factors such as water content and sand – mold material can help manufacturers choose the most suitable frozen sand mold to minimize deformation during clamping and ensure the quality of casting products. For fixture design, choosing rubber – based contact surfaces can effectively reduce the damage to frozen sand molds during transportation, improving production efficiency and product yield.
6.2 Future Research Directions
Although this study has achieved certain results, there are still some aspects that can be further explored. Firstly, the influence of different freezing rates on the clamping deformation of frozen sand molds can be studied. Different freezing rates may lead to different internal structures of the frozen sand mold, thus affecting its mechanical properties during clamping. Secondly, the long – term stability of frozen sand molds during storage and multiple – cycle use can be investigated. Understanding how the mechanical properties of frozen sand molds change over time can help optimize the production process and improve the service life of frozen sand molds. Additionally, research on the coupling effect of multiple factors such as temperature, humidity, and clamping force on the deformation of frozen sand molds can be carried out to provide more comprehensive theoretical support for the development of frozen sand mold casting technology.
7. Conclusion
This paper conducts a series of experiments and numerical simulations on the clamping deformation of frozen sand mold thin – walled parts. Through the establishment of force – balance and strain – plane mathematical models, and the analysis of the influence of different water contents, sand – mold materials, and contact – surface materials on clamping deformation, the following conclusions are obtained:
- When the water content of the frozen sand mold is the same, the friction coefficient between the silica – sand – made frozen sand mold and different contact surfaces is smaller than that of zircon – sand and chromite – sand – made frozen sand molds.
- At the same wall thickness, compared with clamping frozen sand molds with water contents (mass fractions) of 6% and 8%, clamping a frozen sand mold with a water content (mass fraction) of 4% results in the smallest deformation.
- At the same wall thickness, compared with clamping frozen sand molds made of zircon sand and chromite sand, clamping a silica – sand – made frozen sand mold results in the smallest deformation. Compared with using polyurethane plates and steel plates as contact surfaces for clamping frozen sand molds, using a rubber – plate surface results in the smallest deformation.
These research results can provide a theoretical basis for the selection of frozen sand molds and the design of fixtures in the actual production process of frozen sand mold casting, and promote the development and application of frozen sand mold casting technology.
