Numerical Simulation of Liquid-Solid Extrusion Process Based on the Mechanical Model Coupled with Solidification

Abstract

Liquid-solid extrusion process is a combined process of casting and extrusion which can be used to form tubes or bars directly from liquid metal. The performance of products is enhanced through the large deformation and the solidification under pressure with less shrinkage cavity or porosity. Numerical simulation of this process is hard to run for it involves mechanical modeling of the dynamic transition from liquid phase to solid phase. The liquid zone and solid zone were modeled independently for reasons of their different characteristics of deformation. The deformation of liquid zone was described according to the principle of element removal method which eliminates the elemental distortion during the simulation. The solidified zone under elevated temperature was modeled through the hyperbolic sine constitutive equation. The dynamic transitions from liquid phase to solid phase were determined based on the results of thermal analysis. The mechanical model coupled with solidification proposed in this paper was verified through the experiments of liquid-solid extrusion of LY12 alloy.

1. Introduction

Liquid-solid extrusion is proposed to form tubes or bars in a single process which cannot only improve the product quality but also reduce the manufacture cost dramatically [1]. Compared to the squeeze casting process [2], liquid-solid extrusion can cause greater deformation, and thus, the casting defects can be minimized very efficiently. Qi et al. [3] later applied this process to form composite tubes or bars. In the liquid-solid extrusion process of alloy or composite tubes or bars, liquid metal was first poured into the extrusion container and then extruded out of the forming die a little while after liquid metal pouring. This process is hard to be controlled precisely because of the coexistence state of liquid phase and solid phase. If the extrusion is conducted so early, excess liquid metal will be extruded out which will interrupt the continuous extrusion process. On the other side, if the extrusion is conducted very late, liquid metal will be completely solidified into solid metal. The advantage of liquid-solid extrusion process is not preserved any more. Qi et al. [4] had proposed a multiobjective optimization model to optimize the parameters of liquid-solid extrusion forming composites. But this model was based on the abundant expensive experiments.

Numerical simulation provides a powerful means of analyzing various physical phenomena which occurred during liquid-solid extrusion processes. It gives an insight into the details of coupled deformation between liquid zone and solid zone. More importantly, numerical simulations can also help shorten the design process and optimize extrusion parameters to improve the quality of the products. But the liquid-solid extrusion involves large deformation as well as solidification under high pressure, so the simulation of the process is more complex than that of deformation or that of solidification, respectively. It is a considerable challenge to apply the commercial finite element software to simulate the realistic liquid-solid extrusion process. Many obstacles arise during the numerical modeling of thermomechanical behavior which occurred in liquid-solid extrusion process as stated by Qi et al. [5], which include the incorporation and integration of the highly nonlinear viscoplastic constitutive laws, treatment of the liquid/mushy zone, temperature-dependent material properties, and dynamic transition from liquid phase to solid phase.

The literature about the coupled simulation of deformation and solidification under pressure usually deals with the modeling in the thermomechanical way based on the elastic-viscoplastic constitutive equations [6] for both liquid phase and solid phase. Liao et al. [7] simulated the stress field in casting solidification process based on the thermal elastic-plastic model and ignored the effect of phase change from liquid phase to solid phase. Firstly, the temperature field was simulated. Then, the simulation results of the temperature field were added to the stress analysis model as the thermal load to simulate stress field. The hot cracks or dimension change caused by thermal stress can be assessed through the simulation of casting process. But interaction between liquid phase and solid phase during solidification simulation was not considered. Li et al. [8] simulated the temperature, stress, and shrinkage of the billet in the mould during continuous casting by assuming that the material of the billet followed the Von Mises yield criterion with varying material parameters as the temperature of molten steel decreased continuously with phase changes during continuous casting. Ferrostatic pressure was considered, but it was imposed on the internal surface of billet shell as the boundary condition. Koric et al. [9] had chosen the liquid yield stress σ_Y = 0.01 MPa to treat the deformation of liquid and mushy zone in steel casting which approximated the hydrostatic state in liquid zone. Further, Koric et al. [10] created the separate three-dimensional (3D) models of thermomechanical behavior of the solidifying shell, turbulent fluid flow in the liquid pool, and thermal distortion of the mold at the continuum level, and the analyzed results were linked into a coupled thermomechanical model of continuous casting using finite element model in ABAQUS. Hu and Yu [11] and Sun et al. [12] developed a two-dimensional model to simulate heat transfer and solidification phenomena of squeeze cast magnesium alloy. These models mainly predicted the effect of pressure on the temperature distributions, the cooling curves, the shape and position of the solidification front, and total solidification time of a cylindrical squeeze casting but did not consider the mechanical deformation field. Simulation for extrusion coupled with continuous solidification was not reported in the literatures as authors know.

Based on the above literature reviews, the large property variations between the liquid, mush, and solid phases add a significant challenge to thermomechanical simulations for deformation coupled with solidification. More recently, a clear distinction between solid-like and liquid-like constitutive equations for the solidifying alloys has been shown mandatory to treat the deformation of the solidified regions and the liquid hydrostatic effect on the solidified shell in a simultaneous manner. The constitutive equations for solid phase containing liquid phase have been investigated thoroughly in literature. Qi et al. [13] investigated the constitutive behavior of C_sf/AZ91D composites compressed at elevated temperature and containing a small fraction of liquid. Giraud et al. [14] investigated the high temperature compression behavior of a solid representative of the solid phase of the alloy within the solidification range by use of a drained compressive test, and the constitutive equation of the solid phase present in a semisolid 6061 alloy at a given temperature was determined. So the key to the success of simulation of deformation coupled with casting is the treatment to the deformation of the liquid zone. That means to construct the liquid-like model for liquid phase. Qi and Zhou [15] summarize the liquid-like models into two types: one is based on the material modeland the other is based on the element removal. Bellet et al. [16] addressed the computer modeling of pipe formation in metal castings by defining the two types of constitutive equations (liquid-like and solid-like) that were used simultaneously in the mechanical modeling. But Bellet's simulation only accounts for the free solidification without external load being applied. Mori et al. [17] simulated the forming process of solid metal with liquid phase by finite element method. Zhou et al. [18] also simulated the couple deformation behavior of the solid metal with liquid zone. The liquid phase was removed in the simulation in Mori's and Zhou's method which avoided the distortion of elements in the liquid zone. But the simulation on the process of the dynamic transition from liquid phase to solid phase was not considered.

This paper proposes a mechanical model coupled with solidification for the liquid-solid extrusion process which involves the dynamic transition from liquid phase to solid phase and gives an example to illustrate the detailed process of the modeling.

2. Characteristics of Liquid-Solid Extrusion Process

Liquid-solid extrusion process includes the following steps, namely, preheating the mold, pouring the melt alloy into the mold, closing the mold by pushing the punch into the container, keeping pressure to make sure the melt alloy solidify under pressure, and extruding the semisolid metal out of the mold exit. The whole process can be illustrated in Figure 1.

Figure 1:

Illustration of liquid-solid extrusion process: (a) pouring the melt alloy into the container, (b) closing mold and pressure keeping, and (c) extruding—1: punch; 2: container; 3: melt alloy; 4: forming die; 5: liquid phase; 6: solid phase; 7: semisolid; 8: eject pin; 9: extruded bar.

Liquid, liquid-solid, and solid metals coexist in the container at the beginning of extrusion. When the extrusion starts, the melt alloy is poured into the container. The workpiece in the container changes in two ways after the pouring. On the one hand, the workpiece is extruded out of the exit of the forming die. On the other hand, the amount of the liquid phase decreases with the workpiece being solidified. The deformation of the solidified zone and the flow of the liquid zone affect the forming process significantly. The top part of the workpiece crystallizes and moves downward under high pressure with little deformation, whereas the solid and liquid-solid workpiece near the mold exit deforms greatly. The central part of the extruded product would contain liquid metal without the deformed microstructure if the extrusion speed and the temperature are not controlled carefully. The extrusion process will not continue if much liquid metal is contained in the extruded product. This is why we has to do an amount of simulation to find the suitable parameters for better products.

3. Mechanical Modeling Coupled with Solidification

3.1. Mechanical Modeling of Liquid Zone

Hydrostatic fluid elements are provided to model fluid-filled cavities in ABAQUS code. These elements provide the coupling between the deformation of the fluid-filled structure and the pressure exerted by the contained fluid on the boundary of the cavity. As mentioned before, liquid zone and solid zone coexist during liquid-solid extrusion. The workpiece with liquid and solid zones can be regarded as a fluid-filled structure. So the liquid zone can be modeled with the hydrostatic fluid elements.

The hydrostatic fluid elements are based on the following assumptions. (1) The volume of the cavity (V) is equal to the volume of the fluid ( $\bar{V}$ ) in the cavity, which means the cavity is filled completely with the fluid. (2) Temperature gradients and pressure gradients do not exist in the filled fluid. (3) The volume of the fluid is determined by the mass, pressure, and temperature of the fluid; namely; $\bar{V} = \bar{V} (m, p, T)$ .

The assumptions whereby the following constrain condition can be drawn, that is, $V - \bar{V} = 0$ , can be introduced into the virtual work expression through the Lagrange multiplier, λ:

\begin{matrix} δ Π^{*} = δ Π - λ δ V - δ λ (V - \bar{V}) . \end{matrix}

(1)

We can easily conclude that λ is the pressure p in the fluid from the rigid-plastic finite element theory. So the pressure and deformation can be obtained by solving (1) through finite element method.

3.2. Mechanical Modeling of Solid and Semisolid Zone

The hyperbolic sine constitutive equation was used to model the behavior of the solid and semisolid aluminum alloy LY12, as illustrated in

\begin{matrix} \dot{\bar{ε}} = A \cdot {[\sinh (α \bar{σ})]}^{n} \cdot \exp (- \frac{Q}{R T}), \end{matrix}

(2)

where A, n, and α are material parameters, Q is the activation energy, and R is the gas constant.

The compression tests were conducted in the general mechanical test machine CSS-1110 to determine the parameters in the constitutive equations. The constitutive parameters were determined based on the analytical methods, as shown in the Table 1.

Table 1:

Parameters in the hyperbolic sine constitutive equation.

Parameters	1/α/ MPa	n	A/ ×10¹³	Q/ J·mol⁻¹

Values	30	6	2	240000

3.3. Mechanical Modeling of Dynamic Transition from Liquid to Solid

The state of the workpiece before extrusion can be determined by the thermal analysis. The solidified shell was extracted from the thermal analysis results written into the output database. The process of extracting solidified shell was divided into the following three steps: (1) generating the node set in the output database of the thermal analysis: the temperature of the generated node set was above the melting temperature; (2) importing the part from the output database of the thermal analysis through the ABAQUS GUI; (3) after importing the part, deleting the node sets generated in the first step (representing liquid zone) by editing the mesh in the MESH module. Although the fluid-filled cavity had been obtained with this method, the boundary of the fluid-filled cavity was discontinuous as shown in Figure 2 (a). In order to solve this problem, the following steps should be taken:

The first step is extracting the perimeter of the meshed fluid-filled cavity (Figure 2 (a)) generated by deleting the node set of liquid zone with author-compiled Python script, as shown in Figure 2 (b).

Editing the geometry generated in the first step in the sketch module, the discontinuous interface was replaced by the spline generated by connecting the point on the interface, as shown in Figure 2 (c).

Meshing the continuous geometry (as illustrated in Figure 2 (d)) part with fine elements, as shown in Figure 2 (e): the mechanical model was constructed through the above mentioned steps. The whole process of the mesh generation was illustrated in Figure 2 The initial temperature of the mechanical model was read from the result database file of the thermal analysis. At last, we completed the results transfer from the thermal analysis to mechanical analysis.

Figure 2:

Extracting solid part from the thermal analysis and the mesh generation:(a) removing the elements in liquid phase; (b) extracting the geometry based on step (a); (c) smoothing the discontinuous interface between liquid and solid; (d) the revised geometry generated in step (b); (e) meshing the revised geometry.

The hydrostatic elements on the interface between liquid zone and solid zone were generated through in-house Python script procedure. The geometry set were formed for the interface between liquid zone and solid zone as generating the geometry model of mechanical analysis. The node set and element set were also formed under the same name as the geometry set during meshing. The Python script procedure can write a text formatted file which was used to define the hydrostatic elements composed of nodes in the node set of interface. The text formatted file was then incorporated into the model input file which was used to simulate the couple deformation of liquid phase and solid phase.

The coupled mechanical analysis was followed by another thermal analysis. The process of generating the finite element model of sequential thermal analysis was illustrated in Figure 3.

Constructing the geometry model by extracting the perimeter of the imported part (Figure 3 (a)) from the result database of mechanical analysis was shown in Figure 3 (b).

Deleting the boundary of the cavity and closing the perimeter as initial thermal analysis, thus, the geometry model without cavity was obtained, as shown in Figure 3 (c).

Meshing the new constructed geometry part was shown in Figure 3 (d).

Figure 3:

Regenerate the thermal analysis model from mechanical results (a) after the deformation with liquid zone removed; (b) extracting the geometry based on step (a); (c) filling the removed part for further thermal analysis; (d) meshing the generated model in step (c).

After the model of sequential thermal analysis was constructed, the initial temperature of the model should be mapped from the mechanical analysis. With this method, the initial temperature of the liquid zone cannot be mapped because the liquid zone was removed during the mechanical analysis. For the sake of simplicity, the gradient of temperature in the liquid phase was ignored during sequential thermal analysis. The initial temperature of liquid zone in sequential thermal analysis was set to be the same as that of the first thermal analysis. The whole process of generating mesh for the sequential thermal analysis can be illustrated with Figure 3.

The results transferring between thermal and mechanical analysis was illustrated in Figure 4. The predefined temperature in the mechanical analysis was read from the output database of thermal analysis, and the predefined stress state was mapped from the precedent mechanical analysis. The initial temperature in the sequential thermal analysis was mapped from the mechanical analysis.

Figure 4:

Results transferring between thermal and mechanical analysis.

We can summarize the whole process of modeling the dynamic transition from the liquid phase to the solid phase into the flowchart as illustrated in Figure 5. It was a loop circle from the thermal analysis to the mechanical analysis.

Figure 5:

Flowchart of modeling based on the element removal.

4. Numerical Simulation of Liquid-Solid Extrusion Process

4.1. Finite Element Model of Liquid-Solid Extrusion Process

Although the liquid-solid extrusion process includes several steps as illustrated in Figure 1, the numerical simulation is run only for preheating the mold and extruding the workpiece in this paper for simplicity. The thermal analysis was used to simulate the preheating of the mold, and the thermal-mechanical couple analysis was used to simulate the extrusion process.

Figure 6 shows the illustration of preheating the mold and the corresponding finite element model. The finite element model is axisymmetric and includes the right half of experiment setup only since the middle surface of experiment setup is a plane of symmetry. The radiation, heat transfer, and convection were considered in the thermal analysis of preheating mold. The punch, eject pin, insulator, and the shoe plate radiated heat to air through convection and radiation. The convection coefficient and radiation coefficient were 29 N/(s·m·°C) and 0.8, respectively. The sink temperature was set to 25°C. The mold was heated to a set temperature by radiation from the heater. The temperature of heater was set to be constant, 1000°C. The heat was exchanged through conductance among the mold, punch, shoe plate, and the eject pin. The thermal conductance was set to 4000 N/(s·m·°C) as used by Gonzalez et al. [19]. The mold was made of steel 3Cr2W8V with the density of 8350 kg/m³. Other thermal properties of steel 3Cr2W8V were listed in the Table 2.

Table 2:

Thermal properties of steel 3Cr2W8V

Temperature/°C	100	200	300	400	500	600	700	800	900

Conductivity/W·(m·°C)⁻¹	20.1	22.2	—	—	—	—	24.3	—	23.0
Specific heat/J·(kg·°C)⁻¹	468.2	525.5	—	—	685.5	—	—	1262.4	660.4

Figure 6:

Illustration for preheating mold and the finite element model (a) setup for mold heating; (b) finite element model for thermal analysis.

After the initial temperature distribution of the mold was determined from the thermal analysis of preheating the mold, the melt aluminum alloy was poured into the mold. The initial temperature of workpiece was set to be 780°C. For simplicity, the heat transfer and the extrusion process were analyzed through finite element method based on the sequential thermal-mechanical analysis. Figure 7 shows the finite element of sequential thermal-mechanical analysis.

Figure 7:

Finite element of sequential thermal-mechanical analysis.

The properties of aluminum were listed in Table 3. The thermal conductance between workpiece and mold was set to 4000 N/(s·m·°C) also. Extrusion speed was set to be 7 mm/s.

Table 3:

The properties of aluminum LY12.

Properties	Specific heat/J (kg·°C)⁻¹	Density/kg·m⁻³	Conductivity/w· (m·°C)⁻¹

Aluminum alloy LY12	880	2700	228 (solid), 92 (liquid)

4.2. Results and Analysis of Numerical Simulation

Figure 8 shows the distribution of temperature in the workpiece after different pressure-keeping time. The matrix alloy LY12 solidified between 502°C and 638°C. The workpiece with 60 percent liquid phase (587°C, as illustrated in the A area in Figure 8) was regarded to deform in the same way as the pure liquid phase. Figure 8 shows that the liquid phase still exists for about 4 seconds on the condition of the pressure-keeping. The egg-shape of liquid phase A is due to the slow heat transfer in the direction contacting with mold which was heated to elevated temperature before liquid phase pouring, which means that the mold can prevent the heat flowing. Liquid-solid extrusion has to be conducted in a very short time span of 4 seconds for decreasing forming load and improving the products quality. This short operating time would increase the difficulties in controlling the forming process.

Figure 8:

Distribution of the temperature in the workpiece after (a) 15 s, (b) 17 s, and (c) 19 s.

The distribution of the flow stress in the workpiece is illustrated in Figure 9. The shape of liquid zone was determined from the coupled deformation between liquid phase and solid phase. The solid zone deformed under the hydrostatic pressure exerted by the adjacent liquid zone. At the same time, the pressure in the liquid zone became larger and larger with the extrusion proceeding because of the slight reduction in the volume of the liquid zone. When the workpiece was extruded out of the exit of the mold, the liquid zone deformed greatly along the symmetric axial. It is easily concluded that the extruded bar will be broken under the unsuitable conditions, such as too fast extrusion. So we have to control the extrusion speed suitably so as to get sound products with relatively low forming load.

Figure 9:

Distribution of the flow stress in the workpiece.

The comparison of the calculated and measured load is illustrated in Figure 10. They are coincided well. The calculated load increases dramatically at first, and after the peak, the load decreases slowly to the lowest point of the curve. At the last stage of the extrusion, the load increases again. The variation of the load relates to the state of the workpiece. Before the extrusion initiates, the load needs to increase to a threshold force to overcome the resistance from friction and diameter reduction. When the workpiece was extruded out of the exit, the liquid zone approached the exit as illustrated in Figure 9 which made the extrusion more easily. So the load began to decrease after the load peak. When the liquid zone disappeared gradually, the temperature of the extruded part was dropped down continuously which led to the increased load again. Figure 10 shows that the proposed model in this paper is reasonable and can be used to simulate the coupled deformation between liquid phase and solid phase which coexist in the liquid-solid extrusion process.

Figure 10:

Comparison of the calculated and measured load.

5. Conclusions

The hydrostatic elements were used to describe the deformation behavior of the liquid phase contained in the workpiece during the liquid-solid extrusion. The liquid zone need not be meshed through the use of hydrostatic elements which avoided the element distortion in the simulation.

It was the first time to propose the coupled mechanical modeling method for dynamic transition from liquid phase to solid phase in the liquid-solid extrusion process based on the sequential thermal-mechanical analysis. The evolution of the liquid phase and the coupled deformation between liquid phase and solid phase was clearly reflected through the proposed method.

The typical forming process, liquid-solid extrusion process which included the coupled deformation between liquid phase and solid phase, was simulated based on the model proposed in this paper. The analysis results coincided with the experiments which indicate the validity of the model.

Footnotes

Acknowledgments

The authors would like to thank the financial supports from the fund of the State Key Laboratory of Solidification Processing in NWPU (no. SKLSP201103) and the National Natural Science Foundation of China (no. 51275417).

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