Stress-strain state of ceramic shell mold during formation of spherical steel casting in it. Part 1

Introduction

Numerous analytical [1; 2] and theoretical studies [3 – 5] have investigated the crack resistance of a ceramic shell mold (CSM) after liquid metal is poured into it and the solidifying casting is cooled. A glass-shaped shell mold bounded by spherical and cylindrical surfaces was studied. It was established that the most dangerous stresses that arise when liquid metal is cooled in the shell mold are normal tensile stresses on the outer surface of the mold adjacent to the support filler. The researchers determined the optimal external force and temperature influences on the shell mold to guarantee its durability during steel casting production. Additionally, they studied the morphological structures of shell molds that can withstand the thermal stresses of cooled castings and proposed new designs.

Numerous theoretical and experimental studies have been conducted to establish the characteristics of the stress-strain state (SSS) of a ceramic shell mold and the resulting castings in investment casting, and the SSS dependence on factors such as the consumable pattern [6; 7], the shape and geometry of the ceramic shell mold [8; 9], mold wall thickness [10; 11], mold material [12; 13], geometry of castings [14 – 16], and methods of mold strength tests [17; 18].

Other works describe mathematical modeling of these processes, including modeling methods [19], research methods [20 – 22], studies based on numerical modeling [23 – 25], special mathematical models [26 – 28], and software [29; 30]).

Further theoretical studies have shown that the durability of shell molds largely depends on their shape, which is organically related to the geometry of the casting formed in them.

However, practically no studies have investigated the modeling of the crack resistance of a ceramic shell mold based on the quantitative and qualitative indicators of its stress-strain state as a steel casting in the shape of a sphere (ball) is formed in it. This issue is addressed in our study.

We conducted a theoretical study related to steel casting production in a spherical shell mold. A variety of parts have spherical shapes, including spherical joints, which are crucial components in mechanical engineering and robotics.

The first theoretical results of the investigated manufacturing process were published in [31], where it was clearly shown that the stress-strain state in a spherical shell mold is drastically different from that in a cylindrical shell mold when producing a steel casting. However, the paper [31] does not present a mathematical model of the process.

This study shows that high-quality billets for spherical joints can be obtained using casting, which is much cheaper than metal treatment under pressure.

Engineering problem statement

Liquid steel is poured into a spherical mold, where it crystallizes by removing heat through the shell mold walls via the support filler (Fig. 1, а). A spherical shell mold can be monolithic or consist of several layers [1], each with its own physical and mechanical properties. When steel is cooled in a shell mold, temperature stresses arise in the wall due to a large temperature gradient. Under certain external influences, these stresses can lead to wall destruction, resulting in the rejection of the steel casting. Therefore, the objective of this theoretical study is to determine the external factors under which the spherical shell mold will not be destroyed by the temperature stresses arising in it.

Mathematical problem statement

An axisymmetric rotational body is under consideration. The deformable material is considered isotropic, and the movement is deemed slow.

We have a four-component system (Fig. 1, b). The deformable medium includes the solidified metal (region II) and the mold (region III), both of which are isotropic materials. The process is non-stationary. Using the theory of elasticity and the Eulerian reference frame, we write a system of equations for each region:

‒ for region I:

‒ for regions II, III:

where σ_ij are the components of the stress tensor; σ is the hydrostatic stress; ε_ij are the components of the elastic strain tensor; h is the height of the liquid metal column; \({k_p} = \frac{{1 - 2\mu }}{E}\) is the coefficient of volume compressibility; μ is Poisson’s ratio; E is Young’s modulus; G_p is the shear modulus in region p (II, III); α_p is the linear expansion coefficient; a₁ is the temperature conductivity coefficient in the region I; τ is time; θ is temperature; C_p is the specific heat in the region p; γ is density; \(\theta _p^*\) is the initial temperature in the region p; λ = λ(θ) is heat conduction; \({\sigma _{ij,j}} = \frac{{\partial {\sigma _{ij}}}}{{\partial {x_j}}};{\rm{ }}{u_{i,j}} = \frac{{\partial {u_i}}}{{\partial {x_j}}};\) and summation over repeated indices is performed.

In accordance with axial symmetry, we consider the meridian section (Fig. 1, b).

If θ_m ≤ θ_cr (θ_m and θ_cr are the temperatures of the metal and crystallization), as the liquid metal is cooled, its temperature is determined by the thickness of the solidified layer Δ_i from the solution of the interphase transition equation [5].

Initial statement of the problem:

Δ|_τ = 0 = 0 (absence of the metal solid phase);

\(\theta _I^*\)|_τ = 0 = 0 = \(\theta _{\rm{м}}^*\) (temperature of the poured liquid metal);

\(\theta _{III}^*\)|_τ = 0 = θ\(^*\) (initial mold temperature).

Boundary conditions of the problem (Fig. 1, b):

‒ on the axis of symmetry: U₂ = 0; σ₂₁ = 0; q_n = 0;

‒ on surfaces S₁, S₃, S₄

where U_ck is the sliding of the mold material relative to the sand; \(U^*\) is the normalizing displacement; and ψ is a parameter characterizing the friction conditions between the mold and the support filler.

The numerical method was used [32] to solve the system (2). According to this method, the deformation region is divided into a finite number of orthogonal curved elements (Fig. 2, а).

With axial symmetry, we have σ₃₁ = σ₃₂ = 0; σ₁₃ = σ₂₃ = 0; U₃ = 0.

In accordance with [32], the equations (2) and the values ε_ii taking into account axial symmetry, will be written as follows:

where \({U_i} = U_i^1 + U_i^2,{\rm{ }}\Delta {U_i} = U_i^2 - U_i^1,\) (i = 1, 2); \({S_{ij}} = S_{ij}^1 + S_{ij}^2;{\rm{ }}\Delta {S_{ij}} = S_{ij}^2 - S_{ij}^1.\)

The adopted symbols are described in [1; 7].

The equations (4) – (7) are written taking into account that \(\frac{{\partial {U_1}}}{{\partial {x_3}}} = 0,{\rm{ }}\frac{{\partial {\sigma _{3i}}}}{{\partial {x_3}}} = 0;\) i = 1, 2, 3; for rotational bodies ΔS₃₁ = 0; ΔS₃₂ = 0; \(\frac{{\Delta {U_1}}}{{{S_3}}} = 0;{\rm{ }}\frac{{\Delta {U_2}}}{{{S_3}}} = 0;\) U₃ = 0; on the surface x₁x₃: \(S_2^ +  - S_2^ -  = 0;\) on the surface x₂x₃:  \(S_2^ +  - S_2^ -  = 0;\) shift values ε_ij (i ≠ j) shall be written for the node (0) (Fig. 2, d) in the form

where \({S_i} = S_i^1 + S_i^2;{\rm{ }}\Delta {\bar U_l} = {\bar U_{{l_2}}} - {\bar U_{{l_1}}};{\rm{ }}S_i^ +  = S_i^{1 + } + S_i^{2 + };\) \(S_i^ - = S_i^{1 - } + S_i^{2 - };\) \({\bar U_l}\) is the average of the value U_i along the element edges.

The authors of [32] prove that if there are initial and boundary conditions, the difference analogue of the system (4) – (6), taking into account the equation (7), is definable. The order of the system (4) – (6) is significantly reduced when the following operations are performed.

1. The differences (σ_ij – σ_jj) in equations (4) are expressed by formula (5).

2. The mass conservation equation is rewritten in the recurrent form, taking into account expressions (7), in the form \(U_1^2\) = \(U_1^1\) + [A]; where [A] is an operator that does not contain \(U_1^2\); direction of bypassing the area along x₁ (→), along x₂ (↑).

3. Shift expressions ε_ij (i ≠ j) are determined based on internal grid nodes in accordance with formulas (8); i = 1, j = 2.

4. The values σ_ij (i ≠ j) are determined based on the internal grid nodes from the equations of state \(\sigma _{12}^0 = G_p^0\varepsilon _{12}^0.\)

5. The values σ_ij are determined based on the external grid nodes from the boundary conditions, and on the contact surfaces – from the friction law.

6. σ_ij are determined based on the element edges as the average of the values σ_ij at the edges’ nodes.

7. The first equation (4) is rewritten in the recurrent form \(\sigma _{11}^1\) = \(\sigma _{11}^2\) + [Б]; where [Б] is an operator that does not contain \(\sigma _{11}^1\); direction of bypassing the area along x₁ (←), along x₂ (↓).

8. From the system of equations (the second equation in the system (4) and the equation σ₂₂ – σ₁₁ = 2λ(ε₁₁ – ε₂₂)) the values \(\sigma _{22}^1\) and \(\sigma _{22}^2\) for the element are determined, the equations of the form \({F_3} = {(\sigma _{22}^2)_{IJ}} - {(\sigma _{22}^1)_{IJ + 1}} = 0\) composed for the internal edges (where J is the element index on the coordinate x₂).

Thus, if we consider \(X = \left\{ {{U_2},} \right.{\left. {{\rm{ }}{U_1}} \right|_{{x_1} = 0}},{\left. {{\rm{ }}{\sigma _{11}}} \right|_{{x_1}}} = \left. {x_1^*} \right\},\) independent variables, the sequence (1) ‒ (7) can be used to determine the dependent variables through X (\(x_1^*\) is the final value of the coordinate x in the curved region.

The equivalent system of equations looks as follows

where \(U_1^*\)is known from the boundary conditions of displacement U₁ at the boundary of the region (x₁ = \(x_1^*\)); \(\sigma _{11}^*\) is known from the boundary conditions of stress σ₁₁ at the boundary of the region (x₁ = 0).

There are as many equations F₁ = 0 as there are unknown variables \({\left. {{\sigma _{11}}} \right|_{{x_1} = x_1^*}},\) and as many equations F₂ = 0 as there are unknown variables \({\left. {{U_1}} \right|_{{x_1}}}\) = 0.

The coefficients and free terms of the new equivalent system of equations (9) can be found using the following procedure.

Suppose the equivalent system of equations looks as follows

If we assume that all unknown variables are equal to zero (x_i = 0, i = 1, ..., n), based on the above sequence (1) – (7) and the calculation of \({\bar F_i}\) by the formulas (9), we will find the free terms of the new system (10):

\[{\bar F_i}^0 = {b_i};{\rm{ }}i = 1,{\rm{ }}...,{\rm{ }}n.\]

Next, we find the coefficients a_ij . To do so, we calculate x_k = 1, x_i = 0 (i ≠ k; i = 1, ..., n). Using the above sequence, we find the value \(\bar F_i^k\) and a_ik by the following formula:

\[{a_{ik}} = F_i^k - F_i^0,{\rm{ }}i = 1,{\rm{ }}...,{\rm{ }}n.\]

Thus, we determine the entire matrix a_ik of the new equivalent system, which is solved using a standard program. The order of the equivalent system is reduced by approximately 10 times compared to the original one.

The heat conduction equation is solved using a numerical method [1; 32]. Under the method being considered, for each internal k\(^\rm{th}\) element (Fig. 2, а) the heat conduction system is written from the heat balance in difference form, taking into account axial symmetry, and an iterative procedure is constructed, which, given that the heat flow along x₃ is equal to zero, is represented by the iterative formula:

where \(\theta _k^*\)is the average temperature in the k\(^\rm{th}\) element at the beginning of the time step Δτ; λ_k, θ_k, C_k, γ_k are the average heat conduction, temperature, heat capacity, and density in the k\(^\rm{th}\) element at the end of the time step Δτ; \(\lambda _i^ - ,{\rm{ }}\theta _i^ - \) and \(\lambda _i^ + ,{\rm{ }}\theta _i^ + \), (i = 1, 2) are the heat conduction and temperature in the element following the element k on the coordinate x_i in the negative and positive directions x_i; \(S_{21}^ - = S_{21}^{1 - } + S_{21}^{2 - };\) \(S_{21}^ + = S_{21}^{1 + } + S_{22}^{2 + };\) \(S_{ij}^{1 + }\) (i ≠ j; i, j = 1, 2) and \(S_{ij}^{1 - }\) are the length of the arc \(S_{ij}^1\)of the element following the element k in the positive and negative directions on the coordinate x_j.

The authors of [32] proved the iteration procedure convergence (11).

Algorithm for solving the problem

1. The cooling time τ\(^*\) is divided into a finite number of steps: \({\tau ^*} = \sum {\Delta {\tau _n}} ;\) where n is the number of the time step.

2. The region under study is divided into a finite number of orthogonal elements.

3. The initial and boundary conditions for the elements forming the investigated region and the constants of the physical and mechanical properties of the materials are established.

4. The lengths of the elements’ arcs are calculated \(S_{ik}^j\) (i, k = 1, 2; i ≠ k; j = 1, 2).

5. The temperature field at the time step Δτ_n is determined by the numerical solution of the heat equation using the iterative formula (11) if there are initial and boundary conditions at the investigated time step.

6. If the temperature in the region I near the surface S₂ \({\left. \theta  \right|_{{S_2}}} \le {\theta _k},\) the thickness Δ_n of the crystallized crust is calculated [5]. If \({\left. \theta  \right|_{{S_2}}} > {\theta _k},\) the following operation is performed.

7. The system of equations (2) is solved taking into account difference analogues (4) – (7) and the developed procedures [1; 32] described above. The fields of stress σ_ij and displacements U_i (i, j = 1, 2) are determined.

8. Time steps are taken; if \(\sum {\Delta {\tau _n}} < {\tau ^*},\) the operation 4 is performed; if \(\sum {\Delta {\tau _n}} = {\tau ^*}\) he calculation process is completed.

As the temperature problem was solved, boundary conditions of the first kind (3) were used. To determine θ_m(τ) and θ\(^*\)(τ) we will use the experimental data from [1]. Approximation of these values yields the following result:

\[\begin{array}{c}{\theta _{\rm{м}}} = 1550 - 1.666\tau  - \frac{{\tau (60 - \tau )}}{{10 + {\tau ^2}}};\\\theta  \le \tau  \le 60{\rm{ s}};\\{\theta ^*} = 20 + 17.3\sqrt \tau  ;\end{array}\]

where τ is the cooling time, s.

The time τ does not exceed 60 s, since at τ ≥ 60 s the stresses in the shell mold plummet and cannot cause its destruction.

A mathematical model has been developed to determine the stress-strain state and temperature in the shell mold when a spherical casting is cooled in it. This model was used in a numerical solution of the problem on modeling the crack resistance of a spherical shell mold.

Conclusions

The first theoretical attempt was made to formulate and solve the problem of determining external factors at which a spherical shell mold will not fail due to temperature stresses occurring in it.

Based on the fundamental equations of the theory of elasticity and numerical methods, a numerical scheme and algorithm for solving the problem were developed.

The proposed method for modeling the crack resistance of a spherical shell can be recommended for other functional shells.