Effect of annular seams on stress-strain state in cylindrical ceramic shell mold during solidification of a steel casting in it

Introduction

Investment casting is widely used to produce geometrically complex castings while maintaining dimensional accuracy.

However, this process is characterized by a relatively high rejection rate of shell molds (SMs), primarily due to micro- and macrocracking as well as partial or complete mold failure during shape formation and, most critically, during key technological operations such as firing and metal pouring, especially at the early stage of cooling. These failures result from nonuniform heating across the thickness of the shell mold. Consequently, the limited durability of SMs is mainly associated with elevated stress–strain state (SSS) levels within the mold. To mitigate these effects, a range of technological solutions is applied in industrial practice.

The stress–strain state of multilayer shell casting molds has been extensively studied by both domestic and international researchers. In particular, studies [1; 2] examined the influence of shell mold shape and geometry; works [3; 4] focused on wall thickness; investigations [5; 6] addressed the properties of mold materials; and publications [7 – 9] analyzed the influence of casting geometry. Domestic studies dedicated to this problem are presented in [10 – 13]. Similar issues have also been investigated in relation to permanent mold casting [14; 15].

The present study continues the authors’ research on the crack resistance of ceramic SMs used in investment casting. In earlier work, the SSS of cylindrical SMs during pouring with liquid metal was investigated using mathematical modeling. The theoretical analysis identified optimal physical parameters of the SM material and its morphological structure as key factors governing crack resistance. These results formed the basis for the development of new SM designs and configurations, for which several Russian Federation patents for inventions have been granted (including Nos. 2743439 and 2763359).

The theoretical investigations are based on a numerical method [16] applied to the following problem: liquid metal is poured into a multilayer SM, where it solidifies to form a casting; during subsequent cooling, the temperature field and stress–strain state in the cross-sections of the SM are determined.

At the initial stage of the study, the analysis was performed for a casting in the form of a cylinder with a spherical rounding at the lower part, which simulates a riser-type casting model within a shell mold.

Further theoretical investigations of cylindrical SMs focused on quantifying how the force interaction with the supporting filler (SF) and the interlayer friction parameters within the shell mold affect its stress–strain state (SSS) [17; 18]. These results also led to the granting of Russian Federation patents (Nos. 2769192 and 2788296).

As shown by industrial monitoring of casting-mold durability, the most unpredictable casting geometry is spherical (ball-shaped). For spherical SMs, the optimal supporting-filler (SF) coverage angle and its effect on the stress–strain state (SSS) of the SM were determined in [19].

Foreign studies [20 – 22] present numerical-method-based mathematical modeling of these processes, whereas works [23; 24] address modeling of the SSS in a solidifying casting.

The search for technological solutions to reduce critical SSS levels in SMs led to a new ceramic SM design [25]. The proposed design builds on a well-known approach for reducing thermal stresses in castings by introducing so-called stiffening ribs [26].

It was shown that the durability of spherical SMs increases when annular (temperature) seams, or recesses, are introduced on the inner surface of the mold. A similar durability increase associated with such seams has also been reported for metal casting molds.

When steel is poured into a spherical shell mold, the internal normal stresses across the cross-section are fully compressive and reach relatively high values. Accordingly, the theoretical analysis focused on conditions that reduce the absolute magnitude of these stresses. The most effective solution was a shell mold design incorporating annular seams in the inner layer (lining) [25]. In contrast, for cylindrical shell molds, the greatest risk is associated with tensile normal stresses at the mold–SF contact surface.

In the present study, we consider a ceramic shell mold with a cylindrical section and theoretically analyze the effect of temperature seams not only in the outer layer of the shell mold but also on its inner surface. A “rigid” mold configuration is examined, namely a single-layer ceramic mold with a constant shear modulus.

Mathematical formulation of the problem

An axisymmetric body of revolution is considered (Fig. 1), comprising a liquid phase (metal) (I), a solidifying shell (II), a shell mold (III), a supporting filler (IV), circular recesses a_i on the surface of the lining (surface S₂), and circular recesses b_i on the contact surface between the mold (III) and the supporting filler (IV) (surface S₃).

Let A be a finite set of circular recesses a_i on surface S₂; A = {a_i, i = 1, ..., n}; and let B be a finite set of circular recesses b_i on surface S₃; B = {b_k, k = 1, ..., m}. Let C = A \( \cup \) B. As follows from the authors’ numerous studies, the critical stresses during steel pouring into a shell mold are σ₂₂ and σ₃₃. Moreover, during steel cooling in a SM with cylindrical sections, the hazardous stresses are the tensile stresses σ₂₂ on surface S₃, whereas during steel cooling in a shell mold of spherical configuration, the hazardous stresses are the compressive stresses σ₃₃ on surface S₂.

Problem statement. Determine the minimum number and placement of recesses А on surface S₂ of the shell mold and the number of recesses В on surface S₃, as well as their geometric arrangement, such that during metal cooling in the casting mold the maximum (absolute) stresses in the domain Q at τ = τ^* remain within prescribed limits for the axisymmetric problem:

here Q is the meridional cross-section domain; τ^* is the maximum cooling time after which the temperature over the domain Q begins to equalize and the normal stresses σ₂₂ and σ₃₃ start to decrease in absolute value.

The value of τ^* is determined from the function

under the constraint τ ≤ 60 s.

To compute F, we write the governing equations in a Cartesian coordinate system for each subdomain (Fig. 1) using linear elasticity theory:

– domain I:

– domain 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 bulk compression coefficient; μ is Poisson’s ratio; E is Young’s modulus; G_p(θ) is the shear modulus in domain p (p = II, III); α_p is the linear thermal expansion coefficient; a₁ is the thermal diffusivity coefficient in domain (I); τ is time; θ is temperature; C_p is the specific heat capacity in domain (p); γ is the specific weight; \(\theta _p^*\) is the initial temperature in domain (p); λ = λ (θ) is the thermal conductivity coefficient; summation over repeated indices is implied.

During the cooling of liquid metal, under the condition that the metal temperature θ_m ≤ θ_c (where θ_c is the crystallization temperature), the thickness of the solidified layer Δ_i is determined from the solution of the phase-transition equation.

Initial conditions:

\({\left. \Delta  \right|_{\tau  = 0}} = 0\) – absence of solid phase;

\({\left. {\theta _I^*} \right|_{\tau  = 0}} = \theta _{\rm{m}}^*\) – temperature of the poured liquid metal;

\({\left. {\theta _{III}^*} \right|_{\tau  = 0}} = {\theta ^*}\) – initial mold temperature.

Boundary conditions of the problem in orthogonal coordinates (Fig. 1):

– for the axisymmetric problem

– on the axis of symmetry

U₂ = 0; σ₂₁ = 0; q_n = 0; θ = θ_m;

– on surfaces S₁, S₃, S₄

where U_sl is the displacement of the mold material during sliding relative to the SF (sand), U^* is the normalizing displacement; ψ is the friction coefficient between the mold and the supporting filler; τ_s is the conditional yield limit in shear; q_n is the heat flux.

For the thermal analysis, we used Dirichlet (first-kind) boundary conditions. To determine θ_m(τ) and θ^*(τ), the data from Ref. [27] were adopted:

here τ is the cooling time, s; \(\theta _{\rm{m}}^*\) = 1550 °С; θ₁ = 100 °C; θ₀ = 20 °С; θ^* is the temperature on surface S₃; τ₁ = 60 s; τ₁ = 1 s.

The time τ does not exceed 60 s, since for τ ≥ 60 s the stresses in the SM do not pose a risk of failure.

The shear modulus of the SM is taken as

The algorithm for solving system (4) under boundary conditions (5) – (7) is described in detail in Ref. [27].

The calculation yielded the following results:

The solution results are presented in Fig. 2 in the form of stress plots along the cross-section of the shell under consideration. The stresses σ₂₂ and σ₃₃ are highly significant. In the lining, σ₂₂ and σ₃₃ are negative and reach large magnitudes in the cylindrical part of the SM, with |σ₃₃| being approximately 1.5 times greater than |σ₂₂|. In the spherical part, the difference between σ₂₂ and σ₃₃ is smaller; however, as the stresses approach the cylindrical region, the difference becomes pronounced. In the outer layer (at the contact with the supporting filler), the stresses σ₂₂ are positive. In the spherical part, they are approximately equal in magnitude, whereas in the cylindrical part they increase toward the upper portion of the mold. The stresses σ₃₃ on this surface (surface S₃) are comparable to σ₂₂ in the spherical region and are practically zero in the cylindrical region. Fig. 2 shows that the stresses σ₃₃ and σ₂₂ arising during steel pouring into the ceramic shell mold significantly exceed (in absolute value) the limits specified in (1).

Having determined the value of τ^* from (9), we proceed to solving the formulated problem. The process of steel cooling in a ceramic SM with temperature seams (annular recesses) is considered. In contrast to the previous problem, the cross-section Q represents a multiply connected domain. The initial and boundary conditions largely coincide with those of the previous problem. Boundary conditions (6) are supplemented as shown in Fig. 1):

Relationship (7) is also satisfied for the adopted value of the shear modulus given in (8).

Solution algorithm

1. The geometric dimensions of the domain, the final cooling time τ^*, the geometric dimensions of the recesses, and their initial coordinates on surfaces S₂ and S₃ are specified: a_i(0), b_i(0). The cooling time τ^* is divided into a finite number of time steps: \({\tau ^*} = \sum {\Delta {\tau _n}} \) (where n is the time-step index).

2. The investigated domain is divided into a finite number of elements by a system of orthogonal surfaces.

3. The arc lengths of the elements are calculated \(S_{ij}^k\) (i, k = 1, 2, 3; i ≠ k; j = 1, 2).

4. Initial and boundary conditions are specified for the elements forming the considered domain ((5), (6), (10)), as well as the constants of the physical and mechanical properties of the materials.

5. The temperature field at the time step Δτ_n is determined by a numerical solution of the heat conduction equation using an iterative scheme [27], taking into account the initial and boundary conditions at the given time step. The presence of recesses was not taken into account when solving the thermal problem.

6. If the condition \({\left. \theta  \right|_{{S_2}}} \le {\theta _{\rm{c}}}\) for domain (I) at surface S₂, is satisfied, the thickness of the crystallized shell Δ_n is calculated.

7. System of equations (3) and (4) is solved taking in account boundary conditions (6) and (10), finite-difference analogs, and the developed methodology using the Odyssey software package¹. The stress fields σ_ij and displacement fields U_i (i, j = 1, 2) are determined.

8. On surface S₃ , the mold–supporting contact is assessed for each element: if \({\left. {{\sigma _{11}}} \right|_{{S_3}}} > 0 \Rightarrow {\sigma _{11}} = 0,{\rm{ }}{\sigma _{12}} = 0\), the boundary conditions are reassigned and operation 7 is performed.

9. A time step is performed. According to relations (7) the boundary conditions for solving the thermal problem are refined. If \(\sum {\Delta {\tau _n}}  < {\tau ^*}\), operation 5 is performed. If \(\sum {\Delta {\tau _n}}  = {\tau ^*}\), operation 10 is performed.

10. Over the domain Q, the values of σ₃₃ and σ₂₂ are analyzed, and the maximum values (in absolute magnitude) exceeding the constraints (1) are selected. Corresponding matrices are formed \([{\bar \sigma _{22}}],{\rm{ }}[{\bar \sigma _{33}}]\). If constraints (1) are satisfied, operation 12 is performed.

11. From the matrices \([{\bar \sigma _{22}}],{\rm{ }}[{\bar \sigma _{33}}]\), the maximum values are selected, and recesses are introduced in the corresponding cross-sections. Operation 7 is then performed.

12. The calculation process is completed.

Results of the study

Geometric parameters: S = 5 mm; R = 20 mm; h = 50 mm.

Time intervals Δτ_n: 0.01, 0.02, 0.03, 0.04, 0.05, 0.1, 0.2, 0.3, 0.4, 0.5, 2, 5, 5, 5, 3, 3, 5, 5, 5, 5 s.

The following physical parameters of the poured steel at θ > 1000 °C (\(\theta _{\rm{m}}^*\) = 1550 °С) [27]:

G = 1000 kg/mm²; α = 12·10^–6 deg^–1;

λ = 0.0298 W/(mm·°C);

L = 270·10³ J/kg (latent heat of fusion);

C = 444 J/(kg·°C); γ = 7.80·10^–6 kg/mm³; θ_c = 1450 °C.

Physical properties of the ceramic shell mold:

G = 2960 kg/mm²; α = 0.51·10^–6 deg^–1;

λ = 0.000812 W/(mm·°C);

C = 840 J/(kg·°C); γ = 2.0·10^–6 kg/mm³.

Recess dimensions: a_i = 1×2 mm, b_i = 1×3 mm.

Calculations performed using the above algorithm yielded the following results: F = 5; a_i = 2; b_i = 3. The geometric locations of the recesses (a_i, b_i) and the temperature distribution in the cross-section (x₂ = 0) are shown in Fig. 3. The resulting stresses σ₃₃ and σ₂₂ are presented in Fig. 4.

As shown in Fig. 4, all maximum values of the stresses σ₃₃ (in absolute magnitude) and the tensile stresses σ₂₂ satisfy the specified constraints (1), although in some cross-sections they are very close to the limiting values.

Conclusions

An axisymmetric problem was formulated and solved to optimize the cooling of a steel casting in a ceramic shell mold with cylindrical and spherical sections and annular temperature recesses.

The effectiveness of introducing annular recesses on the outer and inner mold surfaces in contact with the cooling metal was demonstrated.

The results can be used to analyze related processes, perform strength calculations, and support optimization studies.