ABSTRACT

In this contribution we discuss the stability of thin, axi-symmetric, shallow bimetallic shells in a non-homogeneous temperature field. The presented model with a mathematical description of the geometry of the system, displacements, stresses and thermoelastic deformations on the shell, is based on the theory of the third order, which takes into account not only the equilibrium of forces on a deformed body but also the non-linear terms of the strain tensor. The equations are based on the large displacements theory. As an example, we present the results for a bimetallic shell of parabolic shape, which has a temperature point load at the apex. We translated the boundary-value problem with the shooting method into saving the initial-value problem. We calculate the snap-through of the system numerically by the Runge-Kutta fourth order method.

1. Introduction

A bimetallic shell is a heat sensitive construction element that changes its geometry under temperature load. The Austrian manufacturer of bimetallic shells, Elektronik Werkstatte from Vienna offers on the market some differrent types of thin shallow axi-symmetric bimetallic shells, Figure 1. In practice, rotational axi-symmetric bimetallic shells of spherical shape are the most common.

At the initial stage, such a shell responds to a temperature load T by changing the flexion curvature while at a given temperature T_p it passes from a stable equilibrium position into a labile one, due to which the shell snaps-through into a new stable equilibrium position. This phenomenon is known in literature under the term ‘snap-through of the system’. The temperature T_p at which the shell snaps-through into a new equilibrium position depends on the material and geometric properties of the shell, and the method of temperature loading. Shells in a homogenous temperature field are the most studied [1-10]. With this supposition, at any given moment, each part of the shell has the same temperature, making the heat flow in the shell equal to zero:

(1)

However, the homogeneous temperature field can only be presumed when the thin shell is heated very slowly and equally across its surface. In the case of fast and spatially varying temperature changes, a mathematical model that can acceptably explain the mechanical and stability characteristics of the temperature loaded shell can no longer be created with this supposition. Hence, in this contribution we will discuss a thin, axi-symmetric, shallow bimetallic shell, which has a constant temperature load at the apex. This type of temperature load that adheres to the apex of the shell can be caused by contact

between the heated part of the device and the shell acting as a heat switch. Because of the thinness of the shell, we will assume that the temperature T across the shell crosssection r is constant at the time t:

(2)

In addition, we will ignore the heat flow between the shell and its surroundings. We will therefore consider that the shell is perfectly insulated, due to which the heat flow will only be present in the shell, running from the apex to the edge for as long as there is a temperature difference in parts of the shell.

2. Geometry of the Shell and Kinematic Equations

Figure 2 shows the rotational curvature for the delineated middle surface of the thin-walled axi-symmetric shallow shell in the initial undeformed state. The shell deforms because of the effects of the temperature load. The point P on the undeformed shell displaces into the point on the deformed shell. Since the shell is of an axi-symmetric shape and under a temperature load at the apex, the displacement of the point P into the point is determined with the displacement vector, which in this case we write with the vector in the tangential direction and vector in the normal direction on the rotational curvature, Figure 2:

(3)

In the Equation (3), the unit vectors in the tangential and normal directions on the rotational curvature are marked with and, Figure 2. The rotational curve of the deformed shell is determined by the function. The position of point on the deformed shell is determined with the coordinates, Figure 2:

(4)

(5)

The discussed bimetallic shell is shallow, so the derivative of the rotational curvature with respect to the variable x is small:

(6)

Strains occurring in the shell due to external loads must not exceed the material elastic limit, therefore, in the formation of a mathematical model we are allowed to introduce the supposition that all strains in the shell are, by absolute value, much smaller than the number one [4]:

(7)

If we consider that the rotations of the shell elements are moderate up to approximately 20 degrees [11]:

(8)

We can under the suppositions (6), (7), and (8) simplify the Equations (4) and (5), that determine the coordinates of point on the deformed shell:

(9)

(10)

With the known equation from differential geometry we also write the length of the elementary part of the undeformed shell on the rotational curvature:

(11)

The length of the elementary part of the shell we write with the displacements and [12, 13], Figure 3:

(12)

The strain of the shell element in the tangential direction is:

(13)

After we develop the Equation (13) into Taylor series and ignore the small nonlinear terms, the equation is:

(14)

Since we are discussing a thin double-layered shell, the displacement field is selected to satisfy the Kirchhoff hypothesis [11]:

1)  Strait lines perpendicular to the shell’s middle surface before deformation, remain straight after deformation;

2)  The transverse normals do not experience elongation;

3)  The transverse normals rotate, so that they remain perpendicular to the shell’s middle surface after deformation.

From Figure 4 it is clear that the displacement u is dependent on the local coordinate r, which is measured from the middle surface of the bimetallic shell in the direction of the unit vector:

(15)

The strain outside the middle surface of the shallow thin-walled bimetallic shell will thus, according to Equation (14), be:

(16)

We determine the strain in the binormal direction on the rotational curvature with Figure 5.

(17)

Outside the middle surface, at a distance r in the direction of the unit vector, this strain, after we consider the Equation (15) for the displacement u, Figure 4, will be:

(18)

3. Equilibrium Equations

Figure 6 shows the cross-section of the elementary part of the deformed shell.

Due to the temperature load, internal forces and moments occur in the shell. In the cross-section that is orthogonal with respect to the rotational curve, the normal and shear forces and the bendingmoment act, while the normal force and the bending moment act in the lateral side of the shell element:

, (19)

(20)

(21)

(22)

(23)

Where and are forces per unit of length, while and are moments per unit of length in the case of thin-walled shallow shells. Stresses in the shell are denoted by and The relation between stresses and strains is determined by Hooke’s law [12]:

(24)

(25)

In the equations for stresses (24) and (25), function denotes the temperature relative to the so-called reference temperature where the stress state throughout the undeformed shell equals to zero:

.

Therefore, the temperature in Hooke's law equations is the difference between the actual temperature and the reference temperature at which the shell is without stresses:

(26)

With this, the given integrals (19), (20), (21), (22) and (23) are:

(27)

(28)

(29)

(30)

where, , and are constants:

(31)

Now we can write the equation for the equilibrium of forces acting in the shell element in the tangential direction at point, Figure 6:

(32)

and similarly in the normal direction, Figure 6:

(33)

Due to the axi-symmetric stress and strains state the equilibrium equation in the binormal direction is identically equal to zero. Let us also write the equation for the equilibrium of moments around the point, Figure 6:

(34)

Let us consider that the differentials of unit forces and moments are:

,

,

and the equilibrium Equations (32), (33) and (34) after arranging:

(35)

, (36)

. (37)

The shear force per unit of length can be expressed from equilibrium equations. For this purpose we multiply the Equation (35) with and add the Equation (36):

.

We can ignore the last term in the equation above as according to the supposition (6):

.

With this, after arranging we obtain:

and after integrating:

(38)

We replace the shear force t_n in the equilibrium Equations (35) and (36) with the Equation (38), and the other forces and moments n_t, n_b, m_t, and m_b with the Equations (27)-(30). We replace the first and second derivative of the rotational curvature with the first and second derivatives of the Equation (10):

(39)

With this, the equilibrium Equations (35) and (36) pass into the final form:

(40)

(41)

The differential Equations (40) and (41) for determining the displacements and should be supplemented with the equations for boundary conditions. If the bimetallic shell is to serve as a thermal switch shutting down a device when it overheats, then it is necessary to assure that the shell can extend unrestrictedly in a horizontal direction [4,7]. So, in continuation, we will discuss a simply supported shell. At the boundary of the simply supported shell, the forces and moments per unit of length are equal to zero:

(42)

We use Equations (27) and (29):

(43)

(44)

The displacement at the apex of the shell is equal to zero in the chosen coordinate system:

(45)

Furthermore the reader should note that the system of Equations (40), (41), (43) and (44) have a symmetry. If the displacement vector is the solution to this system, then the solution is also:

(46)

due to which, it is sufficient that we solve the system of equations only for the positive values in the interval. For the negative, the displacement vector is defined by the Equation (46). The boundary conditions for the displacement vector also follow from Equation (46):

(47)

The remaining conditions at the edges of the shell at are defined with the unit forces and moments in the equations for boundary conditions (43) and (44).

Therefore, the boundary-value problem for the snapthrough of the system of a shallow axi-symmetric bimetallic shell is composed of the equilibrium Equations (40), (41), and the boundary conditions (43) and (44) at the point, and the boundary conditions (47) at the point:

(48)

We will calculate the temperature of the shell in the boundary-value problem (BVP) (48) with the heat equation. Since the discussed shell is thin, we assume that the temperature after the shell cross-section at the time is constant (2). In addition, if we ignore the heat flow between the shell and its surroundings, we can write the net change of heat for the shell element in Figure 7:

(49)

and the difference in heat flows on both edges of the shell:

(50)

where, and are material characteristic of the shell:

: thermal capacity;

: density;

: thermal conductivity;

: a cross-section of the of the shell element.

Since the net change of heat in the shell element is equal to the difference in the heat flow, then:

(51)

or after inserting the Equations (49) and (50), and arranging:

(52)

We supplement the partial differential equation of the second order (52) with two boundary and one initial condition, so that the initial-boundary-value problem (IBVP) for determining the temperature state in the shell is:

(53)

4. Numerical Solution

We first translated the BVP (48) for the shell loaded at the apex with the temperature into the system of five differential equations of the first order, in which the current shell temperature is determined by solution of the IBVP (53).

We solved this system of equations by first translating, with the non-linear shooting method, the solving of the boundary value problem into the solving of the initial value problem, which we then solved with the classic one-step Runge-Kutta 4^th order method [14,15]. The parameter, represents the relation between the height of the deformed shell and the height of the undeformed shell:

(54)

Therefore let us observe the stability conditions during the temperature loading of shells that have the following material and geometric characteristics:

(55)

Figure 8 shows the process of heating the bimetallic shell with an initial height in the undeformed state. At the time we start to heat the shell at the apex with the constant temperature, Figure 8(a). Because of the temperature exerted at the apex of the shell, the temperature of the neighbouring points starts to rise while at the same time the shell deforms. Here, we observed that, at the start, the shell height h increases somewhat, and after a given time, in an actual example, the shell height starts to decrease. The relation of heights at which the shell becomes unstable is denoted by a black dot in Figure 9. At that moment the shell snaps-through into a new stable equilibrium position. The shape of the deformed shell and the temperature profile at the moment of the shell snap-through are shown in Figures 8(b) and 8(c). Figures 10-12, show these characteristics for shells with the initial height of, and.

We also estimated the error e of the described numerical procedure. For this purpose we calculated the shell heating time t when the current height h again becomes equal to the initial shell height. The results at different step size are written in Table 1. The difference between two successive results decreases each time for approximately one half. Thus, we can estimate the error of the numerical procedure with the equation:

(56)

Therefore, the obtained results for the time t are correct to one decimal place even at a relatively large step size at For estimating the error e, it is enough if we reduce the step by half and calculate a new, more accurate value. The difference with the previous result by the Equation (56) does approximately estimate the error e during calculation.

In Table 2, the reader can look for the maximal ratio between the height h of the deformed, and the initial height of the undeformed shell, and the shell heating time t necessary for this deformation. The maximal height h at which a shell can deform is also evident in Figure 9 as the extreme of function.

The moment, at which the shell snaps-through into a new stable equilibrium position, is dependent on the material and geometric characteristics of the shell, as well as temperature load at the apex of the shell. The values for time when the shell snaps-through into a new equilibrium position are written in Table 3. With the increase in the temperature load, the time needed for snap-through to occur is decreased. With the interpolation of results in Table 3 we approximately calculated the values for the snap-through time for the intermediate values of temperature loads. These results are graphically presented in Figure 13.

5. Conclusions

Simply supported thin-walled, shallow, bimetallic shells are distinguished by the property that at a given temperature they become unstable, the result of which is the snap-through of the shell from a convex into a concave shape, or vice versa, and the establishment once more of a stable equilibrium state. Due to this property, such shells are used for safeguarding various devices and machines from excessive temperature loads. With a suitable technical implementation, these shells can be connected to electrical contacts so that they cut-out at the moment when given parts of the safeguarded device overheat. Since the shell snap-through is a reliable, repeatable, and dynamic occurrence lasting around thin-walled, shallow, bimetallic shells are very suitable for the thermal protection of devices, as apart from reliability, at snap-through they also prevent undesired and harmful sparking of electric contacts in the thermal switch. If the

the safeguarded device, and with it the shell as a protective element, is heated slowly, we can legitimately assume that the shell is located in a homogenous temperature field. Therefore, such a temperature load case can be legitimately assumed only when the thin-walled shell is heated very slowly and equally across its surface. If the temperature changes are fast and irregular, we cannot presuppose a homogenous temperaure field. In this contribution, we discussed the problem of heating a thinwalled, shallow bimetallic shell at its apex. The consequence of the current temperature load at the apex of the shell is the increase in height in the initial stage. Only after a given time, does this height start to decrease. The cause of this occurrence, not observed in shells in a homogenous temperature field, is that the area around the apex of the shell heats up very quickly with the temperature load, while the area at the edge of the shell remains cold. Due to this, the shell somewhat ‘swells up’ at the start, all the more the greater the temperature load at the apex of the shell. Within this lies the reason that the shell temperature at snap-through is at each point greater than the snap-through temperature of a shell with the same material and geometric characteristics in a homogenous temperature field.

When using these shells in practice, it is necessary to determine how the heating time (enough for the shells to become unstable and snap-through) is relative to the material and geometric characteristics of the shell and the temperature load at its apex. The larger the temperature load is, the quicker does the shell snapthrough occur, and the  lower is the initial height of the shell. Yet it is important to note, that very shallow bimetallic shells with a low initial height do not even have a snap-through. [9]. In the case of the shell with the characteristics (55), the critical value of the initial height, is when the snap-through of the shell is not possible anymore due to temperature loading. If the temperature load is equal to the snapthrough temperature of a shell located in homogenous temperature field, then theoretically, such a shell should be heated infinitely. On the other hand, the high temperature has as a result, a short time for the shell to snap-through. Apart from the temperature load, and the appropriate initial height, it is also necessary to ensure that the edge of the shell can expand unrestrictedly in a horizontal direction for the snapthrough of the shell to occur. Otherwise the shell cannot serve the function of a thermal switch.

6. REFERENCES