Stability of the Shallow Axisymmetric Parabolic-Conic Bimetallic Shell by Nonlinear Theory

Abstract

In this contribution, we discuss the stress, deformation, and snap-through conditions of thin, axi-symmetric, shallow bimetallic shells of so-called parabolic-conic and plate-parabolic type shells loaded by thermal loading. According to the theory of the third order that takes into account the balance of forces on a deformed body, we present a model with a mathematical description of the system geometry, displacements, stress, and thermoelastic deformations. The equations are based on the large displacements theory. We numerically calculate the deformation curve and the snap-through temperature using the fourth-order Runge-Kutta method and a nonlinear shooting method. We show how the temperature of both snap-through depends on the point where one type of the rotational curve transforms into another.

1. Introduction

The development of machine sciences in recent centuries has led to the manufacture of various devices from relatively simple mechanisms to the very complex mechanical devices used by mankind in the process of transforming material goods. Although modern equipment comes in very different forms, functions, and structure, owing to the importance of smooth, reliable operation and their value, a demand for protection against a number of overloads is expressed. It is especially necessary to provide reliable protection against thermal overload for machines that convert one form of energy into another and heat up in the process. For this purpose, elements are built into devices to serve as a “thermal fuse” turning the machine off as soon as an individual part reaches the maximum permissible temperature. Due to their operational reliability, both line and plane bimetallic structural elements are used in protection against thermal overload, whose operation is based on the known physical fact that bodies expand with the increase of temperature. With a suitable technical act, we can connect a bimetallic construction element, Figure 1, with electrical contacts and make a so-called “thermal switch”. Displacements created on the bimetallic element due to a combination of temperature and mechanical loads turn the device on and off dependent on the temperature 𝑇. For this, it is of course necessary to know the connection between the displacements and loading of the bimetallic construction element. Apart from material properties, this connection is also dependent on the geometric properties of the bimetal, as the line bimetallic construction elements respond with different stresses and deformational states to temperature loads as compared to plane construction elements. In practice, the difference in the stability conditions is the most important. Thin and shallow bimetallic shells with suitable material and geometric properties have the characteristic of snapping-through into a new equilibrium position at a certain temperature. The result of such a fast snap-through of a bimetallic shell acting as a switching element in a thermal switch is the instantaneous shutdown of electric power and the machine. The snap-through of the bimetallic shell is a dynamic occurrence that lasts a very short time and as such prevents the damaging sparking and melting of electric contacts and extends the life expectancy of the thermal switch.

Figure 1 

Some of the different types of bimetallic shells made by the Austrian manufacturer “Elektronik Werkstatte”.

Panov, Timoshenko, Videgren, Witrick, Aggarwala, Saibel, Huai, Vasudevan, Johnson, Keller, Reiss, Brodland, Cohen, Kosel, Batista, Drole, Jakomin, and some other authors have engaged in the research of bimetallic shell elements in a homogenous temperature field. S. Timoshenko [1] researched the problem of the stability of bimetallic lines and plates. The authors Panov [2] and Wittrick et al. [3] researched the problem of the stability for shallow axi-rotational symmetric spherical shells during heating. The related problem of stability of a spherical shell under normal loads has been treated by Keller and Reiss [4]. The problem of thermal stability of a multimetallic strip has been treated by Vasudevan and Johnson in [5]. Aggarwala and Saibel researched the thermal stability of thin spherical shells [6]. The occurrence of the snap-through of an open bimetallic shell was treated by Ren Huai [7] using approximative methods. The problem of finite axisymmetric deflection and snapping of spherical shells which are point loaded at the apex and simply supported at the boundary is analysed by Brodland and Cohen, in [8]. Kosel et al. researched the stability conditions during the temperature and mechanical loading of rotational axisymmetric [9–12] and translation shells [13, 14] with and without an opening at the apex of the shell.

Apart from the spherical and parabolic bimetallic shells, the market also offers shells with more complex initial geometries. “Elektronik Werkstatte” the manufacturer of bimetallic shells from Eichgraben in Austria offers several different types of combined axisymmetric shells (Figure 1, the first and the second shell from the left in the top row) that have an advantage over spherical shells in that the temperature range of the snap-through (the difference in temperature of the upper and lower snap-through) can be changed at the constant of the initial bimetallic shell height. Even with a relatively small initial shell height, using a suitable combination of a parabola and cone rotational curve, it is possible to achieve snap-through of the shell only at very high temperature loads. Due to the smaller initial height, it is important that the stresses in these shells are smaller relative to the stresses in parabolic or spherical shells.

So, this time, we are discussing the stability and deformation conditions for a thin axisymmetric shallow bimetallic shell, composed of a parabola and cone and a plane and parabola. Apart from the temperature, the shell is additionally burdened with a force at the apex and with pressure. When executing a nonlinear mathematical model for the snap-through of a bimetallic translation shell, we will assume a small strain and the moderate rotation of the shell element. In the strain tensor, we will also consider nonlinear terms, while placing equilibrium equations on the deformed part of the shell. For the first time, we will show in both table and graphic form how the temperature of both snap-through depends on the point where the parabolic shape of the rotational curve changes into a cone shape. We will explain the effect of the concentrated force at the apex for the example of a bimetallic shell of the plate-parabola type!

The defined thermoelastic problem will be solved with the following steps:(1)defining the geometry of the undeformed shell,(2)deriving the displacement vector as relation between the geometry of the undeformed and the deformed shell,(3)defining the geometry of the deformed shell,(4)deriving the elements of the strain and stress tensor,(5)introducing the forces and moments per unit of length,(6)deriving the equilibrium equations of unit forces and moments acting in the deformed shell element, and(7)calculating the deformation curve and the snap-through temperature using the fourth-order Runge-Kutta method and a nonlinear shooting method.

2. Geometry of the Undeformed Shell

The axisymmetric shape of the undeformed shell is formed by rotation of the curve 𝑦=𝑦(𝑥), about the 𝑦 axis [15], Figure 2. The middle surface of the undeformed axisymmetric shell in the Lagrange Coordinate System is, therefore, defined by the function 𝑦=𝑦(𝑥). Figure 3 shows the middle surface of a thin rotationally symmetric bimetallic shell.

Figure 2 

Geometry of the undeformed shell and basis unit vectors.

Figure 3 

The axisymmetric shell in a homogeneous temperature field and the connection between the Euler and Lagrange coordinate system.

From Figures 2 and 3, we can obtain the geometric properties of the undeformed shell.

Differential of the length in the meridian direction 𝑑𝑠_𝜓:𝑑𝑠_𝜓=𝑟_𝜓𝑑𝜓.(2.1) Differential of the length in the circular direction 𝑑𝑠_𝜑 is𝑑𝑠_𝜑=𝑟_𝜑sin𝜓𝑑𝜑=𝑥𝑑𝜑.(2.2) Meridian angle 𝜓𝜓=arctan𝑑𝑦𝑑𝑥=arctan𝑦^≈𝑦^.(2.3) The flexion curvature 1/𝑟_𝜓 of the rotational curve 𝑦(𝑥) in the meridian direction is, Figures 2 and 31𝑟_𝜓=𝑦^1+𝑦^2³≈𝑦^.(2.4) The flexion curvature 1/𝑟_𝜑 of the circle formed by rotation of the curve 𝑦=𝑦(𝑥), in the circular direction is, Figures 2 and 3,1𝑟_𝜑=sin𝜓𝑥=sinarctan𝑦^𝑥=𝑥1√1+𝑦^(𝑦^)²≈𝑦^𝑥.(2.5) The simplifications in (2.3), (2.4), and (2.5) are justified since we are discussing a shallow shell, where,𝑦^(𝑥)²≪1,(2.6) and consequently, sin𝜓≈𝜓,sin𝜑≈𝜑,cos𝜓≈1.(2.7)

3. Displacement Vector and Geometry of Deformed Shell

Due to the temperature change, the shell deforms into a new shape defined by the function 𝑌(𝑋) in the Euler Coordinate System. Since we are discussing a thin double-layered shell, the displacement field is selected to satisfy the Kirchhoff hypothesis [16]:(1)straight 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.

The reader should note that the Kirchhoff hypothesis stating that the thickness of the shell before and after deformation remains the same when the shell is subjected to kinematic constraint is not realistic when large strains are admitted in the deformation process [17]. The so-called Zig-Zag theory is known in literature and describes a piecewise continuous displacement field in the direction of the multilayered shell thickness and accomplishes the continuity of transverse stresses at each layer [18–20]. However, given the fact that the ratio between the thickness and length of bimetallic shells, used as safety constructional elements against temperature overheating, is about 1/100, we found the Kirchhoff hypothesis is fully acceptable for our mathematical model.

The new shape of the shell in a deformed state is axisymmetric due to the homogeneous temperature field and axisymmetric mechanical load. Therefore, the displacement 𝑣 in the circular direction as well as other shell properties relative to the angle 𝜑 do not change:𝑣=0,𝜕𝜕𝜑()=0.(3.1)

The displacement vector 𝑢⃗ of any point 𝑃 on the middle surface of an undeformed shell defines the point 𝑃^ on the middle surface of a deformed shell𝑢⃗=𝑢𝑒_𝜓+𝑣𝑒_𝜑+𝑤𝑒_𝑟⃗,(3.2) and with the supposition (3.1),𝑢⃗=𝑢𝑒_𝜓+𝑤𝑒_𝑟⃗.(3.3) Unit basis vectors 𝑒_𝜓,𝑒_𝜑 and 𝑒_𝑟⃗ in the Cartesian coordinate system follow directly from the system geometry, Figure 2,𝑒_𝜓=(cos𝜓𝑒_𝜑cos𝜑)𝑖⃗+(sin𝜓)𝑗⃗+(cos𝜓sin𝜑)𝑘⃗,𝑒_𝑟⃗=(−sin𝜑)𝑖⃗+(cos𝜑)𝑘⃗,=(sin𝜓cos𝜑)𝑖⃗−(cos𝜓)𝑗⃗+(sin𝜓sin𝜑)𝑘⃗.(3.4) These vectors are mutually orthogonal because𝑒_𝜓𝑒_𝜑=𝑒_𝜓𝑒_𝑟⃗=𝑒_𝜑𝑒_𝑟⃗=0.(3.5) Derivatives of these vectors with respect to the curvilinear coordinates 𝜓,𝜑, and 𝑟 are [21]𝜕𝑒_𝜓𝜕𝜓=(−cos𝜑sin𝜓)𝑖⃗+(cos𝜓)𝑗⃗−(sin𝜑sin𝜓)𝑘⃗=−𝑒^𝑟⃗𝜕𝑒_𝜓,𝜕𝜑=(−cos𝜓sin𝜑)𝑖⃗+0𝑗⃗+(cos𝜑cos𝜓)𝑘⃗=𝑒^𝜑𝜕𝑒_𝜓cos𝜓,𝜕𝑟=0,⃗𝜕𝜕𝑒𝜓^𝜑=0,⃗𝜕𝜕𝑒𝜑^𝜑=(−cos𝜑)𝑖⃗+0𝑗⃗+(−sin𝜑)𝑘⃗=−𝑒^𝑟⃗𝜕𝑒_𝜑sin𝜓−𝑒_𝜓cos𝜓,𝜕𝑟=0,⃗𝜕𝜕𝑒𝜓^𝑟⃗=(cos𝜑cos𝜓)𝑖⃗+(sin𝜓)𝑗⃗+(cos𝜓sin𝜑)𝑘⃗=𝑒^𝜓𝜕𝑒_𝑟⃗,𝜕𝜑=(−sin𝜑sin𝜓)𝑖⃗+0𝑗⃗+(cos𝜑sin𝜓)𝑘⃗=𝑒^𝜑sin𝜓,𝜕𝑒^𝑟⃗𝜕𝑟=0,⃗(3) and when simplified due to the supposition(2.7)𝜕𝑒_𝜓𝜕𝜓=−𝑒^𝑟⃗,𝜕𝑒^𝜓𝜕𝜑=𝑒^𝜑,𝜕𝑒^𝜓𝜕𝑒_𝜑𝜕𝑟=0,⃗𝜕𝜓=0,⃗𝜕𝜕𝑒𝜑^𝜑=−𝜓𝑒^𝑟⃗−𝑒_𝜓,𝜕𝑒^𝜑𝜕𝑒_𝑟⃗𝜕𝑟=0,⃗𝜕𝜓=𝑒^𝜓,𝜕𝑒^𝑟⃗𝜕𝜑=𝜓𝑒^𝜑,𝜕𝑒^𝑟⃗𝜕𝑟=0.⃗(3.7) Now, let us observe displacements on a thin shallow axisymmetric bimetallic shell in a homogenous temperature field, Figure 3, which is additionally loaded with the force 𝐹⃗ at its apex.

The point 𝑃 at the position 𝑃(𝑥,𝑦(𝑥)) on the undeformed shell moves into the position 𝑃^ on the deformed shell with the coordinates 𝑃^(𝑋,𝑌(𝑋)). The reader should note that both the Euler (𝑋,𝑌(𝑋)) and Lagrange (𝑥,𝑦(𝑥)) coordinate system have the same origin at point (0,0). So, when a force or temperature load is exerted on the shell the optional point 𝑃 moves, as we have shown in Figure 3, since the reaction force per unit of length 𝑉_𝑒 acts in the opposite direction. The connection between the Euler (𝑋,𝑌(𝑋)) and Lagrange (𝑥,𝑦(𝑥)) Coordinate System is, Figure 3,𝑋⃗=𝑥⃗+𝑢,⃗(3.8) where𝑢⃗(𝑥,𝑦)=csoins𝜓𝜓−scinos𝜓𝜓𝑤𝑢,(3.9) from which we obtain𝑋=𝑌𝑥=+𝑦𝑤−s𝑤inc𝜓os+𝜓𝑢+co𝑢ss𝜓in≈𝜓𝑥≈+𝑦𝑤−𝑦𝑤^+.𝑢,((33..1101)) In (3.10), we also consider that the displacement 𝑢 is small in comparison with the displacement 𝑤, which in turn is small in comparison with the Lagrange coordinates 𝑥 so that the Euler coordinates 𝑋 is approximately𝑋=𝑥+𝑤𝑦^+𝑢≈𝑥.(3.12) From Figure 3, we can also find geometric properties of the deformed shell.

The differential of the length 𝑑𝑠_𝜓 in the meridian direction𝑑𝑠_𝜓=𝑟_𝜓𝑑𝜓=√𝑑𝑋²+𝑑𝑌²=𝑑𝑥√𝑋^2+𝑌^2.(3.13) Meridian angle 𝜓 is𝜓=arctan𝑑𝑌𝑑𝑋≈𝑑𝑑𝑋𝑌≈𝑑𝑑𝑋𝑌𝑑𝑑𝑥𝑥=𝑋𝑌^≈𝑌^.(3.14) Flexion curvature 1/𝑟_𝜓 in the meridian direction is1𝑟_𝜓=||^{𝑋′𝑋𝑌′′𝑌′′}||𝑋^2+𝑌^2³≅𝑌^.(3.15) Flexion curvature 1/𝑟_𝜑 in the circular direction1𝑟_𝜑=sin𝜓𝑋≈𝜓𝑋≅𝑑𝑑𝑋𝑌𝑋1≅𝑌𝑋^.(3.16) With (3.11) and (3.12) for Euler’s coordinates 𝑌 and 𝑋, we finally obtain1𝜓=𝑦^−𝑤^,𝑟_𝜓1=𝑦^−𝑤^,𝑟_𝜑=𝑦^−𝑤^𝑥.(3.17)

4. Strain and Stress Tensor

A shell’s deformation state is shown by the displacement vector 𝑢⃗ in the middle, that is, reference surface. This vector has two components: the displacement 𝑢 in the meridian direction and the displacement 𝑤 in the radial direction. The elements of the strain tensor 𝐄 in the curvilinear orthogonal coordinate system are determined by the Green-Lagrange strain tensor 𝐄 for the middle surface of the shell [16]𝐄=𝐄^𝑇=12∇⃗⊗𝑢⃗+∇⃗⊗𝑢⃗^𝑇+∇⃗⊗𝑢⃗⋅∇⃗⊗𝑢⃗^𝑇,(4.1) where the displacement vector 𝑢⃗ is of the form (3.3). The vector operator ∇⃗ is by definition ∇⃗=𝑒_𝜓𝜕𝜕𝑠_𝜓+𝑒_𝜑𝜕𝜕𝑠_𝜑+𝑒_𝑟⃗𝜕𝜕𝑧=𝑒^𝜓1𝑟_𝜓𝜕𝜕𝜓+𝑒^𝜑1𝑥𝜕𝜕𝜑+𝑒^𝑟⃗𝜕𝜕𝑧.(4.2) The gradient of the displacement vector 𝑢⃗, while keeping in mind the derivatives of the unit basis vectors (3.7) and supposition (3.1), in the Green-Lagrange strain tensor (4.1) isgrad𝑢⃗=∇⃗⊗𝑢⃗=⎛⎜⎜⎜⎝𝑝¹¹,𝑝₁₂,𝑝₁₃𝑝₂₁,𝑝₂₂,𝑝₂₃𝑝₃₁,𝑝₃₂,𝑝₃₃⎞⎟⎟⎟⎠=⎛⎜⎜⎜⎜⎜⎝𝑟1_𝜓𝜕𝜕𝜓𝑢0,+𝑤,10,𝑟1_𝜓𝜕𝜕𝑤𝜓−𝑢𝜕𝑢𝑟_𝜑𝜓(𝑢+𝑤𝜓),0𝜕𝑧,0,𝜕𝜕𝑤𝑧⎞⎟⎟⎟⎟⎟⎠.(4.3) All nine tensor elements are obtained once (4.3) is inserted into (4.1):𝐄=12⎛⎜⎜⎜⎜⎝2𝑝¹¹𝑝₁₂+𝑝²¹¹+𝑝²¹²+𝑝²¹³,𝑝₁₂+𝑝₂₁+𝑝₁₁𝑝₂₁+𝑝₁₂𝑝₂₂+𝑝₁₃𝑝₂₃,𝑝₁₃+𝑝₂₁+𝑝₁₁𝑝₂₁+𝑝₁₂𝑝₂₂+𝑝₁₃𝑝₂₃,2𝑝₂₂+𝑝²²¹+𝑝²²²+𝑝²²³,𝑝₁₃+𝑝₃₁+𝑝₁₁𝑝₃₁+𝑝₁₂𝑝₃₂+𝑝₁₃𝑝₃₃,𝑝₂₃+𝑝₃₂+𝑝₂₁𝑝₃₁+𝑝₂₂𝑝₃₂+𝑝₂₃𝑝₃₃,+𝑝₃₁+𝑝₁₁𝑝₃₁+𝑝₁₂𝑝₃₂+𝑝₁₃𝑝₃₃𝑝₂₃+𝑝₃₂+𝑝₂₁𝑝₃₁+𝑝₂₂𝑝₃₂+𝑝₂₃𝑝₃₃2𝑝₃₃+𝑝²³¹+𝑝²³²+𝑝²³³⎞⎟⎟⎟⎠,(4.4) or in explicit form𝜀_𝜓=1𝑟_𝜓𝜕𝜕𝜓𝑢+𝑤+21𝑟^2𝜓𝑢²+𝑤²+2𝑤𝜕𝑢𝜕𝜓−2𝑢𝜕𝜕𝑤𝜓+𝜕𝜕𝜓𝑢²+𝜕𝑤𝜀_𝜑𝜕𝜓²,=1𝑟_𝜑𝜓1𝑢+𝑤+2𝑟1_𝜑2𝑤²+2𝑢𝑤𝜓+𝜓1²𝑢²𝜀_𝜓𝑟+𝑣²,=𝜀_𝑟𝜓=12𝜕𝜕𝑧𝑢+21𝑟𝜀_𝜓𝜕𝜕𝑤𝜓−𝑢+21𝑟_𝜓𝑤𝜕𝜕𝑧𝑢−𝑢𝜕𝜕𝑤𝑧+𝜕𝜕𝑧𝑢𝜕𝜕𝜓𝑢+𝜕𝜕𝑤𝑧𝜕𝜕𝑤𝜓_𝑟𝑟=,𝜕𝑤𝜕𝑧+21𝜕𝜕𝑧𝑢²+𝜕𝑤𝜀_𝜓𝜑𝜕𝑧²,=0,𝜀_𝜑𝜓=0,𝜀_𝜑𝑟=𝜀_𝑟𝜑=0.(4.5) However, the strains in (4.4) and (4.5) are still cumbersome and can hardly be used in practical computation. On the other hand, some nonlinear terms are relatively small and can be neglected without any significant effect on the accuracy of the results. Let us find the terms to be neglected.

In the case of rotation 𝝎 of the shell element with the length 𝐝𝐬_𝝍 in the direction of the unit vector 𝐞_𝝍, Figure 4, we can express the differential of the displacement vector 𝐝𝐮⃗ with𝑑𝑢⃗=||𝑑𝑢||⃗−cos𝜃𝑒_𝜓+sin𝜃𝑒_𝑟⃗=𝑑𝑥⃗grad𝑢⃗=𝑑𝑠_𝜓𝑝₁₁𝑒_𝜓+𝑝₁₃𝑒_𝑟⃗,(4.6) where 𝜃 is angle that the vector 𝑑𝑢⃗ forms with the basis unit vector 𝑒_𝜓, Figure 4.

Figure 4 

The shell middle surface displacement 𝑢⃗ and displacement 𝑢𝑧⃗ depending on the local coordinate 𝑧 as well as the rotation of the shell element 𝜔.

In the case of small strains |𝜺_𝝍|≪𝟏, we have||𝑑||𝑢𝑑||⃗𝑋≈⃗||=||𝑑𝑥||𝑑||⃗𝑥𝜔||⃗≈1𝑟+_𝜓𝜀_𝜓≈||𝑑𝑥||⃗,𝑑𝜓𝜔=𝑑𝑠_𝜓𝜔,(4.7) where 𝜔 as the rotation of the shell element is expressed in radians. It is also further evident by comparing the coefficients in (4.6) that𝑝₁₁=−||𝑑𝑢||⃗𝑑𝑠_𝜓cos𝜃=−𝜔cos𝜃𝑝₁₃=||𝑑𝑢||⃗||𝑑𝜓||sin𝜃=𝜔sin𝜃.(4.8) If the shell element with the length 𝑑𝑠_𝜓 in the direction of the unit vector 𝑒_𝜓 completes the rotation for the angle 𝜔=20^∘ around the vector 𝑒_𝜑, it follows that:𝑝₁₁≈−𝜋9cos80^∘≈−0.06𝑝₁₃≈𝜋9sin80^∘≈0.34.(4.9) A similar calculation can be performed for the rotation of the shell element with the length 𝑑𝑠_𝜑 in the direction of the unit vector 𝑒_𝜑.

In case of small strains and moderate shell rotations approximately up to 20^∘, the components of the displacement gradient are||𝑝₁₁||,||||𝑝𝑝₂₂||,||𝑝₃₃||≪1,(4.10)₁₃||<1,(4.11) due to which we ignore all nonlinear terms in Green-Lagrange strain tensor (4.4) except the term 𝑝²¹³.

Thus, the displacement of the point 𝑃 due to load is the largest in the direction of the unit vector 𝑒_𝑟⃗, while the displacement in the direction of the unit vectors 𝑒_𝜓 is by absolute value smaller |𝑢|<|𝑤|.(4.12) Further, we consider that the flexion curvature of the shallow shell in the meridian direction is very small due to which it is true that1𝑟_𝜓𝑢≈𝑦^𝑢≪1𝑟_𝜓𝑑𝑤𝑑𝜓=𝑑𝑑𝑠𝑤_𝜓≈𝑑𝑤𝑑𝑥=𝑤^.(4.13) So, the tensor elements (4.4) and (4.5) are finally𝐄=12⎛⎜⎜⎜⎝2𝑝¹¹+𝑝^213,0,𝑝₁₃+𝑝₃₁𝑝₁₃0,2𝑝_22,0+𝑝_31,0,2𝑝₃₃⎞⎟⎟⎟⎠,(4.14) or𝜀_𝜓=1𝑟_𝜓𝜕𝜕𝜓𝑢+𝑤+21𝑟^2𝜓𝜕𝜕𝑤𝜓²,𝜀_𝜑=1𝜀𝑟_𝜑𝜓1𝑢+𝑤_𝜓𝑟,=𝜀_𝑟𝜓=12𝜕𝜕𝑧𝑢+21𝑟_𝜓𝜕𝑤𝜕𝜓,𝜀^𝑟=𝜕𝑤𝜀𝜕𝑧_𝜓𝜑,=0,𝜀_𝜑𝜓=0,𝜀_𝜑𝑟=𝜀_𝑟𝜑=0.(4.15) With the introduction of the local coordinate 𝑧 with the centre of origin on the middle surface of the shell, Figure 4, the elements of the strain tensor 𝐄 are, due to the shell’s curvature, also the function of the coordinate 𝑧. The relation between the displacement 𝑢^𝑧 on the local coordinate 𝑧 and displacement 𝑢 on the middle surface of the bimetallic shell, Figure 4, can be found from the strain tensor element 𝜀_𝜓𝑟, since Kirchhoff hypothesis states that the strait lines perpendicular to the shell’s middle surface before deformation remain straight also after deformation. In other words, the hypothesis declares that the shear strain 𝜀_𝜓𝑟 is zero𝜀_𝜓𝑟=12𝜕𝜕𝑧𝑢+21𝑟_𝜓𝜕𝑤𝜕𝜓=0.(4.16) The solution to this equation with respect to the element 𝑢 of the displacement vector 𝑢⃗ is:𝑢(𝑧)=𝑢^𝑧=𝑢−𝑧𝑟_𝜓𝜕𝑤𝜕𝜓.(4.17) The normal strain 𝜀_𝑟 in the transversal direction is equal to zero according to the second assumption of Kirchhoff’s hypothesis:𝜀_𝑟=0.(4.18) Now, two nonzero elements of the Green-Lagrange strain tensor 𝐄 (4.15) for the middle surface of the shell can be recorded by means of relation (4.13) as𝜀_𝜓=𝑢^+𝑤𝑦^+12𝑤^²𝜀_𝜑,=𝑢+𝑦^𝑤𝑥.(4.19) Note again that when we form the strain tensor 𝐄, we retain the nonlinear term for strain 𝜀_𝜓 in 𝑢^𝑧 the meridian direction according to the third order of the large displacements theory. Namely, it has been confirmed that taking into account this nonlinear term is crucial for the accuracy of the results.

We calculate the strains 𝜀^𝑧𝜓 and 𝜀^𝑧𝜑 at distance 𝑧 from the middle surface of the bimetallic shell in the direction of the unit vector 𝑒_𝑟⃗ with (4.15), where we replace the displacement 𝑢 with the displacement 𝑢^𝑧.

Since we are working on a thin shell, where 𝑧≪{𝑟_𝜓,𝑟_𝜑}, we obtain𝜀^𝑧𝜓=(𝑢^𝑧)^+𝑤𝑟_𝜓+𝑧+12𝑤^²=𝑢−𝑧𝑤^^+𝑤𝑟_𝜓+𝑧+12𝑤^²𝜀_𝑧𝜑≈𝜀_𝜓−𝑧𝑤^,=1𝑟_𝜑+𝑧𝑢𝜓^𝑧+𝑤=𝑟_𝜑1+𝑧𝑢−𝜓𝑧𝑤^+𝑤≈𝜀^𝜑−𝑧𝑤^𝑥.(4.20)

The relation between the strain and stress tensor is determined by Hooke’s law [22]𝜎^𝑧𝜓=𝐸1−𝜇²𝜎𝜇𝜀^𝑧𝜑+𝜀^𝑧𝜓−(1+𝜇)𝛼𝑇_𝑧𝜑,=𝐸1−𝜇²𝜇𝜀^𝑧𝜓+𝜀^𝑧𝜑−(1+𝜇)𝛼𝑇.(4.21) The symbol 𝐸 denotes the Young’s modulus, the symbol 𝜇 denotes the Poisson’s ratio and 𝛼 is the thermal expansion coefficient. The symbol 𝑇 denotes the shell’s relative temperature relevant to the reference 𝑇₀ for which the stress state in the shell is equal to zero:𝜎^𝑧𝜓𝑥,𝑧,𝑇₀=𝜎^𝑧𝜑𝑥,𝑧,𝑇₀=𝜏^𝑧𝜓𝑟𝑥,𝑧,𝑇₀=0.(4.22)

5. Equilibrium of the Forces and Moments

Figure 5 shows an elementary small part of the bimetallic shell with the stresses that arise on the cross-section planes of the shell. The forces 𝑑𝑁_𝜓,𝑑𝑁_𝜑, and 𝑑𝑇_𝜓𝑟 and the bending moments 𝑑𝑀_𝜓, 𝑑𝑀_𝜑 which act upon the 𝐴𝐵𝐶𝐷 planes are𝑑𝑁_𝜓=𝑛_𝜓𝑑𝑋𝑀𝑑_𝜓𝜑,𝑑𝑁_𝜑=𝑛_𝜑𝑟_𝜓𝑑𝜓,𝑑𝑇_𝜓𝑟=𝑡_𝜓𝑟𝑋𝑑𝜑,=𝑚_𝜓𝑋𝑑𝜑,𝑑𝑀_𝜑=𝑚_𝜑𝑟_𝜓𝑑𝜓.(5.1) With 𝑛_𝜓,𝑛_𝜑,𝑡_𝜓𝑟 and 𝑚_𝜓,𝑚_𝜑 in (5.1) the unit forces and unit moments, respectively, are denoted. Since we are dealing with a thin bimetallic shell, these forces and moments have the form𝑚𝑛_𝜓=^{𝛿/2−𝛿/2}𝜎^𝑧𝜓𝑑𝑧,𝑛_𝜑=^{𝛿/2−𝛿/2}𝜎^𝑧𝜑𝑑𝑧,_𝜓=−^{𝛿/2−𝛿/2}𝑧𝜎^𝑧𝜓𝑑𝑧,𝑚_𝜑=−^{𝛿/2−𝛿/2}𝑧𝜎^𝑧𝜑𝑑𝑧.(5.2) The reader should note that the transversal shear force 𝑑𝑇_𝜓𝑟 in (5.1) cannot be expressed by a definite integral since the transversal shear strain 𝜀_𝜓𝑟 is disregarded according to the Kirchhoff hypothesis. However, this assumption does not exclude the full consideration of the shear force 𝑇_𝜓𝑟, which, in continuation, will be expressed with the equilibrium equations. According to the Euler-Bernoulli beam theory assumptions, a similar example occurs for a clamped thin beam when loaded with a transversal shear force.

Figure 5 

Normal and sheer stresses in the element of a deformed shell.

As the shell element in Figure 6 is in equilibrium, we can write three equations for the equilibrium of forces and moments in the meridian, circular and radial directions of the shell. In doing so, let us also remember that we are discussing a shallow shell, where the sinus function of the meridian angle 𝜓 can be replaced by its argument 𝜓, while the cosine of the same angle is very close to onesin𝜓≈𝜓,cos𝜓≈1.(5.3) Equilibrium of the forces in the meridian direction is𝑑𝑁_𝜓+𝑑𝑑𝑁_𝜓−𝑑𝑁_𝜓−𝑑𝑇_𝜓𝑟𝑑𝜓+𝑝𝑟_𝜓𝑑𝜓𝑋𝑑𝜑𝑑𝜓2−𝑑𝑁^𝜑𝑑𝜑=0.(5.4) Equilibrium of the forces in the radial direction𝑑𝑇_𝜓𝑟+𝑑𝑑𝑇_𝜓𝑟−𝑑𝑇_𝜓𝑟+𝑝𝑟_𝜓𝑑𝜓𝑋𝑑𝜑+𝑑𝑁_𝜓𝑑𝜓+𝑑𝑁_𝜑𝑑𝜑𝜓+𝑑𝜓2=0.(5.5) The equilibrium equation of forces in the circular direction cannot be considered due to the axisymmetric deformation. We write the moment equilibrium as𝑑+𝑀𝑝_𝜓+𝑑𝑑𝑀_𝜓−𝑑𝑀_𝜓−2𝑑𝑀_𝜑𝑑2𝜑−𝑑𝑇^𝜓𝑟𝑑𝑋−𝑑𝑇_𝜓𝑟𝜓𝑑𝑌𝑟_𝜓𝑑𝜓𝑋𝑑𝜑𝑑𝑋2+𝜓+𝑑𝜓−𝑑𝑁_𝜓2𝑑2𝑌𝜓𝑑𝑋+𝑑𝑁_𝜓𝑑𝑌+2𝑑𝑁_𝜑𝑑𝜑2𝑑2𝑌=0,(5.6) where𝑑𝑑𝑁_𝜓=𝜕𝑑𝑁^𝜓𝜕𝜓𝑑𝜓=𝑑𝑛^𝜓𝑑𝑑𝑇_𝜓𝑟𝑋𝑑𝜑+𝑛_𝜓𝑑𝑋𝑑𝜑,=𝜕𝑑𝑇^𝜓𝑟𝜕𝜓𝑑𝜓=𝑑𝑡^𝜓𝑟𝑑𝑑𝑀_𝜓𝑋𝑑𝜑+𝑡_𝜓𝑟𝑑𝑋𝑑𝜑,=𝜕𝑑𝑀^𝜓𝜕𝜓𝑑𝜓=𝑑𝑚^𝜓𝑋𝑑𝜑+𝑚_𝜓𝑑𝑋𝑑𝜑.(5.7)

Figure 6 

Unit forces and moments in the element of a deformed shell and external pressure load 𝑝.

After arranging and using (3.17), the equilibrium equations are𝑡𝑛_𝜓𝑥^−𝑛_𝜑−𝑡_𝜓𝑟𝑥(𝑦−𝑤)^=0,(5.8)_𝜓𝑟𝑥^+𝑛_𝜑𝑚(𝑦−𝑤)^+𝑛_𝜓𝑥(𝑦−𝑤)^+𝑝𝑥=0,(5.9)_𝜓𝑥^−𝑚_𝜑−𝑡_𝜓𝑟𝑥=0.(5.10)

6. Solving Equilibrium Equations

The system of thermoelastic equations for a thin shallow axisymmetric bimetallic shell consists of three equilibrium equations (5.8), (5.9), (5.10), the four equations (5.2) for unit forces and moments, two equations (4.21) for stresses in the shell, two equations (4.20) for strains outside the middle plan of the shell, and finally two equations (4.19) which relate strains and displacements. Thus, the system has 13 equations and as many unknowns, namely 𝑛_𝜓, 𝑛_𝜑, 𝑡_𝜓𝑟, 𝑚_𝜓, 𝑚_𝜑, 𝜎^𝑧𝜓, 𝜎^𝑧𝜑, 𝜀^𝑧𝜓, 𝜀^𝑧𝜑, 𝜀_𝜓, 𝜀_𝜑, 𝑢 and 𝑤 and can be reduced further in a following way: first, we insert (4.20) into (5.2). After integration, we have𝑛𝑛_𝜓=𝐴𝜀_𝜓+𝜇𝜀_𝜑−𝑃𝑇,(6.1)_𝜑𝑚=𝐴𝜀_𝜑+𝜇𝜀_𝜓−𝑃𝑇,(6.2)_𝜓=𝐵𝑤^+𝜇𝑤^𝑚𝑥−𝑄𝑇,(6.3)_𝜑=𝐵𝑤^𝑥+𝜇𝑤^−𝑄𝑇,(6.4) where 𝐴,𝐵,𝑃, and 𝑄 are constants as follows [13, 14]:𝐴=𝐸𝛿1−𝜇²,𝐵=𝐸𝛿³121−𝜇²,𝑃=2(1𝐸−𝛿𝜇)𝛼¹+𝛼₂,𝑄=𝐸𝛿²𝛼₂−𝛼₁8(𝜇−1).(6.5) We now multiply (5.8) with (𝑦^−𝑤^) and add (5.9). So, we obtain𝑛_𝜓𝑥^(𝑦−𝑤)^+𝑛_𝜓𝑥(𝑦−𝑤)^+𝑡_𝜓𝑟𝑥^+𝑝𝑥−𝑡_𝜓𝑟𝑥(𝑦−𝑤)^(𝑦−𝑤)^=0.(6.6) We disregard the last nonlinear term of 10⁻³ magnitude order, since𝑦^−𝑤^𝑦^−𝑤^=𝑌^𝑌^=12𝑌^2^≈10⁻³≪1.(6.7)

In this way, we obtain the relation 𝑡_𝜓𝑟𝑥^=−𝑛_𝜓𝑥(𝑦−𝑤)^^−𝑝𝑥=−𝑛_𝜓𝑥(𝑦−𝑤)^+𝑝𝑥𝑑𝑥^,(6.8) and after integration𝑡_𝜓𝑟=1𝑥𝑐−^𝑥0𝑝𝑥𝑑𝑥−𝑛_𝜓(𝑦−𝑤)^,(6.9) where 𝑐 is a constant that depends exclusively on the external force 𝐹 acting on the shell.

The relation (6.9) is inserted into (5.9) and (5.10). 𝑚𝑛_𝜓𝑥^−𝑛_𝜑=0,_𝜓𝑥^−𝑚_𝜑+𝑛_𝜓𝑥(𝑦−𝑤)^−𝑐+^𝑥0𝑝𝑥𝑑𝑥=0.(6.10) We now express the unit forces 𝑛_𝜓, 𝑛_𝜑 and moments 𝑚_𝜓, 𝑚_𝜑 in the equations above with (6.1), (6.2), (6.3), and (6.4), and the strains 𝜀_𝜓, 𝜀_𝜑 wit

h the kinematic equations (4.19). Therefore, the dependent unknown variables by which we solve the differential equations (6.10) are the elements 𝑢 and 𝑤 of the displacement vector 𝑢⃗𝑢+𝑤𝑦^=𝑥𝑢^+(1−𝜇)𝑤^²2+𝑤^𝜇𝑦^𝑐𝑥+𝐵𝑤_+𝑥𝑤^+𝑦^+𝑥𝑢^+𝑤𝑦^+𝑥𝑦^,(6.11)=𝑥×^𝑥0𝑝−𝑥𝑃d𝑇𝑥𝑥++𝐵𝐴𝑤^𝜇−𝑢+𝑤^𝜇−𝑤𝑦𝑦^_+𝑥𝑢^+𝑤^²2+𝑤𝑦^+𝐵𝑥𝑤^.(6.12) We define the integration constant 𝑐 in (6.12) by taking into account the equilibrium of forces on the edge of the shell.

If a shallow shell is simply supported, the supporting force 𝑡_𝜓𝑟 per unit of length at the edge of the shell is directed opposite to and in value equal to the sum of the force 𝐹 and pressure 𝑝 acting on the shell, Figure 7𝑡_𝜓𝑟||_𝑥=𝑎=−12𝜋𝑎𝐹+2𝜋^𝑎0𝑝(𝑥)𝑥𝑑𝑥,(6.13) equation (6.13) is inserted into (6.9) from where the constant 𝑐 is expressed𝑐=−𝐹2𝜋.(6.14) If the bimetallic shell serves as a thermal switch shutting down a device in the case of it overheating, then it is necessary to assure that the shell can extend unrestricted in a horizontal direction [10]. In continuation, we will discuss a simply supported shell. At the boundary of the simply supported shell, the force and moment per unit of length are equal to zero𝑛_𝜓(𝑎)=𝑛_𝜓(−𝑎)=𝑚_𝜓(𝑎)=𝑚_𝜓(−𝑎)=0.(6.15) We use (6.1) and(6.3)−𝑃𝑇+𝐴𝑢^+12𝑤^²+𝜇𝑢+𝑤𝑦^𝑥+𝑤𝑦^|||||_𝑥=±𝑎=0,(6.16)−𝑄𝑇+𝐵𝜇𝑤^𝑥+𝐵𝑤^||||_𝑥=±𝑎=0.(6.17) The displacement 𝑤 at the apex of the shell is equal to zero in the chosen coordinate system:𝑤(0)=0.(6.18) It is also demonstrated that this system of (6.11), (6.12), (6.16), (6.17), and (6.18) has symmetry. Namely, if the displacement vector 𝑢(⃗𝑥)=𝑢𝑒_𝜓+𝑤𝑒_𝑟⃗ is the solution to this system, then the solution is also𝑢⃗(−𝑥)=−𝑢𝑒_𝜓+𝑤𝑒_𝑟⃗.(6.19) due to which it is sufficient that we solve the system of equations only for positive 𝑥 values in the interval [0≤𝑥≤𝑎]. For negative 𝑥, the displacement vector 𝑢⃗ is defined by (6.19). The boundary conditions for the displacement vector 𝑢⃗ also follow from the mentioned symmetry: 𝑢(0)=0,𝑤(0)=0,𝑤^(0)=0.(6.20) The remaining conditions on the edges of the shell at 𝑥=𝑎 are defined with unit forces and moments in the equations for boundary conditions (6.16) and (6.17).

Figure 7 

Loading of a simply roller supported shell with the force 𝐹⃗[𝑁] and external pressure 𝑝[𝑁/𝑚2].

Boundary value problem (BVP) for the snap-through of the system of a shallow axisymmetric bimetallic shell is therefore composed of equilibrium equations (6.11) and (6.12), boundary conditions (6.16) and (6.17) at the point 𝑥=𝑎, and boundary conditions (6.20) at the point 𝑥=0𝑢+𝑤𝑦^=𝑥𝑢^+(1−𝜇)𝑤^²2+𝑤^𝜇𝑦^𝑐𝑥+𝐵𝑤_+𝑥𝑤^+𝑦^+𝑥𝑢^+𝑤𝑦^+𝑥𝑦^,=𝑥×^𝑥0𝑝−𝑥𝑃d𝑥𝑇𝑥++𝐵𝐴𝑤^−𝜇𝑢𝑤+^𝜇−𝑤𝑦𝑦^_+𝑥𝑢^+𝑤^²𝑃𝑇=𝐴𝑢_2+𝑤𝑦^+𝐵𝑥𝑤^,+12𝑤^²+𝜇𝑢+𝑤𝑦^𝑥+𝑤𝑦^|||||_𝑥=𝑎𝑄𝑇=𝐵𝜇𝑤_,𝑥+𝐵𝑤^||||_𝑥=𝑎,𝑢(0)=𝑤(0)=𝑤′(0)=0.(6.21)

7. Analysis of Stability Conditions in a Spherical-Conic Type Bimetallic Shell

In continuation, we will discuss the snap-through and stability conditions during the loading of a parabolic-conic shell composed of a parabola near the apex and of a cone for the rest, Figures 8 and 9, where the rotational curve is determined by the function 𝑦𝑦(𝑥)=𝑘𝑥²𝑘(𝑎,+𝑏)𝑥−𝑎𝑏𝑘,𝑥𝑥>≤𝑏𝑏,,(7.1) with the following material and geometric properties:𝑘=1𝜇=115²mm,𝑎=15mm,𝛿=0.3mm,3,𝑎¹=3.41⋅10⁻⁵K,𝑎₂=1.41⋅10⁻⁵K.(7.2)

Figure 8 

Example of a rotational curve of a parabolic-conic bimetallic shell.

Figure 9 

Example of the full shape of a parabolic-conic bimetallic shell.

7.1. Analytic Solution in the Case of Flattened Parabolic Shell

The parameter 𝑏 in the rotational curve function (7.1) defines the point at which the parabola translates into a straight line. If in (7.1) the parameter 𝑏 is equal to the horizontal radius of the shell 𝑎, therefore, 𝑏=𝑎, we obtain a parabolic shell with the rotational curve 𝑦(𝑥)=𝑘𝑥², Figure 10. Let this parabolic shell be loaded only with the temperature 𝑇. Suppose also that the shell at a certain temperature 𝑇=𝑇_𝑚 is completely flattened, Figure 11.The displacement function 𝑤(𝑥) is in that case𝑤(𝑥)=𝑦(𝑥)=𝑘𝑥².(7.3) We insert (7.3) into the boundary condition (6.17) and express the temperature 𝑇_𝑚 at which the shell is flattened𝑇_𝑚=4𝑘𝛿3𝛼₁−𝛼₂.(7.4) With the displacement 𝑤 as defined by (7.3), the moment equilibrium equation (6.12) is identically satisfied𝐵𝑤^−𝑥𝐵𝑤^−𝑤^−𝑦^−𝑃𝑇𝑥+𝐴𝜇𝑢+𝜇𝑤𝑦^+𝑥𝑢^+𝑤^²≡0,2+𝑤𝑦^+𝐵𝑥𝑤^(7.5) and the equation for the equilibrium of forces becomes2𝑘²𝑥³𝜇−𝑢+𝑥𝑢^+𝑥²𝑢^=10𝑘²𝑥^3,(7.6) with the solution with respect to displacement 𝑢𝑢(𝑥)=𝑘𝑥8𝛼¹+𝛼₂𝛿+3𝑎²𝑘𝛼₁−𝛼₂(𝜇−1)−3𝑘𝑥²𝛼₁−𝛼₂(5+𝜇)12𝛼₁−𝛼₂,(7.7) that at the same time also satisfies the boundary condition (6.16) at the point 𝑥=𝑎 and the boundary condition (6.20) for the displacement 𝑢 at the point 𝑥=0. Therefore, at the temperature 𝑇=𝑇_𝑚 the parabolic shell is completely flattened. The reader should also note that such a state of the shell is only stable with shells that are shallow enough!

Figure 10 

Initial shape of the parabolic shell.

Figure 11 

Deformation of the parabolic shell at the temperature 𝑇=𝑇mp=88.9∘C into a regular circular plate shape.

7.2. Numeric Solution of the BVP

We solved the system of (6.21) using the nonlinear shooting method. First, the BVP (6.21) was converted into the system of ordinary differential equations of the first order 𝑦𝑦^1_2=𝑦₂𝑦^3=𝑦₄𝑦^4=𝑦₅,=12𝑥²2𝑦₁−2𝑥𝑦₂−𝑥𝑦²⁴−2𝑥²𝑦₄𝑦₅+𝑥𝑦³⁴𝜇+2𝑦3𝑦^−2𝑥𝑦₄𝜇𝑦^−2𝑥𝑦3𝑦^−2𝑥²𝑦₄𝑦^𝑦_5−2𝑥²𝑦₃𝑦^,=12𝐵𝑥²2𝑐𝑥+2𝐵𝑦₄−2𝑃𝑇𝑥²𝑦₄+2𝐴𝑥²𝑦₂𝑦₄+𝐴𝑥²𝑦³⁴−2𝐵𝑥𝑦₅+2𝐴𝑥𝑦₁𝑦₄𝜇−2𝑥^𝑥0𝑥𝑝d𝑥−2𝐴𝑥𝑦₃𝜇𝑦^2+2𝐴𝑥²𝑦₃𝑦₄𝑦^𝑦₁+𝑥𝑦^2𝑃𝑇𝑥−𝐴2𝑥𝑦₂+𝑥𝑦²⁴+2𝑦₁𝜇−2𝑦₃𝑦₄𝜇−2𝐴𝑥𝑦₃𝑦^𝑃(0𝑇)==𝐴0,𝑦₂𝑦₃(0)=0,𝑦₄(0)=0,+12𝑦⁴²+𝜇𝑦¹+𝑦₃𝑦^𝑥+𝑦³𝑦^|||||_𝑥=𝑎,𝑄𝑇=𝐵𝜇𝑦⁴𝑥+𝐵𝑦⁵||||_𝑥=𝑎,(7.8) when the substitution is introduced𝑢(𝑥)=𝑦₁,𝑢^(𝑥)=𝑦₂,𝑤(𝑥)=𝑦₃,𝑤^(𝑥)=𝑦₄,𝑤^(𝑥)=𝑦₅.(7.9) We chose the approximate values for 𝑦₂(0) and 𝑦₅(0) and calculated the approximate values for the displacements 𝑢 and 𝑤 by the classic one-step fourth-order Runge-Kutta method [23]. Defining more exact values for 𝑦₂(0) and 𝑦₅(0) was carried out by the Newton method [24] for solving nonlinear equations. We wrote the shell deformation with the ratio 𝜉 between the current height ℎ of the deformed shell and the initial height ℎ₀ of the undeformed shell𝜉=ℎℎ₀=𝑦(𝑎)−𝑤(𝑎)𝑦(𝑎).(7.10) The program code was written in Mathematica package 7.01.0.

As the components 𝑢 and 𝑤 of the displacement vector 𝑢⃗ at the temperature 𝑇_𝑚 for a totally flattened parabolic shell are defined by (7.3) and (7.7), it is possible to calculate how much the numeric results differ from the actual ones. In this manner, we can estimate the accuracy of the chosen numeric method.

7.3. Temperature Loading of a Parabolic Shell

Let us first observe the stability conditions at temperature loading of a parabolic bimetallic shell with 𝑏=𝑎 in (7.1). For this special example of a parabolic-conic shell, we calculated, with the earlier mentioned method, the stability conditions at temperature loading. The obtained results are identical to the results calculated by Wittrick et al. [3] for the parabolic shell.

Figure 12 shows stability conditions for a parabolic shell with the material and geometric properties in (7.2). The graph of the function of dimensionless temperature 𝜏=𝑇/𝑇_𝑚 depending on the ratio of heights 𝜉 represents the stability circumstances during the shell’s temperature load.

Figure 12 

The function 𝜏=𝜏(𝜉) as an example of parabolic axisymmetric shell expressing the phenomenon of a system snap-through between points 𝐴𝐵 in the process of heating up and the points 𝐶𝐷 in the process of cooling.

During the initial state of no temperature load 𝜏=0, at point 𝑂(1,0), the ratio of heights is equal to one. By increasing the dimensionless temperature 𝜏, this ratio decreases. As is clear in Figure 12, the segment on the curve between the point 𝑂 and the point 𝐴(𝜉_𝑢,𝜏_𝑢)=𝐴(0.44,1.51), where the function 𝜏(𝜉) has a local maximum, is the region of stable equilibrium. The upper snap-through of the shell will, therefore, occur at point 𝐴 at the temperature of 𝜏_𝑢=1.51, because the segment between the points 𝐴 and 𝐶(𝜉_𝑙,𝜏_𝑙)=𝐶(−0.44,0.49), where the function has a local minimum, is the region of unstable equilibrium.

After the snap-through, the shell will occupy a new position of stable equilibrium in point 𝐵(−0.91,1.51) at the temperature 𝜏=1,51. With further heating of the shell, the ratio 𝜉 continues to decrease.

During the cooling of the shell, we have the opposite phenomenon and at point 𝐶 at the temperature 𝜏_𝑙=0.49 another lower snap-through. This time the shell snaps-through into the stable equilibrium position at point 𝐷(0.91,0.49) at the temperature 𝜏=0.49. By reheating the shell to the temperature of the upper snap-through 𝜏_𝑢=1.51, we can repeat the complete cycle of the snap-through of the shell. The shape of the shell in the characteristic stages of temperature loading is clear in Figures 13 and 14.

Figure 13 

Rotational curves for an undeformed parabolic shell at the moment of the upper and lower snap-through.

(a)

(b)

(c)

(d)

(e)

Figure 14 

Geometry of a parabolic shell: (a) undeformed, (b) at the upper snap-through, (c) after the upper snap-through, (d) at the lower snap-through, and (e) after the lower snap-through.

From Figure 12, it follows that in a flattened state at the ratio of heights 𝜉=0, the shell cannot endure as the point (0,1) belongs in unstable region between the points 𝐴 and 𝐶. Equation (7.4) determines the temperature 𝑇_𝑚 at which the shell flattens, Figure 11. If at the temperature 𝑇=𝑇_𝑚 the flattened shell deflects downwards so that the ratio of heights is 𝜉<0, then the temperature 𝑇_𝑚 of the shell is too high for an equilibrium state of the shell at the ratio of heights 𝜉<0. The apex of the shell accelerates downwards so that a new stable equilibrium state is established in the position (−0.78,1). It is quite similar if the shell deflects upwards 𝜉>0 except that in this case, the temperature 𝑇=𝑇_𝑚 for the ratio of heights 𝜉>0 is too low to satisfy the conditions for equilibrium and consequently the apex of the shell accelerates upwards so that the new stable equilibrium state is established in the position (0.78,1), Figure 12.

7.4. Temperature Loading of a Conic Shell

If in (7.1) 𝑏=0, then the shell translates into a conic shape, Figure 15. We present an example of a temperature loaded conic shell with material and geometric properties in (7.2) and a rotational curve 𝑦(𝑥)𝑦(𝑥)=𝑘𝑥=115mm𝑥.(7.11) In comparison with the earlier described parabolic shell, the conic shell has a snap-through at a lower temperature. The upper snap-through occurs at temperature 𝜏=1.00 and at the height ratio 𝜉=0.41, Figure 16(b). The shape of the shell after the snap-through is in Figure 16(c). The repeated (lower) snap-through occurs at temperature 𝜏=0.98 and at the height ratio 𝜉=−0.09, Figure 16(d). The shell snaps-through into a new stabile equilibrium position and assumes the shape on the diagram, Figure 16(e).

Figure 15 

Initial shape of the conic shell.

(a)

(b)

(c)

(d)

(e)

Figure 16 

Geometry of a conic shell: (a) undeformed, (b) at the upper snap-through, (c) after the upper snap-through, (d) at the lower snap-through, and (e) after the lower snap-through.

We would expect, like in the case of a flattened parabolic shell, Figure 11, that at a certain temperature the conic shell would also flatten with the function 𝑦(𝑥)=𝑘𝑥. It is verified that it is not possible to identically satisfy (6.12) with the displacement 𝑤(𝑥)=𝑦(𝑥)=𝑘𝑥, thus a flattened state of a conic bimetallic shell is not possible. The shape of the deformed conic shell at the ratio of heights 𝜉=0 is shown in Figure 17.

Figure 17 

The initial and deformed conic shell at the ratio of heights 𝜉=1 and 𝜉=0.

7.5. Temperature Loading of a Parabolic-Conic Shell

We gradually increased the value of the parameter 𝑏 in the equation system (6.21) and calculated the temperature of both snap-throughs for a parabolic-conic type bimetallic shell. The snap-through temperatures 𝑇_𝑢 and 𝑇_𝑙 at the parameter 𝑏 values between 0≤𝑏≤𝑎 are shown in Table 1.

Table 1 

Snap-through temperatures of a parabolic-conic shell relevant to the parameter 𝑏.

We ascertained that the snap-through temperature is dependent from point 𝑏, where the parabolic rotational curve translates into a conic one, Figure 18.

Figure 18 

Snap-through temperatures of a parabolic-conic type of bimetallic shell.

Figure 18 shows how both snap-through temperatures are relative to parameter b. At the extreme point on the left at 𝑏=0, an example of the conic shell is shown, and on the right at 𝑏=15, an example of parabolic shell is shown.

Both curves for the snap-through temperature have a local extreme. The highest temperature at which the shell snaps-through for the first time (upper snap-through) is 𝑇_𝑢=143°C at 𝑏=8.1 mm. The lowest temperature of the return (lower) snap-through is 25°C at 𝑏=5.0 mm. The difference in the temperature of the upper and lower snap-through relevant to the parameter 𝑏 is evident in Figure 19.

Figure 19 

The difference in the temperature of the upper and lower snap-through relevant to the parameter 𝑏.

7.6. Temperature and Force Loading of a Circular Plane-Parabolic Shell

In practice, a shell composed of a circular plane near the apex and a parabola at the edge is frequent, Figure 20. Such a shell occurs with the curve rotation:𝑦(𝑥)=⎧⎪⎨⎪⎩0,1𝑥≤𝑏,𝑎²−𝑏²𝑥²+𝑏²𝑏²−𝑎²,𝑥>𝑏.(7.12) We will again numerically discuss the example of a shell with material and geometric properties in (7.2). The parameter 𝑏 where the shell translates from a plane into a parabola should have a value of 𝑏=7.5 mm. At first, the shell should be loaded only with temperature 𝑇. The upper snap-through of such a shell occurs at the temperature 𝑇=180°C and the height ratio of 𝜉=0.48, Figure 21.

Figure 20 

Undeformed shape of a plane-parabola type bimetallic shell.

Figure 21 

A plane-parabola type bimetallic shell at the moment of the upper snap-through at the temperature 𝑇=180°C and height ratio 𝜉=0.48.

In comparison with a parabolic shell of equal material and geometric properties, Figure 10, this shell has a snap-through at a much higher temperature. A parabolic shell already translates into an unstable equilibrium state at the temperature 𝑇_𝑢=𝜏_𝑢𝑇_𝑚=134°C, when it snaps-through into a new stable equilibrium position. Therefore, with an equal initial height ℎ₀, the combined shell has a higher temperature of the upper snap-through by 46°C or by 34%. Consequently, with an appropriate combination of a circular plate and parabola it is possible to construct a shell with a low initial height that, despite this, will still have a snap-through at a high temperature. Therefore, this is an important advantage of combined shells in comparison with parabolic shells.

The effect of the concentrated force 𝐹[𝑁], exerted at the apex of the shell is clear from Figures 22 and 23. Due to the force, the shell already bends, even when not loaded with temperature, into the shape shown in Figure 23. The concavity is most distinct near the apex. The deformation curve is shown in Figure 22 in blue.

Figure 22 

Deformation curve for a plane-parabola type bimetallic shell loaded with a concentrated force 𝐹=37.7 N.

Figure 23 

The shape of the shell after loading with a concentrated force 𝐹=37.7 N.

If such a shell is to snap-through it should be additionally heated a bit. The instability and snap-through occurs at temperature 𝑇_𝑢=65°C. The shape of the shell at the moment of snap-through is shown in Figure 24, its deformation curve is shown in Figure 22 in red.

Figure 24 

The shape of the shell at the moment of the upper snap-through when loaded with a force 𝐹=37.7 N and temperature 𝑇𝑢=65°C.

8. Conclusion

Simply supported, thin-walled, shallow bimetallic shells of mixed (combined) type have the property to snap-through at a certain temperature into a new equilibrium position. The temperature of the snap-through depends on the material and geometric properties of the shell. As a special example, we analysed the conditions for parabolic, conic, parabolic-conic and plate-parabolic type of shell that have both layers equally thick 𝛿₁=𝛿₂=𝛿/2, and the same Poisson’s ratio 𝜇₁=𝜇₂=𝜇. Two parabolic shells of different rotational curves 𝑦₁=𝑘₁𝑥² and 𝑦₂=𝑘₂𝑥² have at the same temperature load 𝑇 equal relative displacements𝑢₁𝑎₁=𝑢²𝑎₂,𝑤¹𝑎₁=𝑤²𝑎₂.(8.1) If 𝑘₁𝑎₁=𝑘₂𝑎₂,𝛿¹𝑎₁=𝛿²𝑎₂.(8.2) Conic shells with different functions of rotational curves 𝑦₁=𝑘₁𝑥 and 𝑦₂=𝑘₂𝑥 have equal relative displacements (8.1) if the geometry parameters are such that𝑘₁=𝑘₂,𝛿¹𝑎₁=𝛿²𝑎₂.(8.3) A conic shell in comparison with a parabolic shell with equal material and geometric properties snaps-through at lower temperatures, Table 1. With a suitable initial shape of a parabolic-conic type bimetallic shell, we can change the upper 𝑇_𝑢 and lower 𝑇_𝑙 temperature values at which snap-through occurs. By reducing the conic part of the shell at the expense of the parabolic, the temperature of the upper snap-through increases. It is possible to achieve the highest snap-through temperature 𝑇_𝑢 if the shell translates from a conic shape to a parabolic approximately at the middle of the horizontal radius 𝑎. Therefore, for this type of shell, it is possible to influence the upper and lower temperature snap-through by changing the parameter 𝑏, and consequently with it the temperature at which a device would at first shutdown, then after cooling sufficiently start up again. We have ascertained a similar property of changing the temperature of both snap-throughs in parabolic shells with a circular opening in the apex [9]. It is also possible to influence the snap-through temperature by changing the force at the apex of the shell. At a certain force, the shell can snap-through without heating.