1 Introduction

Sensor is the basic element and the information source of internet of things, which is widely used in many fields, such as industrial production, environment protection, medical diagnosis, and bioengineering. As the basic element of the internet of things, it is very important that whether the sensor can gather the signal correctly in the information collection system. The corrugated shallow spherical shell is an important kind of sensor elastic element. Because of the particular metal texture, it can work in high temperature field. The stability problem of corrugated shallow spherical shell in load and temperature field is very important. The sensor component includes corrugated circular plate, corrugated shell, and corrugated pipe. Feodosev^[1], Andryewa^[2-3], Akasaka^[4], and Liu et al.^[5-7] have investigated the nonlinear problem about corrugated plate. Salashiling^[8] discussed nonlinear bending of cylindrical corrugated shell by using equal orthotropic elastic parameter which is provided by Andryewa^[2]. Wang and Liu^[9-11] developed equivalent orthotropic parameter of corrugated shallow spherical shell, and discussed nonlinear dynamic buckling of corrugated shallow spherical shell. Yuan^[12-13] discussed nonlinear vibration of corrugated shell by applying Green's function method. Xia et al.^[14] discussed equivalent model of corrugated panel. Winkler and Kress^[15] discussed deformation of corrugated cross-ply laminates. In this paper, nonlinear stability of corrugated shallow spherical shell in coupled multi-field is studied. With the equivalent orthotropic parameter obtained by the author^[9], the corrugated shallow spherical shell is considered as an orthotropic shallow spherical shell, and the effect of transverse shear deformation is taken into account. The nonlinear governing equations are obtained. The critical load is obtained using a modified iteration method. The effect of temperature variation and shear rigidity variation on critical load is analyzed.

2 Fundamental equations

A corrugated shallow spherical shell as shown in Fig. 1 is considered with thickness h, radius a, radius of curvature R, amplitude H, wavelength l, and center height f under a uniform pressure q and a uniform temperature field T, which can be equal to an orthotropic shallow spherical shell as shown in Fig. 2.

It can be concluded from the axisymmetric condition that the radical, circumferential, and axial displacements of an arbitrary point at a distance z from the middle surface of the shallow spherical shell, which can be written as follows^[13]:

(1)

where u₀ and w₀ are the radical displacement and the deflection of a point on the middle surface respectively, and γ_rz⁰ is the transverse shear strain of the point.

The geometrical equations are shown as follows:

(2)

By substituting (1) into (2), the results are obtained as follows:

(3)

where ε_r⁰ and ε _θ⁰ are the radial and circumferential strains of middle surface respectively, and κ_r and κ_θ are the radial and circumferential curvatures of middle surface respectively.

(4)

According to Andryewa's method^[3], ε_r and ε_θ can be expressed as follows:

(5)

where ε_r1 and ε_θ1 are the radial and circumferential strains caused by stretching, and ε_r2 and ε_θ2 are the radial and circumferential strains caused by bending:

(6)

Because of the dual orthotropic character of the shallow spherical shell, the relation between strain and stress can be shown as follows^[13]:

For the stretching part:

(7)

For the bending part:

(8)

where σ_r1 and σ_{θ 1} are the radial and circumferential stress caused by stretching, σ_r2 and σ_{θ 2} are the radial and circumferential stress caused by bending, E_r1 and E_θ1 are the elastic moduli of the shell material when the shell bears stretching, and E_r2 and E_θ1 are the elastic moduli of the shell material when the shell bears bending. υ_θr1 and υ_θr2 are the Poisson's ratios in the r-direction during tension applied in the θ-direction, and υ_{rθ 1} and υ_rθ2 are the Poisson's ratios in the θ-direction during tension applied in the r-direction.

(9)

and

(10)

where υ is Poisson's ratio of the corrugated shell's material.

The equivalent cylindrically orthotropic shallow spherical shell's elastic moduli can be shown as follows:

(11)

where E is the elastic modulus of the corrugated shell's material, and κ_r1, κ_{θ 1}, κ_r2, and κ_{θ 2} are the equivalent orthotropic parameters about the radical and circumferential rigidity.

According to (6), (7), (8), (9), and (11), the constitutive equations are obtained as follows:

(12)

where

The equivalent shell's transverse shear stress can be determined as follows:

(13)

where G_rz is the shear modulus which determines angle change between the directions of r and z.

By substituting (3) into (13), the expression of τ _rz can be expressed as follows:

(14)

According to Salashiling's method^[8], the expression of G_rz can be shown as follows:

(15)

where G is the material’s shear modulus.

Figure 3 shows the shell element which is cut out by two meridians and two parallel circles. Here, N_r and N_θ are the radial and circumferential membrane forces respectively, M_r and M_θ are the radial and circumferential moments respectively, and Q_r is the radial shear force. The expressions are shown as follows:

(16)

By substituting (12) and (14) into (16), the results are shown as follows:

(17)

where A_ij and D_ij are the stretching rigidity and bending rigidity respectively, G₀ is the shear rigidity, Q_ij' and Q_ij" are same as those in Ref. [7], and α_r and α_θ are the radical and circumferential thermal expanding coefficients of the shell.

The equilibrium equations of the shell element can be shown as follows:

(18)

3 Governing equations

The governing equations^[13] can be obtained as follows according to (16), (17), and (18) :

(19)

(20)

(21)

where

(22)

(19) -(21) can be solved under the edge rigidly clamped boundary conditions:

(23)

(24)

In order to simplify the following calculations, the following dimensionless notations are introduced:

in which λ is the dimensionless thermal membrane force.

With the help of these notations, the nonlinear boundary value problem (19) -(24) may be rewritten in a more convenient dimensionless form:

(25)

(26)

(27)

(28)

(29)

4 Analytical solution

The modified iteration method is used to solve the dimensionless nonlinear boundary value problem (25) -(29). The notation W_m is chosen for the dimensionless center deflection as an iteration parameter.

(30)

For the first approximation, by neglecting the nonlinear term sΦ in (25), the following linear boundary value problem is obtained:

(31)

(32)

(33)

(34)

(35)

By solving (31) -(35) and using (30), the following solutions for the first approximation are obtained:

in which f₁ (ρ), f₂ (ρ), and f₃ (ρ) are the multinomial expressions, which are detailed in Appendix A.

For the second approximation, the following nonlinear boundary value problem is obtained:

(36)

(37)

(38)

(39)

By solving (36) to (39), the solution for the second approximation is obtained:

in which f₄ (ρ) to f₉ (ρ) are the multinomial expressions, which are detailed in Appendix A.

By using (30), the nonlinear characteristic relation of the corrugated shallow spherical shell is obtained as follows:

(40)

where e_i (i=1, 2, …, 6) are the coefficients of multinomial expressions, which are detailed in Appendix A.

In order to find the critical buckling pressure P^* of the shell, the following condition is applied:

(41)

The dimensionless critical center deflection of the shell when buckling occurs is obtained as follows:

(42)

By substituting (42) into (40), the dimensionless critical buckling pressure is obtained:

(43)

5 Numerical example

As a numerical example, a corrugated shallow spherical shell with a=80 mm, l=12 mm, H=1.35 mm, h=0.4 mm, and R=200 mm is considered. According to (40) -(43), the numerical results are shown in the following figures (see Figs. 4-7).

(i) Curves of load versus deflection are shown in Fig. 4. There are two critical points in the figure which represent two critical loads: upper critical load and lower critical load. For example, if the shear rigidity parameter is 1, the upper critical load is 2.392 8 × 10⁴, and the lower critical load is -6.332 2× 10 ⁴; if the shear rigidity parameter is 41, the upper critical load is 3.077 8 × 10⁴, and the lower critical load is -5.143 5 × 10⁴.

(ⅱ) Curves of upper critical load versus thermal membrane force are shown in Fig.5, it can be found that P^* increases as λ increases. If the shear rigidity parameter is 41, the upper critical load increases from 3.077 8× 10⁴ to 3.081 × 10⁴ when the thermal membrane force increases from 0 to 10.

(ⅲ) Curves of lower critical load versus thermal membrane force are shown in Fig.6, it can be found that P^* decreases as λ increases. If the shear rigidity parameter is 41, the lower critical load decreases from -5.143 5× 10⁴ to -5.148 0× 10⁴ when the thermal membrane force increases from 0 to 10.

(ⅳ) Curves of critical load versus shear rigidity are shown in Fig. 7. The critical load P^* increases as the shear rigidity m increases.

6 Conclusions

From the computation and the analysis of the numerical results obtained, the following conclusions can be drawn.

(Ⅰ) The critical load P^* increases as the rigidity m increases.

(Ⅱ) For the designing of engineering structure, the upper critical load is usually taken into account because of the designing value. The upper critical load P^* increases as the thermal membrane force λ increases. In other words, if temperature increases, the shell's stability will be increased.

(Ⅲ) Buckling does not occur on the shell if the shear rigidity is large enough.

(Ⅳ) The result of this paper is the result of thermal stability problem of corrugated shallow spherical shell in coupled multi-field--uniform temperature field and uniform pressure. If there is no temperature field (λ =0), the result reduces to the nonlinear stability problem of corrugated shallow spherical shell under uniform pressure.

Appendix A