Non-Linear Creep of Spherical Container with Fluid Under Increasing Pressure ^†

Abstract

This paper presents an analysis of the time-dependent response of a spherical container to internal pressure that increases over time. The wall of the container is relatively thin, in the sense that the wall thickness is negligible in comparison to the container’s radius. The wall is composed of three layers. The two surface layers of the wall are identical, i.e., they are made from the same material and have the same thickness. One of the most important features regarding the response of the wall layers is the non-linear creep. The stresses and strains are determined, and their relationships with the parameters of the layers are studied.

1. Introduction

The increasing use of containers with fluids (gases, liquids) under pressure in different technical applications has led to a need for the introduction of new engineering materials for the container walls, as well as the development of reliable methods for analyzing the container’s mechanical response to various loading types and influences [1,2,3,4,5,6,7]. Such analyses are challenging and time-consuming, since many issues must be addressed in detail, and the effects of a variety of factors and technical parameters must be properly assessed.

One of the structural material types with a rapidly expanding field of application is layered materials [8,9,10]. These materials have numerous advantages. As a result, they are replacing homogeneous materials at an increasing rate.

This paper addresses the problem of the response of a spherical container—with a wall made of three layers—to internal uniform pressure that increases over time. Under such pressure, the wall layers undergo non-linear creep. Therefore, the response of the container is time-dependent. An analytical procedure is applied to determine the time-dependent parameters of the stressed and strained states of the layers. This procedure exploits the full symmetry that exists in spherical containers under uniform pressure (due to this symmetry, the meridional and the hoop stresses are the same). The relationship between the stresses and the parameters of the wall layers is also studied.

2. Analysis of the Container Wall Response

A spherical container is shown in Figure 1. The container wall has three layers. The surface layers have the same thickness, $\delta$. Thus, the thickness of the middle layer of the wall is $h-2\delta$, where $h$ is marked in Figure 1. The thickness of the container wall, $h$, is negligible compared to the container’s radius, $R$. The container is under an internal uniform pressure of intensity, $q$. The pressure intensity increases with time, $t$, according to Equation (1).

$$q=v_{q}t,$$

(1)

where $v_q$ is a parameter. All symbols are summarized in

Table A1. Symbols and units.

	Symbols and Units
E1i—modulus of elasticity of spring 1 [Pa]	E2i—modulus of elasticity of spring 2 [Pa]	h—container wall thickness [m] Hi—material property [1/Pa]
Li—material property [1/Pa] t—time [s] δ—wall thickness [m] ρm—curvature radius in meridional plane [m] ρp—curvature radius in hoop plane [m] σc—stress in the middle layer [Pa]	q—internal pressure [Pa] vq—parameter of pressure [Pa/s] ηc—coefficient of viscosity of the middle layer [Pa.s] σf—stress in the surface layer [Pa] σp—hoop stress [Pa]	R—container radius [m] vσi—parameter of the model [Pa/s] ε—strain [-] ηf—coefficient of viscosity of the surface layer [Pa.s] σm—meridional stress [Pa]

in Appendix A.

The wall layers behave as viscoelastic bodies under non-linear creep. The creep model is shown in Figure 2.

Figure 2. Creep model.

Figure 2. Creep model.

The stress, ${\sigma }_i$, in the model is as follows:

$${\sigma }_i=v_{\sigma i}t,$$

(2)

where $v_{\sigma i}$ is a parameter. The stress, ${\sigma }_{(nl)i}$, in the non-linear spring, ${\left(nl\right)}_i$, of the model depends on the strain, $\epsilon$, according to Equation (3) [11].

$${\sigma }_{(nl)i}={\displaystyle \frac{\epsilon }{L_i+H_i\epsilon }},$$

(3)

where $L_i$ and $H_i$ are material properties. We investigated the response of the model in Figure 2 to the stress, ${\sigma }_i$, at initial conditions, $\epsilon \left(0\right)=0$, and found the following:

$$\epsilon ={\displaystyle \frac{{\sigma }_i}{E_{2i}t}}\left({\displaystyle \frac{{\eta }_i}{E_{1i}}}-{\displaystyle \frac{{\psi }_{1i}}{E_{2i}}}\right)\left(1-e^{-{\displaystyle \frac{E_{2i}}{{\psi }_{1i}}}t}\right)+{\displaystyle \frac{{\sigma }_i}{E_{2i}}}+{\displaystyle \frac{{\sigma }_i{L}_i}{v_{\sigma i}-{\sigma }_i{H}_i}},$$

(4)

where

$${\psi }_{1i}={\eta }_i+{\displaystyle \frac{E_{2i}}{E_{1i}}}{\eta }_i, {\sigma }_i=v_{\sigma i}t.$$

(5)

The third addend in Equation (4) is found by extracting of $\epsilon$ from Equation (3). We examined the equilibrium of an element of the container wall and obtained the following:

$$qd{s}_{1}d{s}_2-\left[2{\sigma }_{mf}\delta +{\sigma }_{mc}\left(h-2\delta \right)\right]d{s}_{2}d\theta -\left[2{\sigma }_{pf}\delta +{\sigma }_{pc}\left(h-2\delta \right)\right]d{s}_{1}d\phi =0,$$

(6)

where

$$d{s}_1={\rho }_{m}d\theta ,$$

(7)

$$d{s}_2={\rho }_{p}d\phi ,$$

(8)

$$d\theta ={\displaystyle \frac{d{s}_1}{{\rho }_m}},$$

(9)

$$d\phi ={\displaystyle \frac{d{s}_2}{{\rho }_p}}.$$

(10)

where ${\sigma }_{mf}$ and ${\sigma }_{mc}$ are the meridional stresses in the surface and middle layers, ${\sigma }_{pf}$ and ${\sigma }_{pc}$ are the hoop stresses in the surface and middle layers of the container, ${\rho }_m$ and ${\rho }_p$ are the container curvatures radiuses. Due to the full symmetry in the spherical container, we have the following:

$${\sigma }_{mf}={\sigma }_{pf}={\sigma }_f,$$

(11)

$${\sigma }_{mc}={\sigma }_{pc}={\sigma }_c,$$

(12)

$${\rho }_m={\rho }_p=R.$$

(13)

Therefore, from Equation (6), one obtains the following:

$$q-{\displaystyle \frac{2}{R}}\left[2{\sigma }_f\delta +{\sigma }_c\left(h-2\delta \right)\right]\hspace{0.17em} =0$$

(14)

where ${\sigma }_f$ and ${\sigma }_c$ are the stresses in the surface and middle layers.

For ${\sigma }_i={\sigma }_f$, from Equation (4), one has the following:

$$\epsilon ={\displaystyle \frac{{\sigma }_f}{E_{2i}t}}\left({\displaystyle \frac{{\eta }_i}{E_{1i}}}-{\displaystyle \frac{{\psi }_{1i}}{E_{2i}}}\right)\left(1-e^{-{\displaystyle \frac{E_{2i}}{{\psi }_{1i}}}t}\right)+{\displaystyle \frac{{\sigma }_f}{E_{2i}}}+{\displaystyle \frac{{\sigma }_f{L}_i}{v_{\sigma i}-{\sigma }_f{H}_i}}.$$

(15)

For ${\sigma }_i={\sigma }_c$, from Equation (4),

$$\epsilon ={\displaystyle \frac{{\sigma }_c}{E_{2i}t}}\left({\displaystyle \frac{{\eta }_i}{E_{1i}}}-{\displaystyle \frac{{\psi }_{1i}}{E_{2i}}}\right)\left(1-e^{-{\displaystyle \frac{E_{2i}}{{\psi }_{1i}}}t}\right)+{\displaystyle \frac{{\sigma }_c}{E_{2i}}}+{\displaystyle \frac{{\sigma }_c{L}_i}{v_{\sigma i}-{\sigma }_c{H}_i}}$$

(16)

We determined $\epsilon$, ${\sigma }_f$ and ${\sigma }_c$ from Equations (14)–(16) by the MatLab 6.5. This was performed using various values of $t$ in order to study the creep behavior of the spherical container.

Since Equation (6) has fundamental importance for the accuracy of the current investigation, we checked this equation in the following manner. At the following:

$$\delta =0$$

(17)

the wall becomes homogeneous. In view of this, at $\delta =0$, Equation (6) should transform into the equilibrium equation for thin homogeneous containers. In order to verify this, upon substituting $\delta =0$ into Equation (6) and simplifying it, one obtains the following equation:

$${\displaystyle \frac{{\sigma }_m}{{\rho }_m}}+{\displaystyle \frac{{\sigma }_p}{{\rho }_p}}={\displaystyle \frac{q}{h}},$$

(18)

which is the known equation for thin homogeneous containers [12]; this confirms that Equation (6) is right.

Keeping in mind that for spherical containers, ${\rho }_m={\rho }_p=R$, from Equation (16), one can derive the following:

$${\displaystyle \frac{{\sigma }_m}{R}}+{\displaystyle \frac{{\sigma }_p}{R}}={\displaystyle \frac{q}{h}},$$

(19)

which is also a known equation [12].

One of the possible applications of the above-mentioned approach presented by Equations (14)–(16) is to study the change in the response of the container wall layers to increasing internal pressure over time. This application is very useful, as non-linear creep is incorporated in the equations. The approach can also be applied to study how the stresses in the wall layers depend on certain basic parameters of the container, e.g., the relative thickness of the surface layers and the relationships between the material properties of the surface and middle layers.

The study resulted in the dependencies, graphs of which are shown in Figure 3, Figure 4 and Figure 5. The change in $\epsilon$ with normalized time is shown in Figure 3. The dependence of the normalized stress in the surface layer of the wall on the relative thickness of the surface layer is shown in Figure 4 for growing $E_{2f}/E_{2c}$ ratio. The dependence of the normalized stress in the middle layer of the wall on $E_{1c}/E_{1f}$ and ${\eta }_c/{\eta }_f$ ratios is shown in Figure 5.

3. Conclusions

A spherical container subjected to internal pressure that increases over time is analyzed in this study. The container wall consists of three layers that behave as viscoelastic bodies under non-linear creep. The analysis determines the time-dependent wall response in terms of stresses and strains. It is well known in engineering practice that one of the most important questions in the design of containers under internal pressure is how to keep the stresses in the container wall as low as possible. A key conclusion from the current analysis is that the stresses in the surface layers of the container wall can be reduced by increasing the relative thickness of these layers. However, increasing the $E_{2f}/E_{2c}$ ratio results in a rise in stresses within the surface layers. Regarding the behavior of the stresses in the middle layer of the container wall, the analysis indicates that the stresses can be reduced by using materials with lower ${\eta }_c/{\eta }_f$ and $E_{1c}/E_{1f}$ ratios. Concerning the practical implications and limitations of the model, it should be noted that the model is applicable only to spherical containers with thin walls. The practical applicability of such models can ultimately be proven by nonlinear numerical simulations [13,14].