Safety Analysis Using Lebesgue Strain Measure of Thick-Walled Cylinder for Functionally Graded Material under Internal and External Pressure

Abstract

Safety analysis has been done for thick-walled circular cylinder under internal and external pressure using transition theory which is based on the concept of generalized principal Lebesgue strain measure. Results have been analyzed theoretically and discussed numerically. From the analysis, it can be concluded that circular cylinder made of functionally graded material is on the safer side of the design as compared to homogeneous cylinder with internal and external pressure, which leads to the idea of “stress saving” that minimizes the possibility of fracture of cylinder.

1. Introduction

The constantly increasing industrial demand for axisymmetrical cylindrical and spherical components or elements of them has concentrated the attention of designers and scientists on this particular area of activity. The research on the prediction of stresses in thick-walled hollow circular cylinder has never ceased because of the importance of these basic structures in numerous mechanical, civil, electrical, and computer engineering applications. These days in nuclear industry, cylinders subjected to internal and external pressure have become a point of interest due to their application to advanced small and medium-sized light water reactors. For example, steam generator tubes, in which primary coolant flows outside the tubes while secondary water flows inside the tubes are typical examples of cylinders under internal and external pressure. Another example is pipelines under seawater to transport gas, oil, and so forth. Now for design and integrity evaluation of a cylinder under internal and external pressure, one should carefully consider the failure characteristics of a cylinder under internal and external pressure. The failure mechanisms of such type of cylinder might be quite different from those of a typical one under internal pressure. Upon the estimation of load carrying capacity of these thick-walled cylinders under internal and external pressure and combined loading, many numerical and experimental works have been also made to propose relevant design criteria of thick-walled cylinders subjected to internal and external pressure. Plane strain and plane stress analytical solutions of thick hollow cylinder problems in the elastic stress state have been available for many years in standard and advanced textbooks [1–4]. Thick-walled circular cylinder subjected to internal and external pressure is widely used in various industries. In general vessels under high pressure require a strict analysis for an optimum design for reliable and secure operational performance and thus efforts were continually made to increase reliability. Solutions have been obtained either in analytical form or with numerical implementations. The literature includes solutions of Chen [5] who suggested an finite difference approach for the axisymmetric plane strain problem beyond the elastic limit while Durban and Kubi [6] suggested an analytical method for pressurized elastic-plastic tubes in plane strain. Jahed and Dubey [7] proposed a numerical method for solution for elastic-plastic tubes using total deformation theory of plasticity. Parker [8] implemented a numerical procedure to calculate pressure and associated residual stress fields for open cylinder. Dubey et al. [9] obtained solutions for an elastic-plastic work hardening model using piecewise linearization of constitutive law. Olszak and Urbanowski [10] calculate the stresses for nonhomogeneous thick-walled elastic-plastic cylinder subjected to internal pressure while Hodge and Balaban [11] calculated the stresses for rotating cylinder. Sharma [12] analyzed thick-walled cylinders under internal pressure for isotropic nonhomogeneous elastic-plastic states using transition theory. This paper is an extension of Sharma [12] to include the effect of external pressure for functionally graded material because these days cylinders made of functionally graded material under internal and external pressure are an important design consideration in nuclear industry.

2. Generalized Lebesgue Strain Measure

The classical theory of elasticity and plasticity divides the spectrum of deformation of solids into two different states, one of which is elastic and other one is plastic. In classical theory, both field equations are solved separately and later joined together by yield condition. As in the behavior of materials, perfect elasticity and perfect plasticity are extremes, but no one can draw a sharp line between these two states. It is therefore natural to expect that there should be a transition state, and at this transition, a continuum approach necessarily means the introduction of nonlinear measure. But in classical mechanics, the ordinary measure has been found sufficient and so no extension has been made. This is because of the reason that, in classical mechanics, field equation for elastic and plastic regions is calculated separately and then connected by yield criterions which is an assumption. Also, if in a very small interval, the number of fluctuation is very large, the ordinary measure based on Riemann integral fails and measures like that of Lebesgue have been used. This generalized Lebesgue measure gives very satisfactory results in the problems like that of plasticity and creep. The generalized Lebesgue strain measure helps to bridge the gap between microscopic and macroscopic descriptions of physical system and eliminate semiempirical conditions like that of Tresca’s and von-mises, creep strain laws, that is, Nortan’s law which provides a coordination between the theoretical and experimental results. Seth [13] has defined the generalized principal strain measure 𝜀_𝑖𝑖 by taking the Lebesgue integral of the weighted function 𝜀_𝑖𝑖=∫^{𝑒𝐴𝑖𝑖}₀[(1−2^𝐴_𝑒𝑖𝑖)]^(𝑛/2)−1𝑑𝑒^𝐴_𝑖𝑖=(1/𝑛)[1−(1−2𝑒^𝐴_𝑖𝑖)^(𝑛/2)−1], where 𝑛 is the measure and 𝑒^𝐴_𝑖𝑖 is the principal Almansi strain component.

3. Objective

In this paper, our aim is to calculate safety factor for thick-walled cylinder made of functionally graded material under internal and external pressure using Seth’s transition theory [12, 14–16]. The stresses are calculated for transition as well as for fully plastic state. The constitutive equations for both transition and fully plastic states are also derived from the results. Non-homogeneity is taken as the compressibility of material in the cylinder as 𝐶=𝐶₀𝑟^−𝑘, (1) where 𝑎≤𝑟≤𝑏; 𝐶₀ and 𝑘(≤0) are constants.

Results obtained have been discussed numerically and depicted graphically.

4. Mathematical Formulation

Consider a nonhomogeneous thick-walled circular cylinder of internal and external radii 𝑎 and 𝑏, respectively, subjected to internal pressure 𝑝₁ and external pressure 𝑝₂. The non-homogeneity in the cylinder is due to variable compressibility 𝐶=𝐶₀𝑟^−𝑘. The cylinder is taken so large that plane transverse sections remain plane during the expansion, and hence the longitudinal strain is the same for all elements at each stage of the expansion. In cylindrical polar coordinates the displacements are given by [14–16] 𝑢=𝑟1−𝛽;𝑣=0,𝑤=𝑑𝑧, (2) where 𝛽 is a function of 𝑟=𝑥²+𝑦² and 𝑑 is a constant.

The generalized strains are 𝑒_𝑟𝑟=1𝑛1−𝑟𝛽^′+𝛽^𝑛;𝑒_𝜃𝜃=1𝑛1−𝛽^𝑛;𝑒_𝑧𝑧=1𝑛1−(1−𝑑)^𝑛;𝑒_𝑟𝜃=𝑒_𝜃𝑧=𝑒_𝑧𝑟=0, (3) where 𝑛 is the measure and 𝛽^′=𝑑𝛽/𝑑𝑟.

For isotropic material, the stress-strain relation in elastic region is given by 𝑇_𝑖𝑗=𝜆𝛿_𝑖𝑗𝐼₁+2𝜇𝑒_𝑖𝑗;𝑖,𝑗=1,2,3, (4) where 𝐼₁=𝑒_𝑘𝑘; 𝑇_𝑖𝑗,  𝑒_𝑖𝑗 are stress and strain tensors, respectively, and 𝛿_𝑖𝑗 is Kronecker’s delta.

Equations of equilibrium are all satisfied except 𝑑𝑇𝑑𝑟^𝑟𝑟+𝑇_𝑟𝑟−𝑇_𝜃𝜃𝑟=0. (5) A nonlinear differential equation in 𝛽 has been obtained by substituting (4) in (5), as 𝑛𝑃𝛽(𝑃+1)^𝑛−1𝑑𝑃=𝑟𝜇𝑑𝛽_′𝜇−𝐶^′𝐶(3−2𝐶)−(1−𝐶)(1−𝑑)^𝑛1𝛽^𝑛−(1−𝐶)−(𝑃+1)^𝑛×()+𝐶1−𝑃+1^𝑛+𝑟𝐶^′1−2−(1−𝑑)^𝑛1𝛽^𝑛−𝑛𝑃(1−𝐶)+(𝑃+1)^𝑛, (6) where 𝑟𝛽^′=𝛽𝑃; 𝐶=2𝜇/(𝜆+2𝜇).

The transition point of 𝛽 in the previous equation is 𝑃→−1 and 𝑃→±∞.

The boundary conditions are given by 𝑇_𝑟𝑟=−𝑝₁𝑇at𝑟=𝑎;_𝑟𝑟=−𝑝₂at𝑟=𝑏. (7)

In the cylinder, resultant axial force is given by 𝐿=2𝜋^𝑏_𝑎𝑟𝑇_𝑧𝑧𝑑𝑟. (8)

5. Method of Solution

As elastic state can go to plastic state under external loading through a transition state and we consider only the principal stresses. Therefore the transition can take place either through the principal stresses 𝑇_𝑟𝑟 or 𝑇_𝜃𝜃 becoming critical or through the principal stress difference 𝑇_𝑟𝑟−𝑇_𝜃𝜃 becoming critical. It has been shown that the asymptotic solution through the principal stress leads from elastic state to plastic state at transition point 𝑃→±∞. For finding the plastic stress at the transition point 𝑃→±∞, we define the transition function 𝑅 [14–16] in terms of 𝑇_𝑟𝑟 as 𝑅=𝑇_𝑟𝑟−𝜆𝑛𝑘≡2𝜇𝐶𝑛𝐶−𝛽^𝑛(1−𝐶)+(𝑃+1)^𝑛. (9) Taking the logarithmic differentiation of (9) with respect to 𝑟, we get 𝑑𝑑𝑟log𝑅=𝑟𝐶^′1+𝛽^𝑛−𝑛𝑃𝛽^𝑛+1(𝑃+1)^𝑛−1𝑑𝑃𝑑𝛽−𝑛𝑃𝛽^𝑛×(1−𝐶)+(𝑃+1)^𝑛𝜇+𝑟^′𝜇−𝐶^′𝐶×𝐶−𝛽^𝑛(1−𝐶)+(𝑃+1)^𝑛×𝑟𝐶−𝛽^𝑛(1−𝐶)+(𝑃+1)^𝑛−1. (10) Take the asymptotic value 𝑃→±∞ of (10) after substituting  𝑑𝑃/𝑑𝛽 from (6). This on integration gives 𝑅₁=𝐴exp𝑓(𝑟), (11) where 𝐴 is a constant of integration and 𝑓(𝑟)=−∫(𝐶/𝑟)𝑑𝑟.

By using (7) in (9) and (11), we get 𝑇_𝑟𝑟=𝐴exp𝑓(𝑟)−exp𝑓(𝑏)−𝑝₂. (12) Using (12) in (5), we get 𝑇_𝜃𝜃=𝐴(1−𝐶)exp𝑓(𝑟)−exp𝑓(𝑏)−𝑝₂. (13) Equations (4) yields 𝑇_𝑧𝑧=1−𝐶𝑇2−𝐶^𝑟𝑟+𝑇_𝜃𝜃+𝐶𝜆(1−𝐶)3−2𝐶×𝑝2−𝐶₂−𝑝₁/2𝜋−∫^𝑏_𝑎(𝑟𝑐(1−𝑐)/(2−𝑐))𝑇_𝑟𝑟+𝑇_𝜃𝜃𝑑𝑟𝜆∫^𝑏_𝑎(()/(.𝑟𝑐3−2𝑐1−𝑐)(2−𝑐))𝑑𝑟 (14) Taking the non-homogeneity in the cylinder due to variable compressibility 𝑇_𝑟𝑟=𝐴₁𝑐exp⁰𝑟^−𝑘𝑘𝑐−exp⁰𝑏^−𝑘𝑘−𝑝₂;𝑇_𝜃𝜃=𝐴₁1−𝑐₀𝑟^−𝑘𝑐exp⁰𝑟^−𝑘𝑘𝑐−exp⁰𝑏^−𝑘𝑘−𝑝₂;𝑇_𝑧𝑧=1−𝑐₀𝑟^−𝑘2−𝑐₀𝑟^−𝑘𝑇_𝑟𝑟+𝑇_𝜃𝜃+𝜆𝑐₀𝑟^−𝑘1−𝑐₀𝑟^−𝑘3−2𝑐₀𝑟^−𝑘2−𝑐₀𝑟^−𝑘𝑒_𝑧𝑧.Also,𝑇_𝜃𝜃−𝑇_𝑟𝑟=𝑝₁−𝑝₂𝑐₀𝑟^−𝑘exp𝑐₀𝑎^−𝑘/𝑘−exp𝑐₀𝑏^−𝑘𝑐/𝑘×exp⁰𝑟^−𝑘𝑘. (15)(16) It has been observed from (16) that |𝑇_𝑟𝑟−𝑇_𝜃𝜃| is maximum at 𝑟=(𝑒²𝑏^𝑘)^1/𝑘=𝑟₁; that is, yielding starts at 𝑟=𝑟₁; therefore, 𝑇_𝜃𝜃−𝑇_{𝑟𝑟𝑟=𝑟1}=𝑝₁−𝑝₂𝑐₀𝑟₁^−𝑘exp𝑐₀𝑟₁^−𝑘/𝑘exp𝑐₀𝑎^−𝑘/𝑘−exp𝑐₀𝑏^−𝑘/𝑘≡𝑌. (17) Thus pressure required for initial yielding is given by 𝑃_𝑖=𝑒exp−𝑒²𝑅₀^−𝑘−exp−𝑒²𝑘;𝑃_𝑖=𝑝₁𝑌−𝑝₂𝑌=𝑃_𝑖1−𝑃_𝑖2. (18)

For full plasticity (𝐶₀→0), (16) becomes 𝑇_𝜃𝜃−𝑇_{𝑟𝑟𝑟=𝑏}=𝑝₁−𝑝₂𝑘𝑏^−𝑘𝑎^−𝑘−𝑏^−𝑘=𝑌₁𝑃_𝑓=𝑏^𝑘𝑎^−𝑘−𝑏^−𝑘𝑘. (19) Now, we introduced the following nondimensional quantities as 𝑟𝑅=𝑏;𝑅₀=𝑎𝑏;𝜎_𝑟𝑟=𝑇_𝑟𝑟𝑌;𝜎_𝜃𝜃=𝑇_𝜃𝜃𝑌;𝜎_𝑧𝑧=𝑇_𝑧𝑧𝑌. (20) The necessary pressure required for initial yielding in nondimensional form is given by 𝑃_𝑖=𝑒exp−𝑒²𝑅₀^−𝑘−exp−𝑒²𝑘. (21) The transitional stresses are obtained as 𝜎_𝑟𝑟=−𝑃_𝑖exp𝑐₀𝑏^−𝑘/𝑘𝑅₀^−𝑘−1−1exp𝑐₀𝑏^−𝑘/𝑘𝑅₀^−𝑘−1−1−𝑃_𝑖2,𝜎_𝜃𝜃=−𝑃_𝑖exp𝑐₀𝑏^−𝑘/𝑘𝑅₀^−𝑘×−1−11−𝑐₀𝑏^−𝑘𝑅^−𝑘𝑐exp⁰𝑏^−𝑘𝑘𝑅₀^−𝑘−1−1−𝑃_𝑖2,𝜎_𝑧𝑧=1−𝑐₀𝑏^−𝑘𝑅^−𝑘2−𝑐₀𝑏^−𝑘𝑅^−𝑘𝜎_𝑟𝑟+𝜎_𝜃𝜃+𝜆𝑐₀𝑏^−𝑘𝑅^−𝑘3−2𝑐₀(𝑏𝑅)^−𝑘1−𝑐₀(𝑏𝑅)^−𝑘2−𝑐₀(𝑏𝑅)^−𝑘𝑒_𝑧𝑧, (22) where 𝑒_𝑧𝑧=−𝑃_𝑖1−𝑃_𝑖2−2𝜋¹_𝑅0𝑏²𝑅1−𝑐₀(𝑏𝑅)^−𝑘2−𝑐₀(𝑏𝑅)^−𝑘×𝜎_𝑟𝑟+𝜎_𝜃𝜃×𝜆𝑑𝑅_1𝑅0𝑐₀𝑏^−𝑘−1𝑅^−𝑘3−2𝑐₀(𝑏𝑅)^−𝑘1−𝑐₀(𝑏𝑅)^−𝑘2−𝑐₀(𝑏𝑅)^−𝑘𝑑𝑅⁻¹. (23) Also, pressure required for fully plastic state is given by 𝑃_𝑓=𝑅₀^−𝑘−1𝑘;𝑃_𝑓=𝑝₁𝑌₁−𝑝₂𝑌₁=𝑃_𝑓1−𝑃_𝑓2, (24) and fully plastic stresses are obtained as 𝜎_𝑟𝑟=−𝑃_𝑓𝑅^−𝑘−1𝑅₀^−𝑘−1−𝑃_𝑓2;𝜎_𝜃𝜃=𝜎_𝑟−𝑘𝑅^−𝑘−𝑃_𝑓𝑅₀^−𝑘;𝜎−1_𝑧𝑧=𝑘𝜆𝑅^−𝑘−𝑃_𝑓/2𝜋−(1/2)∫¹_𝑅0𝑅𝑏²𝜎_𝑟𝑟+𝜎_𝜃𝜃𝑑𝑅𝑅₀^−𝑘.−1 (25)

Particular Case: Nonhomogeneous Cylinder under Internal Pressure Only. The stresses in fully plastic state are 𝜎_𝑟𝑟=𝑃_𝑓2𝑅^−𝑘−1𝑅₀^−𝑘−1−𝑃_𝑓2;𝜎_𝜃𝜃=𝜎_𝑟𝑟−𝑘𝑅^−𝑘𝑃_𝑓2𝑅₀^−𝑘;𝜎−1_𝑧𝑧=𝑘𝜆𝑅^−𝑘𝑃_𝑓2/2𝜋−(1/2)∫¹_𝑅0𝑅𝑏²𝜎_𝑟𝑟+𝜎_𝜃𝜃𝑑𝑅𝑅₀^−𝑘.−1 (26) These equations in nondimensional form are the same as those obtained by Sharma [12].

6. Numerical Discussion

To observe the combined effect of pressure on a cylinder made of homogeneous and nonhomogeneous material, graphs have been drawn between pressure and radii ratios 𝑅₀ = 0.1 (0.1) 0.5. For a homogeneous (𝑘=0) circular cylinder, yielding starts at internal surface whereas for a circular cylinder made of nonhomogeneous material (𝑘<0, non-homogeneity increases radially), yielding takes place at any radius 𝑟 where 𝑎<𝑟<𝑏 depending upon values of 𝐶₀ and 𝑘. Effective pressure is maximum at internal surface for cylinder made of nonhomogeneous as well as homogeneous material. It is seen from Figure 1 that for homogeneous cylinder, high effective pressure is required for initial yielding than that of nonhomogeneous cylinder. Also, for cylinder made of homogeneous material, effective pressure required for initial yielding is less for highly compressible circular cylinder whereas for circular cylinder made of nonhomogeneous materials, high effective pressure is required for highly compressible cylinder. It is also seen from Figure 2 that pressure (internal/external) is maximum at external surface for cylinder made of nonhomogeneous as well as homogeneous material. It is also seen that homogeneous cylinder requires high pressure for initial yielding than that of nonhomogeneous cylinder. Also, high pressure is required for initial yielding for highly compressible homogeneous cylinder whereas less pressure is required for highly compressible nonhomogeneous cylinder.

Figure 1 

Effective pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder for different compressibility parameters.

Figure 2 

External or internal pressure required for initial yielding for homogeneous and nonhomogeneous circular cylinder (internal or external = 20) for different compressibility parameters.

It has also been observed from Figure 3 that, for homogeneous and nonhomogeneous circular cylinder, effective pressure required for fully plastic state is maximum at the internal surface and for nonhomogeneous material less effective pressure is required for fully plastic state for circular cylinder made of highly compressible material. It is also observed that effective pressure required for fully plastic state is more for cylinder made of homogeneous material than that of nonhomogeneous material. For homogeneous/nonhomogeneous circular cylinder, pressure (internal/external) required for fully plastic state is maximum at the external surface. It has been seen from Figure 4 that for nonhomogeneous cylinder made of highly compressible material, high pressure is required for fully plastic state. It is also observed from Figure 4 that pressure required for fully plastic state is more for cylinder made of nonhomogeneous material than that of homogeneous material.

Figure 3 

Effective pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder for different compressibility parameters.

Figure 4 

External or internal pressure required for fully plastic state for homogeneous and nonhomogeneous circular cylinder (internal or external = 20) for different compressibility parameters.

From Figures 5 and 6, it has been observed that for homogeneous cylinder under external pressure only, circumferential stresses are maximum at internal surface while for nonhomogeneous cylinder, stresses are maximum at external surface. These stresses increase significantly with the increase in external pressure. With internal pressure only, as seen from Figures 7 and 8, circumferential stresses are maximum at internal surface for homogeneous cylinder while maximum at external surface for nonhomogeneous cylinder. Also it has been observed that the compressible circumferential stresses change to tensile stresses. It has also been observed from Figures 9 and 10 that with the increase in pressure, circumferential stresses increases significantly. With the increase in external pressure (greater than that of internal pressure), circumferential stresses increases. It has been observed from Figures 11 and 12 (without internal pressure) that fully plastic stresses are maximum at external surface for cylinder made of nonhomogeneous material and at internal surface for cylinder made of homogeneous material. Also, it has been observed that highly compressible cylinder is having less stress whereas less compressible cylinder is having high stress. With the increase in external pressure, stresses increases significantly. From Figures 13 and 14, it has also been observed that with the introduction of internal pressure (without external), compressible circumferential stresses are maximum at internal surface for homogeneous while at external surface for nonhomogeneous cylinder. As external pressure increase and becomes more than that of internal one, then stresses again increase but are of compressive nature. Also it has been observed that stresses increases significantly with the increase in internal pressure as can be seen from Figures 15 and 16.

Figure 5 

Homogeneous transitional stresses for a thick-walled circular cylinder under external pressure (𝑃2 = 5 and 15).

Figure 6 

Nonhomogeneous transitional stresses for a thick-walled circular cylinder under external pressure (𝑃2 = 5 and 15).

Figure 7 

Homogeneous transitional stresses for a thick-walled circular cylinder under internal (𝑃1=5) and external pressure (𝑃2 = 0 and 15).

Figure 8 

Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (𝑃1=5) and external pressure (𝑃2 = 0 and 15).

Figure 9 

Homogeneous transitional stresses for a thick-walled circular cylinder under internal (𝑃1=10) and external pressure (𝑃2 = 0 and 15).

Figure 10 

Nonhomogeneous transitional stresses for a thick-walled circular cylinder under internal (𝑃1=10) and external pressure (𝑃2 = 0 and 15).

Figure 11 

Homogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (𝑃2 = 5 and 15).

Figure 12 

Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under external pressure (𝑃2 = 5 and 15).

Figure 13 

Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (𝑃1=5) and external pressure (𝑃2 = 0 and 15).

Figure 14 

Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (𝑃1=5) and external pressure (𝑃2 = 0 and 15).

Figure 15 

Homogeneous fully plastic stresses for a thick-walled circular cylinder under internal (𝑃1=10) and external pressure (𝑃2 = 0 and 15).

Figure 16 

Nonhomogeneous fully plastic stresses for a thick-walled circular cylinder under internal (𝑃1=10) and external pressure (𝑃2 = 0 and 15).

7. Conclusions

From the above analysis we can conclude that nonhomogeneous cylinder with internal and external pressure is on the safer side of the design as compared to homogeneous cylinder because nonhomogeneous cylinder requires high pressure for initial yielding as compared to homogeneous cylinder. It has also been concluded that highly compressible nonhomogeneous cylinder is on the safer side of the design as compared to less compressible nonhomogeneous circular cylinder because highly compressible cylinder required high pressure for initial yielding as compared to less compressible nonhomogeneous cylinder, which leads to the idea of “stress saving” that minimizes the possibility of fracture of cylinder.