Creep-fatigue-ratcheting interaction of functionally graded pressure vessel is an important failure mode and their interaction is necessary to be examined to be able to predict service life of the functionally graded pressure vessel.
1.2 What is a Pressure Vessel?
Pressure vessel is typically a container that is subjected to a differential pressure between the internal and external pressure. Theoretically, a pressure vessel can be of any shape, however it is usually an axisymmetric equipment with cylindrical shell closed with hemispherical or formed (ellipsoidal or torispherical) end caps called heads, or may also be a full spherical vessel as complex shapes would be difficult to analyze and construct. The pressure vessel limit is normally up to the first connection of the connection nozzles attached to the cylindrical shell or heads.
Pressure vessel is commonly used in the oil and gas, petrochemical industries, power generation, aerospace and other systems. The pressure vessel is designed based on specific design parameters, namely material, pressure, temperature and cyclic operating condition.
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In the oil and gas production, full well stream fluid from the well is normally at high temperature and pressure. This high pressure and temperature resulted in thick-walled pressure vessel which will be very heavy. In addition the fluid normally is corrosive in nature, and to overcome the corrosion effect, the pressure vessel is normally internally clad with anti corrosion metal liner.
This thick-walled equipment will be very difficult to construct. For thick-walled carbon steel vessel post weld heat treatment is mandatory. With the addition of cladding material and post weld heat treatment requirement, the construction is becoming more difficult. To overcome this construction problem, design optimization has to be carried out while keeping the vessel from surpassing the design stress.
One of the optimization solutions that had attracted numerous researchers' interest is the use of Functionally Graded Material (FGM) as the construction material. Among the existing researchers are N Tutunku and M Ozturk , Jabbari et al. . These researchers had established solutions to analysis of functionally graded cylindrical shell and heads. From their research it can be concluded that FGM may be taken as one of the optimization solution.
ASME Boiler Pressure Vessel Code, ASME Section VIII, British Codes BS PD5500, European codes EN14335 are a few internationally recognized pressure vessel design codes. These codes provide guidelines for the design of the pressure vessel using a specific formula available in the codes, namely known as the design by rules. Alternatively a more optimized design can be achieved using a more rigorous analytical method, such as Finite Element Analysis.
Pressure vessel in most application operates underfluctuation of pressure and temperature. Thesethermal mechanical loadings applied throughout the service life of the equipment will cause significant creep, fatigue and ratcheting phenomena on the pressure vessel. Either of these phenomena on its own reduces the service life of the pressure vessel.
Studies carried out by existing researchers had shown that creep-fatigue-ratcheting interaction is very important in the design and assessment of the pressure vessel.
Studies by numerous researchers also had shown that usage of a properly graded Functionally Graded Material (FGM) has significantly reduced the pressure vessel thickness when it is subjected to high pressure and similar to its metallic material, FGM also exhibits creep, fatigue and ratcheting phenomenon.
1.3 Functionally Graded Pressure Vessel.
Functionally Graded Pressure Vessel is a pressure vessel that is constructed using Functionally Graded Material (FGM). FGM is typically composite materials blend together in which the mechanical properties varies smoothly and continuously over volume. FGM exhibits fully properties of one material on one side of the surface and full properties of other material on the other side while the in-between properties are varied throughout the thickness. FGM is a non homogenous but isotropic material. Since it was first proposed by Japanese Scientist in the 1980's FGM has receive considerable attentions for engineering application, including in pressure vessel application.
Figure 1.2.1 shows a typical appearance of the FGM cross-section. The gradient can be described in many ways. The gradient in most researched has been described as either exponential, power law or linear law; however power law gradient is mostly used. The power law expression is normally expressed as:
E (x1) = E0x1n
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Where E is the material Young's Modulus, x1 is the point along the axis of gradient and n is the exponent that dictates the shape of the property profile.
Material 1 and material 2 graded properties
Typical cross section of FGM
Figure 1.2.1Typical visualization of FGM cross-section
The advantages of using FGM over the traditionally layered composite is that, in FGM the mismatch between properties of the layers disappear while the characteristics of the two materials are preserved. Additionally, there is an increased interfacial strength at the joint and thus the likelihood of debonding is reduced.A layered material produces stress discontinuity and may also exhibit poor adhesion.
E Carrera et al.  in their work concluded that FGM reduces shear and normal stress gradient at the interfaces and provides better buckling protection as compared to two-layered pressure vessel wall.
Graded interface giving a smooth continuous properties change
Layered composite having biomaterial interface giving rise to stress discontinuity giving a smooth continuous properties change
Figure 1.2.2 Visualization of stress comparison between layered composite and functionally graded material.
1.4 Fatigue Failure
Fatigue failure is a metal failure under fluctuating or cyclic stress. If a static load considerably lower than the material tensile or yield strength is applied repeatedly onto the material, it will exhibit fatigue failure. Fatigue failure is brittle-like failure, normally occurs in three stages, i.e. crack initiation, incremental crack propagation and catastrophic failure stages.
There are two types of fatigue situation, the Low-cycle Fatigue (LCF) and the high-cycle fatigue (HCF). When the number of cycles to failure is less than 1,000,000 cycles, it is term as LCF failure. LCF is common in pressure vessel due to the operating condition of the vessel.
1.5 Creep Failure
When a material is subjected to long term influence of high stresses, the material starts to accumulate the strain and slowly moves into permanent deformation and finally into failure. This phenomenon is called creep failure.Creep failure or stress rupture is time-dependent. It does not occur instantaneously upon the application of load instead it is a strain accumulation failure.
In recent years many researchers has shown interest in creep problem on FGM pressure vessel operating under high temperature and pressure. T Singh et al. [4 - 7], concluded that the strain rate in the cylindrical FGM reduces significantly when the distribution of reinforcement matrix of the material is graded. L H You et al. , J J Chen  had investigated the creep effect of functionally graded cylinder. L H You et al.  had investigated the steady state creep of internally pressurized thick-walled spherical pressure vessel.
1.6 Ratcheting Failure
Ratcheting failure occurs when the pressure vessel or component is subjected to asymmetrical cyclic stressing. Ratcheting can be visualized as the cyclic accumulation of inelastic deformation, which exhibit in the permanent change of shape of the component. This change in shape reduces the ability of the component to resist collapse and upon further cyclic loading this will lead to failure.
ASME Boiler Pressure Vessel Code, namely ASME Section VIII Division II, specified that pressure vessel should be protected against failure from plastic collapse, local failure, cyclic loading and ratcheting.
Over the decades, the design of pressure vessel had drawn up interest of many researchers whom had performed their studies either experimentally or numerically. F Z Xuan et al.  studied and performed testing and simulations on Ratcheting-Creep interaction of chrome ferrite steel. K S Lim et al.  studied on fatigue and ratcheting behavior of copper alloy under stress amplitude and mean stress.
Therefore the creep-fatigue-ratcheting interaction of functionally graded pressure vessel is an important failure mode and their interaction is necessary to be examined and to be able to predict service life of the pressure vessel.
2.0 Scope of Study
The scope of this research can be visualized as the following:
To numerically investigate the creep, ratcheting and fatigue behavior of functionally graded pressure vessel and their interaction as an integrated approached.
To investigate the effect of mean stress, stress amplitudes, stress rates and peak stress on ratcheting strain.
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To illustrate the interaction of creep, ratcheting behavior and fatigue failure with the number of operating cycles and strain amplitude.
3.0 Objective of Study
The objective of this study is to study the effect of mean stress, stress amplitudes, stress rate and peak stress on fatigue life of the functionally graded pressure vessel. In achieving this objective, the following will have to be look at and relationship established:
Formulation of a numerical formula for use in analytical analysis in evaluating the failure of FGM at room and high temperature.
The creep curve of the FGM at different temperature
The time-dependent strain cyclic features, which will show the effect of straining rate on cyclic hardening.
The ratcheting behavior of the material in relation to various cyclic stressing at various mean stress amplitude. This will show the relationship between ratcheting strain increase rate with stress amplitude.
Effect of mean stress, stress amplitude, stress ratio and peak stress on the fatigue life of the component.
The interaction of creep-ratcheting-fatigue of material. This will give relationship
Existing studies on creep, fatigue, and ratcheting of functionally graded pressure vessel was carried out considering cylindrical shell and heads in isolation. The set-back of this approach is that the interaction of the creep-ratcheting-fatigue phenomenon of the pressure vessel could not be established. Also the design of the cylindrical shell and heads cannot be optimized. For this particular study it is proposed to carry the study of the pressure vessel as a whole, where the cylindrical section and the end caps is to be modeled as one model.
The study is proposed to be carried under the following sequence:
4.1 Establishing of mathematical formulation and FEM Model
Since there are various formulations and simulations had been performed by various earlier researchers, this study will start with looking into combining the earlier numerical formulas and established a new proposed numerical equation that suit the proposed pressure vessel model.
The creep behavior of FGM will accord the documented creep law based on threshold stress, as utilized by T Singh [4 - 7]
Where the symbols represent:
- Effective creep rate - Structural dependent parameter
- Effective stress - Threshold stress
- True stress exponent - True activation energy
- Temperature dependent Young's Modulus - Gas Constant
- Operating temperature
Finite element model will also be developed using commercially available software ABAQUS or ANSYS. As the FGM properties are varying through the thickness, the thickness will be divided into finite portion and treated like isotropic material. Using the power law properties will be calculated using a self-developed computational program where the properties at the mid-plane of each layer will be used as parameters to the FEM model. It is suggested that shell element model be used in the FEM model.
4.2 Validation of mathematical model and FEM model
This mathematical model and the Finite element model will be validated against the results obtained from experimental studies on creep-ratcheting-fatigue interaction of SS304 stainless steel at room and high temperatureperformed by Zhang Juan et al .As this experiment is conducted using a homogenous material, volume fraction to one of the material in the mathematical formulation has to be set to zero leaving the other material volume fraction as unity. With this modification, the mathematical model will resemble that of homogenous material.
The mathematical model will also be check for consistency with other available experimental studies conducted by other researchers.
4.3 Application to FGM
The model will then be extended to FGM. When extending to FGM the volume fraction that was set to zero earlier will become active and both material will have its own volume fraction, thus making the mathematical model applicable to FGM.
4.4 Data gathering and analysis
Upon validation and extension to FGM, the model will be used to run series of varied variables according to the scope of studies and the data collected.
The data collected will be analyses, tabulated and if necessary plot as graph.
5.0 Expected Finding
The study is expected to enable construction of failure model containing interaction of creep, ratcheting and low-cycle fatigue of functionally graded pressure vessel and applicable to predict the estimated service life of the functionally graded pressure vessel.
6.0 Research Schedule
The research is anticipated to be carried out according to the following schedule:
Established a firm problem statement
Develop a numerical formulation and validation
Numerical analysis , data gathering and data analysis
Report writing and presentation