Elasto-plastic analysis of thick-walled tanks made of targeted materials with linear kinematic stiffness behavior under cyclic loading

Master's Thesis in Mechanical Engineering - Applied Design

Abstract

Elastoplastic analysis of thick-walled tanks made of targeted materials with linear kinematic stiffness behavior under cyclic loading

The analysis of stress and deformation of spherical thick-walled tanks made of flexible materials under internal pressure loading and temperature difference is discussed in this thesis. The material parameters are considered as a function of the radius, which play an important role in the behavior of such materials. In order to determine their significant role, several types of materials with different material parameters have been subjected to temperature and internal pressure gradients and investigated. Also, their difference in the submission of these types of tanks has been identified. To ensure the investigations done, the results obtained in the elastic state have been compared with other articles. Also, by comparing the results related to loadings, it was realized the important role of temperature difference in the analysis of reservoir behavior. Then, the elastoplastic analysis of these tanks was done. For their elastoplastic analysis, linear kinematic stiffness behavior has been followed. According to the investigation and analysis of their behavior under the loading of wheels with temperature difference and constant internal pressure, which led to the determination of the separation diagram of the phenomena, we observed that the spherical tank still remains in the elastic state until the high temperature difference. Also, the elastic region of Shikdan occupies a large area and after that it enters the plastic region of Shikdan. In fact, with this work, we have predicted the behavior of the material before different loadings.

1-1-Preface

For an elastic solid, deformations are reversible after removal of applied loads. In plastic solids, after removing the load, deformations remain in the material and do not return to their original state. These inelastic deformations remain in equilibrium. Their behavior is assumed to be independent of time. As can be seen in Figure 1-1-1, deformation in hardenable elastoplastic solids consists of two parts: elastic deformation and inelastic deformation. When the stress is lower than the yield stress ( ), the plastic strain is zero.

Figure 1-1-1: Hardenable elastoplastic solid

The analogical model of the behavior of this type of material is shown by the developed Saint Venan model.

Various models have been presented to describe the hardenability of solids by deformation. Non-isotropic hardening and kinematic hardening are among them.

Although most materials have non-isotropic hardening, but due to the simplicity of the isotropic hardness model, it is widely used. Especially when the loading is radial, which means that the stress vector in the stress space has a constant direction. In general, a material with isotropic hardness is said to be a material whose elastic zone boundary depends on only one scalar parameter.

The stress-strain curve in tension is symmetrical with the stress-strain curve in compression with respect to the origin (point B in Figure 1-1-2).

The boundary of the elastic zone is symmetrical in all directions, relative to the center O

(A)

(B)

Figure 1-1-2: (a) - Tensile-twist test under isotropic hardness, (b) - Tensile-pressure test In this model, the range of the elastic region remains constant, but this range moves in the stress space. The center of the elastic region (point C in Figure 1-1-3) is called internal stress or return stress. The stress-strain curve in tension and compression is symmetrical around point C. Under a tensile-torsional test, the yield surface is obtained by displacing the initial yield surface by a vector.

(b)

(a)

Figure 1-1-3: (a) - Tensile test - torsion under kinematic stiffness, (b) - Tensile test - compression

Bashinger effect is determined when a compression test is performed after a tensile test. Usually, the tensile test hardens the material in tension (the elastic limit increases), but it softens in the direction of compression. Figure 1-1-4 shows that the elastic limit in pressure is lower than the initial elastic limit in pressure.

Among the two mentioned models, kinematic hardening is closer to reality and provides a better estimate of the Bashinger effect. Hardening properties of most metals and alloys change during testing. Figure 5-1-1 shows the parameters used for a stable cycle of round stresses. Depending on the type of material, temperature and its initial state, hardening and softening occurs. rtl;"> 

Cyclic softening occurs when a stress range decreases during a cycle test under a constant strain range (Figure 1-1-6-(a)) or when a strain range increases during a cycle test under a constant stress range (Figure 1-1-6-(b)).

Cyclic hardening occurs when during a Test laps under a constant stress range, the stress range increases (Figure 1-1-6-(a)) or when in a test laps under a constant stress range, the strain range decreases (Figure 1-1-6-(b)).

If the load of the average stress cycles is non-zero, other effects appear (Figure 7-1-1). This applied asymmetric loading causes the plastic strain to not grow and remain constant in each cycle, or often causes it to grow in each cycle even after the stability of the stress-strain loop. When a range of strain is applied, the release or non-release of the average stress is observed. (Figure 8-1-1)]1[.

(Images and diagrams are available in the main file)

ABSTRACT

Elasto-plastic analysis of thick walled FG pressure vessels with linear kinematic hardening behavior under cyclic loading

BY

In this thesis, thick-walled spherical tanks made of functionally graded materials under internal pressure and temperature gradient are studied.  Material parameters have been considered as a function of radial coordinate which serves an important function on the behavior of such materials. To clarify their significant role, several types of materials with different material parameters under internal pressure and temperature gradient have been studied. Furthermore, their effects on yielding have been determined. The elastic results have been compared with other papers to validate the investigations. Moreover, comparing the results associated with loadings has revealed the importance of temperature gradient in analysis of vessels behavior. Afterwards, considering linear kinematic hardening behavior, elasto-plastic analysis has been conducted. According to investigations and study of their behavior under thermal cyclic loading and constant internal pressure which results in determining phenomena separation diagram, we observed that spherical vessels remain in elastic region under a high temperature gradient. Also, a large area has been allocated to the elastic shakedown region, and outside this region, the material enters the plastic shakedown region. As a matter of fact, using this procedure, we will be able to predict the behavior of the material in advance.