A study on the effect of creep on strength of nuclear reactor vessel during severe accident

Melt pool in the lower head and Reactor Pressure Vessel failure .... The hypothetical scenario of severe accident in a nuclear power plan with core meltdown and formation of a melt pool

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

-

NGUYEN VAN THANH

A STUDY ON THE EFFECT OF CREEP ON STRENGTH OF NUCLEAR REACTOR VESSEL

DURING SEVERE ACCIDENT

Field: ENGINEERING MECHANICS

MASTER OF SCIENCE THESIS

ENGINEERING MECHANICS

Scientific Supervisor Asso, Pro, NGUYEN VIET HUNG

Table of contents

Table of contents i

Acknowledgement iii

Declaration iv

Nomenclature v

List of acronyms vii

List of figures viii

List of tables x

Abstract xi

Chapter 1: Introduction 1

1.1 Motivation 1

1.2 Severe accident in a Light Water Reactor 2

1.2.1 Basics of Light Water Reactor nuclear reactor 2

1.2.2 In-vessel accident progression and phenomena 4

1.2.3 Melt pool in the lower head and Reactor Pressure Vessel failure 6

1.3 Objectives of thesis work 8

Chapter 2: Thermal hydraulic calculation of homogeneous melt pool 10

2.1 Basics of natural convection heat transfer 10

2.2 Heat transfer simulation of homogeneous melt pool 15

2.2.1 Modeling of natural convection 15

2.2.2 Temperature results and discussion 19

Chapter 3: Numerical creep modeling of reactor pressure vessel lower head 22

3.1 Creep mechanism theory 22

3.1.1 Background on creep 22

3.1.2 Creep equation and hardening theory 24

3.2 Linking a Time-hardening creep model to ANSYS 29

3.2.1 Creep calculation in ANSYS 29

3.3.2 The use of User Programmable Features 33

3.2.3 Linking a Time-hardening creep model into ANSYS 34

3.3 Creep simulation of reactor pressure vessel 40

3.3.1 SA533B1 material properties 40

3.3.2 Geometry model and Mesh 41

3.3.3 Boundary condition 42

3.3.4 The results and discussion 45

Chapter 4: Summary and outlook 49

References 51

Appendix 52

A1 Usercreep.F routine 52

A2 Compiling and linking UPFs procedures 55

A3 SA533B1 temperature - dependent properties 59

Acknowledgement

I would like to express great thank to Associate Professor Nguyen Viet Hung - the director of International Computational Science and Engineering Institute - Hanoi University of Science and Technology (ICSE, HUST), for his supervision during the time of implementing this thesis work

My special thanks go to MSc Le Anh Tuan and other colleagues in the ICSE for their support and assistance throughout the duration of this thesis

I would like to take this opportunity to thank Advanced Technology Joint Stock Company (http://advantech.vn/) for providing the licensed ANSYS software and other relative documents

Lastly, I would also like to thank my family and friends for their encouragement throughout the time of taking Master course and doing this thesis as well

Hanoi, November 2014 Nguyen Van Thanh

Declaration

I, the undersigned, Nguyen Van Thanh, hereby declare that the work entitled “A study

on the effect of creep on strength of nuclear reactor vessel during severe accident” is

my original work under the supervision of Associate Professor Nguyen Viet Hung I

do not copy from any other sources except where due reference or acknowledgement

is made explicitly in the text, nor any part is written for me by another person

Hanoi, November 2014 Nguyen Van Thanh

E Elastic modulus, Mpa

 Equivalent stress, MPa

R Boltzmann gas constant

k Coefficient of thermal conductivity, W/m.K

h Film coefficient in natural convection heat transfer, W/m2.K

T Temperature of the surface, K

 Density of the quiescent fluid sufficiently far from the surface, kg/m3

List of acronyms

LWR Light Water Reactor

PWR Pressurized Water Reactor

BWR Boiling Water Reactor

RPV Reactor Pressure Vessel

LOCA Loss of Cooling Accident

CFD Computational Fluid Dynamic

FEA FEA: Finite Element Analysis

IAEA International Atomic Energy Agency

SMART System-Integrated Modular Advanced Reactor

List of figures

Figure 1.1 PWR reactor design 2

Figure 1.2 BWR reactor design 3

Figure 1.3 Physical phenomenon during the severe accident [IRSN and CEA-2007/83-351] 5

Figure 1.4 Core meltdown accident progression overview 5

Figure 1.5 (a) Illustration of melt pool formation; (b) Layer separation of melt pool 7

Figure 1.6 SMART PWR design reactor assembly 8

Figure 1.7 Two considered hypothetical scenarios, (a) Scenario 1 and (b) Scenario 2 9 Figure 2.1 (a) Force acting on a differential control volume in natural convection; and (b) typical velocity and temperature profiles for natural convection flow over a hot vertical plate 11

Figure 2.2 Natural convection model induced by internal heating 15

Figure 2.3 (a) 2-D axisymmetric geometry of melt pool and vessel wall, (b) Mesh configuration 17

Figure 2.4 Temperature field in the whole of fluid volume and vessel wall at 6h in (a) scenario 1; and (b) scenario 2 19

Figure 2.5 Temperature field in vessel wall at 6h in (a) scenario 1; and (b) scenario 2 20

Figure 2.6 Temperature over time at the highest temperature point of vessel wall 20

Figure 3.1 Three stages of creep curve [1] 22

Figure 3.2 [1] Creep (a), and Stress relaxation (b) 23

Figure 3.3 Creep strain history prediction from Time Hardening theory [Kraus, 1980] 27

Figure 3.4 Creep strain history prediction from Strain Hardening theory [Kraus, 1980] 28 Figure 3.5 Step by step procedure to start working with UPFs on Window platform 34

Figure 3.6 Experimental SA533B1 creep curve at T = 1150K and Stress = 26.5 MPa (J L Rempe, 1993, [1]) 35Figure 3.7 Verification of usercreep.F routine 39Figure 3.8 (a) 2-D axisymmetric geometry, (b) mesh with six layers of elements over wall thickness, (c) 2-D - Quad Plane 183 element geometry 41Figure 3.9 Graph of mesh independent investigation 41Figure 3.10 Process of importing temperature field from CFD to Mechanical APDL 43Figure 3.11 Mechanical boundary conditions applied to both of scenarios 44Figure 3.12 Graph of temperature distribution at internal surface of vessel wall at 4.3h 46Figure 3.13 Equivalent creep strain at 4.3h in (a) scenario 1, and (b) scenario 2 47Figure 3.14 Equivalent creep strain at 6h in (a) scenario 1, and (b) scenario 2 47Figure 3.15 (a) equivalent creep strain at 4.3h with creep effect; (b) total mechanical and thermal strain at 4.3h without creep effect 48

List of tables

Table 1.1 Core melt accident progression 6

Table 2.1 Some correlations of internally heated fluid volume 16

Table 2.2 Main thermal properties of homogeneous melt pool [3] 18

Table 3.1 Stress-dependent function 24

Table 3.2 Time-dependent function 25

Table 3.3 13 available implicit creep models in ANSYS [1] 32

Table 3.4 SA533B1 temperature-dependent coefficients of Time hardening creep model given in Equation (3.41) 40

Abstract

Severe accidents in a Light Water Reactor (LWR) have been a subject of interest for the last few decades The research in this area tends to reach the understanding of the possible physical processes and phenomena during the hypothetical severe accidents, with the ultimate goal of developing solutions for safety analysis and predicting the events liable to occur during severe accidents in nuclear reactors

The hypothetical scenario of severe accident in a nuclear power plan with core meltdown and formation of a melt pool in the lower head of a reactor pressure vessel (RPV) can result in the creep failure of RPV and discharging of the melt to the containment The thermal interactions of core melt pool with the reactor vessel depend strongly on the heat transfer ability of the core melt pool itself In the case of inadequate cooling, the excessive heat would drive the structure overheating, and hence govern the vessel wall failure mode and timing That is the reason why the calculation of transient melt pool heat transfer in the reactor lower head and numerical prediction of the creep failure of vessel wall are of paramount importance for severe accidents analysis in nuclear safety

The main purpose of the present thesis work is to apply the ANSYS numerical simulation tools for determine the effects of creep phenomenon on the strength of RPV under an assumed hypothetical core meltdown accident To achieve the goal, firstly the Computational Fluid Dynamic (CFD) with module Fluent has been used to describe heat transfer in an internally heated volume of homogeneous melt pool The temperature field within the whole of melt pool and the vessel wall are calculated Based on the calculated temperature field in the vessel wall, the creep mechanics simulation are then performed The main task of the numerical creep simulation in this thesis is to using ANSYS User Programmable Features (UPF) to link a Time hardening creep model [2] into ANSYS Solver for more accurate describing the behavior of vessel wall material under severe conditions of temperature in an assumed hypothetical accident

Keywords: creep mechanics, ANSYS UPFs, numerical analysis, CFD simulation, natural convection heat transfer

Chapter 1: Introduction

1.1 Motivation

Nuclear power is a safe and efficient measure to provide the pure power resource guaranteeing the sustainable development in demanding of electrical energy which is strongly increasing in the developing countries The future nuclear power plants demand that there are no consequences for the environment, even in any conceivable accident scenario

Vietnam is currently on the project of two nuclear power plants located in the NinhThuan province Thus, it is crucial to increase knowledge about nuclear safety analysis, especially in the severe accident management after Fukushima accident The motivation for doing this thesis work is to reach a basic understanding of possible phenomena in severe accident in nuclear power plants The most two questions must

be answer here are the physical phenomena of natural convection heat transfer in the assumed homogeneous melt pool in reactor lower head and the time of reaching the tertiary creep stage of vessel wall material

1.2 Severe accident in a Light Water Reactor

According to the definition of IAEA, severe accident is the accident conditions more severe than a design basic accident with involving significant core degradation due to violent core disruption or slow core melting The significant feature of severe accident

by IAEA definition is that whatever the accident initiating condition, the outcome of accident is significant core degradation

1.2.1 Basics of Light Water Reactor nuclear reactor

LWR comes in two distinct types, the PWR and the BWR In both cases, water is used

as both of the coolant and neutron moderator In the PWR (Figure 1.1), water is kept liquid in the core, preventing it boiling and turning into steam by maintaining the pressure vessel at high operating pressure It passes through a steam generator for creating steam, which runs the turbines In the BWR (Figure 1.2), water is allowed to boil into steam within the pressure vessel This steam is then passed through the turbines, and back into the reactor

Figure 1.1 PWR reactor design

Figure 1.2 BWR reactor design

There are basically three dangers for a LWR reactor in the accident of LOCA Firstly, the risk of a steam explosion The pressure vessel of a reactor is basically a boiler, if the pressure goes high enough and pressure valves fail, it could burst

The second major risk is that of a hydrogen explosion At very high temperatures and

in the presence of molten reactor fuel and cladding material the water in the core can break down forming to hydrogen and oxygen If concentrations of hydrogen and oxygen get high enough there is an increasing risk of an explosion This is a particular risk if containment fails and air intrudes into the pressure vessel

The third major danger is the core meltdown accident That is generally initiated by a break in the reactor coolant system or one of the connecting water lines Such a break would cause loss of coolant in the primary system, discharging in-vessel pressure and core uncovered The fuel will be heated up by the decay heat which leads to cladding oxidation and temperature increase If no coolant measures are applied, the cladding, the fuel as well as the reactor internal structures will be melted That leads to molten pool formation in the lower head of RPV of a LWR

1.2.2 In-vessel accident progression and phenomena

a) Core heat up

When an accident involving a break in the coolant primary system of LWR occurs, with continued water-boiling due to decay heat power, the core eventually becomes uncovered The heat transferred to steam generator is reduced leading to a temperature increase in the fuel

b) Cladding oxidation

At temperature above 1300 K, the Zircaloy (Zr) in the fuel cladding is oxidized by the steam This exothermal reaction plays the important role because the amount of heat energy released can further increase the cladding temperature

c) Formation of melt pool on RPV lower head

If the temperature is high enough, fuel melts from top to bottom That leads to a melt pool formation in the RPV lower head In case of formation of a melt pool in the lower plenum, heat transfer in the pool is governed by turbulent natural convection, which characterized by internal Rayleigh number Rayleigh number of a decay-heated melt pool is a function of melt property, internal heat generation and the pool height The height of the formed melt pool depends on the melt mass relocated into the lower plenum

As the summary of this session, Figure 1.3 schematically presents the major physical phenomena liable to occur during severe accident in PWR Figure 1.4 shows the overview of the progression of core melt accident with the detailed processes listed in Table 1.1

Figure 1.3 Physical phenomenon during the severe accident [IRSN and

CEA-2007/83-351]

Figure 1.4 Core meltdown accident progression overview

Table 1.1 Core melt accident progression

1 Initiation of loss of coolant accident

2 Core uncovered

3 Volatile fission products released to upper part of vessel and containment

4 Lower vessel internal structures fails

5 Core debris interaction with residual coolant in vessel lower head

6 Vessel lower head fails

7 Core debris interaction with reactor cavity

8 Core debris and/or activity release from cavity

9 Containment leakage

10 Containment failure

1.2.3 Melt pool in the lower head and Reactor Pressure Vessel failure

a) Melt pool in the reactor lower head

In the stabilized melt pool, there may be separation between pure metals and its oxides, resulting in two layers, a metal layer floating above an oxide pool In such a case, if the upper layer is thin, it may result in a local concentration of heat flux The heat generation in the metallic layer is insignificant and the main heat source is coming from the oxide pool below In the metallic layer, natural convection heat transfer is induced by bottom heating and top cooling In the oxide layer, natural convection is induced by internal heating There are two ways of removing the heat from melt pool, the first one is radiation heat transfer from the top surface of metallic layer and the second one is the conduction heat transfer to the vessel wall

In this thesis work, without loss of generality of the natural convection heat transfer process in the internal-heated melt pool, the melt pool is assume to be homogeneous, this can be explained by the limitations of using CFD simulation for such a complex phenomenon described above, for instance, the Rayleigh number of stratified melt

pool is very high (1015-1017) result in the turbulent complex flows, long transient of severe accident progression, complex 3D geometry, etc

Figure 1.5 (a) Illustration of melt pool formation; (b) Layer separation of melt pool b) RPV failure

The vessel failure time as well as the failure location are considered as the key element because of their role not only in the core melt retention but also in ex-vessel accident progression During severe accident, vessel integrity may be threatened by various phenomena, which are mentioned above Especially after the time of the stabilized melt pool formation, the lower head of vessel wall will be violently heated by the direct contact with the pool at high temperature The failure location depends essentially on temperature distribution inside the vessel wall, and the position likely to fail first is the hottest area Vessel failure can be the result of either creep mechanics or plasticity Plasticity occurs when the equivalent stress existing in the vessel thickness

is greater than the ultimate material strength, which decreases significantly at higher temperatures Creep, which is an active mechanics of deformation, generally occurs at temperatures higher than 0.5 Tm, where Tm is the melt temperature of considered material When higher temperature reaches throughout the vessel wall thickness, creep may occur even if internal pressure remain low The real recent experiments prove that the temperature within the vessel wall is the paramount important factor which affects

to the mechanical strength of reactor vessel and governs the timing and location of RPV lower head failure

1.3 Objectives of thesis work

The scope of this thesis work is to examine the effect of creep phenomenon on the strength of nuclear reactor vessel during a hypothetical of core meltdown accident In order to conduct this objective, two aspects of thermal-hydraulic and creep mechanism are considered

In the thermal-hydraulic aspect, the natural convection heat transfer of assumed homogeneous melt pool is modeled and simulated by means of ANSYS CFD The results of this section provides the temperature field within the whole of vessel wall Those temperature fields are then imported to the ANSYS Mechanical as the input load of creep modeling

In the creep mechanism aspect, in order to estimate the effect of creep on the strength

of RPV, the main task of this section is to use ANSYS User Programmable Features to compile and link a Time hardening creep model into ANSYS Solver for more accurate describing the behavior of vessel wall material under severe condition of temperature

in an assumed hypothetical accident

This thesis work considers the simplified geometry model which represents the prototype of SMART (System-Integrated Modular Advanced ReacTor) PWR design

of KAERI, Korea ([3] and Keung Koo Kim, 2014)

Figure 1.6 SMART PWR design reactor assembly

To estimate the effect of passive water cooling measure, this thesis work considers two accident scenarios: In the scenario 1 (Figure 1.7a), the external cooling pump does not work, so that the outside of vessel wall contacts with air in containment; In the scenario 2 (Figure 1.7b), the external cooling pump works properly so that the outside

of RPV is fully flooded by water

Figure 1.7 Two considered hypothetical scenarios, (a) Scenario 1 and (b) Scenario 2

Chapter 2: Thermal hydraulic calculation of homogeneous melt pool

2.1 Basics of natural convection heat transfer

This section will start with a discussion of physical mechanism of natural convection The process of establishing the governing equations, Grashof number and Nusselt number of natural convection are then implemented in detail

Natural convection is a mechanism type of heat transfer, in which the fluid motion is not generated by any external source but only by the density differences in the fluid occurring due to temperature gradients [6] In natural convection, fluid surrounding a heat source receives heat, becomes less dense and rises The surrounding, cooler fluid then moves to replace it This cooler fluid is then heated again and the process continues, forming a convection current The driving force for natural convection is buoyancy force, a result of differences in fluid density

In gravitational field, the magnitude of the buoyancy force is equal to the weight of the fluid displaced by the body That is,

.g.V

buoyancy fluid body

F  (2.1) wherefluid is the average density of the fluid, g is the gravitational acceleration, and

Vbody is the volume of the portion of the body immersed in the fluid In the absence of other forces, the net vertical force acting on a body is the difference between the weight of the body and the buoyancy force That is,

hot vertical plate

Consider a differential volume of element of height dx, length dy, and its unit depth in the z direction for analysis The forces acting on this volume element are shown in Figure 2.1a Newton’s second law of motion for this control volume can be expressed

as,

m a x F x

  (2.5) where m  dx dy .1is the mass of the fluid element The acceleration in the x direction is obtained by taking the total differential of u x y , ,

The net surface force acting in the x direction

Gravitatinal force Net pressure force

Net viscous force 2

Substituting Equation (2.7) and Equation (2.6) into Equation (2.5) and dividing by

dx dy .1 gives the conservation of momentum in the x direction,

2 2

2 2

The momentum equation involves the temperature, and thus the momentum and energy equations must be solves simultaneously The set of three partial differential equations (the continuity, momentum and the energy equations) that govern natural convection flow over vertical plates can be reduced to a set of two ordinary nonlinear differential equations by the introduction of a similarity variable

In natural convection, the Reynolds number is no longer characterizes the flow regime Instead of it, the Grashof number governs the flow regime in natural convection, which represents the ratio of the buoyancy force to the viscous force acting on the fluid volume,

  3 2

The Grashof number provides the main criterion in determining whether the fluid flow

is laminar or turbulent in natural convection For a vertical plate, for example, the critical Grashof number if observed to be about 109 Therefore, the flow regime on a vertical plate becomes turbulent at Grashof numbers greater than 109

The complexities of fluid motion make it very difficult to obtain simple analytical relations for heat transfer by solving the governing equations of motion and energy, especially in the problem with complex geometry and multi phases So that, with exception of some assumptions, heat transfer relations in natural convection are based

on experiment The simple empirical correlations for the average Nusselt number in natural convection are of the form,

For the uniform internal heating, the modified Rayleigh number is defined as,

2.2 Heat transfer simulation of homogeneous melt pool

This section is divided into two parts The first part basically describes the model of natural convection of homogeneous melt pool, which is used for the CFD simulation

In the second part, the results of temperature field in the whole of melt pool as well as the vessel wall of both scenarios are given and discussed in detail

2.2.1 Modeling of natural convection

As mentioned above, the stratified melt pool of metallic layer and oxide pool below is assumed to be a homogeneous melt pool of mixture, so that the model of heat transfer

is simplified to natural convection of a melt pool induced by internal heating and external cooling

Figure 2.2 Natural convection model induced by internal heating

Considering the natural convection model of Rayleigh-Benard in Figure 2.2a , which is heated at the bottom and cooled at the top, the Nusselt number is as follow,

 , a and b are the constants

The homogeneous melt volume is heated by the internal heat generation and cooled by conduction heat transfer via the vessel wall Figure 2.2b presents the heat transfer model of melt pool in CFD simulation Nuup represents to the heat flow released into

the steam in reactor, Nudown represents to the heat flow which is transferred into the vessel wall This bottom heating and top cooling causes the natural convection heat transfer in the homogeneous melt pool The Nusselt number in this model is,

Table 2.1 Some correlations of internally heated fluid volume

Author Correlation Ra’ range Geometry

This thesis work uses a 2D axisymmetric model, which represents the SMART PWR design of KAERI, Korea [3]

Two accident scenarios are considered here: In the scenario 1, the external cooling pump does not work, so that the outside of vessel wall contacts with air in containment; In the scenario 2, the external cooling pump works properly so that the outside of RPV is fully flooded by water

The overall process of CFD simulation is implemented in ANSYS Workbench including Meshing and Fluent

Figure 2.3 (a) 2-D axisymmetric geometry of melt pool and vessel wall, (b) Mesh

configuration

Figure 2.3a and Figure 2.3b shows the 2-D axisymmetric geometry model and mesh configuration of the whole of homogeneous melt pool and vessel wall Figure 2.3b also illustrates the cooling surfaces of two scenarios The top surface of melt pool is a cooling surface by convection and radiation heat transfer to the steam in reactor The

outside surface of vessel wall, which is contacted with the air in containment in scenario 1 and saturated water in scenario 2, is also set as the convection and radiation heat transfer surfaces

Main thermal properties of homogeneous melt pool used for heat transfer simulation in the CFD are listed in the Table 2.2 below

Table 2.2 Main thermal properties of homogeneous melt pool [3]

Thermal properties Values

Density  2500 kg m/ 3

4.7

V  m Specific heat C p 2200 /J kg K.

Heat generation G3,93.10 W /5 m3

2.2.2 Temperature results and discussion

Figure 2.4 Temperature field in the whole of fluid volume and vessel wall at 6h in (a)

scenario 1; and (b) scenario 2

In Figure 2.4a the temperature field in scenario 1 is shown for the whole of fluid volume and vessel wall after 6h (21600s) of simulation time The temperature ranges from 453K at the top of cylindrical section of the vessel wall to nearly 1293K in the top upper of the melt pool Figure 2.4b shows the temperature field in scenario2 Due

to the significant cooling effect at the outside of vessel wall, the temperature ranges from 373K at the top of cylindrical section of the vessel wall to approximate 1025K in the top upper of the melt pool

Figure 2.5 Temperature field in vessel wall at 6h in (a) scenario 1; and (b) scenario 2

Figure 2.6 Temperature over time at the highest temperature point of vessel wall

Figure 2.5 shows the temperature field only in the vessel wall for a cleaner view In Figure 2.5a maximum temperture in vessel wall in scenario 1 is 1136K In the figure 2.5b, the cooling effect at the outside of vessel wall can be easily recognized as the max temperature on the vessel wall is limitted by some 777K

The graph in Figure 2.6 shows the temperature over time of a point at the hot - focused region in the vessel wall The black line indicates that the relative stable temperature in scenario 1 is reached after nearly 5000s (1.38h) After that time, temperature is almost stabilized The red line represents to scenario 2 It is obviously to see that temperature reaches the highest value at the time of nearly 2400s (0.67h) After that, temperature tends to decrease over time without stabilization

The thermal hydraulic by means of CFD simulation in this thesis work were implemented by a simplified heat transfer model and without taking into account the possibility of some complex phenomena such as the solidification of bottom layer of melt pool and the air gap between the melt pool and the vessel wall, etc So that there are some discrepancies between the results of temperature configuration in this thesis work and experiments This could be improved in the future studies

As mentioned above, the temperature field within the vessel wall in two scenarios, which are obtained from this section, are then transferred into the numerical creep mechanicsm modeling in the flowing section by mean of ANSYS Mechanical APDL

Chapter 3: Numerical creep modeling of reactor pressure vessel lower head

3.1 Creep mechanism theory

3.1.1 Background on creep

Creep of materials is associated with time-dependent plasticity under a fixed stress at high temperatures, often greater than roughly 0.5T m

In crystalline materials, such as metals, creep mechanism is linked to diffusional flow

of vacancies and dislocation movement Vacancies are point defects, and it tend to favor grain boundaries that are normal, rather than parallel, to the applied stress Vacancies tend to move from regions of high to low concentrations Diffusional flow can occur at low stresses but usually require high temperatures Dislocations in grains are line defects The movement of dislocations tend to be activated by high stresses, although it may also occur at intermediate temperatures

Creep and viscoplasticity are the same from a material standpoint In engineering usage, creep is generally used to describe a thermally-activated process with low strain rate Rate-independent plastic and implicit creep strains are treated in a weakly coupled manner Conversely, viscoplasticity constitutive models are used to describe high-strain-rate application Inelastic strains are treated in a strongly coupled manner

Figure 3.1 Three stages of creep curve [1]

The Figure 3.1 shows the uniaxial strain - time behavior of creep under constant load

It is clear to see the three stages of creep curve In the primary stage, the strain rate decreases with time This tends to occurs over a short period The secondary stage has

a constant strain rate associated with it In the tertiary stage, the strain rate increases rapidly until failure The creep strain rate may be a function of stress, strain, temperature, and/or time For engineering analysis, the primary and secondary stages

of creep are usually of greatest interest Meanwhile, tertiary creep stage if usually associated with the onset of failure and short-lived Hence, tertiary creep is not modeled in the numerical software The strain rate in the primary creep is usually much greater than those in the secondary creep However, the strain rate is decreasing

in the primary stage whereas it is usually nearly constant in the secondary stage Also, primary creep tends to be of a shorter period than secondary creep

There are two sort of creep in general (Figure 3.2) Creep is the term, which is used as strain increases under constant applied stress The other sort of creep is stress relaxation, which is typically occurs in the plastics or polymers material when the stress decreases under constant applied strain

Figure 3.2 [1] Creep (a), and Stress relaxation (b)

The creep phenomenon in material is the time-dependent process, so that the most different between creep calculation and typical stress-strain calculation is the presence

of time in the creep-calculating equations

3.1.2 Creep equation and hardening theory

During creep process, a tensile specimen under a constant load will continually deform with time This deformation depends on three main parameters, which are stress, time and temperature The creep most general creep equation is therefore,

cr f t T

   (3.1) For the application of creep calculation, the dependency of creep deformation on stress, strain, time and temperature are generally modeled of a creep strain rate form similar to the one below,

     

   (3.2) The use of separation function of stress, f1   , emerged from early studies of secondary creep in 1930s

Stress-dependent function

Creep strain is usually stress-dependent The function f1   is chosen in many different ways Table 3.1 below gives a summary of the most common forms [Kennedy]; of course, there are others

Table 3.1 Stress-dependent function