Experimental and Numerical Investigation of Ratcheting Effects in 316L Stainless Steel - The Two-Rod approach.

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Master’s Thesis in Solid Mechanics Second level, 30.0 ETCS credits Stockholm, Sweden 2014 KTH Department of Solid Mechanics Inspecta Nuclear AB

Gustav Eklund Numerical Investigation

of Ratcheting Effects in

316L Stainless Steel –

The Two-Rod Approach

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Experimental and Numerical

Investigation of Ratcheting Effects in 316L Stainless Steel –

The Two-Rod Approach

Experimentell och Numerisk Undersökning av Ratchetingeffekter för 316L Rostfritt Stål –

Tvåstångsmetoden Gustav Eklund

Master’s Thesis in Solid Mechanics

Second level, 30.0 ETCS credits Stockholm, Sweden 2014 Conducted at Inspecta Nuclear AB and

KTH Department of Solid Mechanics

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Sammanfattning

Det här examensarbetet utfördes under våren 2014. En experimentell och numerisk undersökning genomfördes på det austenitiska rostfria stålet 316L. Huvudområdet för studien var att undersöka fenomenet ratcheting (progressiv plastisk deformation).

Experimentellt var huvudfokus på det så kallade tvåstångstestet, vilket tidigare inte hade utförts.

Tvåstångstestet utgör en struktur och ett lastfall vari ratcheting kan skapas, samtidigt som strukturen är mer renodlad än de som undersökts i tidigare studier för samma ändamål. Dessutom är spänningstillståndet enaxligt i strukturen. Utöver tvåstångsprovning gjordes även ytterligare provning för att karaktärisera materialet. Utgående från resultat från enaxligt dragprov och fullt reverserad töjningsstyrd cykling anpassades fyra materialmodeller efter materialet. Dessa fyra materialmodeller var



Bi-linjär kinematiskt hårdnande modell



Multilinjär kinematiskt hårdnande modell (Mróz)



Armstrong-Frederick icke-linjärt kinematiskt hårdnande modell



Chaboche icke-linjärt kinematiskt hårdnande modell med tre superponerade back stress- tensorer.

En FEM-modell över tvåstångsprovet användes för att simulera de olika materialmodellernas respons. Resultaten från dessa jämfördes sedan med resultaten från tvåstångsprovningen. Målet, bortsett från att karaktärisera ratchetingeffekterna i 316L-stålet, var att utvärdera materialmodellernas förmåga att återskapa resultaten från tvåstångsprovningen.

Resultaten från jämförelsen mellan simuleringarna och tvåstångsprovningen pekar på att den bi- linjära och den multilinjära materialmodellen förmår återskapa provresultaten bättre än Armstrong- Frederick-modellen och Chaboche-modellen. De två sistnämnda materialmodellerna predikterade i de flesta fall konstant ratchetinghastighet, vilket inte överensstämde med provresultaten från tvåstångsprovningen. Även om predikteringen av tvåstångsprovningen med den bi-linjära och multilinjära materialmodellen överlag var bättre än för de icke-linjärt hårdnande materialmodellerna predikterade den bi-linjära och multilinjära materialmodellen i vissa fall plastisk shakedown, vilket inte sågs i provresultaten.

Införandet av isotropt hårdnande i de icke-linjärt kinematiskt hårdnande materialmodellerna kan ha förbättrat simuleringarnas överensstämmande med provresultaten då materialet visar på omfattande plastiskt hårdnande, både i monotont dragprov såväl som cykliskt hårdnande.

Metoden som utvecklades för tvåstångsprovningen visade sig robust och pålitlig. En slutsats som

kan dras är att effekter från materialratcheting förmodligen är små i jämförelse med effekter från

strukturratcheting i tvåstångsprovningen. Dessutom kan från jämförelsen mellan simuleringarna och

tvåstångsprovningen sägas att en mer avancerad materialmodell inte nödvändigtvis resulterar i en

prediktering som överensstämmer bättre med provningen.

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Abstract

This Master’s Thesis was conducted during spring 2014. An experimental and numerical investigation was conducted on the austenitic 316L stainless steel. The main focus of the study was the investigation of ratcheting effects.

Experimentally, the main focus was the two-rod test, which had not been conducted previously. The two-rod test resembles a structure and a load case where ratcheting effects may be produced, although being less complicated than structures used in prior studies. Furthermore, the stress state in the structure is uniaxial. Other tests were also performed to characterize the material. Based on results from uniaxial tensile tests and fully reversed strain cycling of 316L, four material models were calibrated. The four material models were



Bi-linear kinematic hardening model



Multilinear kinematic hardening model (Mróz)



Armstrong-Frederick non-linear kinematic hardening model



Chaboche non-linear kinematic hardening model with three superimposed back-stress tensors.

The two-rod test was then numerically simulated with different material models. The results from the FE simulations were then compared to the test results obtained from the two-rod tests. The goal, apart from investigating the ratcheting effects in 316L steel, was to evaluate the material models’

ability to reproduce the two-rod test results.

The results from the comparison suggest that the bi-linear and the multilinear material model agreed with the test results to a larger extent than the Armstrong-Frederick and Chaboche model. The two non-linear hardening material models predicted in most cases a constant ratcheting rate which did not agree with the test results. Even though the predictions of the two-rod tests with the bilinear and the multilinear models generally was better than predictions with the two non-linear hardening material models, the bilinear and the multilinear models predicted plastic shakedown in certain cases which was not observed in the tests. The employment of an isotropic part in the non-linear kinematic hardening material models might have improved the simulations’ agreement to experimental results.

The setup for the two-rod test proved robust and reliable. The results suggest that structural

ratcheting effects dominate the two-rod test results. Furthermore, the comparison between

simulations and the two-rod tests suggest that a more advanced material model does not necessarily

yield in a better prediction.

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Acknowledgements

Throughout the project I have been lucky to have been backed up by dedicated persons who have been very helpful. First, I would like to thank Martin Öberg and especially Hans Öberg at the Department of Solid Mechanics lab for help and advice in connection with the experimental part of the study. Hans Öberg has been very helpful with the programming of the two-rod test controlling program. Without his help the two-rod test would probably not have been as functional as it turned out to be.

Furthermore I would like to thank the people at Inspecta Nuclear for help, advice and support. I would especially like to thank Pär Ljustell for his sharing of experience concerning 316L and valuable ideas regarding modeling and testing.

Last but not least, I would like to thank my supervisor Peter Segle at Inspecta Nuclear. He has been

dedicated to help and have been invaluable in all parts of the study, from modeling, testing and

calibration as well as writing and providing theoretical expertise. I am very thankful to have been his

supervisee during this project which would have taken a lot more time without his help.

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Nomenclature

Parameter Explanation

Initial area of virgin specimen

Specimen measurement in uniaxial tensile test

Specimen area at failure (ductility measurement in uniaxial tensile test)

Armstrong-Frederick material model back-stress parameter

Chaboche material model back-stress parameter Specimen diameter parameter

Specimen diameter parameter Specimen diameter parameter

Young’s Modulus

Young’s Moduli in multilinear material model Force in rod 1

Force in rod 1

Desired primary force in two-rod test

Force tolerance in two-rod test

Hardening modulus in bi-linear material model Uniaxial tensile test length measurement

Uniaxial tensile test length measurement Specimen length parameter

Specimen length parameter

Multilinear material model parameter

Maximal engineering stress in uniaxial tensile test

Proof stress (0.2 %)

Deviatoric stress tensor

ASME Class 1 parameter

Tangent modulus in bilinear material model Time at N:th half-cycle in two-rod test

Area reduction at failure (ductility measurement in uniaxial tensile test)

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Total strain

Strain in rod 1 (two-rod test) Strain in rod 2 (two-rod test) Plastic strain

Corrective strain term in two-rod test Effective plastic strain

Plastic strain tensor

Initial plastic strain

Multilinear material model parameter

Specimen radius of curvature

^Stress

First principal stress Second principal stress Third principal stress Effective stress

Stress tensor

Mapping stress of multilinear material model Yield surface radius

Yield stress

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1. Introduction

1.1. Presentation of Inspecta

Inspecta Group is one of the leading inspection companies in Northern Europe, present in Sweden, Finland, Norway, Denmark and in the Baltic region. They are specialized in testing, inspection, technical consulting and certification and employ approximately 1400 persons in total. Inspecta Nuclear AB is accredited for inspecting nuclear power plants and validating constructions in accordance with present regulations in the area, as well as being involved in modernizing and uprating projects within the nuclear industry in Sweden.

1.2. Background

Producing electric energy from nuclear energy is attractive since the amount of energy per fuel unit is extremely high in comparison to for example oil or biodegradable fuel. Also, there are no emissions of CO

2

, which is the case with energy from fossil sources. The first nuclear reactor started in 1954 in the USSR, and the first reactor in Sweden (R1 in Stockholm) started in the same year [1]. Although the existence of nuclear power plants has been intensively debated over the years, many countries still rely heavily of this source of energy.

Currently there are two types of nuclear reactors in Sweden, namely boiling water reactors (BWR’s) and pressurized water reactors (PWR’s) where the former is the most common in Sweden. When the power plant is running, energy is released through fissile reactions which heats water. Steam drives the turbines which in turn creates electric energy through generators. In the process the steam creates thermal loads and internal pressure, creating a need for judicious designing and careful choices of material. Due to the massive consequences of a leakage or other failure, high requirements are put on structural reliability. Substantial proactive actions against failures have to be taken and several redundant safety systems are often utilized. A nuclear power plant is a construction where the safety has the highest priority.

For controlling purposes of nuclear power plants, frameworks of regulations have been created, both specific for Sweden and from the European Union. Among other things, these regulations emphasize the control of design, demanding utilization of codes of conduct.

1.2.1. ASME code on nuclear power plant components

ASME (American Society of Mechanical Engineers) is a nonprofit membership organization which over the years has built an extensive code framework which among other things aims at providing guidelines for a wide variety of engineering areas. For structures designed for use in a nuclear power plant, different classes of components are identified by ASME. The components closest to the reactor core are classified as Class I components with higher requirements of safety factors than for example Class II and III components. In order for a component to meet the requirements, the loaded structure needs to meet certain criteria. The analysis needed can be done in different ways where elastic analysis is the easiest and most commonly used. This method includes only the elastic behavior, and by safety factors, stresses the any plastic deformation are held within the allowed bounds.

If the elastic analysis is proven insufficient, for example if the behavior of the component in reality differs much

from the analysis or the behavior includes some unexpected phenomena, a more refined approach towards

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plants there is a need for a more refined process of evaluating components, but still fulfilling ASME’s requirements in plastic analysis.

The implications of developing a more refined method for structural verification is that more advanced material models need to be utilized. This is especially the case when cyclic stress is present as oppose to monotonous or static stresses. The development from elastic-ideal plastic models includes linear hardening models and non-linear hardening models in various forms suitable for predicting non-monotonous loading situations. When encountering cyclic plasticity in low cycle fatigue, there have been problems when assessing the adequacy of different material models. This has especially been the case when ratcheting behavior is observed (explained below) in pressurized piping. For example, Hassan [3] provided an evaluation of different material models, but the results were not possible to replicate by using linear hardening models (Bi-linear, Mróz) or non-linear hardening models (Chaboche, Ohno-Wang). Since there are rigorous safety regulations involved in construction of nuclear power plants, it is crucial to investigate the applicability of different models, and how well they replicate the behavior in experiments.

1.3. Definitions

1.3.1. Ratcheting

The concept ratcheting (also, cyclic creep [4] or ratchetting) is commonly used to denote a situation where the following criteria are fulfilled:

 The component is subjected to cyclic loading,

 plastic deformation occurs in every cycle,

 the mean stress is non-zero, and

 the plastic strain (averaged over a cycle) grows with increasing cycles.

One situation in which ratcheting occurs is when a specimen is subjected to a cyclic load with prescribed stress

amplitude with a non-zero average stress [4], as can be seen in Figure 1.

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Figure 1.An example of ratcheting in a specimen subjected to prescribed stress amplitude with a non-zero mean stress [5].

The ratcheting phenomenon can be seen on a structural scale (structural ratcheting) or on a micromechanical scale (material ratcheting). The two types of ratcheting are related, although simulating structural ratcheting requires less advanced material models than simulating material ratcheting [6].

1.3.2. Primary and secondary stress

The stress that arises in components subjected to different loading is sometimes (e.g. ASME) divided into two categories, namely primary and secondary stresses. Primary stress is referring to a stress that does not change as the geometry changes due to plastic deformation. Primary stress is not self-limiting, and arises from situations where the load is prescribed rather than the displacement (load-controlled stress). Primary stresses that “exceed the yield stress considerably will result in failure or, at least, in gross distortion” [ASME III div. 1 NB-3213.8].

Secondary stress is distinguished from primary stress in the way that it is self-limiting. These stresses arise

primarily from displacement-controlled situations such as thermal loading or local stress at a structural

discontinuity. The reason for distinguishing these two stresses is that primary stresses are more serious concerning

the failure risk of the structure. Secondary stresses per se are not as serious as primary stress but may be

dangerous when present together with primary stresses.

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2. Thesis’ objective

The objective of this study was first and foremost to study the ratcheting phenomena in 316L stainless steel through material testing and numeric simulations. This included

 Design and development of the so-called two-rod test,

 Perform a test series in two-rod testing for 316L material,

 Characterize 316L by material testing to calibrate material models, and

 Simulate the response of the two-rod tests with the calibrated material models in FEM.

Four different material models were evaluated:

 Bi-linear kinematic hardening model,

 Multilinear kinematic hardening model (Mróz),

 Armstrong-Frederick non-linear kinematic hardening model, and

 Chaboche non-linear kinematic hardening model with three superimposed back-stress formulations.

The first main goal was to evaluate the different material models’ ability to reproduce the two-rod tests’ response in FEM. As the two-rod test has not been conducted previously, the second main goal was also to investigate the specifics concerning this test, and the potential of the test to evaluate the ratcheting phenomena.

2.1. Delimitations

As this study was conducted during a limited period of time, delimitations had to be made accordingly. Firstly, the experimental investigation was carried out on an austenitic stainless steel in the 316-family. As the properties of

‘steel’ vary due to many factors the results observed in this study may have been different if some other type of steel was used.

Secondly, only kinematic hardening material models were employed in the numerical investigation. The material

models were furthermore chosen for its popularity in the industry as they are present in commercially available

FE software. There are more advanced material models that can be employed, but these generally are cumbersome

to implement and requires more testing.

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3. Theoretic background

3.1. Yield surfaces, hardening models and the Bauschinger effect

A stress state in a material point lies on the yield surface if the so-called flow function [7] is fulfilled. This can be written as

(

)

(1)

If also

(2)

then plastic deformation is introduced in the material point. There are several yield criteria, forms of (

), developed specially for different materials, e.g. Tresca, von Mises, Drucker-Prager, Gurson-Tvergaard, and Hill etc. For metal alloys von Mises yield criterion has proven useful and as such will be used exclusively in this study. The von Mises criterion has also been adopted as standard in most material models presented below. In the von Mises yield criterion the effective stress, σ

e

in Equation (1) can be written as

[

] [ (

)] (3)

which corresponds to a circle in the synoptic plane

¹

, and a cylinder in the principal stress space. It implies that the material is insensitive to hydrostatic pressure, which has turned out to give a fairly good representation of most metal alloys far above the yield stress [6, 8]. Chaboche and Lemaitre argued for this to be true for metals up to 3,000 MPa isostatic pressure [8, p. 16]

In order to explain what happens when a stress state fulfils Equation (1) and (2), plasticity models are needed.

Plasticity theory has a number of basic models which enlightens important concepts such as isotropic hardening, kinematic hardening, mixed hardening and the Bauschinger effect.

Isotropic hardening denotes the behavior when the yield surface expands as the loading exceeds the initial yield condition. For von Mises yield condition, isotropic hardening is equivalent with increasing radius of the circle in the synoptic plane, as in Figure 2 (right). For an isotropic hardening model, the yield criterion is slightly changed as

( )

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where is a function describing the hardening behavior. The hardening function can be modelled in several ways and can either be a linear function or be described as a non-linear function. The choice determines the ability of the model to capture different material phenomena, but as the model gets more advanced the number of parameters that need to be determined often increases, which lead to a tedious process of material testing.

1

The synoptic plane is the plane in the principal stress space [

] with normal[ ]. [6]

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Figure 2. Kinematic hardening (left) implies a translation of the initial yield surface in the synoptic plane whereas isotropic hardening (right) implies an expansion of the initial yield surface [8].

When the material yields and deforms plastically, the yield surface of a kinematic hardening model translates in the synoptic plane. A fundamental parameter for these models is sometimes referred to the back stress and denotes the translation of the center of the yield surface in relation to the origin. The back stress tensor is here denoted

and can be seen in Figure 2 (left). Kinematic hardening models denote theories in which the yield surface of a material is allowed to move from its reference point in the synoptic plane of the three-dimensional principal stress space. Mixed hardening is (intuitively) the case where both isotropic and kinematic hardening effects can be noticed. Most materials display mixed hardening to some degree [9]. It is possible to incorporate mixed hardening in most elastic-plastic material models.

The Bauschinger effect denotes the effect where an initial plastic deformation in one direction lowers the yield stress in the reversed loading direction. This is a consequence of kinematic hardening effects, and cannot be observed with a purely isotropic hardening model. The Bauschinger effect can only be seen when cyclic loading is present and the material yields in both positive and negative direction in a cycle. A monotonous loading show the same response when alternating between isotropic and kinematic hardening.

3.2. Cyclic plastic deformation

When a specimen or a structure is subjected to cyclic loading, the strains may be either fully elastic or display

some plastic deformation. In for example a fully reversed strain cycling test, one can identify the concept cyclic

plastic deformation in which the structure is deforming plastically in each cycle. The prerequisite is that the

material yields during each cycle (at least in the first few cycles). This is drastically changing the course of action

when the material behavior is modelled. The hardening properties on a material subjected to monotonic loading

may differ from what is observed in reversed loading. This creates a need for more advanced models capable of

modeling cyclic plasticity.

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Figure 3. Cyclic plastic deformation curve is collected by fitting the maximum points of each saturated hysteresis curve. It is here shown together with the monotonic stress-strain curve of the material [10].

The behavior of a specimen or structure subjected to cyclic plastic deformation can be divided into a few broad phenomenological categories. Furthermore, a distinction can be made whether the test is conducted using prescribed stress amplitude or prescribed strain amplitude.

Cyclic hardening and cyclic softening can be observed both when conducting tests with prescribed stress span and prescribed strain span. The effects which can be visualized through hysteresis curves in Figure 4 and Figure 5 below is not to be mixed up with strain hardening and softening effects in the plastic region of the material curve.

Of course, the monotonous loading stress-strain response is linked to the behavior in cyclic plastic deformation, but a material which exhibits strain hardening in the plastic region may very well display cyclic softening. The behavior in the plastic region depends on the material characteristics, any plastic deformation prior to testing, temperature and load [8].

When the strain span is prescribed the stress span may increase or decrease. This is denoted cyclic hardening, or cyclic softening, respectively as seen in Figure 4.

Figure 4. Cyclic softening (a) and cyclic hardening (b) when prescribed strain span, zero mean strain

[11]. Note that the elastic strain has been excluded from the graphs.

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Figure 5. Cyclic hardening (c) and cyclic softening (d) when prescribed stress span, zero mean stress [11].

Note that the elastic strain has been excluded from the graphs.

When the test is conducted using prescribed stress, the maximum strain for each cycle may decrease or increase depending on material properties and the loading specifications. This corresponds to cyclic hardening and cyclic softening, respectively and is shown in Figure 5.

These behaviors are observable primarily when the stress or strain (depending on the prescribed entity) is purely alternating with mean value zero. When however an initial tensile or compressive stress (or strain) is superimposed the cyclic load, other categories may be observed. This loading situation is sometimes referred to as unsymmetrical [6] or nonsymmetrical [8]. As can be seen in Figure 6, when prescribing the strain span, one may observe the mean stress of the hysteresis to approach zero, which usually is denoted mean stress relaxation. If this is not observed, the cycling is stable both within the stress range and the strain range. Prescribed stress and prescribed strain can be linked to the ASME distinction between primary and secondary stress, respectively.

Figure 6. Mean stress relaxation at unsymmetrical prescribed strain [6].

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If the stress is prescribed, with an initial means stress superimposed, two main categories can be observed, namely so-called shake-down or ratcheting. Shake-down, which can be seen in Figure 7, can further be divided into elastic and plastic shake-down. Elastic shake-down refers to the case when the cycles after stabilizing are purely within the elastic range of the material. Plastic shake-down refers to the case when the stable-state cycling still lies partly within the plastic range of the material. In Figure 7, plastic shakedown is depicted.

Ratcheting, which can be seen in Figure 7, is observed when the maximum strain for each cycle increases without reaching a steady state as was the case with shake-down. Both shake-down and ratcheting are associated with primary stresses but the former are not as serious of a threat to failure as the latter. Ratcheting can furthermore be divided into two mechanisms, namely material ratcheting and structural ratcheting depending on what mechanisms causing the ratcheting.

Figure 7. Ratcheting (upper) and plastic shakedown (lower) [12]. Note that the elastic strain has been

excluded from the graphs.

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isotropic.

If a grain is perfectly arranged in a crystal-like pattern, it would only be able to account for elastic deformation and brittle fracture (non-ductile fracture). Briefly, defects in the grains (or in the intergranular borders) allow the atoms (or grains) to slip irreversibly, which give rise to plastic deformation. Whether the dislocations are intra- or intergranular (within the grain or at grain borders) depends on manufacturing process and alloy composition [13].

Chaboche and Lemaitre [8, pp. 13-16] explain plastic hardening through slip barriers and other restrictive forces within the Frank networks to contribute to an increased resistance as the plastic strain increases on the macroscopic level. Also, Chaboche and Lemaitre argue for an additional mechanism, anisotropy induced by permanent deformation. They state that “As permanent deformations differ from one crystal to the next, the compatibility at grain boundaries is assured only by elastic microdeformations; these remain partially locked when the load is removed resulting in self-equilibrated microscopic residual stresses” [8, pp. 16-17]. As the grains are deformed plastically, the compatibility restricts the motions and the grains might interlock each other.

A detailed study of how different alloys and annealing affects the behavior in plastic deformations lies outside of the scope of this study.

3.4. Material ratcheting

Metal alloys may display ratcheting effects without the loading characteristics described above (combination of primary stress and alternating secondary stress), and during single specimen tests display ratcheting. This behavior needs to be distinguished from structural ratcheting, and has been denoted material ratcheting. Another way to put it is that material ratcheting is a response in a point of the structure, whereas structural ratcheting requires a structure. Prior studies on the subject have found the reason for material ratcheting to lie at the micromechanical level [14, 8]. The common denominator for material ratcheting seems to be unsymmetric loading [15] with accumulation of plastic deformation in one direction. The ratcheting behavior can be observed in uniaxial testing, but also in more complicated loadings such as bi-axial tests as examined in for example [16].

In order to capture material ratcheting in FE modeling, the material model needs a kinematic hardening part which is nonlinear.

Many studies have been conducted to evaluate the material ratcheting of different alloys, mostly through uniaxial

ratcheting. For this type of experiment, mean stress and stress amplitude are the primary parameters, but

temperature is also of importance. Material ratcheting is material-specific and a generalization on how these

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Figure 8. Ratcheting strain of virgin specimens, typical curve with primary and secondary ratcheting. Taken from [15]

increasing cycles, as shown in Figure 8. This early stage is often referred to as primary ratcheting whereas the following part where the ratcheting rate change decays is referred to as secondary ratcheting

To further complicate the matter, material ratcheting in uniaxial tests has proven to be strongly history-dependent.

For example, Yiang and Zhang [5] showed that if the specimen is loaded in tension prior to cycling, the specimen may display reversed ratcheting (as shown in Figure 9) even though a virgin specimen would ratchet in the other direction.

Figure 9. Reversed ratcheting after initial monotonous tensile loading. [5]

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The original problem has in recent years been extended and is often referred to when primary stress (often internal pressure) and secondary stress (bending, torsion or thermal load) are acting simultaneously on a hollow pipe.

When primary stress is superimposed by a cyclic secondary strain (as for example the Bree problem) the material can behave in two different ways, with shake-down or ratcheting.

Another visualization of the structural ratcheting phenomenon is the two-rod test. The load case is described in Figure 10. The structure consists of two rods that together are subjected to a constant load, P. At Load Step 1, the first rod is elongated whereas the second rod is compressed with

in deformation control. If the stress is large enough, this causes the first rod to yield. In Load Step 2, which is the reverse of Load Step 1, the second rod will yield. Thus, through Load Step 1 and 2 both rods will elongate slightly through plastic deformation. When repeated, the rods will elongate using each other as dollies creating what is commonly denoted structural ratcheting. The same load can also be constructed by thermal loads, i.e. cyclically increasing and decreasing temperature in one rod or both rods as was conducted in [18].

Figure 10. Load case setup of two-rod testing.

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3.6. Similarities between material and structural ratcheting

As mentioned above, there are many similarities between material and structural ratcheting. However, as the

mechanisms leading to these behaviors are different, a ratcheting behavior of a structure may be due to material,

structural or both types of ratcheting. Which type of ratcheting that is dominating depends on the structure, the

material and the loading. When looking at the response of a structure subjected to a certain load, for example the

elongations of the rods in the two-rod test in Figure 10, one cannot be sure whether the ratcheting response comes

from material or structural ratcheting, or a combination of the two. This can however be evaluated using different

material models when simulating. Non-linear hardening material models are capable of simulating material

ratcheting, whereas material models with linear hardening properties are not capable of capturing material

ratcheting [6]. If a simulation is conducted using some of the linear hardening models only the structural

ratcheting is observed.

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4. Two-rod testing

The two-rod test consists of two rods that together holds a constant load but their contribution varies in time. It is illustrated in Figure 11. The variation is conducted by changing the displacement of the rods back and forth. This is controlled by prescribing the displacement difference of the rods at the end of each half-cycle (load step in Figure) whereas holding the total load constant.

The effects on the specimens are that they through cycling get progressively longer, i.e. the structure experiences structural ratcheting. Also, as the mean stress in each one of the rods is non-zero, material ratcheting can also be expected. Thus, the two-rod test is a structure in which material and structural ratcheting can occur simultaneously.

The test was set up by simultaneously controlling two test machines, as shown in Figure 12 and described below.

They were controlled by one computer running a PASCAL-written routine specifically written for the test.

Figure 11. Two rod test cycling scheme.

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Figure 12. Left: The two machine setup for the two-rod test. Right: Close-up picture on the specimen fixture and extensometers.

4.1. Test setup

The testing was conducted at KTH, at the Department of Solid Mechanics. The testing was performed on MTS312.21 load frames with a 100 kN load cell, and INSTRON 8500 controls recorded by a computer. The specimen strain measurements were made by two 12.5 mm extensometers fastened on opposite sides of each specimen, using the mean value for recording. All tests were conducted at room temperature.

The specimens were clamped using a ring (2) and a wedge (3), as depicted in Figure 12 and illustrated in Figure 13. This fixture is used for support in both compression and tension. When the bolts are tightened against the machine head, the fixture ring (2) slides relative to the fixture wedges (3) which clamp the specimen.

The specimens used in the study were round, with varying length and diameter. The specimen geometry was chosen according to ASTM E606 standard [19]. The general appearance of the specimens is shown in Figure 14.

The values of the parameters are presented in the context of each test. The varying parameters in the study were mainly the test length, l

test

, and the diameter, d

0

, which also affected the radius of curvature, ρ. All specimens were made uniform within 0.01 mm diameter tolerance throughout the test length, l

test

.

Figure 13. Test specimen fixture.

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Figure 14. General test specimen appearance.

4.2. Controlling the test machines

When a test was running, the computer controlled the test machines’ strain through the extensometers. A primary routine governing the strains in the machines was responsible for that in the end of each half-cycle, the strain difference between the two specimens were as desired (denoted

). The primary routine had the following outline (for half-cycle N running from to

) :

 Strain rate for rod 1: ̇

 ∫ ̇

 ∫ ̇ .

Another routine was responsible for holding the sum of the two loads constant. This routine consisted of a loop which was running without stopping, slightly altering the output signal and . The outline of the loop is described in Figure 15.

The calculation of the corrective term

was done proportionally to the error (P-control). Due to a delay in the

response time, the corrective term was bounded to a maximal correction. As apparent in the scheme above, if the

difference of the forces and the desired force was within a tolerance,

, no alteration was made. This tolerance

was set to 90 N. Since these two regulations were conducted independently, the rods were allowed to elongate

successively with increasing cycles.

(34)

Figure 15. Scheme over regulation algorithm controlling the sum load in two-rod tests.

At the end point of each half-cycle, data was recorded. This included

 Piston position for machine 1 and 2,

 Extensometer position for rod 1 and 2,

 Piston force readings for machine 1 and 2.

Between the end of each half-cycle, no data was recorded. However when the test was running, the force in machine 1 and 2 were visualized in an oscilloscope in real-time. Furthermore, an electric circuit was made to show the mean force of machine 1 and 2 in real-time.

4.3. Experiments

The specimen geometry is specified in Table 1.

Table 1. Parameter values for two-rod test specimens.

Parameter

Value [mm] 6 25 30 12 11 75 10 13 5

𝜀

_{𝑐𝑜𝑟𝑟}

<

𝜀

_{𝑐𝑜𝑟𝑟}

elseif 𝐹 𝐹 𝐹

_{𝑝𝑟𝑖𝑚𝑎𝑟𝑦}

𝐹

_𝑡𝑜𝑙

else

end 𝜀 𝑡 𝜀 𝑡 𝜀

_{𝑐𝑜𝑟𝑟}

𝜀 𝑡 𝜀 𝑡 𝜀

_{𝑐𝑜𝑟𝑟}

3. Alter output signal

(35)

All tests were conducted at a strain rate of 0.005 %/s. The different tests in the test series were characterized by a sum load causing a primary stress,

, and a secondary stress range,

. The entities were linked to the ASME material parameter , which in this study was set to two thirds of the yield stress

²

. For 316L, the value of was set to 195.3 MPa. Three levels of primary stress were tested, i.e. 0.5 , and 1.25 . For each level of primary stress, different secondary stress ranges were tested. The secondary stress range was introduced by controlling the displacement,

, over the specimen’s length between the extensometer edges,

. The ranges were calculated by

(5)

where

was linked to . The test plan is summarized in Table 2.

Table 2. Two-rod test combinations of primary and secondary stress. = 195.3 MPa.

Primary stress levels (

) Secondary stress range (

)

3 , 4.5 , 6 , 9

2 , 3 , 4.5 , 6 , 9

1 , 2 , 3 , 6

4.4. Data extraction from two-rod tests

At each measurement point recorded from the tests, the mean strain in the two rods was calculated. This was then used as the characteristic curve for each test. An example of a curve is showed in Figure 16 together with the strain measurement points in both specimens as a function of cycles. This was done for all combinations of primary and secondary stress.

Figure 16. Strain measurements in both rods, and mean strain as a function of cycle number.

2For details concerning , the reader is referred to the uniaxial tensile test in the material characterization section below.

0 5 10 15 20 25

-0.01 0 0.01 0.02 0.03 0.04 0.05 0.06

Cycle number

Log strain [ ]

Strain evloution rod 1 Strain evloution rod 2 Mean strain evolution

(36)

Figure 17. Two-rod test results (curves show mean strain evolution), divided into subplots with the same primary stress.

The results suggest interesting details. The general appearances of the curves suggest the presence of primary/secondary ratcheting, mentioned in the plastic deformation theory section. The primary ratcheting denotes the first cycles of each test (with higher ratcheting rate). After this, the curve stagnates into the secondary ratcheting where the ratchet increment decreases, and in some cases approaches zero (elastic shakedown). If the tests with the same secondary stress range are examined (for example all solid red curves in Figure 17) higher primary stress yields in higher ratcheting rates, primarily in the initial part of each test. Furthermore, the mean strain evolution is dependent on the secondary stress range. If for example the middle subplot is examined, higher secondary stress range increases the ratcheting rate, both initially and in the secondary ratcheting domain.

Following ASME’s guidelines, the strain in any direction in a pressurized Class I component must not exceed 5 % during the lifetime of the component. This limit is in some cases achieved within only a few cycles (indicated by dotted line in Figure 17). This means, if for example a pressure vessel or a pipe experiences a load case leading to these levels of stress, the component will exceed the 5 % strain limit rapidly. Some load combinations are indeed

0 50 100 150

0 0.05 0.1 0.15

Log strain [ ]

Cycle number

0 50 100 150

0 0.05 0.1 0.15

Log strain [ ]

Cycle number

0 50 100 150

0 0.05 0.1 0.15

Log strain [ ]

Cycle number

5 %

(37)

5. Characterization of 316L material

To evaluate different aspects of the material properties and calibrate the material models, tests were performed including

 uniaxial tensile tests

 fully reversed strain cycling tests,

 uniaxial ratcheting test,

 strain rate dependency test in plastic cyclic deformation, and

 uniaxial tensile tests with periodic unloading.

Using the same material batch as an earlier study [10], the results were compared, when possible, to strengthen the reliability of the results. Full access to the test data from [10] were given to this study which facilitated the verification process. In this study, all specimens were manufactured from the same plate, partially used in [10].

In the material characterization tests, three different specimen geometries were used. These are referred to as Geometry 1, 2 and 3 in the sequel. The specimen geometries’ parameter values are presented in Table 3.

Table 3. Parameter values for Geometry 1, 2 and 3 used for material characterization tests [mm].

Geometry 1 6 30 35 12 11 80 10 13 5

Geometry 2 7 18 26 12 11 75 10 13 6,08

Geometry 3 6 18 26 12 11 75 10 13 6

5.1. Material composition

One material was used exclusively in this study, namely the stainless steel alloy 316L (austenite). Type 316/316L (EUROCODE: X1 CrNiMo 17 12 2 / X3 CrNiMo 17 12 2) is a chromium-nickel stainless steel with good heat resistance and high corrosion resistance. It is suitable for use in the presence of corrosives and is amongst other areas used in nuclear reactor components.

Type 316L, which is the low-carbon version of 316, is less prone to grain boundary carbide precipitation

(sensitization). Carbide precipitation causes the material to be more susceptible to corrosion at grain boundaries,

which if present in a component affects the durability and mechanical properties negatively. The austenitic

structure also gives very good toughness properties as well as creep strength. The composition limits of 316L are

shown in Table 4 [20].

(38)

The purpose of the tensile tests was to obtain the stress-strain relationship during monotonous loading, along with characteristic material parameters such as Young’s modulus and yield stress. The rolling direction from manufacturing the plates used for specimens was unknown. Therefore two uniaxial tensile tests were performed with specimens cut out perpendicular to each other in the plate to evaluate if the plate had similar properties in the two directions, denoted Longitudinal and Transversal direction. Specimen geometry 1 (cf. Table 3) was used for this test.

Since the material in this study proved to be very ductile, the extensometers could not be used to capture the full stress-strain curve. The piston displacement data were therefore used to extrapolate the extensometer data. Details on how this was conducted are presented in Appendix A. The stress-strain response of the material is presented in Figure 18.

0 10 20 30 40 50 60 70

-500 0 500 1000 1500

Log strain [%]

True stress [MPa] Transversal Longitudional

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

0 100 200 300 400

True stress [MPa]

Zoomed in at elastic domain

Transversal Longitudional Comparison between Transversally and Longitudionally cut specimens

(39)

The specific averaged parameters for the test are presented in Table 5. The yield stress, , was chosen to the

value. Furthermore, material parameter is the maximal engineering stress in the test. Characterizing parameter was calculated as

(6) where marks a distance on the specimen that initially is five times the diameter, and

is the length between these marks at failure. Characterizing parameter is a measure of the area reduction at failure, and is calculated as

(7) where

was the smallest area of the specimen at failure.

Table 5.Material parameters (averaged) from uniaxial tensile test.

Parameter

Value 197 GPa 293 MPa 614 MPa 80% 89%

The tensile test of the longitudinal and transversal specimens did not differ significantly, and as such, no anisotropy could be detected from these tests. The material proved to be very ductile, with large elongation at failure and area reduction at failure. Values of

and correspond well with material certificate given in [10].

5.3. Fully reversed strain cycling test

Fully reversed strain cycling tests of the material was conducted to evaluate the cyclic behavior. The tests were

conducted at three different prescribed strain ranges, namely 0.5%, 1% and 2 %, all at a strain rate of 0.01 %/s and

zero mean strain. The primary interest in this test was the shape of the saturated hysteresis loops. These are the

stress-strain response after any cyclic hardening of softening effects had subsided. Initial tests showed problems

with buckling, and the specimen geometry was altered slightly. Specimen geometry 2 (cf. Table 3) was used for

this test. In Figure 19, full tests for each strain range are presented.

(40)

Figure 19. Fully reversed strain cycling at the three strain ranges: 0.5%, 1% and 2%.

The tests suggest an extensive cyclic hardening, especially at 2 % strain range. The results show good correspondence with the test results from [10].

The saturated cycles (after cyclic hardening effects) were extracted and later used for material model calibration.

The three hysteresis loops are presented in Figure 20. The maximum and minimum stress values during a saturated cycle are presented in Table 6.

Table 6. Maximum and minimum stress during saturated cycles for each of the three levels.

Test at 0.5 % strain range Test at 1 % strain range Test at 2 % strain range Maximum stress in

saturated cycle [MPa] 276.9 337.8 409.0

Minimum stress in

saturated cycle [MPa] -268.8 -339.3 -408.0

-1 -0.5 0 0.5 1

-400 -300 -200

Log strain [ ]

Tru

-1 -0.5 0 0.5 1

-400 -300 -200

Log strain [ ]

Tru

-1 -0.5 0 0.5 1

-400 -300 -200

Log strain [ ]

Tru

(41)

Figure 20. Collection of saturated hysteresis loops for the three cyclic strain amplitudes. Upper: Stress versus plastic strain. Lower: Stress versus total strain.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-500 -400 -300 -200 -100 0 100 200 300 400 500

Log Plastic Strain [-]

True Stress [MPa]

Prescribed strain cycling at  0.25 % Prescribed strain cycling at  0.5 % Prescribed strain cycling at  1 %

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-500 -400 -300 -200 -100 0 100 200 300 400 500

Log Strain [-]

True Stress [MPa]

Prescribed strain cycling at  0.25 % Prescribed strain cycling at  0.5 % Prescribed strain cycling at  1 % Extracted saturated cycles from low-cycle fatigue: Total strain and plastic strain

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 2 in stress amplitude.

This meant a cycling between 488 MPa and -293 MPa, with a mean stress of 97.7 MPa. The test result is presented in Figure 21. Here the stress versus strain plot for the test is shown. For each cycle, the mean strain was calculated and the strain progression is also shown below.

Figure 21. Uniaxial ratcheting test. Upper: Stress versus strain plot. Lower: Mean strain progression with increasing cycles.

0 5 10 15 20 25 30 35 40

-300 -200 -100 0 100 200 300 400 500

Eng strain [%]

Eng Stress [MPa]

Ratcheting test

0 50 100 150 200 250

0 5 10 15 20 25 30 35 40

Mean strain progression over cycles

Cycle number [N]

Mean strain over cycle N [%]

(43)

5.4.2. Strain rate dependency

As the control system of the two-rod test did not allow a strain rate higher than 0.005 %/s and the fully reversed strain cycling tests were conducted at a strain rate of 0.01 %/s, it was of interest to investigate the strain cycling rate dependency. Therefore the strain rate dependency was evaluated over the interval 0.01-0.0025 %/s. Specimen geometry 2 (cf. Table 3) was used for this test. The saturated cycles for the strain rates are presented in Figure 22.

As can be seen in the figure, strain rate within 0.0025 %/s and 0.01 %/s has negligible influence in the stress- strain response. As no trend can be seen, the differences can instead be used as a measure of the repeatability in the fully reversed strain cycling tests.

Figure 22. Strain rate dependency test.

5.4.3. Uniaxial tensile test with periodic unloading

This test was similar to the tensile test, but the specimen was unloaded at regular increments. The outcome of the test was compared to the monotonous stress-strain response from the uniaxial tensile test. The purpose of the test was to see if the hardening properties of the material changed if the specimen was unloaded regularly, as this type of loading is similar to what a specimen could experience during a two-rod test. Specimen geometry 1 (cf. Table 3) was used for this test.

The specimen was first loaded up to 0.2 % strain. Then, the specimen was unloaded by approximately one yield stress. After this, the specimen was loaded up to 0.4 % and again unloaded by one yield stress. This was repeated

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

-300 -200 -100 0 100 200 300

Strain [%]

S tr e s s [ M P a ]

Strain rate dependency test: three strain rates cycled at



0.5%

1e-4

0.5e-4

0.25e-4

(44)

Figure 23. Comparison between monotonous tensile tests and tensile test with periodic unloading.

0 2 4 6 8 10 12 14 16 18 20

0 100 200 300 400 500

True stress [MPa]

Log strain [%]

(45)

6. Material models and calibration

To simulate the response of a structure subjected to cyclic plastic deformation, higher demands are put on the material model. During the last decades, the development in computational power has allowed the usage of larger and more complex numerical models. It has also become possible to utilize more complicated material models, and many have been implemented in commercially available FE software. In this study the focus was on four material models, with varying degree of complexity. These were

1. A bi-linear kinematic hardening model, 2. A multi-linear kinematic hardening model,

3. The Armstrong-Frederick (AF) non-linear kinematic hardening model, and

4. The Chaboche model with multiple superimposed kinematic hardening A-F models.

In this study, Ansys was used as primary software for FE modeling. For a more detailed derivation regarding the models the reader is referred to the manual [21] and reference literature.

6.1. Bi-linear kinematic hardening model

The bi-linear kinematic hardening model is, as the name suggests, assuming linear elastic properties up to the yield stress. The plastic region has linear hardening or no hardening at all (ideal plastic). In this study the von Mises yield condition, Equation (3), is employed. It also has a back stress formulation as proposed by Melan (1928) and Prager (1955) and is written as

̇

(8)

or in integrated form

∫ ̇

(9)

where H is constant. A duly characterization of this material model can be found under BKIN in Ansys manual [21]. A bilinear response is shown in Figure 24.

Figure 24. Cyclic response of a bi-linear kinematic hardening model [18].

(46)

Figure 25. Bilinear material model fit (red curve) to uniaxial tensile test (black curve).

Table 7. Parameter values for the bi-linear material model.

Parameter Young’s modulus Yield stress Hardening modulus

Value 197 GPa 315 MPa 2000 MPa

0 2 4 6 8 10 12 14 16 18 20

0 100 200 300 400 500 600 700

Log strain [%]

S tr e s s [ M P a ]

Bilinear material model (red curves) calibration versus material response (black curves)

(47)

6.2. Multi-linear kinematic hardening model (Mróz)

This material model is fitted to the monotonous stress-strain curve with multiple linear segments. It was first suggested by Mróz [22] and is sometimes referred to as the sub-layer or overlay model. The name sub-layer comes from the assumption that it is “composed of various portions (or sub-volumes), all subjected to the same total strain, but each sub-volume having a different yield strength” [21]. In other words, the response is built up by a superposition of several elastic-ideal plastic material models, each with different Young’s moduli and different yield stress. The tangent modulus at each segment of the monotonous response is thus depending on how many of these sub-layered models that has yielded and how many that has not. The response of such a material model is shown in Figure 26.

Although the yield condition can be visualized by several circles in the synoptic plane (von Mises) there is no non-linear effect in the back stress evolution. The back stress can be seen as a superposition of several forms of Equation (9), or equivalently

∑

(11)

where

̇

̇ (

) (12)

Here, is a positive coefficient, the plastic multiplier and M the point where the N:th yield surface touches the N+1:th yield surface at the mapping point. Consequently,

is called the mapping stress. A thorough mathematical derivation can be found in [6]. Ansys provides two kinds of kinematic multilinear models, referred to as MKIN and KINH, where the latter allows up to 40 different sub-layered models that can be temperature- dependent if desired. The multilinear model was fitted by picking points along the stress-strain curve, as illustrated in Figure 27. For this, the true stress-logarithmic strain response was used.

Figure 26. Visualization of a Mróz multi-linear (kinematic) material model.

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Figure 27. Fitting of the multilinear kinematic hardening model. Illustration taken from [21].

The material model was built up by 17 stress-strain points as indicated in Figure 28, picked at certain stress levels of the tensile test curve. The stress-strain pairs selected for the multi-linear model are presented in Table 8. As can be seen in Figure 28 and Table 8, the stress levels were chosen at certain levels and the strain measurements were picked from the uniaxial tensile test. The point sampling was denser at the start and sparser at higher stress levels.

0 0.2 0.4 0.6 0.8

0 200 400 600 800 1000 1200

Stress [MPa]

Full test

Uniaxial tensile test points for the Mroz model

0 0.005 0.01 0.015 0.02

0 50 100 150 200 250 300 350 400

Stress [MPa]

Zoomed in at elastic range Uniaxial tensile test points for the Mroz model

(49)

Table 8. Pairs of stress-strain values for the multi-linear model.

Strain [-] Stress [Pa]

2,3288E-04 4,9852E+07 5,1649E-04 1,0120E+08 8,2745E-04 1,5093E+08 1,1729E-03 1,9965E+08 1,6585E-03 2,5063E+08 2,0969E-03 2,7497E+08 3,3407E-03 3,0018E+08 7,5970E-03 3,2493E+08 1,6368E-02 3,5002E+08 2,7294E-02 3,7501E+08 3,8979E-02 4,0000E+08 9,1852E-02 5,0009E+08 1,5722E-01 5,9997E+08 2,3032E-01 6,9997E+08 3,1078E-01 8,0001E+08 3,9900E-01 9,0001E+08 4,7978E-01 9,8001E+08 The fit of the model is presented in Figure 29.

Figure 29. Multilinear material model fit (red curve) to uniaxial tensile test (black curve).

0 2 4 6 8 10 12 14 16 18 20

0 100 200 300 400 500 600 700

Log strain [%]

S tr e s s [ M P a ]

Mroz material model (red curves) calibration versus material response (black curves)

(50)

grows. This model will further be referred to as the AF model.

The Armstrong-Frederick model fitting process starts in extracting a saturated half hysteresis loop from the cyclic testing. The elastic strains are then excluded through

(14)

During loading from the start point of maximal compression the plastic increment is (for uniaxial stress state)

̇ ̇ (15)

Thus, employing (13) and (15), ̇ yields

∫

̇ (16)

which yields

̇ |

[

]

(17)

Thus,

∙ (

)

⁽ ⁾

(18) and the stress through the evolution of the back stress is

(19)

where is the yield stress of the model. It is seen from (18) that the maximum value of

is

which sets a maximum allowed stress in the model to

.

Through variation of the parameters ,

and

the response of the model was fitted to the hysteresis loops at

(51)

levels. The parameter values for the AF model are summarized in Table 9. The material model fit to the saturated hysteresis loops at the three levels is presented in Figure 30.

Table 9. Parameter values for AF material model.

Parameter

Value 197 GPa 240 MPa 280 4.5e10

Figure 30. AF material model fit (red curves) to saturated hysteresis loops from fully reversed strain cycling tests (black curves).

6.4. Chaboche model

The Chaboche model is a development of the AF model. The basic idea of the model is to superimpose several AF models to make the representation of the material more flexible than it would have been if just one was used.

It is noteworthy that AF models are sometimes referred to as single Chaboche models.

The back stress of the Chaboche model for superimposed AF models can be formulated as

∑

(20)

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-500 -400 -300 -200 -100 0 100 200 300 400 500

Log plastic strain [%]

S tr e s s [ M P a ]

AF material model (red curves) calibration versus material response (black curves)

(52)

(22)

With the same reasoning as for the AF model, the maximum allowed stress in the Chaboche model is . The calibrated parameter values are summarized in Table 10.

Table 10. Parameter values for Chaboche material model.

Parameter

Value 197 GPa 145 MPa 160 1.85e10 800 0.8e11 3500 1.85e11 Similarly to the AF model, the Chaboche material model fit to the saturated hysteresis loops at the three levels is presented in Figure 31.

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1

-500 -400 -300 -200 -100 0 100 200 300 400 500

Log plastic strain [%]

Stress [MPa]

Chaboche material model (red curves) calibration versus material response (black curves)

Experimental and Numerical Investigation of Ratcheting Effects in 316L Stainless Steel - The Two-Rod approach.

Master’s Thesis in Solid Mechanics Second level, 30.0 ETCS credits Stockholm, Sweden 2014 KTH Department of Solid Mechanics Inspecta Nuclear AB

Gustav Eklund Numerical Investigation

of Ratcheting Effects in

316L Stainless Steel –

The Two-Rod Approach

Experimental and Numerical

Investigation of Ratcheting Effects in 316L Stainless Steel –

The Two-Rod Approach

Experimentell och Numerisk Undersökning av Ratchetingeffekter för 316L Rostfritt Stål –

Tvåstångsmetoden Gustav Eklund

Master’s Thesis in Solid Mechanics

Second level, 30.0 ETCS credits Stockholm, Sweden 2014 Conducted at Inspecta Nuclear AB and

KTH Department of Solid Mechanics

Sammanfattning

Det här examensarbetet utfördes under våren 2014. En experimentell och numerisk undersökning genomfördes på det austenitiska rostfria stålet 316L. Huvudområdet för studien var att undersöka fenomenet ratcheting (progressiv plastisk deformation).

Experimentellt var huvudfokus på det så kallade tvåstångstestet, vilket tidigare inte hade utförts.

Bi-linjär kinematiskt hårdnande modell

Multilinjär kinematiskt hårdnande modell (Mróz)

Armstrong-Frederick icke-linjärt kinematiskt hårdnande modell

Chaboche icke-linjärt kinematiskt hårdnande modell med tre superponerade back stress- tensorer.

Införandet av isotropt hårdnande i de icke-linjärt kinematiskt hårdnande materialmodellerna kan ha förbättrat simuleringarnas överensstämmande med provresultaten då materialet visar på omfattande plastiskt hårdnande, både i monotont dragprov såväl som cykliskt hårdnande.

Metoden som utvecklades för tvåstångsprovningen visade sig robust och pålitlig. En slutsats som

kan dras är att effekter från materialratcheting förmodligen är små i jämförelse med effekter från

strukturratcheting i tvåstångsprovningen. Dessutom kan från jämförelsen mellan simuleringarna och

tvåstångsprovningen sägas att en mer avancerad materialmodell inte nödvändigtvis resulterar i en

prediktering som överensstämmer bättre med provningen.

Abstract

This Master’s Thesis was conducted during spring 2014. An experimental and numerical investigation was conducted on the austenitic 316L stainless steel. The main focus of the study was the investigation of ratcheting effects.

Bi-linear kinematic hardening model

Multilinear kinematic hardening model (Mróz)

Armstrong-Frederick non-linear kinematic hardening model

Chaboche non-linear kinematic hardening model with three superimposed back-stress tensors.

The two-rod test was then numerically simulated with different material models. The results from the FE simulations were then compared to the test results obtained from the two-rod tests. The goal, apart from investigating the ratcheting effects in 316L steel, was to evaluate the material models’

ability to reproduce the two-rod test results.

The setup for the two-rod test proved robust and reliable. The results suggest that structural

ratcheting effects dominate the two-rod test results. Furthermore, the comparison between

simulations and the two-rod tests suggest that a more advanced material model does not necessarily

yield in a better prediction.

Acknowledgements

Furthermore I would like to thank the people at Inspecta Nuclear for help, advice and support. I would especially like to thank Pär Ljustell for his sharing of experience concerning 316L and valuable ideas regarding modeling and testing.

Last but not least, I would like to thank my supervisor Peter Segle at Inspecta Nuclear. He has been

dedicated to help and have been invaluable in all parts of the study, from modeling, testing and

calibration as well as writing and providing theoretical expertise. I am very thankful to have been his

supervisee during this project which would have taken a lot more time without his help.

Table of Contents

Nomenclature

1. Introduction

1.1. Presentation of Inspecta

1.2. Background

Producing electric energy from nuclear energy is attractive since the amount of energy per fuel unit is extremely high in comparison to for example oil or biodegradable fuel. Also, there are no emissions of CO

For controlling purposes of nuclear power plants, frameworks of regulations have been created, both specific for Sweden and from the European Union. Among other things, these regulations emphasize the control of design, demanding utilization of codes of conduct.

1.2.1. ASME code on nuclear power plant components

If the elastic analysis is proven insufficient, for example if the behavior of the component in reality differs much

from the analysis or the behavior includes some unexpected phenomena, a more refined approach towards

plants there is a need for a more refined process of evaluating components, but still fulfilling ASME’s requirements in plastic analysis.

1.3. Definitions

1.3.1. Ratcheting

The concept ratcheting (also, cyclic creep [4] or ratchetting) is commonly used to denote a situation where the following criteria are fulfilled:

 The component is subjected to cyclic loading,

 plastic deformation occurs in every cycle,

 the mean stress is non-zero, and

 the plastic strain (averaged over a cycle) grows with increasing cycles.

One situation in which ratcheting occurs is when a specimen is subjected to a cyclic load with prescribed stress

amplitude with a non-zero average stress [4], as can be seen in Figure 1.

Figure 1.An example of ratcheting in a specimen subjected to prescribed stress amplitude with a non-zero mean stress [5].

The ratcheting phenomenon can be seen on a structural scale (structural ratcheting) or on a micromechanical scale (material ratcheting). The two types of ratcheting are related, although simulating structural ratcheting requires less advanced material models than simulating material ratcheting [6].

1.3.2. Primary and secondary stress

Secondary stress is distinguished from primary stress in the way that it is self-limiting. These stresses arise

primarily from displacement-controlled situations such as thermal loading or local stress at a structural

discontinuity. The reason for distinguishing these two stresses is that primary stresses are more serious concerning

the failure risk of the structure. Secondary stresses per se are not as serious as primary stress but may be

dangerous when present together with primary stresses.

2. Thesis’ objective

The objective of this study was first and foremost to study the ratcheting phenomena in 316L stainless steel through material testing and numeric simulations. This included

 Design and development of the so-called two-rod test,

 Perform a test series in two-rod testing for 316L material,

 Characterize 316L by material testing to calibrate material models, and

 Simulate the response of the two-rod tests with the calibrated material models in FEM.

Four different material models were evaluated: