ratcheting evaluation in two-rod testing878326/... · 2015. 12. 8. · chaboche non-linear...

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

Experimental and

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


Conducted at Inspecta Nuclear AB and


Experimental And Numerical Investigation of Ratcheting Effects in 316L Stainless Steel – The Two-Rod Approach

Inspecta Nuclear AB and KTH Department of Solid Mechanics 5

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.



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.



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.



Table of Contents

1. Introduction ....................................................................................................................................................................... 15

1.1. Presentation of Inspecta ........................................................................................................................................... 15

1.2. Background ............................................................................................................................................................... 15

1.2.1. ASME code on nuclear power plant components............................................................................................. 15

1.3. Definitions ................................................................................................................................................................. 16

1.3.1. Ratcheting ........................................................................................................................................................ 16

1.3.2. Primary and secondary stress .......................................................................................................................... 17

2. Thesis’ objective ................................................................................................................................................................ 19

2.1. Delimitations ............................................................................................................................................................. 19

3. Theoretic background ....................................................................................................................................................... 21

3.1. Yield surfaces, hardening models and the Bauschinger effect ................................................................................. 21

3.2. Cyclic plastic deformation ......................................................................................................................................... 22

3.3. Micro mechanics and physical mechanisms of plastic deformation .......................................................................... 26

3.4. Material ratcheting .................................................................................................................................................... 26

3.5. Structural ratcheting .................................................................................................................................................. 28

3.6. Similarities between material and structural ratcheting ............................................................................................. 29

4. Two-rod testing ................................................................................................................................................................. 31

4.1. Test setup ................................................................................................................................................................. 32

4.2. Controlling the test machines .................................................................................................................................... 33

4.3. Experiments .............................................................................................................................................................. 34

4.4. Data extraction from two-rod tests ............................................................................................................................ 35

4.5. Two-rod test results .................................................................................................................................................. 36

5. Characterization of 316L material ..................................................................................................................................... 37

5.1. Material composition ................................................................................................................................................. 37

5.2. Uniaxial tensile test ................................................................................................................................................... 38

5.3. Fully reversed strain cycling test ............................................................................................................................... 39

5.4. Supplemental tests ................................................................................................................................................... 42

5.4.1. Uniaxial ratcheting test ..................................................................................................................................... 42

5.4.2. Strain rate dependency .................................................................................................................................... 43

5.4.3. Uniaxial tensile test with periodic unloading ..................................................................................................... 43

6. Material models and calibration ......................................................................................................................................... 45

6.1. Bi-linear kinematic hardening model ......................................................................................................................... 45

6.2. Multi-linear kinematic hardening model (Mróz) ......................................................................................................... 47

6.3. Armstrong-Frederick model ...................................................................................................................................... 50

6.4. Chaboche model ....................................................................................................................................................... 51

7. Two-rod FE model ............................................................................................................................................................. 53

8. Numerical results .............................................................................................................................................................. 55



8.1. Material model behavior in fully reversed strain cycling test and uniaxial tensile test ............................................... 55

8.2. Two-rod FE simulations – comparison ...................................................................................................................... 57

8.2.1. Bi-linear model ................................................................................................................................................. 58

8.2.2. Multi-linear model ............................................................................................................................................. 58

8.2.3. AF model .......................................................................................................................................................... 59

8.2.4. Chaboche model .............................................................................................................................................. 59

9. Discussion ......................................................................................................................................................................... 61

9.1. Development of the two-rod test ............................................................................................................................... 61

9.2. Characterizing 316L .................................................................................................................................................. 62

9.3. Material model calibration ......................................................................................................................................... 62

9.4. FE simulations’ agreement to two-rod test results .................................................................................................... 63

10. Conclusions ................................................................................................................................................................. 65

10.1. Recommendations for future studies .................................................................................................................... 65

11. Bibliography ................................................................................................................................................................. 67

Appendix A – Extrapolating extensometer data ......................................................................................................................... 69

Appendix B – FE simulation comparison to two-rod test ............................................................................................................ 71



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



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






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)



Back stress

Back stress tensor

N:th back stress tensor

Initial value of back-stress

Armstrong-Frederick material model back-stress parameter

Chaboche material model back-stress parameter

Secondary displacement difference in two-rod test

Secondary strain difference in two-rod test

Secondary stress difference in two-rod test

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



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 CO2, 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



designing the component is needed. One approach is then a plastic analysis, in which the plastic behavior also is

taken into account. This is more time-consuming approach but sometimes necessary in order to ensure that the

component meets the requirements.

Since a plastic analysis provides a more detailed description of the component’s behavior, ASME gives more

freedom in this kind of analysis than when relying solely on an elastic analysis. One requirement is however that

the maximum accumulated strain at any point in the component must not exceed 5.0 % at any stage in the life of

the component. Studies collecting experimental data [2] on fatigue of pressure vessels suggest that the mode of

failure for such components is different from plastic collapse, as is accounted for in elastic analysis and limit load

analysis. These findings put the design procedure in new perspectives and earlier conservative estimates designing

pressure vessels might need reconsideration in certain cases. As modernizing actions are taken in some nuclear

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.



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.



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.



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, σe in Equation (1) can be written as

[

]

[

(

)]

(3)

which corresponds to a circle in the synoptic plane1, 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

( ) (4)

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]



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.



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.



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].



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.



3.3. Micro mechanics and physical mechanisms of plastic deformation

On the micromechanical level, metal alloys are far from homogenous. Metal components are generally

manufactured from cooling the material from a liquid state to a solid state. As the material cools, crystals are

formed in a repeated parallelepipedic pattern, or lattice. These lattices’ exact structure depends on the alloys’

composition, but possesses in general directionality and is thus in general anisotropic. They are also randomly

oriented in space. As the material cools down further the crystals grow and at a certain point the crystals meet,

creating boundaries. The final structure is thus polycrystalline, built up by grains oriented randomly. The sizes of

the grains range between a few micrometer up to a few millimeter depending on alloy, cool rate, and the

constituent elements of the alloy [8]. Hence, although on a microscopic level most materials display anisotropic

properties the macroscopic properties (where the grains can be considered small) can often be regarded as

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

entities interact is hard to make. Often the ratchet increment is greater in early stages of the test and decays with



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]



However, the mechanics giving rise to material ratcheting has mostly been qualitatively examined [14] [15]. This

is due to the complexity of material ratcheting and in large differences between different alloys. Since this effect

is complex in its appearance, it needs sophisticated material models to adequately resemble the behavior in

simulations [14].

3.5. Structural ratcheting

When progressive deformation of a structure is observed, it is often labeled structural ratcheting (or just

ratcheting), as it is the structure as a whole whose deformation is progressively increasing. One classical example

is the Bree problem [17] in which a hollow cylinder is subjected to primary stresses and intermittent heat-fluxes.

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.



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.



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.



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, ltest, and the diameter, d0, which also affected the radius of curvature, ρ. All specimens were

made uniform within 0.01 mm diameter tolerance throughout the test length, ltest.

Figure 13. Test specimen fixture.



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:

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.



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

Repeat

1. Read the load in each of the two machines

(𝐹 and 𝐹 ), add them and compare to desired

value ( ∙ 𝐹𝑝𝑟𝑖𝑚𝑎𝑟𝑦)

𝜀𝑐𝑜𝑟𝑟

𝜀𝑐𝑜𝑟𝑟 <

𝜀𝑐𝑜𝑟𝑟

2. Calculate correction term 𝜀𝑐𝑜𝑟𝑟

if 𝐹 𝐹 < 𝐹𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝐹𝑡𝑜𝑙

elseif 𝐹 𝐹 𝐹𝑝𝑟𝑖𝑚𝑎𝑟𝑦 𝐹𝑡𝑜𝑙

else

end

𝜀 𝑡 𝜀 𝑡 𝜀𝑐𝑜𝑟𝑟

𝜀 𝑡 𝜀 𝑡 𝜀𝑐𝑜𝑟𝑟

3. Alter output signal



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 stress2. 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.

2 For 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 s

train

[ ]

Strain evloution rod 1

Strain evloution rod 2

Mean strain evolution



4.5. Two-rod test results

The result from the two-rod tests is presented in Figure 17. Here, the result is divided into tests with the same

primary stress. For each primary stress level, the different tests with varying secondary stress range are compared

(cf. Table 2).

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

extreme, but can occur in reality if the component is subjected to for example large temperature variations.

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

Log s

train

[ ]

Cycle number

Primary stress = 0.5 Sm

P05 - S3

P05 - S4,5

P05 - S6

P05 - S9

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

Log s

train

[ ]

Cycle number

Primary stress = Sm

P1 - S2

P1 - S3

P1 - S4.5

P1 - S6

P1 - S9

0 50 100 1500

0.05

0.1

0.15

0.2

0.25

Log s

train

[ ]

Cycle number

Primary stress = 1.25 Sm

P1.25 - S1

P1.25 - S2

P1.25 - S3

P1.25 - S6

5 %



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].



Table 4. 316L constituents.

Chemical composition

C Mn Si P S Cr Mo Ni N Fe

Wt. % min - - - - - 16.0 2.0 10.0 - rem.

Wt. % max 0.03 2.0 0.75 0.045 0.03 18.0 3.0 14.0 0.1 rem.

5.2. Uniaxial tensile test

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.

Figure 18. Stress-strain comparison between the two perpendicularly cut out specimens.

0 10 20 30 40 50 60 70-500

0

500

1000

1500

Log strain [%]

Tru

e s

tress [

MP

a]

Transversal

Longitudional

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

100

200

300

400

Log strain [%]

Tru

e s

tress [

MP

a]

Zoomed in at elastic domain

Transversal

Longitudional

Comparison between Transversally and Longitudionally cut specimens



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.



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

-100

0

100

200

300

400

Log strain [ ]

Tru

e s

tress [

MP

a]

Fully reversed strain cycling at 0.25 %

-1 -0.5 0 0.5 1

-400

-300

-200

-100

0

100

200

300

400

Log strain [ ]

Tru

e s

tress [

MP

a]

Fully reversed strain cycling at 0.5 %

-1 -0.5 0 0.5 1

-400

-300

-200

-100

0

100

200

300

400

Log strain [ ]

Tru

e s

tress [

MP

a]

Fully reversed strain cycling at 1 %



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 [-]

Tru

e S

tress [

MP

a]

Prescribed strain cycling at 0.25 %


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 [-]

Tru

e S

tress [

MP

a]



Prescribed strain cycling at 1 %

Extracted saturated cycles from low-cycle fatigue: Total strain and plastic strain



5.4. Supplemental tests

5.4.1. Uniaxial ratcheting test

To evaluate the material ratcheting effects, a test was performed where a specimen was cycled between two stress

levels, with a non-zero mean stress. Specimen geometry 3 (cf. Table 3) was used for this test.

For this test, the parameter was used to determine the stress levels. The levels chosen for the test was

0.5 in mean stress

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 S

tress [

MP

a]

Ratcheting test

0 50 100 150 200 2500

5

10

15

20

25

30

35

40Mean strain progression over cycles

Cycle number [N]

Mean s

train

over

cycle

N [

%]



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 [%]

Str

ess [

MP

a]

Strain rate dependency test: three strain rates cycled at 0.5%

1e-4

0.5e-4

0.25e-4



up to approximately 20 % total strain. The stress-strain curve of the test along with the results from the uniaxial

tensile test without unloading is presented in Figure 23. As seen in the figure, periodic unloading within the elastic

range does not show any significant hardening differences when compared to monotonous tensile tests.

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

600

700

Tru

e s

tress [

MP

a]

Log strain [%]

Uniaxial tensile test w/ unloading

Longitudinal

Transversal



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].



Results from the uniaxial tensile tests were used to fit the bilinear model. In this model the material response is

characterized by the Young’s modulus, , the yield stress, and the Hardening modulus, . Above the yield

stress, the hardening is constant with a slope, , which is linked to Young’s modulus and Hardening modulus as

(10)

The fit of the model to the tensile test is presented in Figure 25, and the calibrated parameter values are presented

in Table 7.

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 18 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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



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.



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.

Figure 28. Pairs of stress-strain values for the multi-linear model indicated as red circles on the uniaxial

tensile test.

0 0.2 0.4 0.6 0.80

200

400

600

800

1000

1200

Strain [%]

Str

ess [

MP

a]

Full test

Uniaxial tensile test

points for the Mroz model

0 0.005 0.01 0.015 0.020

50

100

150

200

250

300

350

400

Strain [%]

Str

ess [

MP

a]

Zoomed in at elastic range

Uniaxial tensile test

points for the Mroz model



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 18 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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



6.3. Armstrong-Frederick model

The Armstrong-Frederick (1966) model constitutes a development in the classic kinematic hardening models. The

difference lies in the back stress increment formulation, which can be formulated (for uniaxial stress state) as

(13)

where and are material dependent constants, is the effective plastic strain rate, and is the

accumulated back stress tensor defining the accumulated translation of the yield surface. The last term in Equation

(13) serves as a “recall term” for the yield surface. Conceptually, the yield surface is increasingly obstructed in

translating from its original location in this formulation, making the yield surface more resistant to movement as

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

0.5%, 1% and 2%, where the goal was to get as good fit as possible with the same parameters to all three strain



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 [%]

Str

ess [

MP

a]

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



and the back stress evolution law as

(21)

where the parameters are defined above.

The Chaboche fitting process is similar to the fitting process of the AF model. For the uniaxial case with three

back stress tensors , and , corresponding parameters , , , , and , have to be fitted. During

each plastic loading the stress is given by

(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.

Figure 31. Chaboche material model fit (red curves) to saturated hysteresis loops from fully reversed

strain cycling tests (black curves).

-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


Str

ess [

MP

a]

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



7. Two-rod FE model

In order to evaluate the performed two-rod tests with the material models described above, an FE model for

simulations of the two-rod test was created. The model is presented in Figure 32. The two specimens were

modeled with 2-D axisymmetric elements. The axisymmetric modelling was made to shorten the FE analysis

time. Different mesh configurations were used. The mesh in Figure 32 was for the case where a small curvature

was initiated along the length of the specimens, disturbing the homogeneity of the stress distribution.

Due to software restrictions the axis of rotation for the axisymmetric 2D elements cannot be arbitrary; it is locked

to the y-axis. Thus, the two rods need the same axis of rotation and lie as shown in Figure 32. The nodes of the

bottom of the two rods were coupled (as indicated in green) and the force was applied to the hub node (located at

root of red arrow). A prescribed displacement was applied on the upper ends. The specimens were made straight

and did only model the specimens between the extensometer edges. The elements used in the FE model were 8-

node 2D elements3. The element is shown in Figure 33. The quadratic element shape was used exclusively in this

study.

Figure 32. FE model setup. Left: 2D axisymmetric model with boundary conditions. Right: 3/4 expanded

view of the model for visualization.

3 Denoted PLANE183 in Ansys.



Figure 33. Element 183 variations. Taken from [21].

When conducting FE simulations, Ansys was called in batch mode through DOS commands from MATLAB.

From MATLAB the desired combination of primary stress and secondary strain range could be specified, as well

as number of cycles and the material model. The solution was looped in Ansys and the displacement reversed in

each half-cycle until the desired number of cycles had been reached. When a converged solution had been

finished, the data was transferred to MATLAB. The elongations in the rods the end of each half-cycle were used

for conversion to strains.



8. Numerical results

8.1. Material model behavior in fully reversed strain cycling test and

uniaxial tensile test

To gain a deeper understanding of the material models, their response in a fully reversed strain cycling test and

during a uniaxial tensile test was simulated. In Figure 34 to Figure 37, the response is presented.

Figure 34. Simulated response of the bilinear model in fully reversed strain cycling (three strain levels)

and tensile test.

-1 -0.5 0 0.5 1-500

-400

-300

-200

-100

0

100

200

300

400

500


Str

ess [

MP

a]

0 5 10 15 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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



Figure 35. Simulated response of the multilinear model in fully reversed strain cycling (three strain levels)

and tensile test.

Figure 36. Simulated response of the AF model in fully reversed strain cycling (three strain levels) and

tensile test.

-1 -0.5 0 0.5 1-500

-400

-300

-200

-100

0

100

200

300

400

500


Str

ess [

MP

a]

0 5 10 15 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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

-1 -0.5 0 0.5 1-500

-400

-300

-200

-100

0

100

200

300

400

500


Str

ess [

MP

a]

0 5 10 15 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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



Figure 37. Simulated response of the Chaboche model in fully reversed strain cycling (three strain levels)

and tensile test.

From the simulations with the different material models, it is apparent that calibrating towards one test does not

mean a good fit on the other. The linear kinematic hardening models, which are fitted to the uniaxial tensile test

gives a poor prediction of the saturated hysteresis loops in the fully reversed strain cycling test. The non-linear

kinematic hardening models on the other hand are fitted to the fully reversed strain cycling test, but does not

capture the uniaxial tensile test curve. It is also noteworthy that the non-linear kinematic hardening models behave

as ideal-plastic at a certain stress level, as seen in the uniaxial tensile test simulation. This is expected since these

material models have an upper stress limit depending on yield stress and back stress parameter values (cf.

Calibration section).

8.2. Two-rod FE simulations – comparison

In Figure 38 to Figure 41, the FE simulations are compared to the corresponding two-rod tests. The comparison is

divided on material model. In this section, for simplicity only selected combinations are presented. Simulations

for all combinations of primary and secondary stress are presented and compared to test results in Appendix B.

-1 -0.5 0 0.5 1-500

-400

-300

-200

-100

0

100

200

300

400

500


Str

ess [

MP

a]

0 5 10 15 200

100

200

300

400

500

600

700

Log strain [%]

Str

ess [

MP

a]

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



8.2.1. Bi-linear model

Figure 38. FE simulation with bilinear material model (red curves) – comparison to test results (blue

curves)

8.2.2. Multi-linear model

Figure 39. FE simulation with multilinear material model (red curves) – comparison to test results (blue

curves)

It is apparent when examining the simulations for the bilinear and multilinear material models is that both of these

models’ response stagnates after a number of cycles. This is true for all combinations of primary stress and

secondary stress range, but for higher secondary stress range the stagnation is more accentuated.

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

= 2 Sm

Real test

Simulation

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

=3 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= 1.25 Sm

- SEC

=6 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

= 2 Sm

Real test

Simulation

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

=3 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= 1.25 Sm

- SEC

=6 Sm



8.2.3. AF model

Figure 40. FE simulation with AF material model (red curves) – comparison to test results (blue curves)

8.2.4. Chaboche model

Figure 41. FE simulation with Chaboche material model (red curves) – comparison to test results (blue

curves)

The AF model and the Chaboche model generally give similar simulation responses to each other. One of the

cases for these models is where the ratchet increment observed is constant throughout the simulations, which

happens for higher secondary strain ranges. For lower secondary stress ranges the two models’ response does not

resemble the observed curves from the tests very good, as the response flattens far below the real test. The

secondary stress level at which the transition to constant ratcheting occurs differs slightly between the two

material models.

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

= 2 Sm

Real test

Simulation

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

=3 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= 1.25 Sm

- SEC

=6 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

= 2 Sm

Real test

Simulation

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= Sm

- SEC

=3 Sm

0 50 100 1500

0.05

0.1

0.15

Cycles

Log s

train

[-]

PRIM

= 1.25 Sm

- SEC

=6 Sm



9. Discussion

9.1. Development of the two-rod test

As this type of testing has not been conducted before, a lot of planning and trial tests were conducted before the

actual testing was conducted. The first concept idea is shown in Figure 42, where a single test machine pulled the

two test specimens with a force (primary stress), and one person controlled the strain range by manually

displacing a lever beam some degree range (secondary stress).

After initial tests had been performed it was concluded that the concept worked and structural ratcheting was

achieved, but the low accuracy of the manual secondary stress controlling had to be developed. One idea was to

control the lever beam displacement with a second test machine, but the concept was further developed into

having one rod in each test machine. This development, later used as final setup, meant higher accuracy and

reproducibility of the secondary stress, fewer moving parts and better strain rate control.

In the development of the two-machine setup of the two-rod test, there was a problem with regulating loop

responsible for keeping the sum load constant. Throughout the process different designs of the load regulating

loop were tested including a PI controller (Proportional-Integrative), a PD controller (Proportional-Derivative)

and a constant correction factor. Due to a built-in delay in the test machine controls when controlling the test

machine from the computer, instability in the sum load continued to occur. The final solution was to lower the

initially determined strain rate (0.01 %/s) to 0.005 %/s and simplify the controller to a purely proportional for

faster loop time. Since all material characterization tests were performed at a strain rate of 0.01 %/s - but the two-

rod test could not be performed at this strain range - the strain rate dependency tests were performed in order to

ascertain the rate independence. With these test specifics, there were some fluctuations of the sum load but never

more than 1-2% of the total load during testing. The average load over time was very close to desired.

Figure 42. Initial conceptual design of two-rod test. Left: manufactured test fixture. Right: Conceptual

design sketch.



9.2. Characterizing 316L

In this study several different tests were performed to characterize the material. The uniaxial tensile test and the

fully reversed strain cycling test were used for calibration whereas the purpose of the three supplemental tests was

to gain a deeper understanding of the material properties, outside what was taken into account in the material

models.

One property of the material that was noted was the high ductility. In the uniaxial tensile test, the material

displayed an area reduction at failure of 89 % which is relatively high. Related to the area reduction, the material

showed extensive hardening properties. This was apparent in the stress-strain curve in which the specimens

peaked at a true stress of around 1000 MPa. This can be compared to the yield stress of around 290 MPa. As

stainless steels generally are ductile (especially austenitic stainless steel) this behavior was expected.

The tensile test and the fully reversed strain cycling test were previously conducted on the same plates as used as

raw material in this study. The results could therefore be compared for increased reliability to the results. This

revealed problems with buckling during the compressive part of the fully reversed strain cycling test, and the

specimen geometry was changed. After the geometry change, the results showed better agreement.

9.3. Material model calibration

When calibrating the material models, the final parameter choice required extensive calibration work. Especially

with the Chaboche model, where the fitting process to the saturated hysteresis loops required a lot of manual

reasoning. The fitting of a Chaboche model is very difficult to automate, since the optimization of the parameters

leads to a non-linear optimization problem. Ansys has provided a section for automated calibration of the

Chaboche model, but emphasizes the importance of the initial guesses of parameters. A small change in initial

parameters might lead to completely different optimal parameter set. This is often the case for a so-called non-

convex optimization problem, and it is as such very cumbersome to obtain optimal parameter values. The

calibration of the Chaboche and the AF model was therefore done manually, and the parameter choice was

verified through FE simulations (see Figure 36 and Figure 37).

The parameter set of the Chaboche and the AF models did not allow simulation of the uniaxial ratcheting test. As

an effect of the material models being calibrated to a fully reversed strain cycling test of max 2 % strain range, the

models have a maximum allowed stress, as can be seen in the simulated tensile tests in Figure 36 and Figure 37,

where the stress-strain response stagnates at roughly 400 MPa. After this, the models behave as an ideal-plastic

model. In the uniaxial ratcheting test, the maximal stress is (approximately 490 MPa) which for the AF

and Chaboche models lead to strains growing towards infinity. However, by using an isotropic hardening part or

adding a linear term to the Chaboche model (a back-stress tensor where is set to zero) larger stress would be

allowed in the models. Also, calibrating the AF and the Chaboche model to fully reversed strain cycling tests at

larger strain ranges may also increase the maximal allowed stress in the models.

The bi-linear and the multilinear material models are, relative to the Chaboche model, fast and easy to calibrate.

The multi-linear material model is constructed from the stress-strain response, which is straight-forward and only

needs one good uniaxial tensile test. The bi-linear material model needs two parameters (yield stress and

hardening modulus), which requires a fitting process that relatively easy can be automated. For a material such as

316L, which does not have a distinct yield stress, the yield stress of the bilinear model may be lowered for a

conservative estimate of the material behavior.



The behavior in fully reversed strain cycling showed extensive cyclic hardening. This can be modeled by adding

isotropic hardening to the material model, but this was not included in the study. However, the AF and Chaboche

models were calibrated based on the saturated hysteresis loops where cyclic hardening indirectly was considered.

9.4. FE simulations’ agreement to two-rod test results

The predicted response of the linear kinematic hardening material models can be explained through the back

stress evolution in the models during a test. If the secondary stress range is larger than (the diameter of the

yield surface) the back stress increases during simulation up to a point where the back stress is equal to the

primary stress. This corresponds to a certain mean strain level. From this point on, the response of the model

transcends into a plastic shakedown where no additional ratcheting occurs. The yield surface moves back and

forth, but no net strain increment is observed.

If on the other hand the secondary stress level is lower than of the material model, the response of the model

transcends into an elastic shakedown. As the simulation continues from this point, the yield surface does not move

back and forth as was the case in the plastic shakedown. The mean strain level where the elastic shakedown

occurs depends on the primary stress and secondary stress range in the test, but the terminal mean strain level is

lower than for the case where plastic shakedown occurs.

Before shakedown occurs during simulation, the bi-linear material model shows better agreement to test curves

when the primary stress and/or the secondary stress is higher. For low primary stress and low secondary stress

range the simulation with the bilinear model gives a higher ratcheting rate than observed in the tests. The

multilinear model on the other hand gives a systematic lower predicted ratchet rate than observed in the tests,

especially for the lower secondary stress ranges. Since the basic formulation of these material models is the same

(linear hardening) the reason for the differences has to lie in the material model stress-strain response, and at what

stress plastic strains are initiated. Since the multilinear material model follows the stress-strain response of the test

better than the bilinear material model, plastic strains are initiated at a lower stress for the multilinear model than

for the bilinear model.

The non-linear kinematic hardening material models produce similar predictions of the two rod test. For tests with

higher secondary stress range the material models predict constant ratcheting rate, and for tests with lower

secondary range the simulation flattens far below the test curve. The latter behavior is due to predicted elastic

shakedown in the simulations. Because the nonlinear material models are calibrated to the saturated hysteresis

loops and the material displays cyclic hardening, the prediction of the strain evolution at the low secondary stress

ranges are an underestimation compared to the two-rod test results. When none of the rods yield in compression,

the cyclic hardening effects are not visible (cf. results from uniaxial tensile test with periodic unloading in Figure

23). AF and Chaboche models are developed for cyclic plasticity, they do not predict the behavior very well in

such a stress-strain history (see Figure 36 and Figure 37). However, at higher secondary stress ranges the cyclic

response of the two rods resembles cyclic plasticity. When the rods in the two-rod test yield in compression, the

prediction by the AF and the Chaboche models are closer to the test curve.

Similar to the linear kinematic hardening material models which predict plastic shakedown when the secondary

stress range exceeds the yield surface diameter, the AF and Chaboche models’ predict constant ratcheting rate at

higher secondary stress ranges. The point at which the non-linear kinematic hardening material models goes from

predicting elastic shakedown to predicting constant ratcheting rate partly depends on the yield stress, as was the

case with the linear kinematic hardening material models. Also, as the AF and Chaboche models behave as ideal-



plastic after the back stress tensors have reached its maximum value (see Figure 36 and Figure 37) this sets the

boundary for which stress the material model can handle. Furthermore, if the combination of primary stress and

secondary stress range in the two-rod test yields in a stress higher than the maximum stress allowed in the

material model, the absence of plastic hardening causes the ratcheting increment per cycle in the rods to be

constant for the remaining part of the simulation. Since the yield stress and maximum allowed stress differs

between the AF and Chaboche model, the transition from predicting elastic shakedown to predicting constant

ratcheting rate occurs at different load combinations.

The fact that linear hardening material models cannot capture material ratcheting effects but still were able to

predict the two-rod tests fairly good (up to a certain strain level) suggest that structural ratcheting effects

dominates the response over material ratcheting in the two-rod tests. As material ratcheting effects are linked to

the cyclic behavior, whereas the structural ratcheting effects are linked to the uniaxial tensile hardening, the

uniaxial tensile hardening properties plays a larger role in the material model’s ability to predict the two-rod test.

For the Chaboche model, uniaxial tensile hardening can be accounted for by adding a linear back-stress term (as

noted above), which may improve the Chaboche model’s ability to predict the two-rod tests.



10. Conclusions

The final setup of the two-rod test proved to be reliable and robust, and the ratcheting effects seemed to be

produced as desired. The outcome in this study suggests that the two-rod test can very well serve as an alternative

method to the test methods for evaluating ratcheting effects used in earlier studies. The results in the two-rod tests

suggest that structural ratcheting effects dominate the response over material ratcheting effects.

When simulating the two-rod tests, the employment of a more advanced material model does not necessarily yield

in a better prediction. Furthermore, using more back-stress tensors in a non-linear hardening model does not

necessarily produce a better prediction of the two-rod test even though the fit to the hysteresis loops are better.

This is because the non-linear kinematic hardening material models were not calibrated for the stress levels

encountered in some of the two-rod tests. Neither does the multilinear material model produce better prediction

the two-rod test than the bilinear material model, even though the fit to the uniaxial tensile test is better.

10.1. Recommendations for future studies

As the test setup of the two-rod test proved to function properly, further testing of different materials may increase

the understanding of the test and how it can be used as a complement to other tests. In order to gain a deeper

understanding of the two-rod test, the strain and stress in the two rods could be recorded continuously rather than

at the end of each half-cycle, as was done in this study.

The primary interest in further testing of two-rod tests would probably be how the structural ratcheting is affected

by different materials, as this effect seemed to dominate over material ratcheting effects. As a test series of two-

rod tests have been performed, ratcheting effects in an arbitrary component could be evaluated by FE simulations

where the presence of primary and secondary stresses are investigated.

As noted above, the 316L material displays significant cyclic hardening at fully reversed strain cycling tests,

especially at higher strain ranges. The consequences of calibrating the kinematic AF and Chaboche material

models to the saturated hysteresis loops were that they produced poor predictions of the start of the fully reversed

strain cycling test. The models also produced poor predictions of the uniaxial tensile test, which affected the

ability to simulate the two-rod tests. For a material such as 316L it would therefore be an improvement to add an

isotropic hardening part, and possibly also a back-stress tensor with linear hardening in the Chaboche material

model. It could also be of relevance to conduct strain cycling tests at higher strain ranges, such as conducted in

[10]. At higher strain ranges fully reversed strain cycling tests may experience bucking or specimen slipping in

fixtures (as experienced in [10]), therefore only partially reversed strain cycling test may be performed. In these

tests one can investigate the hardening at higher strain ranges and the isotropic hardening properties at higher

strain ranges, which might improve the material model’s predictions of the two-rod tests.



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Fluxes with Application to Fast Nuclear Reactor Fuel Elements," Journal of Strain analysis, no. 2, pp. 226-

238, 1967.

[18] K. Steingrimsdottir, "ANALYSIS OF PLASTIC DEFORMATION IN COMPONENTS SUBJECTED TO

CYCLIC LOADING," Inspecta Technology and Royal Instiytute of Techology, Stockholm, 2009.

[19] ASTM, E606 - Standard Test Method for Strain-Controlled Fatigue Testing, West Conshohocken: ASTM

international , 2012.

[20] Penn Stainless Products, "http://www.pennstainless.com," Penn Stainless Products, 2013. [Online].

Available: http://www.pennstainless.com/stainless-grades/300-series-stainless-steel/316l-stainless-steel/.

[Accessed 11 02 2014].

[21] ANSYS Inc., Ansys Help release 14.5., SAS IP, Inc. , 2012.

[22] Z. Mróz, "On the Description of Anisotropic Workhardeninng," Journal of the Mechanics and Physics of

Solids , vol. 15, pp. 163-175, 1967.

[23] P.-L. L. H. Ö. Bo Alfredsson, Testing Techniques in Solid Mechanics - Course Compendium, Stockholm:

Institutionen för Hållfasthetslära - KTH, 2012.



Appendix A – Extrapolating extensometer data

The extensometers used in this study had a displacement limit of ±2.5 millimeters. The initial length between the

edges was 12.5 mm, which gives a possible strain measurement range of ±20% (±18.2% in logarithmic strain).

Fore ductile materials the strain region of interest may be larger than this. In order to extend the data available

from a test, one can use the piston displacement to extrapolate the extensometer curve.

Since the test machine also deforms when testing, one cannot use the piston displacement as it is. To get around

this, a common assumption [23] is to regard the deformations in the machine as linear elastic, with a compliance

C. Furthermore, the extensometer does not cover the whole length between the test machine clamps. Therefore it

is assumed that the strain within the extensometer measurement range, with initial length , is the same

throughout the length between the test machine clamps, with initial length . To extrapolate the extensometer

curve with the piston displacement curve, one uses

(A-1)

where is the length of the extensometer, is defined in Figure 14, is the compliance of the test machine,

and is the piston force. The result from such an operation is presented in Figure A1, where a uniaxial tensile test

data has been used.

Figure A1. Extensometer data extrapolated by using piston displacement.

0 1 2 3 4 5 6 7 8 9 100

2

4

6

8

10

12

14

16

18

20Piston data fitting to extensometer data

Displacement (mm)

Forc

e [

kN

]

Piston displacement data fitted to extensometer data

Extensometer data



Appendix B – FE simulation comparison to two-rod test The simulations are divided on material model in Figures B1-B4. In each figure, each row has the same primary

stress whereas all columns have the same secondary stress range. The load combination is specified in each plot.

Figure B1. FE simulation with bilinear model (green curves) – comparison to test results (blue curves).

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

6 S

m

Bili

near

mate

rial m

odel (

gre

en c

urv

es)

com

pare

d to test re

sults

(blu

e c

urv

es)



Figure B2. FE simulation with multilinear model (green curves) – comparison to test results (blue curves).

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

6 S

m

Mro

z m

ate

rial m

odel (g

reen c

urv

es)

com

pare

d to test re

sults

(blu

e c

urv

es)



Figure B3. FE simulation with AF model (green curves) – comparison to test results (blue curves).

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

6 S

m

AF

mate

rial m

odel (g

reen c

urv

es)

com

pare

d to test re

sults

(blu

e c

urv

es)



Figure B4. FE simulation with Chaboche model (green curves) – comparison to test results (blue curves).

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

0.1

5

Cycle

s

Log strain [-]

P

RIM

= 0

.5 S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

4.5

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

6 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= S

m -

SE

C=

9 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

Sm

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

2 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

3 S

m

050

100

150

0

0.0

5

0.1

Cycle

s

Log strain [-]

P

RIM

= 1

.25 S

m -

SE

C=

6 S

m

Chaboche m

ate

rial m

odel (g

reen c

urv

es)

com

pare

d to test re

sults

(blu

e c

urv

es)



Inspecta Nuclear AB

ratcheting evaluation in two-rod testing878326/... · 2015. 12. 8. · chaboche non-linear...

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