Parametric Study for the Determining of the Yield Surface of a Metal Using Notched Strip Specimens

Texto completo

(1)

METAL USING NOTCHED STRIP SPECIMENS

Dissertation submitted as part requirement for the Degree of Master of Science in

Steel Construction

By:

Mariela de los Angeles Mendez Morales

Supervisor: Dr. Jurgen Becque

The University of Sheffield

(2)

i

DECLARATION STATEMENT

Mariela de los Angeles Mendez Morales certifies that all the material contained within this document is her own work except where it is clearly referenced to others.

(3)

ii

Supervisor: Dr. Jurgen Becque Mariela de los Angeles Mendez Morales

Parametric Study for the Determining of the Yield Surface of a Metal Using Notched Strip Specimens

ABSTRACT

Hitherto, determining the yield surface of a material by experimental means has been a rather complex task. Theoretical yield criterions, often based on one – dimensional observations, are commonly accepted when estimating the plastic characteristics of a metal in a multidimensional stress space.

Researchers have been trying to find a simple way of conducting such experiment by pulling in tension metal strips with oblique notches. This specific shape forces a neck to develop between the roots of the notches, in which plastic deformations can be found. Despite some success, no reliable results have been reported yet.

It is believed that the geometry of the piece plays and important role in this experiment and that it might be responsible for part of the variability in the results. By means of the finite element method, a parametric study was completed to evaluate if this is the case, and to which degree does it extends. Statistical tools were also used to support the findings made from the finite element analyses.

Confirming the initial hypothesis, variables such as the neck’s width and the radius of the root demonstrated to have a large impact on the results obtained. Whereas others, like the neck angle or the opening angle of the notch, had very little or even no influence at all.

(4)

iii

ACKNOWLEDGEMENTS

I would like to express my gratitude to my father and my brother, who have been there during every moment of this experience to encourage me to give my best effort. I would also like to thank my family and friends back home, as well as the ones that I found here, for supporting me and making this one of the most enjoyable times of my life.

Appreciation is also due to Dr. Jurgen Becque, for his patient guidance and advice throughout my time as his student and, Professor Luca Susmel for his suggestions on which variables were most likely to have a role on this investigation and ideas on how to reduce them.

I am indebted to the Ministry of Science, Technology and Telecommunications (MICITT) and the National Council for Scientific and Technological Investigations (CONICIT), both institutions of the Government of Costa Rica, who kindly agreed to fund my postgraduate studies.

(5)

iv

CONTENTS

Declaration Statement ... i

Abstract ... ii

Acknowledgements ... iii

Contents ...iv

List of Tables ...vi

List of Figures ...vi

1. Introduction ... 1

1.1. Background... 1

1.2. Justification and Value of Research ... 2

1.3. Specific Problem ... 3

1.4. Aims and Objectives ... 3

1.5. Delimitation of the problem ... 4

1.5.1. Research scope ... 4

1.5.2. Limitations ... 4

1.6. Methodology ... 5

2. Literature Review ... 7

3. Theorethical basis ... 10

3.1. Strength of Materials ... 10

3.2. Yield Criterions ... 12

3.2.1. Maximum Shearing Stress ... 12

3.2.2. Maximum Distortion Energy ... 12

3.2.3. Hill’s Quadratic Anisotropic Yield Criterion ... 13

3.3. Plastic State and Necking on Ductile Isotropic Metals ... 14

3.4. General Anisotropic Case of the Plastic State ... 17

3.5. Experimental application ... 20

4. Finite Element Modelling ... 22

(6)

v

4.2. Basics of the Finite Element Modelling ... 23

4.3. Material Modelling ... 25

4.4. Mesh Study and Calibration of the Model ... 29

4.5. Validation of Results for the Mesh Study ... 33

4.6. Measuring Points Study ... 34

4.7. Definition of the Parameters Under Investigation ... 35

5. Finite Element Analysis Results and findings ... 37

6. Statistical Analysis ... 43

6.1. Descriptive Statistics ... 43

6.1.1. General Case ... 43

6.1.2. Grouped by Geometric Variable ... 44

6.2. Analysis of Variance (One – Factor) ... 46

6.3. Analysis of Variance (Two – Factor) ... 48

7. Discussion ... 51

7.1. Ratio between the neck’s length and the total width of the strip (R1) ... 51

7.2. Neck angle (θ) ... 52

7.3. Opening angle (α) ... 52

7.4. Ratio of notch’s root radius and thickness of the strip (R2) ... 53

8. Conclusions ... 54

9. Recommendations ... 56

10. References ... 57

11. Bibliography List ... 59

12. Appendices ... 60

(7)

vi

LIST OF TABLES

Table 4.1. Description of the materials used in the models. ... 26

Table 4.2. Hill's quadratic yield criterion parameters. ... 29

Table 4.3. Yield stress ratios. ... 29

Table 4.4. Results of the mesh study done for the SSA. ... 30

Table 4.5. Summary of the mesh study for the SSF. ... 30

Table 4.6. Results from the study on the location of the measuring angle. ... 34

Table 4.7. Variables of the parametric study. ... 36

Table 5.1. Results for the relative velocity angle (ψ, measured in degrees)... 38

Table 6.1. Descriptive statistics for H (General case). ... 43

Table 6.2. Descriptive statistics (Grouped by R1). ... 44

Table 6.3. Descriptive statistics (Grouped by θ). ... 45

Table 6.4. Descriptive statistics (Grouped by α). ... 45

Table 6.5. Descriptive statistics (Grouped by R2). ... 46

Table 6.6. One factor analysis of variance. ... 47

Table 6.7. Two factor analysis of variance (R1 as the main factor). ... 50

Table 7.1. Selected results from the study for SSF. ... 53

Table 8.1. Suggested dimensions for test pieces. ... 55

LIST OF FIGURES

Figure 1.1. Diagram of the methodology followed on this research. ... 5

Figure 3.1. Idealised engineering stress – strain curve for a ductile material. ... 10

Figure 3.2. Representation of a yield surface in two and three - dimensional space. ... 11

Figure 3.3. Tresca and Von Mises criterion in the two - dimensional stress space. ... 13

Figure 3.4. Specimens studied by Bijlaard and Hill, respectively. ... 14

Figure 3.5. Yield surface for an isotropic metal on the deviatoric plane. ... 16

Figure 3.6. Principal directions from strain rate. ... 17

Figure 3.7. Mohr's circles for (a) plastic strain rate and (b) stresses in the neck... 18

Figure 4.1. Geometry used in the calibration of the model (thickness equal to 2 mm). . 23

Figure 4.2. Stress - strain curve for the SSA. ... 27

Figure 4.3. Variation in the maximum load with the number of FE’s. ... 31

Figure 4.4. Variation in the relative velocity angle according to the mesh study. ... 31

(8)

vii

Figure 4.6. Path of stress perpendicular to the neck (SSF). ... 32

Figure 4.7. Variation in the percentage of error for the parameter H. ... 33

Figure 4.8. Location of the measuring point for analysis. ... 34

Figure 4.9. Graphical representation of the parameters to be evaluated. ... 35

Figure 5.1. (a) FE model for C6-402 (b) Von Mises stress for the deformed shape (c) Total displacement for the deformed shape. ... 37

Figure 5.2. Measuring of the angle of relative velocity. ... 39

Figure 5.3. Variation of the parameter H. ... 40

Figure 5.4. Variation of the stress along the neck for different values of R2. ... 40

Figure 5.5. Variation of the stress perpendicular to the neck, R2 variable. ... 41

Figure 5.6. Variation in the stress perpendicular to the neck for a different R1 values. 41 Figure 5.7. Variation in the stress perpendicular to the neck for a constant R1 and R2 values. ... 42

Figure 6.1. Histogram for the parameter H. ... 44

Figure 6.2. Estimated marginal means for H in terms of R1 and θ. ... 48

Figure 6.3. Estimated marginal means for H in terms of R1 and α. ... 49

Figure 6.4. Estimated marginal means for H in terms of R1 and R2. ... 49

Figure 7.1. Path of stress perpendicular to the neck for a constant α. ... 52

Figure 8.1. Geometry of a notched test piece. ... 54

(9)

1

1.

INTRODUCTION

1.1.

Background

Classical theory of plasticity has tried to explain the behaviour of metals under different stress combinations through various yield criterions, being Tresca’s and Von Mises’s, without a doubt, the most popular ones so far. Other researchers have tried refining those ideas and produce solutions for even more general cases. Such is the case of Hill (1950), who presented a potential equation that parametrises Von Mises yield criterion, allowing to account for phenomena such as anisotropy.

The interest in determining the yield surface of a metal lays in its bare definition. This locus describes the boundary in which non – elastic behaviour starts to take place in the stress space, in other words, when yielding starts. The concept is fundamental in almost any area of theoretical or applied science that deals with the behaviour, and particularly the strength, of a metal.

In practice, researchers such as Taylor and Quinney (1931) or Schmidt (1932) succeeded at experimentally determining the yield surface of a metal by applying an internal pressure in combination with torsion to a cylindrical – shaped specimen. Whereas others like Lode (1926), Davis (1943) and Fraenkel (1948) managed to do so by combining axial loading with torsion, also using a cylindrical – shaped specimen.

In both scenarios, specialised test equipment was used so the different stress combinations could be achieved. In addition to the latter, manufacturing cylindrical specimens requires extra work on cold forming and welding. The manipulation of the material will add a set of variables into the process that, most likely, will alter the final results. Hence, it is relatively straightforward to conclude that traditional testing methods have a high level of complexity and a lot of precautions need to be taken in order to obtain good results.

This issue was first addressed by Hill (1953), who proposed a new method for determining the yield surface by using obliquely notched or grooved strips pulled in tension. According to Hill, the plastic potential and the yield criterion could be derived by only measuring the load and the relative displacement of two points on either side of what he called the distortion zone.

(10)

2

Becque et al. (2014). However, a collection of issues has been experienced and it has not been possible to obtain reliable results through experiments. Early studies reported difficulties in measuring the relative velocity angle and the load, due to a wide spectrum of factors that go from complications in making the measurements, to problems with the metals used because of rounded stress – strain curves that made it impossible to determine the load that initiated the necking or, when succeeding in both of the latter, obtaining results with a high level of scatter.

Yet, on each one of these occasions, significant improvement has been achieved and interesting findings were made. Most recently, technological advances have allowed to take precise readings of the displacements and theoretical developments now enables to determine the yield locus without the necessity of measuring the applied load (Becque

et al., 2014).

1.2.

Justification and Value of Research

When a slight increase in stress above the elastic limit results in a permanent deformation, it is said that the material has yield. In other words, the yield stress marks the onset of plastic deformation. This property is completely material dependent as it relates directly to its molecular mobility.

In ductile materials, if yielding is never reached, failure or even damage is not likely to take place ever, since all the deformations will remain in the elastic range and can be reverse with no consequences. Therefore, when designing a structure, it is of vital interest to understand this phenomenon in order to ensure safety and make the most efficient use of the materials at the same time.

Since structures are subjected to a variety of forces in terms of direction and magnitude, engineers must be able to estimate when will yielding occur in general multiaxial stress states. In the multidimensional space, yielding is not defined by a single point but by a surface, i.e. the yield surface. Several yield criterions have been developed to predict the shape of this surface by analysing experimental evidence as to what is the exact stress state that causes yielding.

(11)

3

Nowadays, the use of relatively new mathematical tools, such as the finite element method (hereafter referred as FE) has opened the possibility for researchers to explore a wide range of scenarios in a very efficient manner. Materials can be modelled in detail and experiments can be recreated in a controlled environment. The use of these aids also helps minimizing the use of resources, as solutions can be obtained, with a high level of accuracy, from FE modelling.

Hence, a FE parametric study of the geometric variables that are believed to have an influence on the experimental determination of the yield surface could lead the efforts on the topic into the right direction. An understanding on how the shape of the specimens can influence the data obtained could significantly help to finally achieve Hill’s ideas and eventually, standardise a simple way of performing this test.

1.3.

Specific Problem

In spite of the evident development that the initial ideas proposed by Hill have experienced, a simple and reliable testing method to predict the yield surface of a metal using notched strips pulled under uniaxial tension, has not been found yet.

Overall, high scatter in the results have made it impossible to obtain reliable conclusions. It is believed that part if this variability is due to the geometry of the metal specimen itself.

1.4.

Aims and Objectives

 Determine how and until which degree does the geometry affects the results of a uniaxial tensile test on a notched specimen when determining a metal’s yield surface.

o Use FE analysis software to model the test procedure of oblique notched strips.

o Validate the FE model by replicating results obtained in previous experiments.

o Define which geometric variables may have an impact on the desired results and design a set of cases that best represents them and their possible interaction.

o Perform a statistical parametric study on the results obtained.

(12)

4

 Set the basis for a standard test that would facilitate the use of notched strips specimens in determining the yield surface of a metal.

o Decide on which geometric variables must be taken into account when conducting such test.

o Establish a range for those geometric variables to ensure the most precise and accurate results.

1.5.

Delimitation of the problem

1.5.1.

Research scope

This study will focus on the next topics:

 The foundations of this research are based on Hill (1953) ideas for determining the yield surface and plastic potential of a metal while using only notched specimens.

 The influence of the neck width, the neck and opening angles and the radius of the root in the notch will be carefully assessed.

 The geometric variables under investigation are believed to be the most relevant ones and a total of 144 different interactions will be analysed.

 The project is largely based on the behaviour of stainless steel since it brings the opportunity to use recent experimental data to validate the results.

 FE software will be used for the evaluation of all the pre – defined cases.

 Statistical tools are going to be employed to validate the results and support analytical conclusions that can be derived from comparing theory with FE results.

1.5.2.

Limitations

The next statements must the taken into consideration when going through this research:

 Only obliquely notched specimens will be investigated, in spite of knowing that equivalent results can be obtained by using grooved strips.

 No laboratory test will be carried out to determine the stress – strain curve of the materials, numerical models will be used for this purpose.

 The FE modelling fails to represent if fracture will occur during the loading of the strip, however, this research in only interested in the behaviour of the specimen at the moment in which yielding initiates.

(13)

5

 No laboratory experiments will be carried out to test the findings of the analytical and parametric studies.

1.6.

Methodology

This investigation will be conducted in a linear manner in which each step needs to be completed in order to continue with the next one, as it can be seen from the diagram in Figure 1.1. The starting point will be a literature review to gain a better comprehension on the subject and recognise which investigators have worked on the topic before, how did their projects concluded and identify where could improvements be made.

Once the academic grounds were set, FE modelling will be carried out using the software Abaqus/CAE 6.14-1 (Dassault Systèmes, 2014). The modelling of the material will be done by deducing its plastic and anisotropic behaviour from numerical models and previous experiments. In terms of the geometry, a standard shape successfully used in past experiments will be replicated.

To calibrate the model, a mesh study will be performed for two different types of alloys of stainless steel. Several considerations will be made when choosing the correct mesh size and finally the results are going to be compared with those obtained by (Becque et al., 2014) in order to achieve validation.

Figure 1.1. Diagram of the methodology followed on this research.

With the intention of standardise the procedure as much as possible, another study will be made to determine if the location of the measuring points (referenced as MP from now on) above and below the neck have any influence on the measuring of the relative velocity angle itself.

Literature review FE modelling FE mesh study MP's study

Design of the cases

Abaqus v.6.14-1

scripting FE processing

Collection of results

Statistical analysis

Discussion of

(14)

6

After having a model that successfully replicates experimental data, the cases to be analysed will be design, considering all the geometric variables that are believed to have an impact on the measurement of ψ and its possible combinations. Since a large number of cases is expected from this step, before continuing with the FE processing, an Abaqus v.6.14-1 script will be develop. This computational aid will make possible to perform a large number of simulations making the most efficient use of the resources available. The aim will be to construct a script that allows the user to easily modify the geometry of each model and automatized the extracting of the results.

When focusing on the collection of the results, four outcomes are necessary: the displacement of the MP above and below the neck and the path of stresses along and perpendicular to the neck. With the displacements of the measuring points, the angle of relative velocity can be easily calculated. Successfully obtaining the value of ψ will make possible to determine the parameters of Hill’s quadratic yield equation and the anisotropic yield ratios.

As a large number of results is expected, a statistical analysis will be executed to finally determine which of the geometric parameters have an influence on the variability of the results and if there is any kind of interaction between them. This is planned to be done by using descriptive statistics to compare the results with experimental data and, on a later stage, by conducting several analyses of variances of one and two factors using the geometry as an independent variable (ANOVA as it is commonly referred).

The findings will then be discussed considering both theory and statistics. At this stage, the researcher hopes to determine which geometric variables are the most important ones in the aims of obtaining precise and accurate results, as well as how or in what degree they affect those results.

Finally, for the conclusion, a range of values in which the test will perform as best as possible is expected to be determined. The basis for an standardise test are expected to be established at this point of the research.

(15)

7

2.

LITERATURE REVIEW

The experimental determination of the yield surface in a multidimensional stress space has been the focus of researchers for almost a century now, or at least accordingly to well registered investigations. As it was stated before in this work, some of them succeeded at this task, but the proposed methods were too complex or costly, narrowing the determination of a metals yield characteristics to theoretical criterions that, in some cases, were developed based on one – dimensional observations.

Both Bijlaard (1940) and Hill (1953) suggested a method in which the yield surface was determined by applying uniaxial tension to a metal strip that had a weakened zone, forcing yielding to occur around this area. Both theories will be explained with greater detail in Chapter 3.

Despite their efforts to find a simpler test to determine the yield locus of a metal, it was not until Hundy and Green (1954) that this innovative idea was put to practice. With the support of the British Iron and Steel Research Association in Sheffield, the researchers used notched strips of copper, zinc, stainless steel and mild steel to try and determining the yield characteristics of each one of these materials.

For the copper, zinc and stainless steel, a well – defined, straight and narrow neck was developed, allowing the investigators to determine Lode’s parameters and discovering that the metals had very little anisotropy. Unfortunately, no yield criteria could be established as the load – displacement curves were so rounded that it was impossible to determine the load at which the neck started to yield. Hundy and Green suggested that this might have happened because of a fluctuant yield point on the metals and that a possible solution could be achieve by pre – straining the specimens, but they also warned that this procedure may lead to the incorporation of anisotropy, at least at a small level.

In the case of mild steel, the results were even less promising. No neck was ever developed, and fracture occurred by a crack spreading in a right angle from the tip of the notches, until finally joining the roots in an S – shaped path. The researchers attributed this behaviour to a fast work – hardening of the material in which the yielding of the neck is very diffuse instead of being sharp and localized.

(16)

8

On a second opportunity, Lianis and Ford (1957) tried to follow Hill’s concepts but this time using pure aluminium specimens. The investigators succeeded in determining a yield locus and when the results were plotted against the Von Mises criterion they compared in a favourable manner, being scatter on both sides of the proposed yield surface and presenting a maximum variation of less than 3%.

The aluminium used for this research was rolled at the temperature of liquid are in order to have a material as isotropic as possible. Therefore, as an auxiliary test, several aluminium specimens were cut in different angles with respect to the rolling direction and by calculating Lode’s parameters, it was then demonstrated that the rolling process of the aluminium indeed had not included significant anisotropy to the material. Hence, the use of Hill’s necking theory was validated through the experimental data.

Ellington (1958) focused his investigation of the plastic potential of ductile metals on annealed brass and copper. Following Hundy and Green (1954) conclusions, Ellington used grooved specimens, arguing that notched specimens were not suitable for materials with high rates of work hardening since they presented higher chances of developing a diffuse neck between the notches.

In spite of the latter, the yield locus was also not determined. Ellington stated that the accuracy of the method relied on the accuracy of measuring the load that started the neck’s yielding, task rather difficult for materials that do not have a defined yield point.

It was possible though, to prove that the brass had a strain – rate proportional to the deviatoric stress, meaning that it had a Von Mises potential in the sense that was fairly isotropic (μ = ν). On the other hand, results for copper showed a large anisotropy level.

Years later, Baraya and Parker (1963) conducted the same test on aluminium, using notched strip specimens. Unlike previous investigators, the material was found to differ significantly from the Von Mises potential, i.e. μ ≠ ν.

As an attempt to overcome this result, efforts were made to obtain the yield surface by means of the same procedure but, in this case, using pre – strained specimens, as Hundy and Green (1954) had suggested before. Again, the results were unsuccessful, and it was not possible for the investigators to reach any conclusion, as experimental data differ considerably from anything expected.

(17)

9

in the roots – are just some of them. They concluded that the proposed experiment was extremely sensible to the rolling direction of the material, and that the method was only to be used as an ancillary procedure to determine Lode’s parameters and establish if a metal was isotropic or not.

The inability to succeed on this task led researchers to drop the efforts on experimentally determining the yield surface of a metal for nearly fifty years. It was not until recently that Becque et al. (2014) returned to the topic by conducting several laboratory experiments based on Hill’s original ideas using two types of stainless steel alloys; one austenitic grade and one ferritic grade.

On this last research, two major advances were incorporated; (1) the developing of a parametrised equation based on Hill’s quadratic yield criterion that is general enough to account for any anisotropy on the material and does not require the measuring of the necking load, and (2) the implementation of digital image correlation techniques to refine the measurement of the displacements above and below the neck.

Likewise, most of the test apparatus and machinery has developed significantly in the past years, allowing the specimens to be cut using laser technology. Thus, reducing the chances of having undesired residual stresses in the root of the notches.

Additionally, three different geometries were tested. Results allowed to satisfactorily determine the yield surface of both types of stainless steel, although with a high scatter in the measurement of the angle of relative velocity and therefore, on the parameters of Hill’s quadratic yield criterion. Also, some of the specimens were reported to have fractured before yielding by the S – shape type of failure.

It was concluded that the large variability in the angle of relative velocity appears to be inherent of the test method, rather than the precision of the measuring technique, and that it may be due to only very localized areas of the material actually yielding, making local defects and spatial variations in the strength significantly important. To account for this variability, a large number of tests should be performed.

(18)

10

3.

THEORETHICAL BASIS

3.1.

Strength of Materials

The strength of a material is most likely one of the most fundamental unknowns’ engineers have had over history, how much can a material withstand without breaking or deforming so much that it cannot be used anymore? This mechanical property, inherent to the material itself, is the basis for many engineering applications.

Traditionally, the strength of almost any ductile material has been determined by experimental tests in which a standardised piece is subjected either to tension or compression in one direction. The results are then plotted in terms of a stress – strain curve, similar to the one shown in the subsequent figure.

Key:

1 : proportional limit

2 : elastic limit

3 : yield stress

4 : ultimate stress

5 : fracture stress

Figure 3.1. Idealised engineering stress – strain curve for a ductile material.

When a material is being loaded, initially, the stress is proportional to the strain. The material is said to be linear – elastic until it reaches the proportional limit (σpl). After

reaching this point, it will continue to deform to the elastic limit; in many metals, the proportional and elastic limit are so close to each other that are often taken as the same. If the load is removed at this point, the material will return to its original shape.

This proportionality between stress and strain during the elastic behaviour was first discovered by Robert Hook in 1676 and is referred as Hook’s law. It is mathematically expressed as:

= Eq. 3.1

(19)

11

On the other hand, if the stress is increased beyond the elastic limit, a phenomenon in which deformations can no longer be reversed will take place. This stage is called yielding and the stress at which it occurs is the yield stress (σy). When yielding takes

place, the material will continue to elongate without any increase in the stress and deformations will now be referred as plastic deformations.

After yielding, strain hardening is likely to occur. In this stage, the material’s strength will rise up to a maximum point or ultimate stress (σu) and then drop until the specimen

breaks at the fracture stress (σf). This final behaviour is usually referred as necking, as a

result of the material experiencing a large reduction of its cross section, due to excessive deformations, until it breaks.

All of the above is true for ductile materials, this means that, because of its capability to absorb energy, large strains can be withstood before fracture occurs when being overloaded. On the contrary, brittle materials have little or no yielding at all before fracture takes place. Most materials can have either a ductile or brittle behaviour depending on its chemical composition or the temperature at which it is being tested.

Also, the explanation given above is specific for a one – dimensional stress state . When applying stress in two or three – dimensions, a large number of stress states could induce plastic deformations; hence, yielding is not defined anymore by a single point but by a surface. A graphical explanation of this concept is shown in the figure below.

(20)

12

3.2.

Yield Criterions

Since the execution of two or three – dimensional tests is considerably more complex and expensive than one – dimensional tests, auxiliary theories, often based on one – dimensional experiments, were developed to estimate the yield surface in the multidimensional stress space. These theories are what we call today yielding criterions.

3.2.1.

Maximum Shearing Stress

Broadly known as Tresca’s yield criterion, it is based on experimental observations that suggested that plastic deformations only occur when the shearing stress reaches its maximum value.

This maximum shear stress (τmax) takes place at a 45° angle to the direction of the

maximum (σ1) and minimum (σ3) principal stresses and has a value equal to:

= −

2 Eq. 3.2

According to Mohr’s circle, the maximum possible value for the shear stress is half of the yield stress, hence, Tresca concludes that for the case in which σ1

> σ

2

> σ

3, no yielding

will develop if

− < Eq. 3.3

It has been proven that this very simple equation is a good approximation and therefore, it is commonly used in practice.

3.2.2.

Maximum Distortion Energy

Von Mises proposed another theory which is stated that plastic behaviour will start when the maximum distortion energy is the same as the one that would have been obtained at yielding in a uniaxial tensile test. In terms of the principal stresses:

( − ) + ( − ) + ( − ) = 2 Eq. 3.4

In a two – dimensional space, this equation represents an ellipse, as it can be seen from Figure 3.3. And would take the form of:

( − ) + + = 2 Eq. 3.5

(21)

13

Figure 3.3. Tresca and Von Mises criterion in the two - dimensional stress space.

The figure above illustrates the two criterions explained in a two – dimensional stress space. For any stress combination that lies in the grey area, no yielding will be developed, and all deformations will remain within the elastic range.

3.2.3.

Hill’s Quadratic Anisotropic Yield Criterion

Both Tresca and Von Mises criterions for yielding are focused on the behaviour of isotropic materials. However, this is a simplification since every material has a certain level of anisotropy. In the case of metal, as an example, forming processes such as rolling, drawing or extrusion impregnate the material with some anisotropy that is difficult to eliminate, though, it can be significantly reduced by heat treatments.

Hence, Hill (1950) proposed a generalised yield criterion based on Von Mises’ idea of the plastic potential. In this case, no single shear stress can occur linearly. Making the assumption that the reference axes coincide with the principal orthogonal axes of anisotropy, the resulting equation can be expressed as follows:

= ( − ) + ( − ) + ( − ) + 2 + 2 + 2 Eq. 3.6

From now on, F, G, H, L, M and N will be referred as Hill’s parameters, while

σ

0 represents

the reference stress, usually taken as the yield stress parallel to the rolling direction. Hill’s parameters describe the current anisotropic condition for a material.

In the case of plane stress, Eq. 3.6 can be simplified to:

= + + ( − ) + 2 Eq. 3.7

(22)

14

= 0.5 + 0.5 + 0.5( − ) + 3 Eq. 3.8

Meaning that for an isotropic material, F = G = H = 0.5 and N = 1.5.

3.3.

Plastic State and Necking on Ductile Isotropic Metals

As explained before, experimentally determining the yield surface of a metal has been a relatively complex and expensive task. Therefore, several researchers have tried to develop new methods to achieve this goal.

In his paper, called Theory of Local Plastic Deformations, Bijlaard (1940)proposed the hypothesis that minor plastic deformations occur on a localised weakened line when applying a certain load to the specimen. This weakened area was the result of cutting an oblique groove across the metal strip.

While studying the modes of necking in plastic – rigid sheets deformed in its plane using notched tension strips with wedge – shaped or circular roots, Hill (1952) came with the idea of using the geometry of unsymmetrical oblique notches to determine the yield characteristics of the metal.

By pulling the material in tension and forcing it to yield along a narrow path that connects the notches, commonly referred as the neck (Hill, 1953), a state of combined shear and tension is induced and the plastic characteristics of the material could be determined. A representation of both Bijlaard and Hill’s proposed geometries is showed in Figure 3.4.

(23)

15

Both investigators developed the theory that supports this experiment in a successful manner, either by considering that the weakened zone was achieved by the means of grooving the material or cutting asymmetric notches. The two made three basic assumptions in developing this theory: (1) the principal direction of stress and plastic strain rate coincide, i.e. the material was considered to be isotropic, (2) the behaviour is that of a perfectly rigid – plastic metal and (3) after the developing of the neck, each half slides relative to the other as a rigid body.

Two very important variables are the neck angle (θ) and the angle of relative velocity (ψ). The first one is measured between the neck and the longitudinal axis of the specimen. Hill (1952) determined that it can be varied between 55° and 90°, being 55° the correspondent neck angle that a specimen without notches would develop under a pure tension test. On the other hand, ψ represents the angle between the neck and the vector of relative velocity (v), which is the result of each side of the neck sliding relative to each other once yielding starts.

Having all of the above in mind and taking P as the load that causes the neck to yield, t as the material’s thickness and

L as the length parallel to the neck, the ratio of principal

stresses in the neck can be expressed as:

=

sin( − ) + =sin( − ) − Eq. 3.9

Similarly, the ratios between the principal strain rates are given by:

̇

1 + =

̇

−1 + =

̇

−2 Eq. 3.10

Considering a state of plane stress, Lode’s parameters can be calculated from Eq. 3.9 and Eq. 3.10 as follows:

= − +

− = −

3 − sin ( − )

+ sin ( − ) Eq. 3.11

=2 ̇ − ̇ − ̇

̇ − ̇ =

3(1 − )

1 + 3 Eq. 3.12

Hill (1950) demonstrated by simple geometry that

= −√3 Eq. 3.13

(24)

16

Where β is an angle determined by μ in the stress space, it represents the angle between the stress vector and the direction of pure shear (σ2

= -σ

1) in the deviatoric plane. At the

same time,

γ is the angle defined by

ν between the plastic strain rate vector and the

direction in which μ = 0. The next figure represents these concepts in a graphical manner.

Figure 3.5. Yield surface for an isotropic metal on the deviatoric plane.

As presented in Figure 3.5, when an isotropic metal follows the Von Mises criterion, it has been demonstrated that μ = ν, meaning that the deviatoric stress vector (represented in colour red) and the plastic strain vector (displayed in colour blue) are aligned, this is also commonly referred as having a Von Mises potential. Thanks to the associativity flow rule, normal and tangential vectors to the yield surface can also be known (marked in colour green).

In this case, the length of the stress vector in the deviatoric plane can be expressed as:

= 2

3( − + )

. Eq. 3.15

Substituting Eq. 3.9 into Eq. 3.15, yields:

= 2

3 ( ( − ) + 3 )

. Eq. 3.16

(25)

17

pure shear, taking advantage of the symmetry in the deviatoric plane about the σ1, σ2 and

σ

3 axes.

3.4.

General Anisotropic Case of the Plastic State

The assumption that any metal will behave as an isotropic material is rather strict and limiting, narrowing the chances of using the theory explained above to a relatively small number of scenarios. To overcome this restriction, Becque et al. (2014) developed the idea of generalising the yield surface by a parametric equation and to determine these parameters by lining up the normal vector to the yield surface with the plastic flow vectors, determined from experiments through Eq. 3.10.

In this particular case, Hill’s quadratic yield criterion for plane stress (Eq. 3.7) was chosen to estimate the yield locus. However, any other parametrised expression could be used for this purpose.

On the onset of necking, two principal directions of strain can be identified, one perpendicular to the relative velocity vector and another one along the neck (detailed in colour red in Figure 3.6). Along these two axes, the normal straining is equal to zero. Mohr’s circle of plastic strain – see Figure 3.7 (a) – can be constructed to derive the principal directions of the plastic strain rate, in spite of not having a plane strain rate condition.

Figure 3.6. Principal directions from strain rate.

Since the radius of Mohr’s circle in Figure 3.7 (a) can be defined as:

=

2 Eq. 3.17

(26)

18 ̇ =

/ Eq. 3.18

Hence, the plastic strain rates in the x – y reference system are:

̇ =

2 [ − sin(2 − )] Eq. 3.19

̇ =

2 [ + sin(2 − )] Eq. 3.20

̇ =

2 cos(2 − ) Eq. 3.21

Being ψ the angle of relative velocity and θ the neck’s angles.

(a) (b)

Figure 3.7. Mohr's circles for (a) plastic strain rate and (b) stresses in the neck.

(Becque et al., 2014)

Making use of the associativity of the flow rule, it is possible to obtain:

̇ = = [2 − + 2 ] Eq. 3.22

̇ = = [2 − 2 − ] Eq. 3.23

̇ = = [2 ] Eq. 3.24

Being f the Hill’s quadratic yield surface, λ a proportionality factor and F, G, H and N, Hill’s parameters.

(27)

19

′ = Eq. 3.25

′ = Eq. 3.26

Using Mohr’s circle for stresses along the neck, detailed in Figure 3.7 (b), it is possible to determine the stresses in the x – y coordinate system so it is possible to make use of the expressions previously derived in Eq. 3.22, Eq. 3.23 and Eq. 3.24, resulting in:

= − [ − cos(2 + )] Eq. 3.27

= − [ + cos(2 + )] Eq. 3.28

= sin (2 + ) Eq. 3.29

Being

α the angle the angle between the two directions in which the rate of normal

straining is zero (Figure 3.6). In the case of an isotropic material αi

= π/2 – ψ.

When deriving two ratios from Eq. 3.22, Eq. 3.23 or Eq. 3.24 and replacing with Eq. 3.27, Eq. 3.28 and Eq. 3.29, the load that causes necking and the width of the neck can be overlooked, as:

̇ + ̇

̇ − ̇ =sin (2 − )=

( + ) cos( + ) + ( − ) cos (2 + )

( − ) cos( + ) + ( + + 4 ) cos (2 + ) Eq. 3.30

̇ + ̇

̇ =

2

cos (2 − )= −

( + ) cos( + ) + ( − ) cos (2 + )

(2 + ) Eq. 3.31

Using the two latter equations, α can be removed and the results yields to:

+ +

(1 + 2 ) − 2 − 2

= ( − 1) − ( + 1) + 2 2

( − 1)(1 + 2 ) + ( + 1) 2 + 8 Eq. 3.32

Where;

=

sin (2 − ) Eq. 3.33

= 2

(28)

20

3.5.

Experimental application

In the case of a uniaxial tensile coupon test in an anisotropic material, the stresses in the

x – y directions are given by:

= ∙ Eq. 3.35

= ∙ Eq. 3.36

= ∙ Eq. 3.37

Where s is the applied tensile stress and α0 the angle with the rolling direction.

Substituting Eq. 3.35, Eq. 3.36 and Eq. 3.37 into Hill’s quadratic yield criterion (Eq. 3.7) yields:

= + + 2 +1

2 2 Eq. 3.38

It has been demonstrated that to generate four valid equations that allow for the parameters F, G, H and N to be determined, at least three uniaxial tensile tests in different directions with the rolling angle and one test involving shear and tension are required (Becque et al., 2014). In this case, the notched strips pulled in tension are proposed as the experiment combining tension and shear stresses.

When cutting simple tensile coupons at angles of 0, 90° and 45° with respect of the rolling direction, and remembering that

σ

0 is to be taken as the yield stress in the rolling

direction, Eq. 3.38 can be simplified, respectively, as:

+ = 1 Eq. 3.39

+ = Eq. 3.40

+ + 2 = 4 Eq. 3.41

Using Eq. 3.39, Eq. 3.40 and Eq. 3.41 in Eq. 3.32, it is possible to obtain a quadratic equation in terms of H, as follows:

(29)

21 Being;

= − − + ∙ Eq. 3.43

= 2 − 2 − 1 + ∙ 2 Eq. 3.44

= + + ( + 2 2 − ) Eq. 3.45

= 2 + 1 − 2 + (8 − 2 − 2 − 1) Eq. 3.46

= + + ∙ Eq. 3.47

= 1 + 2 − 2 + Eq. 3.48

= ( − 1) − ( + 1) Eq. 3.49

= 2 − 2 − 1 + 1 + 2 + 2 Eq. 3.50

(30)

22

4.

FINITE ELEMENT MODELLING

As it was stated before in the methodology of the project, in order to determine which geometric variables are the ones who have the largest impact in the determining of a metal’s yield surface by using notched strips, 144 FE models where constructed and analysed using the software Abaqus/CAE 6.14-1 (Dassault Systèmes, 2014).

Once the main variables of the FE model were defined, a mesh study was carried out to assess in which degree the size of the FE affects the measuring of the relative velocity angle. A validation of the chosen mesh was later performed by comparing the results obtained with those found by Becque et al. (2014), when laboratory test were conducted. These steps were completed for both an austenitic and a ferritic grade of stainless steel, details on the materials will be provided later in 4.3. Material Modelling.

After calibrating the model, a second sensibility study was completed. In this opportunity, the aim was to evaluate the influence that the location of measuring points has over the reading of the displacements.

The next step in the process was to define which geometric parameters were going to be investigated and in which degree, therefore concluding that 144 different simulations needed to be modelled and analysed.

In the next pages, each one of these stages will be explained in detail and the most important results of the sensibility analysis are going to be presented and discussed.

4.1.

Description of the Geometry Used in the Calibration of the Model

The geometry used to calibrate the FE model was a scaled down version of the one first used by Hundy and Green (1954) and Lianis and Ford (1957) and in recent years by Becque et al. (2014). This shape has proven to give good results in past researches, it was decided to be scaled down to optimise processing time in the FE software and also, to eventually make a more efficient use of resources in the case of conducting the experiment, as less material and heavy machinery would be required. A drawing of this geometry is presented in Figure 4.1.

(31)

23

Figure 4.1. Geometry used in the calibration of the model (thickness equal to 2 mm).

The notched strip consists of a 280 mm long and 45 mm wide metal piece with a thickness of 2 mm. Each pair of notches is located 20 mm away from the centre of the strip, and both; the opening angle (α) and the neck angle (θ) are set to 60°. Notice that this configuration forces the neck to be perfectly align with the notches. Another characteristic worth mentioning is that the neck width is one third of the total width of the specimen.

To optimise processing time, only half of the specimen was constructed in the FE software, and mirror boundary conditions were applied to the symmetry axis line, denoted in Figure 4.1 as SY-A. Is worth remembering that, in reality, the imposed displacement on the specimen is applied by pulling both ends of the strip simultaneously.

It is also important to clarify that in the FE model, axis x, y and z correspond to numbers one, two and three respectively. In this case, the y axis is aligned with the length of the strip while the x axis corresponds to the width and the z axis to the thickness.

4.2.

Basics of the Finite Element Modelling

Even though the thickness of the strip is relatively small in comparison to the other two dimensions and a plane stress simplification has been made, suggesting that a shell element could be suitable for this problem, the three – dimensional deformable model was set to be an extruded solid. Solid elements are known to be more robust since they belong to a more general case of the FE method.

Also, the use of solid elements will allow for the detailed investigation on the variation of the thickness or other parameters related to it, such as its interaction with the radius of the notch’s root.

(32)

24

solution. This artificially increased stiffness is due to an inability of the shape functions to describe certain deformations modes. Locking can be avoided by controlling the size of the elements and assuring that the thickness of each FE is larger than the two other sides.

A single static – general step was created to define the sequence in which the imposed vertical displacement was going to be applied to the model. The controls for geometric nonlinearities were turned on for this step. This means that, in the case of large displacements, nonlinear geometric effects will be considered by the software as they may have an important effect on the model.

In terms of boundary conditions, with the aim of minimizing the processing time, symmetry conditions were applied in the model such as it was explained before in Figure 4.1. Also, in order to replicate the effects of gripped ends in a laboratory test, a cell was created on top of the metal strip and the displacements in the horizontal (U1) and out of plane (U3) directions were restricted, as well as all the rotations (UR1, UR2, UR3). Finally, an imposed displacement of 5 mm was applied in the vertical direction (U2) to represent a uniaxial tensile experiment.

Moving forward to the mesh controls, swept hexahedral elements were assigned to the model. Swept meshing is usually recommended for solids with complex shapes, like in the case of this research. The elements on the mesh were defined as quadratic twenty – node brick elements with reduced integration (this means 2x2x2 integration points), C3D20R according to Abaqus/CAE 6.14-1.

These elements have proven to behave well and to be really adequate for general purposes, rarely exhibiting hourglassing in spite of the reduced integration. The hourglass mode is another phenomenon related to the FE method as a consequence of not using enough integration points for the numerical integration and therefore, opening the door for elements to undergo any deformation whatsoever, resulting in exaggerated displacements but correct stress fields.

To finish, four essential outputs were expected from each simulation:

 The displacement in the x and y direction of a measuring point located on the upper side of the neck (hereafter referred as MP1).

(33)

25

 The path of stress along the neck in the time step were the relative velocity angle is maximum.

 The path of stress perpendicular to the centre of the neck, also in the time step in which the relative velocity angle is maximum.

In addition to these four outputs, the reaction force in the y direction was also registered for each one of the models. To standardise the retrieving of the information, the assembly of the part was created as an independent instance and a specific partition sketch was design to force the mesh to create nodes in the points of interest, i.e. in the exact location of the measuring points, along and perpendicular to the neck. These nodes were used to create sets and later on, to request each individual history outputs.

4.3.

Material Modelling

The material modelling was carefully done since it is the main focus of this research. Density, elastic, and plastic anisotropic properties were defined in the model for two different alloys of stainless steel.

These two different types of stainless steel were chosen because they were used in the most recent studies on this subject (Becque et al., 2014) and will serve as verification for the process followed on this research.

The first type of stainless steel is an austenitic alloy (SSA). It is characterised as a non – magnetic steel that owes its name to the fact that, when adding sufficient amounts of nickel, the crystal structure changes to that of austenite. Usually, is composed by 18% chromium and 8% nickel. This combination enhances its corrosion resistance, which makes it one of the most commonly used alloys of stainless steel. According to European steel designation system, the specific austenitic alloy used in this project is formally referred as type 1.4307 (British Standards Institution, 2014) and as type 304L according to ASTM International (2001).

(34)

26

As expected, both alloys have the same density, Young’s modulus and Poisson’s ratio. Differences start to be relevant when it comes to proof stress, roundness parameter and the yield stress under different angles relevant to the rolling direction of the metal. The properties used in this research as fundamental input data are summarised in Table 4.1.

Table 4.1. Description of the materials used in the models.

SSA SSF

European designation (1) 1.4307 1.4316

ASTM designation (2) 304L 430

Density (1) ρ 7900 kg/m3 7900 kg/m3

Young’s modulus (1) E 200 GPa 200 GPa

Poisson’s ratio (3) ν 0.3 0.3

Proof stress (4) σ0.2 245 MPa 280 MPa

Roundness parameter (4) n 7.0 7.5

Yield stress in the rolling direction (5) σ0 290 MPa 315 MPa

Yield stress perpendicular to the rolling

direction (5) σ90 290 MPa 345 MPa

Yield stress at an angle of 45° with the

rolling direction (5) σ45 292 MPa 340 MPa (1) (British Standards Institution, 2014)

(2) (ASTM International, 2001) (3) (Brinson and Brinson, 2015) (4) (Becque and Rasmussen, 2009) (5) (Becque et al., 2014)

The proof stress and the roundness parameter are of special importance because the material stress – strain curves, and therefore its plasticity, was modelled using the Ramberg – Osgood equation, detailed next;

= + 0.002

. Eq. 4.1

Where,

=

=

=

. =

(35)

27

Both stress – strain curves are shown in Figure 4.2. As it can be seen from it, the two materials have a fairly similar stress – strain curve, founding the biggest difference in the proof stress, just as it would be expected from Table 4.1 and Eq. 4.1.

Figure 4.2. Stress - strain curve for the SSA.

To mimic the anisotropic yielding behaviour of the material, Abaqus/CAE 6.14-1 potential function was selected. This function allows the user to define the stress ratios on each principal direction, conveniently for this research, this can be done accordingly to Hill’s yield criterion (Eq. 3.6). Each one of the anisotropic yield ratios is defined as:

= Eq. 4.2

= Eq. 4.3

= Eq. 4.4

= Eq. 4.5

= Eq. 4.6

= Eq. 4.7

Where,

=

0 100 200 300 400 500 600

0 0.05 0.1 0.15 0.2 0.25

St

ress

[MP

a]

Strain [%]

(36)

28

= ℎ ℎ

=

= √3

In the case of this project, and can be easily obtained from the yield stresses measured parallel and perpendicular to the rolling direction. However, in order to determine the rest of the stress ratios, another set of equations must be included. By using Hill’s criterion (Eq. 3.6), it is possible to establish a relationship between the F, G,

H, L, M and N parameters with the measured stress values and the yield stress ratios.

= 2

1

+ 1 − 1 =1

2 1

+ 1 − 1 Eq. 4.8

= 2

1

+ 1 − 1 =1

2 1

+ 1 − 1 Eq. 4.9

= 2

1

+ 1 − 1 =1

2 1

+ 1 − 1 Eq. 4.10

=3

2 =

3

2 Eq. 4.11

=3

2 =

3

2 Eq. 4.12

=3

2 =

3

2 Eq. 4.13

Since F, G, H and N are known from experimental data, from any of Eq. 4.8, Eq. 4.9or Eq. 4.10, it is possible to derivate an equation to obtain . In the case of using Eq. 4.8, would be calculated as:

= 1

2 − 1 + 1 Eq. 4.14

can be obtained from reformulating Eq. 4.13;

= 3

(37)

29

Because of the plane stress assumption, the parameters L and M were omitted from the calculations. On the other hand, the stress ratios and were simplified as the average of the yield ratios in the principal directions, as shown in Eq. 4.16 and Eq. 4.17.

=1

2( + ) Eq. 4.16

=1

2( + ) Eq. 4.17

The parameters F, G , H and N were then taken from Becque et al. (2014). These values and the correspondent yield ratios are shown below in Table 4.2 and Table 4.3.

Table 4.2. Hill's quadratic yield criterion parameters.

F G H N

SSA 0.521 0.521 0.479 1.452

SSF 0.540 0.378 0.460 1.266

Table 4.3. Yield stress ratios.

R11 R22 R33 R12 R13 R23

SSA 1.00 1.00 0.98 1.02 0.99 0.99

SSF 1.00 1.10 0.90 1.09 0.95 1.00

The material orientation in the model was set so the main or rolling direction is parallel with the y axis.

4.4.

Mesh Study and Calibration of the Model

Continuing on to the mesh examination, two studies were carried out using the geometry previously shown in Figure 4.1, one for each alloy under investigation. Like in the experimental data, the measuring points were located at 4 mm above and below the centre of the neck.

(38)

30

of elements was double since now the depth of the specimen had to be composed by two separate FE.

On each one of the models, the maximum load, the displacements of the MP’s (later used to calculate the angle of relative velocity), the path of stress along and perpendicular to the neck was recorded. The results obtained from this study are shown in the next tables and figures.

Table 4.4. Results of the mesh study done for the SSA.

Test Number of

elements

Max element size

[mm] Mesh/thickness

Max Load [kN]

ψ [°]

AMS-01 137 10.00 5.000 13.65 24.89

AMS-02 187 7.50 3.750 13.29 23.26

AMS-03 363 5.00 2.500 13.49 25.80

AMS-04 1237 2.50 1.250 13.15 25.62

AMS-05 1991 2.00 1.000 12.99 24.29

AMS-06 2033 1.50 0.750 12.96 23.91

AMS-07 4822 1.25 0.625 12.96 23.75

AMS-08 6630 1.00 0.500 12.93 23.64

AMS-09 9328 0.85 0.425 12.90 23.78

AMS-10 15891 0.75 0.375 12.89 23.78

Table 4.5. Summary of the mesh study for the SSF.

Test Number of

elements

Max element size

[mm] Mesh/thickness

Max Load [kN]

ψ [°]

FMS-01 110 10.00 5.000 14.07 28.55

FMS-02 166 7.50 3.750 13.87 26.95

FMS-03 319 5.00 2.500 13.88 28.49

FMS-04 1211 2.50 1.250 13.58 27.25

FMS-05 1826 2.00 1.000 13.73 27.22

FMS-06 3214 1.50 0.750 13.66 26.90

FMS-07 4896 1.25 0.625 13.65 26.88

FMS-08 6498 1.00 0.500 13.65 26.88

FMS-09 7950 0.85 0.425 13.64 26.88

FMS-10 15048 0.75 0.375 13.64 26.88

(39)

31

Nevertheless, despite having a much larger number of FE, the difference in the measuring of the maximum load or relative velocity angle is not significant, being 0.2% or less for both materials. This conclusion can also be drawn by looking at Figure 4.3 and Figure 4.4.

Figure 4.3. Variation in the maximum load with the number of FE’s.

Figure 4.4. Variation in the relative velocity angle according to the mesh study.

As it can be seen from the information above, both studies display a very similar variation, having a considerable jump when it comes to a mesh of elements of maximum 7.5 mm size. It can be noticed that the maximum load and relative velocity angle tend to stabilise when the number of elements is around 2000, this corresponds to the case in which the ratio between the maximum element size and the thickness of the strip is equal to 1.00 (AMS-08 and FMS-08 on the tables shown above). Between this case and the more refined ones, the difference in the relative velocity angle is less than 1°.

12.80 13.00 13.20 13.40 13.60 13.80 14.00 14.20

0 2000 4000 6000 8000 10000 12000 14000 16000

Max

lo

ad

[k

N]

Number of elements

SSA SSF

22.00 23.00 24.00 25.00 26.00 27.00 28.00 29.00

0 2000 4000 6000 8000 10000 12000 14000 16000

Angle of relative

ve

locity

]

Number of elements

(40)

32

On the other hand, when looking at the path of stress along and perpendicular to the neck (Figure 4.5 and Figure 4.6, respectively), there is a clear difference between the firsts, more coarse cases, and the more refined ones. This divergence is particularly obvious in the path of stress perpendicular to the neck, where the variation in shape and maximum stress in quite large. The stress path becomes more uniform and stable when the thickness of the specimen is divided in at least two different elements.

Figure 4.5. Path of stress along the neck (SSF).

Figure 4.6. Path of stress perpendicular to the neck (SSF).

Even though there is not a large alteration in the maximum load applied and the relative velocity angle when the number of elements is greater than 2000, there are visible and important differences when observing the performance of the specimen. With a higher

400 500 600 700 800 900 1000 1100 1200 1300 1400

0 2 4 6 8 10 12 14 16 18

Von M

ises

st

ress [MPa

]

Neck lenght [mm]

FMS-01 FMS-02 FMS-03 FMS-04 FMS-05

FMS-06 FMS-07 FMS-08 FMS-09 FMS-10

200 300 400 500 600 700 800

-15.00 -10.00 -5.00 0.00 5.00 10.00 15.00

Von M

ises

st

ress [MPa

]

Distance from the centre of the neck [mm]

FMS-01 FMS-02 FMS-03 FMS-04 FMS-05

(41)

33

level of refinement, the path of stresses along and perpendicular to the neck show a behaviour that resembles theory in a more accurate way.

4.5.

Validation of Results for the Mesh Study

To assure that the modelling process replicates real life experiments as closely as possible, the meshing study was also evaluated in terms of the parameter H. Each one of the measuring angles obtained in the last step was used to calculate this parameter and then compared with the experimental ones (see Table 4.2). The results, expressed in terms of the percentage of error in accordance to the experimental value, are shown in the figure below.

Figure 4.7. Variation in the percentage of error for the parameter H.

As it can be seen from this figure and in conjunction with the analysis made by only comparing the outcomes between different mesh sizes, the best results were found when the element size did not exceed 0.85 mm at any moment. The difference between this mesh and the more refined one is not of relevance, but the processing time is reduced considerably. Therefore, this was the maximum mesh size used throughout the rest of the investigation.

Another aspect worth mentioning is the considerably large error in the case of the SSF, which takes a constant value of 10% when the mesh is sufficiently refined. This error seems to be quite large, especially when compared to the SSA values, that never surpasses 4%. This may be due to the more anisotropic characteristics that the SSF has (see Table 4.3), however it should not consist of an obstacle for the FE software, which has the tools to replicate this characteristic behaviour.

0.0% 5.0% 10.0% 15.0% 20.0% 25.0%

0 2000 4000 6000 8000 10000 12000 14000 16000

Error

[%]

Number of elements

(42)

34

Yet, at this point of the research, it is not possible to determine if the error is due to the modelling process or due to the geometry itself. The detailed investigation that follows will most likely help to determine which is the case.

4.6.

Measuring Points Study

Another significant variable in the estimation of the relative velocity angle is where exactly to locate the measuring point, above and below the neck. Previous researches have shown that measuring points too close to the ends of the neck can display exaggerated results with large variability, most likely due to the stress concentrations around the root of the notches (Becque et al., 2014).

Taking into account the experimental observations, a study on the location of these points was carried out using the FE method once again. Four different locations where registered for the same geometry shown in Figure 4.1, in this case, only for the ferritic alloy. The pairs of measuring points were all placed in the centre line of the specimen, as shown in Figure 4.8.

Figure 4.8. Location of the measuring point for analysis.

The variance in the relative velocity angle is detailed in the table below, as well as its effect on the parameter H.

Table 4.6. Results from the study on the location of the measuring angle.

ID Distance from

neck’s centre [mm] ψ H Error [%]

MP-03 3 26.10 0.431 6.3%

MP-05 5 26.15 0.430 6.5%

MP-10 10 29.82 0.338 26.5%

(43)

35

From Table 4.6 it is evident that the variation between the 3 mm and 5 mm test is not of significance but it becomes quite large when it comes to the 10 mm and the 15 mm test. In spite of MP-03 having a smaller percentage of error than MP-05, when compared with the experimental results obtained for the parameter H, bearing in mind that the aim is to finally standardise the test, the rest of the study will be carried out locating the measuring points exactly at 5 mm above and below the centre of the neck.

4.7.

Definition of the Parameters Under Investigation

With the preliminary steps of the research completed, four different geometrical variables where chosen for the parametric study. These variables are represented in Figure 4.9 and are explained in the paragraphs below.

Figure 4.9. Graphical representation of the parameters to be evaluated.  Neck angle (θ): following the path lead by past researches, different neck angles

will be modelled in Abaqus/CAE 6.14-1 accordingly to the limits previously determined by Hill (1953). The angle

β shown in Figure 4.9 is the neck angle

measured with respect to the horizontal axis.

(44)

36

 Opening angle of the notch (α): Becque et al. (2014) explored the influence of this variable by manufacturing a specimen with a very close and sharp angle. Through the FE analysis, several combinations of α and θ will be evaluated.

 Ratio between the thickness of the specimen and the root radius of the notch

( 2 = / ): theory and experimental data suggests that a more rounded root

could relief stress concentrations in the edge of the notch, hence, lowering the chances of having a brittle or a straight S – shape fracture in the neck. However, a very large root can also lead to the development of a diffuse neck, making it harder to measure the displacements of the measuring points.

Table 4.7 summarises all the different variations of the parameters explained above. To evaluate if there is a particular relation amongst these variables, they were combined in different scenarios, leading to a total of 144 different models.

Table 4.7. Variables of the parametric study.

R1 θ [°] α [°] R2

1/4 60 20 0.0

1/3 70 40 1/4

1/2 80 60 1/2

- - 90 1.0

In the case of future reference, each model was identified as follows:

To optimise the modelling and analysis process, a Python 3.7 (2018) script was developed where all the variables cited above were easily defined and most of the process was done automatically.

Figure

Actualización...

Referencias

Actualización...

Related subjects : parametric study