determination of the paris curve for a twip steel for body-in-white applications

Determination of the Paris curve

for a TWIP steel

for Body-in-White applications

Matteo de’Giovanetti

Politecnico di Torino

A thesis submitted for the degree of

Bachelor of Science in Automotive Engineering

2012 07

mailto:[email protected]

http://www.something.net

http://www.polito.it

1. Reviewer: Ing.Paolo Matteis

Day of the defense: 07-24-2012

ii

Abstract

The purpose of this thesis is to outline the results obtained in fatigue

experiments performed on a TWIP steel, a material expected to be

soon employed in Body-in-Whites.

It is indeed compulsory to explain why such data are needed, and so

a discussion on fracture mechanics and fatigue behaviour is included.

An insight on the steels currently employed for Body-in-Whites works

as an introduction to a more detailed disquisition on TWIP steels.

Finally, results are presented, discussed and compared with literature,

where relevant.

Research is what I’m doing when

I don’t know what I’m doing.

Wernher von Braun

To my family

Contents

List of Figures v

1 Introduction 1

1.1 Steels in automotive industry . . . . . . . . . . . . . . . . . . . . 3

1.2 TWIP steels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Paris curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

1.4 Purpose of the study . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Steels for automotive application: an overview 9

2.1 Most common steels . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Treatments and coatings . . . . . . . . . . . . . . . . . . . . . . . 17

3 TWIP steels 21

3.1 Microstructure and strain hardening mechanism . . . . . . . . . . 21

3.2 Production processes . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.3 Fracture and fatigue . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.4 Employment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

4 Fracture mechanics 29

4.1 The energy balance approach . . . . . . . . . . . . . . . . . . . . 30

4.2 The stress intensity approach . . . . . . . . . . . . . . . . . . . . 35

4.3 Fracture and R-curve . . . . . . . . . . . . . . . . . . . . . . . . . 40

4.4 Fatigue and Paris curve . . . . . . . . . . . . . . . . . . . . . . . 42

iii

CONTENTS

5 Standard test methods 49

5.1 ASTM E-647: Standard test method for measurements of fatigue

crack grow rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

5.2 Recommended data reduction techniques . . . . . . . . . . . . . . 55

6 The experiment 57

6.1 Preparation of the equipment . . . . . . . . . . . . . . . . . . . . 57

6.2 Precracking and crack growth . . . . . . . . . . . . . . . . . . . . 58

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

7 Conclusions 65

A Experimental values 69

B Influence of material selection on chassis performance and man-

ufacturing 73

References 77

iv

List of Figures

1.1 BMW 5-series Body-in-White . . . . . . . . . . . . . . . . . . . . 3

1.2 Volvo XC-90 Body-in-White . . . . . . . . . . . . . . . . . . . . . 5

2.1 Volvo V60 Body-in-White . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Comparison in engineering stress-strain curves between three steels 15

2.3 TRIP Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.4 Steel properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.5 Galvannealing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.1 TWIP Microstructure . . . . . . . . . . . . . . . . . . . . . . . . . 23

4.1 Fracture diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4.2 Sample for fracture . . . . . . . . . . . . . . . . . . . . . . . . . . 33

4.3 Plastic zone in a specimen . . . . . . . . . . . . . . . . . . . . . . 39

4.4 Example of R-curve . . . . . . . . . . . . . . . . . . . . . . . . . . 42

4.5 Example of Paris curve . . . . . . . . . . . . . . . . . . . . . . . . 45

5.1 C(T) sample in 3D . . . . . . . . . . . . . . . . . . . . . . . . . . 54

5.2 C(T) sample in 2D . . . . . . . . . . . . . . . . . . . . . . . . . . 56

6.1 Paris curve for second TWIP sample . . . . . . . . . . . . . . . . 62

6.2 Partial Paris curve for third TWIP sample . . . . . . . . . . . . . 63

6.3 Combined Paris curve . . . . . . . . . . . . . . . . . . . . . . . . 64

7.1 Data Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

7.2 Paris Curve Comparison . . . . . . . . . . . . . . . . . . . . . . . 68

v

LIST OF FIGURES

vi

1

Introduction

When reading a press release for a new car from a major automotive manufac-

turer, there is a high possibility that the most used word is lightweight: reducing

the mass of vehicles, keeping unaltered general performance and ride comfort, or

even improving them, is arguably the biggest challenge in contemporary automo-

tive design.

This new pathway is being widely adopted in car industry, and it seems that

lightness will be the main goal for a long time: this happens because pollutions

regulations are becoming increasingly stringent (without a real reason in much

cases, but the motivations are not to be discussed here).

In last ten-fifteen years, both petrol and diesel engine have been thoroughly

refined, thanks especially to our much improved understanding in combustion

processes and consequent technological amelioration, i.e. direct injection, turbo-

chargers with virtually no lag, and engines efficiency has been stretched, maybe

even to its limits. As of today, drivetrain development alone is not capable to

make cars meet the emissions standards set by lawmakers: see, for example, the

massive reduction in NOx required for diesel engines by EURO6 with respect to

EURO5.

Car makers have even been intensely studying homologation cycles, in order to

obtain the highest efficiency out of their cars exactly in such conditions, and, at

least to some extent, this is the reason why such amazing improvement in toxic

emissions reduction took place in last years (see, for example, the criticism on

1

1. INTRODUCTION

NEDC1).

Nowadays, the best way to reduce pollution and match incoming regulations is

to decrease the mass of new cars, in order to fit them with smaller engines (the

much cited downsizing) without penalizing straight-road performance; it is obvi-

ous that mass reduction carries with it a positive effect on behaviour in cornering,

especially if these modifications mean a more refined design for sprung and un-

sprung masses.

A number of formulae has been proposed to set a correlation between mass re-

duction and fuel consumption: in general, a 10% diminution in weight is matched

to a 5% decrease in fuel consumption (1), with obvious advantages on emissions.

Several new materials have been introduced in car industry in the last period:

aluminium employment is spreading, new types of plastics are being utilized in

highly innovative ways, carbon fibre, despite its cost, is starting to be used also

in middle to high-range saloons, beside its consolidated application in supercars.

In this process, it is beyond question that safety must stand as the first issue to

take care of, and no other aspect should be regarded as prominent. Front and

side crash have to be considered both in part design and material choice, in order

to prevent fatal harm to occur and, not by happenstance, earn a good score in

safety tests such as the notorious EuroNCAP, which can be a tie-breaker for the

customer.

Other requirements in chassis design are to be found in vibrational comfort and

acoustics (see, for example, the importance of eigenfrequencies in chassis design)

and in the values of torsional and bending stiffness2.

Steels, which went hand in hand with automotive industry since its early

birth, were supposed to be at their best and expected to be technologically sur-

passed in a short time, especially by 5000 and 6000-series of aluminium alloys for

BIW applications (2), and generally overlooked by car manufacturer craving for

improvements in materials.

1NEDC: New European Driving Cycle, the official pattern to assess car fuel consumption

in the European Union.2See also Appendix B.

2

1.1 Steels in automotive industry

Paradoxically, some research showed that it is not true that lighter materials carry

with them, automatically, a better environmental impact: life cycle assessments

stated that non-steel materials, such as carbon composites and magnesium, have

a similar or even worse repercussion if we consider the entire story of production

(3).

In more recent years, advance in steel composition and processes, both probably

unexpected, shed a new light on them: in the ranks of modern BIW employment,

we find hot-formed, dual-phase, TRIP (TRansformation Induced Plasticity) and

TWIP (TWinning Induced Plasticity) steels.

Figure 1.1: BMW 5-series body (E60). Aluminium parts in blue, steel in grey.

c©BMW AG.

1.1 Steels in automotive industry

Steels are the main component of almost every vehicle produced: steels employ-

ment in a BIW can reach up to 100%, while only a few cars currently in production

benefit from a body made only of aluminium or carbon fibre. In general, high-

end cars tend to show a hybrid design in which Al is used for a bunch of relevant

parts.

It is neither easy nor completely correct to give a classification of steels used in

BIWs; indeed, it is without any doubt useful to try to subdivide them with re-

spect to yield strength (Y S) and ultimate tensile strength (UTS), with the latter

commonly used also in commercial names to identify the different alloys.

• Very high strength steels (also Higher strength or Ultra-high strength).

3

1. INTRODUCTION

– Dual Phase (DP) steels.

– TRIP steels.

– Complex Phase steels.

– Hot-rolled ferrite-bainite steels.

– Steels for hot stamping

• High strength steels

– High strength low alloy steels (HSLA).

– Bake hardening steels.

– Interstitial free (IF) steels.

– Solid solution steels.

• Deep Drawing steels.

Another important feature is the coating applied on the steel and the correspon-

dent process: for example, zinc coatings, applied by means of either electrogalva-

nization or galvanannealing, to prevent corrosion, can also affect the mechanical

properties of the starting alloy.

Moreover, innovative processes permit less stringent and more effective design,

allowing to have different properties in varying zones of the same piece: for ex-

ample, AUDI is now using on the new A6 a new type of form-hardening (partial

quench and temper) on B-pillars, which gives optimal protection in case of side

impact by shifting deformation in places where it can give no harm to the pas-

sengers (4).

In order to assess material suitability is also significant to know how the piece has

to be worked and formed, and understand the consequent response to afterward

applied external forces, for example in case of impact.

1.2 TWIP steels

TWIP steels are currently under great attention in automotive industry, as a

consequence of their combination of very high UTS (even greater than complex

4

1.2 TWIP steels

Figure 1.2: Example of steel employment in a luxury car (Volvo XC90, P2 plat-

form). c©Volvo Personvagnar AB.

phase) and ductility, with values close, under this category, to mild steels. High

manganese austenitic steels were first introduced by Sir Robert Hadfield in 1888

(5), but experienced no fame, at least in automotive field, for almost one cen-

tury1, with only a few remarkable publication during this period. In the early

90s, an increased interest in TWIP steels led to different patents from Japanese

steelmakers (Kobe Steel, Sumitomo and Nippon Steel) and Korean Posco Cor-

poration (6). Today, the strain hardening mechanism shown by these steels is

persuading carmakers to follow their development; this is aimed at the introduc-

tion of new process technologies in order to produce lighter parts.

Nevertheless, so far just a handful of car parts made of TWIP steel have been

commercialized, inasmuch as some properties would not exactly fit in the desired

application.

Although there is not an exact definition of TWIP steel, up-to-the-minute re-

search suggest the following rules for a FeMnC steel to exhibit in order to be

considered as such:

• Respect the Schumann Equation: wt%Mn = −20wt%C + 32 (7).

• Display a carbon content smaller than 1.2% (8).

1High-Mn content steels have been extensively made use of for safe-deposit boxes produc-

tion.

5

1. INTRODUCTION

• A lower limit for Mn of 15-17%, to avoid ductile-to-brittle transition.

• A Mn content lower than 35%, to avoid intergranular fracture (9).

1.3 Paris curve

One of the most important elements to assess the suitability of steels to the de-

signed role is their fatigue behaviour, i.e. the maximum force they can withstand

at a theoretical condition of infinite cycles and their reactions to cracks; all this

is focused on preventing dramatic failures and consequently fatal outcomes.

Regarding crack formation and growth, the most used model is the Paris equa-

tion (also known as Paris-Erdogans equation), which correlates crack growth rate

under fatigue to stress intensity factor, thus putting forward a relationship with

tensile stress; this equation allows, on the condition that some specific parame-

ters of the materials are known, to predict the number of cycles remaining before

failure (10).

The usual representation of Paris equation is the following:

da

dN= C∆Km

i

Actual meaning of the symbols and implications of this theory will be deeply

examined in the appropriate section.

1.4 Purpose of the study

The introductory paragraphs shed light on the aim of this experimental research:

provide a acceptable prediction of TWIP steels behaviour under determined fa-

tigue conditions, by determining the Paris curve.

Crucially, experiments will be carried out on specimens not conforming to the

minimum thickness prescribed by standard test methods; nevertheless, the result

will still be relevant, since the parts that could be theoretically employed in the

automotive field would have a thickness close to the specimen; it is, that being

the case, intelligible why a test run in conditions abiding by the standard would

be, on the flip side, futile or even non-conservatively misleading.

6

1.4 Purpose of the study

At first, the most used type of steels will be enumerated and explained in their

distinctive features; next, a deeper insight on TWIP steels, including both pro-

duction and employment informations will be presented.

A theoretical approach on fracture mechanics is essential to fully and correctly

interpret the experiment, and will be the subject of the subsequent section.

A brief recall of the test method will work as introduction to the actual experi-

ment report; a conclusion on obtained results and comparison with previous ones

will end the dissertation.

7

1. INTRODUCTION

8

2

Steels for automotive application:

an overview

2.1 Most common steels

As discussed before, a whole lot of different steels is currently used in automotive

industry; hereafter, a significant and almost comprehensive selection is presented,

and each type is briefly analyzed.

Figure 2.1: Middle-sized saloon (Volvo V60 Y20 platform), materials and yield

strength. c©Volvo Personvagnar AB.

9

2. STEELS FOR AUTOMOTIVE APPLICATION: AN OVERVIEW

2.1.1 Low carbon, Interstitial Free and mild steels

As suggested by their name, these are carbon steels with a generally accepted C

content lower than 0.15%. They are used where high ductility and formability are

needed, and notable concern for cost reducing plays a part in selection criteria;

UTS can reach values up to 350 MPa, with a typical reduction if the steel is cold

rolled.

Moreover, these types of steels can be heat treated (but not quenched and tem-

pered) to increase strength and ductility, reducing the influence of cold embrit-

tlement, thus boosting welding possibilities (11).

Given these fundamental premises, it is not unfathomable the reason why low

carbon steels have seen their contribution dropped from close to totality, this

happening still in the eighties for small family cars, to a figure of 20% or less in

cost-competitive vehicles, to almost not being used in middle-sized saloons. A

non-acceptable level of passive safety represents the main issue.

Deep drawing steels account for a branch of low carbon steels: they have similar

compositions but undergo different treatments in order to make deep drawing and

extra deep drawing easier, therefore allowing to obtain precise pieces of complex

shape. Obviously, none energy-absorbing or specific load-carrying part is made

from these steels.

Hereafter the core properties to correctly recognize deep drawing steels are de-

picted: C content lower than 0.1%, Mn presence up to 0.5%, Si contained up to

0.1%. They can be both hot and cold rolled and, in the former case, reach a UTS

of 400-450 MPa.

The so-called Interstitial Free (IF) steels are also used for deep drawing applica-

tions: their peculiarity is an almost negligible C content; the remotion of carbon

atoms is performed in oxygen steelmaking by means of two main techniques: top

blown or bottom blown. Both ensure C to be removed using gaseous carbon

monoxide or dioxide. Often, after this process, a vacuum degasser is employed to

reach the requested C percentage, which is usually close to 0.005.

Among the many characteristics, it is worthwhile to remember the high strain

hardening coefficient which, along with a low YS/UTS ratio and a high strain

ratio, insures excellent deep-drawing usability. Depending on different grades,

10


the main applications shift from visible parts to more convoluted and significant

ones, but usually not in deformation structure, because of a UTS not higher than

500 MPa.

In order to increase mechanical properties, Niobium and Titanium can be added,

thus further improving drawability and enhancing coating adhesion (12), showing

somehow unexpectedly a reduction in planar isotropy (13).

These additions provide hardening as a consequence of the improved precipita-

tions of alloy carbides.

2.1.2 High Strength Low Alloy steels

High strength low alloy (from now on HSLA) steels show a low carbon content and

are enhanced thanks to the addition of interstitial elements in low-to-moderate

quantities, and obtained by both hot and cold rolling.

Beside the interstitial elements, Manganese can be added in higher quantities as

solid solution; hardening is usually performed by precipitation and consequent

grain size refinement, in a range of values between 5 and 10 µm. Because of the

obtained impact strength, weight can be reduced even in important structural

components such as longitudinal crossmembers and energy-absorbing parts, and

reinforcements are seldom needed; indeed, UTS can climb up to 750 MPa if the

HSLA steel is hot rolled.

A secondary phase can be obtained also by adding Niobium as micro-alloying

element, thus obtaining higher strength as a result of Nb(CN) carbonitrides

precipitation. Attention must be paid to Nb percentage, in order to achieve a

sufficient level of austenite conditioning to form fine enough ferrite grains (14).

Good fatigue strength makes them suitable also for different sub-assemblies, such

as suspensions and wheels; cost-effectiveness is guaranteed by remarkable cold

forming performance and low-temperature brittle fracture strength.

Another important advantage of HSLA steels is the absence of grain coarsening

and weld zone softening in hot-spot welding.

Major drawback is represented probably by the extended elastic-plastic transition

zone which sets some issues in practical application, especially if some specific

requirements from safety point of view are involved.

11


2.1.3 Ferritic-Bainitic steels

This type of steels provides some advantages, with respect to HSLAs, in strain

hardening exponent and total elongation: this makes FB steels specifically suit-

able where Stretch Flangeable (SF) or High Hole Expansion (HHE) properties

are required, in order to obtain an improved edge stretch capability.

FB steels enjoy a bainitic second phase hardening, and can also be upgraded

by grain refinement. For normal applications, FB steels are hot-rolled and cold

drawn.

Because of said properties, along with a very good weldability, FB steels are em-

ployed where good performance are required even when holes are drilled: a clear

example is represented by a suspension lower arm, which has to accommodate coil

spring and damper mountings without compromising the stiffness of the entire

sub-assembly.

2.1.4 Martensitic steels

Martensitic steels are not widely used in automotive industry; nevertheless, when

very high UTS (up to 1700 MPa) is prescribed, and brittle failures do not stand

as an issue, they can be employed. As can be easily agreed-upon, such a high

strength is detrimental from a ductility point of view, but recently introduced

post-quench tempering treatments allow to obtain quite an acceptable ductility

even at these UTS values, with a very little loss of martensitic structure, which

would happen in conventional tempering.

Martensitic steels are obtained with a very fast-paced cooling, which enables

austenite to be completely turned in martensite during quenching; the obtained

martensitic matrix steel encapsulates small amounts of ferrite and bainite. Com-

plex Phase steels follow a similar cooling pattern, but customarily retain more

austenite and do not show the typical fine precipitate characterizing Martensitic

steels. It is possible to add carbon to martensitic steels to improve hardenability,

along with small quantities of Mn, Si, Cr, B, V.

Martensitic steels are generally used were very good crash performance is needed,

for example in anti-crash rods (15).

12


2.1.5 Hot Formed Boron steels

Hot formed steels play a key role in automotive industry, especially where com-

plex shapes are required in association with optimal mechanical properties, but

no springback issue is allowed to arise. Desired strength is obtained through

quenching and austenitization, a process demanding temperatures higher than

850 ◦C and a high cooling rate; furthermore, the addition of very small quantities

of boron (not more than 0.005%) can led to UTS above 1350 MPa, and facilitate

the quenching operation.

Two main hot-forming processes exist, direct and indirect, but they do not affect

in a significative way the final mechanical properties, except for some differences

in elongation and ductility.

Overall, this type of steel could represent a major innovation and is already

widespread in automotive industry for some load-carrying parts: conceivably,

boron steel are among the materials which can guarantee the highest weight re-

duction, as a result of the improved as-delivered hardness, as well as increased

toughness and weldability (because of limited carbon content).

Boron steels can be also convenient from an economic point of view: usually they

are cheaper than medium carbon alloys they replace (16).

Nonetheless, they suffer from some recurrent problems, briefly explained in the

following paragraphs.

First of all, given the specific properties, it is almost impossible to drill them by

conventional means.

They also suffer from huge sensibility to heat, which makes them definitely weaker

when heated, therefore not allowing galvanization to be performed: as a re-

placement, a covering Al-Si layer is applied before drawing, in order to improve

strength.

Yet, the major issue is probably represented by the total impossibility of being

repaired in any way if damaged, which would make a car extremely expensive to

be fixed after a crash if all safety components were made of boron steel.

13


2.1.6 DP steels

Dual phase steels are, in all likelihood, the most widely employed material for

BIW and chassis application, by virtue of their unique union of cost-effectiveness,

proper weldability, strength and drawability, almost unmatchable in car environ-

ment; arguably, the only major drawback of these steels is to be found in the

springback effect.

They consist of a hard martensitic (in rarer cases bainitic) phase, with a shape

recalling circular islands, immersed in a soft ferritic matrix (up to 90%). The size

of the second phase can be controlled, thus allowing a variation of the resulting

strength of the steel.

The soft and continuous ferritic phase contributes to excellent ductility; drawa-

bility is enhanced by good strain redistribution capability, which derives from

an excellent strain hardening capacity. Work hardening rates give birth to steels

with a much higher UTS at similar YS, with respect to conventional ones, thereby

eventually leading to more appropriate YS to UTS ratios, even if compared with

HSLA steels (17).

Concerning the production of these steels, two main procedures exist today: one

is the controlled cooling from the austenitic phase obtained after hot-rolling; the

second is from a biphasic ferritic-austenitic region, attained in continuously an-

nealed cold-rolled and hot-dip coated products. The former process insures an

elevated resistance to stretching on a blanked edge by means of significant quan-

tities of bainite.

Customarily, the choice is based upon final destination, and related thickness, of

the sheets: cold rolling is employed for BIW parts, where hot rolling is preferred

for rims.

Dual phases are reported with a 20% higher dent resistance than conventional

HSS, allowing a weight reduction around 15%.

DP steels are particularly indicated for usage where strain hardening is consid-

ered; if not, they are not in position to determine a clear advantage. Nevertheless,

a new range of DP steels is being studied to provide good performance even in

simple parts.

14


Figure 2.2: Comparison in engineering stress-strain curves between three steels

(18).

2.1.7 TRIP steels

TRansformation Induced Plasticity (TRIP) steels are seeking for increasing em-

ployment in automotive industry, as a result of their acclaimed properties, and

despite their cost. The structure is composed of islands of residual austenite

(minimum 5% wrt volume, caused by high silicon and carbon content), with

carbide-free bainite sheathed in a soft ferritic matrix; from up-to-date evidence,

it seems that during the bainite transformation is essential to prevent carbide

precipitation, which can be avoided by means of silicon and aluminium addition,

thereby increasing the residual austenite content.

The main advantage of TRIP steels configuration lies in the distinctly increased

work hardening rate: much like in dual phase (see later), dispersion of second

phases opposes the dislocation; conversely, in TRIP steels the residual austenite

transforms progressively into martensite, thus increasing work hardening rate at

high levels of strain; as a result, the achieved elongations associated with the dif-

ferent phases account for the remarkable combination of strength and ductility.

Residual austenite can affect TRIP steels in two ways: if the carbon content is

relatively low, the deformation impacts very early the transformation, and the

residual austenite is transformed in martensite, hence increasing, even in a pre-

liminary phase, work hardening rate and positively influencing formability. At

higher carbon contents residual austenite persists even after sheet forming, being

15


eventually led to martensitic transformation only in event of a crash: it is then

self-evident why parts which have not been heavily formed are preferred in struc-

tures with energy-absorbing target as primary.

Work hardening rates make possible to design a part taking into consideration the

as-formed properties and capitalizing them, since they persist at extreme strains:

as a result, they show a response that is remarkably better than Dual Phase (see

later), especially in the most severe stretch forming.

TWIP steels exhibit outstanding performance on the whole. Ultimate Tensile

Strength can be as high as 900 MPa if the carbon content is of 0.25%, with a

matching YS of 550 MPa; elongation and n coefficient are middle-range, with

values respectively of about 23% and 0.18 accepted as minima. Welding is pos-

sible with multiple techniques: resistance-spot, MAG arc and laser are the more

widely used. Fatigue strength is also exceptional, with values that can almost

double the ones of conventional steels (19)

Figure 2.3: TRIP Microstructure (18).

2.1.8 Complex Phase steels

In the range of CP steels it is possible to find different structures; the most com-

mon one is represented by a ferrite/bainite structure with variable amounts of

martensite, bainite and retained austenite contained in it. In order to secure high

UTS and good ductility, a very precise grain refinement is promoted by retarded

crystallization or micro-alloying elements precipitation (Ti, Cb), thus also im-

proving energy-absorbing capability.

16

2.2 Treatments and coatings

By virtue of their inherent properties, this type of VHSS is suitable for safety

components which require almost no deformation in production process.

Both hot-spot and laser welding can be performed without significant problems.

Summarizing, CP steels have very good characteristics which make them partic-

ularly satisfactory for weight reduction; unfortunately, the intricate process used

for production makes them also quite expensive (20).

Figure 2.4: Comprehensive diagram on steels main properties (18).


Sometimes, to improve steel behaviour in determined conditions, some treatments

are performed or coatings applied, especially where obtaining same properties

with different methods would imply a considerable modification in composition

and production techniques.

Here, the most important are presented.

17


2.2.1 Aluminized steels

Aluminized steels are covered with an Al coating, resistant to heat, corrosion

and high temperature oxidation (up to 800 ◦C); such a treatment is virtually

perpetual. This coating is used for thermal insulation applications, where other

materials are discarded for weight or price reasons, and it is suitable also in

corrosive atmosphere.

These steels are covered with two layers of 90% Al and 10% Si: the inner ternary

layer works as interface, while the overlay of binary Al-Si is the main coating.

Thickness ranges from 10 to 35 µm. Usually deep drawing capabilities are also

enhanced, allowing the employment for fuel filters and tanks.

2.2.2 Bake hardening

Bake hardening is a controlled aging phenomenon designed to insure a remarkable

increase in YS during heat treatment. Considering BIW applications, said heat

treatment coincides with paint curing. Higher strengths can be achieved, espe-

cially for DP steels, with a gain quantified in at least 50 MPa, licensing smaller

parts to be used, thereby further decreasing weight.

Bake hardening mechanism can be described as the action of carbon atoms block-

ing dislocations movement during and after forming, thus increasing YS. This

allows to have good formability and, at the same time, good mechanical strength

when in action.

BH mechanism is also carried out during aging, where aluminium and nitrogen

combine to form aluminium nitride.

2.2.3 Zinc coatings

Zinc coatings are applied on visible and non-visible parts to improve corrosion

resistance and surface quality. Several types exist, each varying for procedure,

results and performance.

Hot dip galvanizing is probably the best way to apply a zinc coating: in the

wake of multiple stages of optimization and annealing, a cold-rolled steel stripe

is continuously moved into a bath of lead-free, cyanide-free molten alkaline zinc.

18


Special controls and inspections during all phases can lead to exceptional quality

and performance.

The zinc layer thickness is then adjusted and iron atoms are moved from substrate

to zinc layer by means of heat treatment; iron presence affects the hardness

and the drawability: under particularly severe deep drawing operations the so-

called powdering effect can happen. As a general consideration, all the normal

welding and adhesive bonding processes can be accomplished without significant

drawbacks.

In order to reduce costs and simplify post-galvanizing treatments, up to a value

of 5% of Al can be added to the molten zinc bath; the other side of the coin is

represented by a slightly decreased resistance to corrosion.

Electrogalvanization is also possible: a layer of pure zinc is deposited on the

steel, and no iron atoms diffusion takes place; as a consequence, the steel offers

superb corrosion resistance even if damaged. Principal drawback is in the enlarged

friction coefficient, important especially for deep drawing.

If particular weldability and formability are required, thin organic coatings can

be employed: a thinner zinc layer is covered with an organic layer of about 5 µm,

including lubricants and metallic particles to improve respectively drawability

and weldability (21).

Figure 2.5: Galvanannealing process (22).

19


20

3

TWIP steels

TWIP steels could represent a cardinal breakthrough in automotive industry;

still, recent results call for additional research, in order to solve the problems that

today preclude these steels from being employed. Hereafter, main characteristics

and an insight on relevant future applications are put forward.

3.1 Microstructure and strain hardening mech-

anism

TWIPs are austenitic steels with addition of variable quantities of Manganese

(17-30%); carbon is present depending on the initial steel while Al can be seldom

added to improve behaviour in tensile stress values.

In 1936, a formula was proposed to ensure fully stabilization of austenitic phase

(23):

wt.%Mn+ 13wt.%C ≥ 17

What is absolutely special with these steels is their astonishing strain hardening

mechanism, which is encountered in no other type of steel. A good number of

theories to explain them has been developed in recent research programs, and the

outstanding mechanical properties seem to be well described only by the action of

twins as obstacles to glide dislocations: this outcome derives from the fact that in

absence of twinning no special property is observed, while even in case of binary

alloys with twinning taking place in the structure, a remarkable strain-hardening

21

3. TWIP STEELS

is secured.

Twinning mechanism is believed to happen in certain favourable conditions: for

example, a temperature low enough to allow this process to be energy-competitive

with respect to dislocation glide; nonetheless, the temperature must be high

enough not to let ε-martensite form. However, the latter is a minor problem,

since an extrinsic stacking fault is needed as a nucleus, and the energy required

for this process to occur is 1.5 times greater than the energy needed for an extrinsic

stacking fault leading to twinning. (24).

A low stacking fault energy (SFE) seems then to be an important parameter for

twinning: this explains the required high content of Mn, which lowers SFE, and

the necessity of avoiding high carbon contents that would increase SFE; anyway,

no twinning effect has been observed with C content lower thank 0.3% (25).

Besides SFE influence, other parameters play an important role: for example,

dislocation glide must be thermally activated and so requires an excess in flow

stress that makes TWIP (or even TRIP) more energetically efficient; moreover,

at a first stage, Dynamic Strain Ageing (DSA) was supposed to play another

important part in induced plasticity, especially because the so-called A-type PLC

band appearing in stress-strain diagram are consistent with DSA; lately, a much

reduced influence for DSA has been suggested, with an induced increment, for

UTS, in the range of 20/30 MPa (26).

Solid solution hardening is also believed to exhibit more powerful twins, able to

obstacle dislocation movement and therefore further increasing TWIP effect.

3.2 Production processes

TWIP steels can definitely benefit from different techniques, principally to in-

crease yield strength.

Probably, the easiest and most employed one is grain refinement: using the usual

Hall-Petch relationship, and set as demanded a value of YS of 700 MPa, the grain

size required is about 1 µm; unfortunately, industrial processes, specifically cold

rolling and controlled annealing, only make possible to reach a grain size of 2.5

µm, leaving the YS limited to a maximum value in the range of 450 MPa, thus

restraining practical applications.

22

3.2 Production processes

Figure 3.1: EBSD image of the microstructure of a TWIP steel (27).

Pre-straining by rolling could also be used, but it is almost ignored because of two

main problems: a dramatic reduction in strain hardening coefficient, which nul-

lifies all the TWIP steel bonuses, and the appearance of remarkable anisotropy.

Strain hardening has a considerable effect on the stress-strain curve, and has a

variable effect depending on strain rate, with also mixed results on density of

deformation twins (28).

Recrystallization looks as an optimal solution, along with a milder recovery, for

pre-strained steels. Annealing, in fact, has been shown not to be a factor in grain

growth, but dislocations are far more rarified in recrystallized grains with respect

to un-crystallized areas; the main interest is therefore in finding the correct tem-

perature which allows stability for existent twins and, at the same time, prevents

cementite formation.

Experimental results manifest an optimal temperature of 500 ◦C and 600 ◦C; in

addition, highly pre-strained steels seem to better respond to this process: for

23

3. TWIP STEELS

example, for some recrystallization processes, a YS higher than 1000 MPa can be

kept, with a correspondent elongation of 15%. As a drawback, TWIP steels show

a high sensibility to temperature variations, and great attention has to be made

in order not to induce complete recrystallization.

In order to fully grasp recrystallization mechanism, a brief digression on TWIP

steel prominent texture is essential. After cold-rolling process three main textures

are present: Brass (typical of deformed low SFE materials), Goss and Copper; the

recrystallization process is not considered to induce modifications, since structures

appears to be the same, with comparable intensities. This led to the following

explanation: recrystallization depends only on the nuclei orientation after rolling,

a mechanism called oriented nucleation.

These separate textures cause an anisotropic behaviour in the steel, which can

be unquestionably detrimental.

Precipitation strengthening has been considered, but it gives rise to several

problems: iron carbides precipitation can happen, as well as pearlitic phase for-

mation, thus decreasing stacking fault energy (see ε-martensite) and lowering

toughness and ductility.

Main precipitating elements are Vanadium, Niobium and Titanium. The latter

is the quickest way to improve YS of about 150 MPa; however, it tends to sat-

urate after this empirically found limit, causing large TiC precipitations. Nb

precipitations have coarse grain, meaning an almost neglectable strengthening is

obtained. Current research is directing its attention towards Vanadium, which

has much higher solubility in austenite, allowing finer precipitates as well as a YS

increase in the 400 MPa range, without noticeable loss in ductility (8).

The actual mechanism that rules precipitation and twinning interaction is, in-

deed, yet to be studied and completely understood, distinctly at high strains;

hence we will not draw any conclusion on precipitation effects.

3.3 Fracture and fatigue

Arguably, the foremost reason TWIP steels are not widespread is to be searched

in their fracture ad fatigue features.

24

3.3 Fracture and fatigue

Actually, under tensile deformation, TWIP steels happen to systematically fail

before necking, in the uniform deformation zone; no thinning takes place, con-

trarily to what could be expected, and the specimen fails in a manner similar to

Magnesium and Aluminium alloys. As of today, neither explanations nor solu-

tions exist, hampering automobile applications.

Several methods have been introduced to explain this behaviour, from a simple

and, to some extent, simplistic, Tresca criterion, to more complex ones, taking

into account also shear stress applied before failure.

Uniaxial tension failures can misleadingly bring to the conclusion a Portevin-

LeChatelier (PLC) band is the main cause. Here, needless to say, another suc-

cinct digression is needed.

PLC effect is the theory describing a jerky stress flow in a material undergoing

plastic deformation, and is influenced greatly by negative strain rate sensitivity.

This effect starts at a critical strain and produces a strange wavy stress-strain di-

agram, obviously not compatible with normal measurements. Parameters known

to affect this effect are strain rate, temperature and grain size, while three main

types of bands (waves) exist: A, B and C, depending on what can be defined

as the frequency and the amplitude of the excursion. PLC effect can slightly

increase strength, but brings with itself major problems regarding ductility, also

introducing brittleness and even producing an increased roughness surfaces (8).

The results of this effect gave, at first, the impression of perfect matching with

experiment on TWIP steels; rather surprisingly, biaxial tension tests show no

PLC bands, suggesting that PLC effect cannot be considered as a comprehensive

explanation. Moreover, the exact correlation between negative strain rate sensi-

tivity and PLC bands is not unambiguous; in any case, no conclusions on PLC

bands have been drawn until now, and debate is still open.

Some researches evince a localized necking to occur before fracture, by means of a

microvoid convalescence. This conclusion is supported by very up-to-date X-ray

microtomography experiments (29).

Other authors refuse this explanations, arguing that primary carbides make pos-

sible the nucleation of large voids and this causes failure. Although no final

interpretation is already possible, it is almost certain that twinning, dynamic

25

3. TWIP STEELS

strain aging and strain rate sensitivity all play a fundamental role in this be-

haviour.

Literature on manganese austenitic steels fatigue behaviour is not extensive.

As it could be expected, a pre-straining treatment is beneficial to fatigue resis-

tance, and it is possible to set a trade-off between strength and part life thanks

to monotonic pre-straining in axial tension-compression or tension-tension. The

proportion between fatigue limit and UTS is much similar to stainless steel one,

fluctuating from 0.40 to 0.50.

What is really fascinating in TWIP steels fatigue is that, although no intense

twinning is present in cyclic loading test, they can reach as high as a 400 MPa

fatigue limit, with no apparent explanation.

Recent TEM analysis showed again that twin formation and growth is not, in

any way, affected by cyclic stress, even with fine grains, which can considerably

increase fatigue limit.

3.4 Employment

All the considerations made up to now belong to a quite theoretical territory;

it is therefore well-founded to analyze where TWIP actual development lies in

the pattern of industrial specifications and demands. As stated before, scarcely

any automotive part is currently employing this steel: too many issues arise from

the uncommon behaviour in tensile stress, which is still not justified, with the

possibility of dramatic failure when in use, and almost no room for prediction.

TWIP steels, moreover, present very high post-forming residual stresses, which

make them very sensitive to hydrogen-induced fracture and corrosion cracking,

while anti-corrosion coating effect has not been explored yet. Further attention

is needed since TWIP steels are reported to contain significant percentages of

phosphorous.

Concerning formability, response to stretch forming is still under investigation,

but it can be safely stated that the behaviour, especially in hole expansion, is

not as predictable and as desirable when compared to IF steel. This is reported

to be an effect of missing post-uniform strain (30). Little studies have been done

26

3.4 Employment

on welding: it is possible to summarize them saying that the spot-welding range

is generally reduced with respect to the normal one (in terms of kA), being the

melting temperature lower, and excellent results have been obtained for cross

tensile resistance.

Studies on reaction to welding adhesive bonding are not known by the author to

be in progress and will probably be delayed until the whole set of characteristics

is clear.

Last but definitely not least in importance, Zn coatings can be applied, taking

special care as MnO surface layer can be formed in hot-dip galvanizing. Probably,

electrogalvanization could be a better method to apply Zn coating on TWIP

steels.

27

3. TWIP STEELS

28

4

Fracture mechanics

Historically, mechanical parts design and dimensioning has been based on the

belief materials, and correspondent manufacturing, were ideal, thus leading to

the analysis of perfect parts that were expected to strictly obey to even the most

uncomplicated formulae. As time went by, this approach proved itself to be com-

pletely wrong, especially considering all the unexpected failures that happened

in most stressed components.

At the beginning of Twentieth century, scientists and engineers began to be con-

cerned about geometrical discontinuities, which are obviously inherent in every

blueprint; further inquiry displayed an anomalous stress concentration where very

sharp radii were featured, both on a 2D and 3D basis. This led to crack formation

in the most stressed zone and, eventually, to dramatic failure, even if unpredicted

by conventional dimensioning methods.

Additional studies showed that crack presence decisively modifies local stresses,

making usual elastic stress analysis completely ineffectual and, therefore, com-

pletely misleading. At a certain crack length, which varies according to several

factors, even a low macroscopic stress could induce high local stress and conse-

quently result in fracture.

Another factor that influences response to solicitation is the microstructure of

the material taken into consideration, especially if steel: inhibiting the dislo-

cation motion is preferable for high stresses but, normally, leads to unwanted

brittleness; this causes very quick and unannounced crack propagation. Indeed,

one main attribute of TWIP steels is their outstanding combination of UTS and

29

4. FRACTURE MECHANICS

elongation, and proving it on an experimental and practical basis, allowing a di-

rect comparison with competitors, and even quantifying it, could end in a leap in

knowledge.

Given these premises, it is undoubtedly crucial to study behaviour of parts when

cracks are present. The discipline of solid mechanics which analyses the presence,

influence and reaction of crack growth is called Fracture Mechanics. Its main ob-

jective is to find mathematical relations between crack length, material intrinsic

resistance to crack growth and critical stress, in order to prevent failure. For sake

of precision, Fracture Mechanics has been always divided in two branches: linear

elastic (LEFM) and elasto-plastic (EPFM). The former applies to brittle-elastic

materials such as HSS and concrete, while the latter better fits the behaviour of

low-carbon and stainless steels, some aluminium alloys and polymers.

Most recent understanding of fracture mechanics induced a proper revolution in

aeronautical design, introducing the so-called damage tolerance design methodol-

ogy and philosophy; nowadays, mainly because of economical reasons, this prin-

ciples are extending also to automotive industry. A brief history of fracture

mechanics, with the most germane results, follows1.

4.1 The energy balance approach

One of the very first approaches to fracture mechanics was the one theorized by

Inglis in 1913, regarding elliptical holes under plain stress. He concluded that the

only factor affecting noteh root stress was the the value of the radius of curvature,

which was proportional to the stress concentration factor to be applied. Inglis´

computation exhibited, however, a major problem: at the theoretical limit of a

perfectly sharp crack, the stresses tend to infinity, implying a material strength

close to nought. This, as can be easily seen, bears no resemblance to reality.

A.A. Griffith played an outstanding role in fracture mechanics, with a break-

through intuition: he applied the energy-balance principle, in addition to Inglis

results, to crack growth theory, suggesting that residual elastic potential energy

in the material can be discharged to enlarge the crack surface while maintaining

1All the remaining part of the chapter was written with reference to the following

books:(31)(32)(33)(34)(35).

30


equilibrium.

The first approach was to define the work performed per unit time (i.e. the aver-

age energy applied) as equal to the rates of change of elastic and potential energy,

kinetic energy of the whole body, and energy used for crack propagation:

W = Ue + Up + K + Γ

Assuming the crack growth is small enough to allow kinematic energy change to

be considered null, it is possible to derive said equation with respect to the crack

area

−δΠδA

=δUpδA

+δΓ

δA

with

Π = Ue −W

If the material is brittle, the energy change due to plastic deformation can be

neglected. Furthermore, introducing the concept of energy used to form new

material surface γ, it is possible to state

−δΠδA

=δΓ

δA= 2γ

and then to introduce a sort of fictitious crack extension force G

−δΠδA

= G

Back to the initial system, the total energy can be defined as

Utot = (−W + Ue) + Π

The Clapeyron

’

s theorem of linear elastostatics says that a body under constant applied load

follows

W = 2Ue

leading therefore to

Utot = −Ue + Γ

31


Griffith plugged the results obtained by Inglis in the equation he derived, obtain-

ing

Utot =−πa2σ2B

2+ 4aBγ

where a is the crack halfwidth, B is the plate thickness, σ is the constant stress

applied, E is the Youngs modulus, and γ is the energy per unit area used to

expand crack surface (considered to be a material property). Said function is a

Figure 4.1: Utot vs. crack length

linear combination of two other functions, a parabola and a line. It is straight-

forward, from mathematical analysis, to state that such a function has an only

maximum.

This means that after the maximum value, the energy is lowered by the crack

growth, and so the crack would enlarge until total rupture.

Deriving with respect to a, it is possible to write both the critical stress given a

defined crack

σc =

√2γE

aπ

and the critical crack half-length known the applied stress

ac =2γE

σ2π

32


These results carry with themselves two important informations: the critical

stress reduces with crack length, and directly depends on the material.

It should be noted that these outcomes apply faultlessly only to systems charac-

terized by linearity; even so, for our purposes, steel can be considered as such.

It is noteworthy to point out that both critical values of crack and applied stress

are absolute, independent of length, thickness and width of the specimen; this

means that, in some limit cases, narrower pieces can be more resistant simply

because they are not large enough to contain a critical-length crack.

An additional term can be introduced to better understand crack behaviour: the

compliance, i.e. the displacement per unit load and, consequently, the reciprocal

of stiffness. Compliance is further defined (referring to 4.2) as

C =u

P

with δu as the infinitesimal increment in the distance between the two points load

application.

In order to find the total strain energy it is mandatory to know the load condi-

Figure 4.2: General shape of a sample.

tions, i.e. how displacement and load vary with time.

33


Two common situation are present in literature, constant load and constant dis-

placement. The former is governed by

−δΠ =1

2Pδu

while the latter by

δUe =1

2uδP

In any case, if δA tends to nothing, as it is, both formulae, which express a

potential energy release rate, can be rewritten, with some manipulations, as

δUe =1

2CPδP

suggesting similar behaviour under stated conditions.

The compliance of a specimen can be measured as a function of crack length.

Differentiating the curve of compliance vs. crack length, we obtain

G =1

2P 2 δC

δa

It is also significant to introduce the concept of critical strain energy release rate

Gc =1

2P 2 δC

δa

∣∣∣∣a=ac

Now, known the exact relative arrangement of crack and stress, it is possible to

obtain analytical expressions for deflection. For example, assuming the sample

to be a double cantilever beam, it is possible to recall

u

2=Pa3

3EI

which leads, if we substitute in the critical strain energy release rate formula (also

assuming linearity), to

Gc =12Pc

2a2

b2h3E

which finally relates the critical release of energy with only geometrical and phys-

ical properties.

34

4.2 The stress intensity approach


What has been analyzed until now is just one condition for a load to be applied

on a cracked body, more specifically what is called normal opening. In literature,

other two modes are presented, both implying shear sliding: in-plane, and out-

of-plane tearing.

Westergaard provided an analytical solution using the so-called semi-inverse func-

tion; the stress function is

Φ = Re(Z)

+ yIm(Z)

The solution, obtained by trial-and-error, has been found to be

Z(z) =σ√

1−(az

)2

In a neighbourhood close enough to the tip, given polar coordinates are adopted,

the following solution has been analytically derived and experimentally proven:

Z =σ√a√

2re

−3iΘ2

For a general crack in a theoretically infinite plane, the stress can be expressed

as

σij =K√2πr

fij (Θ)

while the displacement is

ui =K

2µ

√r

2πg (Θ)

K is the stress intensity factor; subscripts indicate the modes of stress (Ki, Kii,

Kiii). It includes in itself the geometrical and loading conditions, and the depen-

dance on configuration can be expressed as

K = Y σ√πa

where Y is the geometric factor, equal to 1 in case of finite center crack in an

infinite plane. Geometry, as said, induces modifications in the stress field. The

35


geometric factor can be considered as a function of both crack length and piece

width

Y = f (a, w)

Values of Y are found mainly either by means of numerical methods, chief among

which the finite elements method, or combining together experimental evidence

with semi-theoretical computations. Tables exist for the most poles apart config-

urations.

During the design process, Kic (critical stress intensity factor) value is introduced.

It can be considered as a meter of the material toughness, and it is the maximum

stress intensity a material can withstand, above which microstructural mobility,

especially in ductile materials, does not guarantee crack extension: the crack zone

cannot broaden farther and the energy constraint loses its capability to stop crack

growth.

In order to find this value we know that

σf =Kic

α√πa

We also recall from energy balance approach

σf =

√EGc

πa

Equating the two formulae we obtain

Kic2 = EGc

for plane stress. Plane strain implies also the presence of Poisson ratio

Kic2 = EGc

(1− ν2

)which, indeed, generates no major modification in metals. Heading back to pre-

vious DCB example, it is within the realms of possibility to correlate also K to

only geometrical and load terms, by replacing G:

K = 2√

3P√

1− ν2

a

Bh32

36


for plane strain conditions. If dependance on displacement is required

K =

√3E

4−√

1− ν2

uh32

a2

This last formula shows how the crack growth would, at a definite value, cease

to increase were the displacement kept constant.

When the stress formulae are used to predict values very close to crack tip they

tend to infinity, which is of course not possible in reality. It is then needed to

find a different way to define stress in these points. It has been experimentally

shown that the limit for stress is Kic itself, and it is applicable for

r

a≤ 0.02

The zone where this approximation holds is called singularity-dominated. This

sparked off new theories regarding plasticity that will be later elucidated.

Just as a hint, it has to be underlined that the assumptions of linearity allows

the system to be applied superposition on, both in case of heterogeneous modes

or different stresses.

4.2.1 Plastic yielding at crack tip

All the methodologies suggested until now imply a not relevant plasticity. In real

conditions, however, local plastic yielding at crack tip and ensuing elastic-plastic

deformation must be inspected, for better comprehension of fracture mechanism,

especially concerning steels.

One approach is the Irwins model, which assumes that the material has a perfect

separate elastic plastic behaviour. The local perpendicular stress is

σyy =Ki√2πr

as derived before. The zone affected by yield can be found substituting in the

equation an equivalent stress obtained by means of a determined yield criterion.

Considering plane stress, the equivalent stress is the uniaxial yield strength; the

area impacted by yielding is assumed circular with a radius

ri =Ki

2

2πσ2ys

37


Energy conservation principle implies stress exceeding yield must redistribute in

the material, therefore forcing the plastic zone to enlarge. A second order ap-

proximation enables a force balance, assuming that the elastic stress distribution

has the same magnitude through time

σysrp =

∫ 1

0

Ki√2πr

dr

Accordingly,

rp =Ki

2

πσ2ys

The same result can be obtained considering a fictitious crack extending to the

centre of the plastic zone. Plane stress and plane strain conditions allow different

values of applied stress before yielding: in the configuration used until now, the

first permits some 14% of the yield stress, while the last allows until 35%.

The strip yield model

Another framework proposed to assess the size of the plastic zone is the strip

yield model, firstly introduced by Dugdale and Barenblatt.

A strip yield plastic zone is defined with theoretical dimensions of a and ρ (see

4.3), with the latter chosen such that the stress singularity vanishes at the end of

the effective crack.

Kρ +Kσ = 0

At the two different ends of the crack, if the reference frame origin is set in the

center of the fracture, the intensity factors are

Ka,b =P√πa

√a± xa∓ x

which are equal for opposite values of x. The infinitesimal closure force is

P = −σysdx

Consistently with our initial hypothesis, we set the end of crack opening at a

value c, such that it represents the singularity-affected zone limit. This value is

then equal to a+ ρ. Integrating to obtain the total closure stress intensity

Kρ = − σys√πc

∫ c

a

(√c− xc+ x

+

√c+ x

c− x

)dx = −2

√c√πσys cos−1

(ac

)

38


Figure 4.3: Plastic zone on specimen.

It is known that this value is equal in magnitude and opposite in sign to the

remote tensile stress induced stress intensity.

Kρ = −Kσ = −σ√πc

Equating the two results, then approximating to a first order series development,

the requested value of ρ is then

ρ =πK2

8σ2ys

if σ is low enough; anyway, this is exactly the condition we are taking into con-

sideration, and then such an outcome is still worth of attention.

Comparison with Irwin´ s method shows that the obtained value is higher of

about 23%, resulting in a more conservative approximation.

Correction factors are introduced in these formulae if the condition is of plane

stress, plane strain or finite thickness. Different conditions usually overlap when

concurring to the definition of the shape of the plastic zone, but this assumption

does not hold in all cases.

39


4.3 Fracture and R-curve

In almost every automotive application where crack growth is of relevance, the

actual stress exceeds the yield stress, and plasticity cannot be neglected.

What is far more concerning is the critical stress and strain directly at the crack

tip, i.e. when only slight plastic deformation is allowed, because of geometrical

or micro-structural reasons, this inducing brittleness in the material.

The resistance to fracture, and consequently the toughness of the material, has

been shown to be strongly dependent on thickness, in a non-linear manner.

The concept of a critical stress intensity depending on load conditions is intro-

duced with the symbol Km,c, where m indicates the different mode, as explained

before.

If the material is considered to fail when the stress intensity reaches the critical

one, the residual strength of the component being

σ =Kc

Y√πa

where Y is the usual geometry correction factor. Fracture mechanics derived

design methodologies customarily assume the conservative case of Kc = Ki,c.

The geometry correction factor is a function of crack length, and equations to

derive its value can only be solved by means of numerical iteration. A formula

commonly employed to know the value for crack length, declared the load, is

ac =1

π

(KcW

σY ac

)2

All the formulae and solution provided until now hold valid if LEFM is still

involved, i.e. the net section stress level is far below the material yield stress;

if not, the failure will be more probably caused by plastic collapse; the value of

plastic collapse stress limit, from undemanding geometrical considerations, is

σpc =W − 2a

Wσys

Two failure modes are then possible: brittle fracture and plastic rupture; expec-

tations are based on toughness, crack length and width of the specimen.

The two modes can be drawn as curves with the ratio between fracture length

40

4.3 Fracture and R-curve

and piece width as the independent variable, and residual strength as dependent.

This leads to better knowledge of different modes prevalence by inspecting the

intersection point. As a result, plastic collapse is dominant when

W − 2a

Wσys >

Kc

√πa√

sec(πaW

)Be that as it may, the rigorous superposition of these effects is better investigated

by means of EPFM.

Up to this point, had a brief conclusion to be drawn from the fracture me-

chanics overview presented, the most remarkable result is the concept of a general

external energy that has to overcome the material capability of energy absorption

in order to enlarge the crack.

The surface energy needed to open the crack is, according to the previously men-

tioned Clayperons theorem

W = 2Ue

Plasticity must be taken into account for engineering materials, leading to the

following equation, which is expressed in modern terminology as

G = 2Wf = R = 2 (γp + γf )

where G is the energy required for crack growth, γp and γf are respectively the

plastic work per unit area and the surface energy, with the latter almost neglected

in case of metal alloys. R is purely introduced for the sake of clarity.

To gain better comprehension of material behaviour, it is possible to plot G and

R values versus crack size, obtaining, in order, a resistance curve (R-curve) or

a driving force curve. This last is dependent on structure geometry and loading

conditions, while the R-curve is also affected by temperature, geometry, load

conditions and other parametres.

The R-curve is especially useful to find the fracture toughness because, in case

of slow-stable crack, growth (obtained with finite stepped increments in load),

this is equal to the material resistance found experimentally. The R-curve thus

defines the resistance of a material as the plastic zone extends. There are two

41


graphically different types of R-curves: the ones obtained for brittle materials

and the ones for ductile materials (see 4.4). The crack grows when

dG

da>

dR

da

which guarantees that the external energy cannot be absorbed by the material,

and

G ≥ R

which guarantees that the driving force overcomes the resistance of the material.

Graphically, the conditions are satisfied at the tangent line.

Figure 4.4: R-curve for brittle (left) and ductile (right) materials.

4.4 Fatigue and Paris curve

4.4.1 Constant amplitude load

All the discussion presented until now is based on instantaneous fracture, i.e.

finding the conditions under which the sample fails, no matter what he withstood

before. Nevertheless, also smaller stress can cause rupture, if they are constantly

applied. One of the simplest fatigue modes is when a constant range of cyclic

42


stress is applied; the parametres defining each cycle are two: the stress range

∆σ = (σmax − σmin)

and the stress ratio, which can be found in literature either as

R =σminσmax

or

R =σmaxσmin

Image 4.5 makes use of the latter definition, while the rest of this thesis is con-

sistent with most recent papers, and uses the former.

The type of loading can be expressed also as the range of stress intensity factors

related to the two limit stresses

∆K = (Kmax −Kmin)

This means that, exactly as fracture can exist with a value of the maximum stress

well below the yield of the material, it takes place with a value of Kmax lower

than Kic too. Assuming that crack growth occurs only in the crack tip under

plasticity deformation, fatigue crack growth can be related to ∆K.

When the infinitesimal crack growth with respect to number of cycles is plotted

against the stress intensity factors range, we obtain a Paris curve; usually, for the

sake of clarity, both axis are in logarithmic scale.

This approach assumes that all the loading conditions can be actually summed

up in an unique and univocal parametre; as a matter of fact, this technique seems

to work properly.

A relationship is now sought, in order to correctly relate ∆Kto crack growth. In

the second portion of the graph, the linear section can be expressed, given the

plastic zone is little enough to obey completely to elastic singularity zone rules,

asda

dN= C∆Km

where C and m are functions of multiple agents, such as material, external con-

ditions and stress rate.

43


If ∆K is low enough, crack will not grow anymore, in a condition similar to

the maximum stress for infinite life encountered in Wohler curve; the value un-

der which no fracture occurs is called Kth; it is still not clear the reason of this

phenomenon, but two main explanations have been made: the first is that the

induced compressive stress are able to screen the crack tip from the rest of the

material, disabling crack growth; the second, less famed but still valid, is that

the crack tip plastic deformation cannot penetrate micro-structural barriers.

Another of the actual challenges is to correctly relate K to σ also in case of com-

pressive stress; today, the most common solution in this case is to consider the

minimum stress intensity as null.

More comprehensive equation are now studied, in order to achieve better correla-

tions between theoretical computations and experimental data. One of the most

interesting development is in the Pridle equation, a more complicated and wide-

ranging version of the Paris equation, in which the crack growth is a function of

more variables.

da

dN= f (∆K,R,Kc, Kth) = C

(∆K −Kth

Kc −Kmax

)mThe actual application is to be found in the determination a priori of the number

of cycles before failure, known material, load and geometrical conditions, given

an initial crack length a0

Nf =

∫ af

a0

da

C (∆K)m=

∫ af

a0

da

C [Y (a)∆σ√πa]

m

Assuming the geometry correction factor constant within the interval,

∆K = ∆K0

√a

a0

= Y (a0)∆σ√πa

√a

a0

Solving in the interval

Nf =2a0

(m− 2)C (∆K0)m

[1−

(a0

af

)m2−1]

If m = 2, as it can sometimes happen, the singularity result of the integral is

Nf =a0

C (∆K0)mln

(afa0

)

44


In any case, this provides only an approximated result, biased by the actual vari-

ation in the geometry correction factor. Only numerical methods allow precise

predictions. Another interesting modification was introduced by Elber, who noted

that for asymmetric loading a plastic zone smaller than expected is formed, prac-

tically decreasing the effect of the external load. The concept of effective stress

intensity factor range is introduced as

∆Keff = Kmax −Kop

where Kop is the highest factor for which two faces do not part one from another.

Effective stress ratio U is defined as

U =∆Keff

∆K

and it is usually found in literature as a linear equation depending on R.

Figure 4.5: Example of Paris curve.

45


4.4.2 Variable amplitude load

In real-life cases constant amplitude load is nigh on inexistent: external loads

vary on an almost randomly basis, and real behaviour parts systematically from

the expected one.

However, fluctuating loads are not the only discrepancy from reality: also crack

retardation due to overload, acceleration due to underload and highly variable

conditions must be counted as important factors.

Constant amplitude load approximation can still be kept valid when similitude

conditions apply, i.e. when load configuration does not part significantly from

theory. In cases of variable load spectrum, simple linear combinations are used to

predict part life; these outcomes have been demonstrated to be very conservative

projections, often leading to unnecessary repairs or even overhauls, remarkably

increasing costs.

The basics of the models used today date back to 70s; further developed through

the years, now the model is considered quite accurate, also by virtue of com-

putational numeric techniques. Anyway, the Paris equation has proven itself

essentially right, with some modifications adopted to better fit experimental re-

sults.

Studies on plastic wake and crack closure stress have led to the following formula

da

dN= C

{[1− S0

Smax(1−R)

](∆K)

}mwhich, as can be seen, closely resembles in structure Paris equation; S is the

applied stress, R is the stress ratio, and S0 stands as the only unknown, and

represents the crack opening stress level.

A good trade-off between simplicity and accuracy is embodied in the Wheeler

model; it deals with overload, and starts off with the known equation for plastic

zone size

rpo =1

π

(K0

ασys

)2

while the rupture would be reached, in constant load, at

rpc =1

π

(Kmax

ασys

)2

46


where α is a coefficient depending on type of loading.

Wheeler assumed that an overload retardation effect exists, unknown in mag-

nitude; he also supposed that this vanishes when the newly induced plastic zone

reaches the boundary of the previous one. Two parametres put into numbers this

concept

λ1 =∆a+ rpcrpo

;λ2 =rpc

rpo −∆a

Retardation factor is thus defined as

φ = λγ

where γ is a fitting parametre depending on material properties and loading

spectrum, and can actually be obtained only experimentally. The new crack

growth rate is now defined as(da

dN

)OL

= φ

(da

dN

)0

The main disadvantage of this approach is in its blindness: being highly sensitive

to load spectrum and variations, it is mandatory to calibrate every experiment

on actual usage, thus requiring a reliable and accurate prediction of the state of

operation.

Despite all the difficulties presented, nowadays design methodologies are widely

based on damage tolerance, especially in aerospace environment. This requires

more complicated computations and theories, and prescribes strict inspections of

all the parts; nonetheless, great advantages are achieved in life prediction and

money saving.

47


48

5

Standard test methods

Every study we mentioned has probably been carried out on ever changing sam-

ples: scientists were not really concerned with a qualitative uniformity, usually

working with the samples they were provided with; a sort of continuation and

correlation in ideas and procedures was pursued and considered to be relevant,

instead of a standard method.

Indeed, when it all came down to practical applications, the necessity to have

consistent and homogeneous results quickly arose, and the definition of standard

methods was perceived as a priority.

Nowadays, the most important standards in metal testing are set by ASTM

(American Society for Testing and Materials), which foundation dates back to

1898. Other organization involved in imposing regulations are BSI (British Stan-

dards Institute), DIN (Deutsches Institut fur Normung), AFNOR (Association

Francaise de Normalisation) and ANSI (American National Standards Institute);

the last three are active members of ISO, the International Organization for Stan-

dardization, which is probably the most important and famous organization for

standardization. Anyhow, in most cases ISO itself relies on ASTM for regulations

concerning the field which is dealt with here. In the following sections, standards

method for the definitions of Paris curve will be briefly explained and the main

features listed.

49

5. STANDARD TEST METHODS

5.1 ASTM E-647: Standard test method for mea-

surements of fatigue crack grow rates

As suggested by the title, said regulations cover the definition of crack growth

rates in the central part of the Paris diagram, i.e. from the threshold stress to

the instability region.

An excerpt from the test method follows, outlining its purpose 1:

This test method involves cyclic loading of notched specimen which have been ac-

ceptably precracked in fatigue. Crack size is measured, either visually or by an

equivalent method, as a function of elapsed fatigue cycles and these data are sub-

jected to numerical analysis to establish the rate of crack growth. Crack growth

rates are expressed as a function of the stress intensity factor range, ∆K, which

is calculated from expressions based on linear elastic stress analysis.

Of course, for every specific doubt or clarification it is preferable to refer directly

to the whole text of the test method; in any case, for the sake of clarity, main

points are hereinafter introduced and discussed. The specimen is, as a very first

presumption, assumed to be sufficiently planar to guarantee that elastic defor-

mation is predominant; the second prescription is related to the first one, and

requires the sample to be thick enough to preclude buckling.

The test method than counsels to keep in the correct perspective the influence

of temperature, environment, geometry, crack closure and residual stresses; it is

also recalled that results holding for long cracks can differ noticeably from the the

ones of small cracks: factors having a strong impact on the definition of small are

microstructural dimension and scale of local plasticity; this considerations fail to

apply when crack is shorter than 1 mm and thus considered physically small; this

memoranda are included in section 5, which encompasses a broader definition of

the purpose of the directive.

Section 6 is a succinct indication on the apparatus needed for the experiment:

the importance of good gripping and perfect alignment is emphasized; it is also

prescribed the employment of rigid, non-rotating joints to reduce lateral motion,

along with an optimization of the force train length, to reduce the arm of the

1In this section, all the parts in italic are direct quotes from the test method (36).

50

5.1 ASTM E-647: Standard test method for measurements of fatiguecrack grow rates

eventual resulting force.

Section 7 establishes the basic configuration, size and preparation for both C(T)

and M(T) types of sample. At this stage, only theoretical advices are given, and

for practical prescriptions reference must be made to Annex I.

Of great importance is the adoption of geometrical expedients to prevent the

residual stress acting perpendicular and parallel to the direction of crack growth

from biasing results. Symmetrical specimen are generally a satisfactory measure

to avoid residual-stress-induced clamping on crack growth measurement; peculiar

shapes allow to steer clear off crack curvature and irregularities.

For high-strain hardening materials, a new effective yield stress can be introduced

to meet the uncracked ligament requirement. It is defined as flow strength:

σfs =σys + σuts

2

This modification tends towards conservative predictions, which is anyway better

than overestimation. Further indications are provided, regarding notch prepa-

ration procedure, which can be achieved by various means: electric discharge

machining, mill, broach, grind, sawcut.

The most substantial part, concerning the actual procedure, is the following sec-

tion, namely number 8.

The first advice is to perform the highest number of tests possible, in order to min-

imize, and eventually neglect, the influence of external factors, especially when

studying low crack growth rate (da/dN ≤ 10−8 m/cycle).

Paragraph 8.3 is an extensive description of precracking, which objective is to

provide a sharpened fatigue crack of adequate size and straightness which ensures

that 1) the effect of the machined starter notch is removed from the specimen

K-calibration, and 2) the effects on subsequent crack growth rate data caused by

changing crack front shape or precrack load history are eliminated.

Fatigue precracking must be conducted with the specimen already fully heat

treated and with an equipment that guarantees the force distribution to be sym-

metrical with respect to the machined notch, and can be applied with any known

loading frequency that enables a ±5% accuracy to be achieved. Supplementary

indications follow, regarding the K-decreasing test condition and the symmetry

of the crack extension during the test; this part will be investigated in the section

51


concerning the actual experiment, marking any difference that should arise.

Paragraph 8.5 is a series of detailed indications for constant-force-amplitude test;

this type of test is suggested to be carried out at constant force range and at

fixed set of loading variables (i.e. stress ratio and frequency); anyway, when sev-

eral informations are needed and only a few samples are available, force range

can be modified, provided a set of rules are applied to avoid undesired factors to

affect the outcome: transient grow rates should be avoided by reducing relative

incremental increase and steering clear of force range shrinking; sufficient crack

extension must be allowed when environmental effects are remarkable; last, long-

duration interruptions must be avoided since they could result in unacceptable

relaxation or modification in the material.

Paragraph 8.6 deals with K-Decreasing procedure, valid when da/dN assumes a

value under 10−8 m/cycle. This method is only applicable in such a case because,

if not, prior leading history could alter near-threshold fatigue crack growth rate

behaviour. Test should be conducted until the lowest ∆K or crack growth rate

of interest is achieved; the test may then continued, if demanded, to obtain com-

parison data.

Force shedding in K-Decreasing procedure is more fitting if continuous; if stepwise

reduction is adopted, effort must be made to preclude anomalous data coming

from a not gradual diminution. Should not these indication be respected, specific

hints on the course of action are presented. Additionally, 8.7 gives a guideline

on alternative K-control test procedures; this is not of much interest for our case

and therefore is not further discussed.

An important paragraph on measurement of crack size follows.

It is, as first, required the use of an instrument capable of resolving extensions

of 0.10 mm or 0.002W , whichever greater. Reference marks, grids or scales are

highly recommended for more precise values and is preferable to have the possi-

bility of measuring without stopping the experiment.In case the cyclic loading is

interrupted, strict care must me taken to avoid the introduction of any significant

extraneous damage or transient crack extension; the suspension must be limited

in time and, in case a static force is maintained for the purpose of enhanced crack

tip resolution, it should be carefully controlled.

Measurements must be executed in order to have data nearly evenly distributed

52

5.1 ASTM E-647: Standard test method for measurements of fatiguecrack grow rates

with respect to ∆K.

A minimum ∆a of 0.025mm is recommended; in any case, the minimum ∆a must

be at least 10 times the crack size measurement precision; the data to be put

into computations are the average of two measurements, on the two sides of the

sample; should the crack deviate from the straight line of a value higher than

20 ◦, test must be considered invalid for non-meeting of symmetry conditions.

Branching crack could occur; this condition is cleared for report utilization, pro-

vided it is explicitly declared.

Section 9 deals with calculation and interpretation of results; at the end of the

test, it is prescribed to examine the fracture surface in, at least, two points, to

determine the extent of through-thickness crack curvature (also called crack tun-

nelling). The difference obtained between the average through-thickness crack

size and the actually measured crack size is called crack curvature correction; in

case this value results in a greater than 5% difference in calculated stress-intensity

factor at any crack size, then this is the value to be used when analyzing recorded

data; it is also suggested to use a linear interpolation if the magnitude of crack

curvature correction increases or decreases with crack size.

Paragraphs 9.2, 9.3 and are about the determination of, respectively, crack growth

rate, stress-intensity factor range and fatigue growth threshold; being highly sig-

nificant topics, they will be analyzed in a later subsection.

Section 10 gives instruction and recommendations on report drafting. Key data

of the sample must be compulsorily included, such as geometric features, ma-

terial characterization and treatment, mechanical properties; regarding the in-

struments, description of test machine and general equipment must be provided,

including the measuring devices and their resolution. Test loading variables are

obviously needed: R, ∆K, ∆P , initial crack size, cyclic frequency and waveform.

As previously observed, environmental variables can affect the outcome of the

test; consequently, crucial values (temperature, humidity, pressure) should be re-

ported, especially in the case they vary appreciably from normal STPs.

Further information must be supplied regarding the analysis method and the in-

strument calibration. As a final indication, any anomalous situation occurred

during the test must be accurately detailed. The last section covers precision and

bias; it is clear that a small error in the measurement can reflect in climacteric

53


errors when determining the Paris curve. The test method reports an average

reproducibility of 27%, and a range from 13% to 50%, with the repeatability

at 32%; these values are standard errors based on residual standard deviations

about the mean response determined from regression analysis. This variability is

believed to derive from random crack size measurement errors, given the high

homogeneity of the employed material (10 Ni steel); near threshold regime show

much lower values.

Studies on variance and covariance of Paris curves have been accomplished (37),

but are more mathematical assessments than physical deductions.

Finally, it is recalled that the highest part of variability of da/dN is due to inher-

ent material variability, i.e. variations in chemical composition, microstructure,

etc.

There are not any indications on bias, because no accepted standard value for

da/dN versus ∆K exist.

Various annexes follow; where relevant, they will be mentioned in the section

regarding the actual experiment.

Figure 5.1: 3D representation of the specimen.

54

5.2 Recommended data reduction techniques

5.2 Recommended data reduction techniques

The appendix X1 (along with sections 9.2 and 9.3, as previously mentioned) of

the test method E-647 gives information on how to derive valid crack growth

rates from known measures.

The first one is known as secant method, and it is specifically indicated for a K-

decreasing procedure: this technique, also known as point-to-point, simply pre-

scribes the computation of the slope of the straight line connecting two adjacent

data points on the a versus N curve:(da

dN

)a

=

(a1+i − aiN1+i −Ni

)∆K is usually calculated using the average crack size

a =1

2(a1+i − ai)

The other technique that can be employed is the incremental polynomial method;

this procedure involves fitting a second-order equation to sets of (2n+ 1) consec-

utive points, where n is usually between 1 and 4. The local fit is

ai = b0 + b1

(Ni − C1

C2

)+ b2

(Ni − C1

C2

)2

where

−1 ≤(Ni − C1

C2

)≤ +1

and

C1 =1

2(Ni−n +Ni+n) ;C2 =

1

2(Ni+n −Ni−n)

Regression parametres b0, b1 and b2 are to be determined by means of least squares

method. The rate of crack growth is the derivative of said parabola, namely(da

dN

)ai

=b1

C2

+ 2b2

(Ni − C1

C22

)Usually computer programs are used to compute those values and plot them.

55


Figure 5.2: 2D representation of the specimen, with dimensions.

56

6

The experiment

6.1 Preparation of the equipment

The composition of the TWIP steel samples is in Table 1 (average of 6 measures).

The sample arrives blurred and smeared, and needs to be properly prepared for

the actual measurements to be correct and reliable.

As first operation, the specimen is smoothed down with a highly abrasive mag-

netic disk; this grinding operation is carried out in order to remove the very

superficial film of dirt and dust that covers the sample.

It is of appreciable importance to be both cautious and meticulous at this stage,

because the useful area for crack growth is here defined and it is not possible to

enlarge it later on.

The second step employes the usual non-magnetic soft disks for grinding and pol-

ishing, again utilized on a rotating platform, at the two allowed velocities of 125

and 250 rpm; here, only water is used as lubricant.

The lapping disks are usually made of silicon carbide; zirconia alumina-based

disks are also available for sale, but are far more aggressive and were not used.

The abrasive material is generally coated with resins and latex additive to ensure

long-lasting cutting capabilities.

Different grit size were used: from 400 to the finest available, 4000 (FEPA scale,

correspondent to a 1200 U.S. Industrial Mesh), going through 800, 1200 and 2500.

The number refers to the meshes in a linear inch featured by the filter, and con-

sequently to the maximum size of the grit, the finest of which has a nominal

57

6. THE EXPERIMENT

Table 1: Composition of TWIP steel samples

C Si Mn P S Cr Mo Ni Al Cu V Sn Fe

0.66 0.05 17.25 0.025 0.011 0.03 0.01 0.30 0.60 0.03 0.04 0.11 80.89

diameter value of 5 µm.

The polished finish is obtained by means of diamond paste, applied together with

a lubricant; it consists in powder diamond particles diluted in a water-based bind-

ing agent; 3 and 1 µm pastes were used.

The same treatment was applied to both surfaces.

Before mounting the specimen it was necessary to check, with a microscope, that

it did not show any evidence of corrosion or pre-cracking.

The employed machine is a hydraulic-operated device capable of a maximum

force of 250 kN, which is unquestionably over-dimensioned for the purpose of this

experiment.

In order to improve sensibility at low forces, the 50 kN load cell was deployed.

The force was applied through Clavis-type grips and a chain of threaded reducers.

6.2 Precracking and crack growth

Precracking, as said before, is aimed at the creation of a sufficiently long crack

to be enlarged in the following passages; such procedure is necessary in order to

insure that the crack growth is independent of residual plastic stress, and to find

the threshold ∆K.

A growth of 0.5 mm is generally considered enough, but also the influence of the

initial plastic radius must be taken into account when deciding to move on with

actual crack growth cycles.

To be foolproof, a total crack length of 1.5 mm was set as a lower bound to give

start to the actual cracking procedure.

In the light of time needed and data required, it was decided to perform a K-

increasing test, with 8% steps.

In order to predict the load Pmax needed to elongate the crack of 0.5 mm, the

58

6.2 Precracking and crack growth

following formula was used

Pmax =(1 + 0.08)∆Kavg,prev

0.9(KP

)avg

with

∆K = (Pmax − Pmin)1√Wb

(2 + α)−5.6α4 + 14.72α3 − 13.32α2 + 4.64α + 0.886

(1− α)32

where b and W are respectively the thickness (1.8 mm) and the length (64 mm)

of the sample, and

α =aavgt

Rearranging, we find

∆K = f(a, t)∆P

Wt

√W√a

√a

which leads backs to the known formula

∆K = f(a, w)σnom√a = Y σnom

√a

The pre-set number of cycles was a speculation too, defined as

Ncycles =0.5(

dadN

)new

with (da

dN

)new

=

(da

dN

)prev

((KP

)avg,prev(

KP

)avg,new

)m

The value of m was assumed to be 3 for the first two samples, whereas it was de-

fined as 3.3 for the last specimen, on account of results obtained with the second

piece.

The loading ratio R was set at 0.1, a normal value for this type of experiment,

and the input signal was sinusoidal..

Optical measurement was performed, with a microscope capable of a maxi-

mum magnification of 230x, and resolving lengths as small as 0.01 mm at the

highest enlargement. This allowed to remain in the margins prescribed by the

test method.

59

6. THE EXPERIMENT

Needless to say, using such a method the experiment had to be stopped for the

crack growth to be evaluated; the highest effort was made in trying to minimize

these discontinuations and keep the cycles as close as possible, for the sake of

precision.

The sample was secured to the machine structure through two gudgeon pins,

which longitudinal movement was restricted by hairpin cotter pins, and transver-

sal motion was restricted by four thick washers.

For higher loads, an additional structure was mounted on the sample, consisting

of two bulky steel plates bolted together, separated by two greased Teflon sheets

in which to insert the sample. This helped to reduce lateral vibration (Kii mode)

and avoid specimen slanting.

The main, and eventually only, problem encountered during the experiment

was the tuning of the parametres of the machine: the two most relevant were the

convergence rate and the so called P-gain. The former accounts for the percent-

age of the error (betwixt the command and the response) the control unit tries

to correct each cycle, while the latter represents, briefly, how quick the machine

reaches the correct Pmax.

As obvious, a high value for both parametres would be preferable; nonetheless,

this induces a high instability in the machine, leading to huge drifts from desired

conditions, and finally to insoluble troubles. Too low values would have trig-

gered, on the other side, wrong loading conditions, in particular a much different

R. Compromise was found at the expenses of the first sample: the values set for

the following pieces were a P-gain of 3 and a convergence rate of 8%.

When a high number of cycles (hundreds of thousand) was set, no big attention

was paid to the initial ones, allowing for a drift from the desired load conditions

in the starting 10% of the cycles, at a fixed frequency of 40 Hz. When higher

precision was needed, and a number of cycles in the order of hundreds was pre-

dicted, much lower frequencies (2 Hz) were utilized. Another approach was tried,

consisting in continuously increasing frequency, but was later abandoned when

different frequencies were found to be unstable (17 and 27 Hz among all).

60

6.3 Results

6.3 Results

Unfortunately, it was possible to obtain only one complete Paris curve, this owing

to the fact that the machine gave rise to multiple hurdles.

The cracks always propagated on a linear trajectory and the difference between

the two sides was never remarkable.

The first sample experienced an overload which resulted in a distortion, thereby

eliminating the possibility of further tests. Moreover, problems in early calibra-

tion suggest that the few obtained data are not reliable.

The second sample encountered no substantial difficulties (just a second pre-

cracking was needed at about half the procedure): the conditions and procedure

completely comply with the standard test method, and the correspondent Paris

curve can be considered correct (6.1).

The third sample was on a similar path when a sudden overload (in the ranks of

5 kN) took place, almost destroying it. Even so, some relevant data have been

registered and can be used to better understand behaviour at low values of ∆K

(6.3).

A final data analysis, performed on the maximum and minimum values of each

cycle, showed a negligible variance in loads. Therefore, the actual load ratio R

can be safely considered to be equal to the nominal one.

61

6. THE EXPERIMENT

Figure 6.1: Paris curve for second TWIP sample.

62

6.3 Results

Figure 6.2: Partial Paris curve for third TWIP sample.

63

6. THE EXPERIMENT

Figure 6.3: Paris curve comparison: second sample in green, third sample in

purple.

64

7

Conclusions

The Paris curve is more of a functional and helpful result, than an experiment

to anon draw any sort of definitive concept from. Much like the Wohler curve,

or other fatigue diagrams, its usefulness can be fully appreciated under operative

conditions only.

Natheless, it is still possible to glean some enlightenment from our research.

First, it is definitely material, being our subject what it is, to write down some

meaningful digits.

Software-based interpolation provides the following values, as regards Paris curve

coefficients, for the second sample

C = 1.56 ∗ 10−9mm;m = 3.325;Kic = 93.352MPa√m

and for the third

C = 2.22 ∗ 10−9;m = 3.266

with R2 (coefficient of determination) values of 0.9907 and 0.9699, forsooth ac-

ceptable ones.

The combined Paris curve has the following parametres:

C = 2.16 ∗ 10−9;m = 3.241

These outcomes lead to the already widespread belief that twinning mechanism

is not relevant in case of fatigue.

Withal, as a consequence of the limited thickness of the specimens, it is ticklish

65

7. CONCLUSIONS

to collate this results with known figures; this notwithstanding, some comparison

can be carried out.

The m value, for instance, resembles the behaviour of austenitic stainless steels,

oftentimes reported in literature to lie close to 3.25 (38), exceeding known data for

martensitic (m ≈ 2.25) and ferrite-perlite steels (m ≈ 3). Existing investigation

on TWIP (39) outlines slightly different characteristics, with m included between

2.4 and 2.7, and C around 2.0 ∗ 10−8 mm, with similar thickness involved. This

is somehow puzzling, and makes it taxing to come up with a convincing clarifi-

cation.

A partial explanation can be tracked down to the very distinct operation con-

ditions of the CT sample: as a matter of fact, the loading takes place only in

the plastic zone, and the local evolution hinders the formation of new twins,

simultaneously expediting expansion of low to intermediate dislocations. The de-

pendance on twin presence can help to elucidate the differences spotted.

Eke, the dissimilarities in composition can account to some extent: the apparent

brittleness can be easily explained by the relatively low percentage of Mn con-

tained.

In all likelihood, the most important comparison which could be made is the one

with a steel the TWIP is expected to replace. One germane example is, doubt-

less, the DP1000, a dual phase steel employed where high strength and weight

reduction are ineludible priorities. Literature (40) makes a Paris curve available

for comparison (7.1, 7.2).

Data for DP1000 Paris curve are

C = 8.85 ∗ 10−9mm;m = 2.88;Kic = 74.75Mpa√m

The Paris exponent for the dual phase is lower, thus suggesting a more prominent

brittle behaviour; yet, the TWIP stress intensity fracture at rupture is consid-

erably higher, and the crack growths, if we set the two curves side by side, are

well-nigh analogous, with a slight advantage for the TWIP steel especially at low

∆K.

As an all-embracing conclusion to this dissertation, it is possible to safely state

that no evidence points in the direction against TWIP employment, at least for

66

what concerns fatigue crack growth. As it happens, the behaviour seems to be

as predictable and as suitable as other steels currently utilized for critical Body-

in-White parts.

Provided other crucial and inescapable issues are found a solution to, TWIP steel

can indeed represent a resource for the automotive industry.

Figure 7.1: Paris curve data for two steels: DP1000 in blue, TWIP in red.

67

7. CONCLUSIONS

Figure 7.2: Paris curve (interpolation) for two steels: DP1000 in blue, TWIP in

red.

68

Appendix A

Experimental values

In the following two pages, it is possible to find the exact values used to plot the

two Paris curves in chapter 6. Formulae for computation of ∆K can be found

there too.

69

A. EXPERIMENTAL VALUES

Plotted values of ∆K and da/dN , sample number 2

∆K da/dN

MPa√m mm/cycle

12.686 9.45E-6

13.729 1.15E-5

14.760 1.34E-5

18.261 1.95E-5

20.921 3.02E-5

22.448 5.10E-5

24.290 5.57E-5

26.301 6.62E-5

28.614 1.09E-4

31.097 1.46E-4

33.595 1.87E-4

36.414 2.24E-4

39.370 2.83E-4

42.530 3.67E-4

46.102 4.50E-4

49.843 5.61E-4

53.579 7.07E-4

57.749 1.02E-3

62.402 1.34E-3

67.328 1.70E-3

72.838 2.09E-3

79.253 2.77E-3

85.143 4.67E-3

93.352 1.01E-2

70

Plotted values of ∆K and da/dN , sample number 3

∆K da/dN

MPa√m mm/cycle

12.488 7.70E-6

14.978 1.24E-5

16.354 2.42E-5

17.684 3.08E-5

19.083 3.96E-5

20.554 4.14E-5

22.332 5.23E-5

24.689 8.17E-5

26.908 9.04E-5

71

A. EXPERIMENTAL VALUES

72

Appendix B

Influence of material selection on

chassis performance and

manufacturing

In the introduction, we mentioned as selection criteria safety and weight reduc-

tion. Of course, these two are the most relevant, but a bunch of others exists,

and it is interesting to shortly investigate why and to which extent they influence

material selection.

From a physical point of view, the main task of the chassis is to carry loads,

being them external (both instantaneous and fatigue) or internal. On a bounce,

vertical loads can be up to seven times the weight of the car, and the chassis must

be able to undergo uneven and not constant cycles for at least 300000 km (41).

Nowadays, even lateral acceleration can reach up to 1 g, subjecting the chassis

and subassemblies also to torsion and bending. It is also essential, for a chassis

designer, to make a good weight balance, approaching as much as possible the

desired 50:50 distribution. For this reason, lighter materials are more likely to be

used in overhangs and deformation structures.

From a dynamic point of view, the most valuable parametre is the stiffness,

declined in torsional and bending, depending on the application of the force:

when substituting steel with other materials, it is necessary to make sure the

73

B. INFLUENCE OF MATERIAL SELECTION ON CHASSISPERFORMANCE AND MANUFACTURING

characteristics are similar; if not, the car will roll (or pitch) much more, making

it almost impossible to drive.

Moreover, parting for a moment from pure science, a great role is played by the

handling characteristics: the response the driver feels changes dramatically when

materials are changed, for example when a steel suspension strut is substituted

with an aluminium one.

In case of sports cars, the buyers’ choice is based partially, if not for the higher

part, on driving emotions, and it is fundamental to understand how they are

modified by different materials.

In the last 20 years, comfort has become of considerable importance also for

medium-to-low size cars, and is now a priority for all new projects. The suppres-

sion of vibrations and the acoustic isolation are probably the two main aspects

taken into account by potential purchasers during a test drive, and for sure the

easiest to evaluate.

Topological optimization is aimed at increasing torsional eigenfrequencies, reduce

acceleration levels at acceleration points and keeping the forces as low as possible,

i.e. in the crossmembers.

Another important area is in the mountings, where a large impedance gap is re-

quired in order to filter the input from driveline and suspension.

Material selection also has an impact on the shapes that can be obtained, both

from a aesthetic and functional perspective.

From the manufacturing and assembly view point, materials play a great role

both for cost and reliability.

First of all, stamping and drawing possibilities for the different materials should

be taken into account during the design process.

Second, the selection of the joining techniques to be employed strongly depends

on materials: case in point, where welding two steel plates is easy, connecting

a steel and an aluminium plate is likely to require riveting, or adhesives. This

impacts the manufacturing line too, if a machine switching is demanded, thus

requiring money to be invested. Furthermore, the plant will require more skilled

workers, or will have to hire new ones, able to correctly exploit the technological

74

improvements.

In the last decade, the consumers have become to be concerned about the sus-

tainability of the cars they buy, and the regulations about recycling have become

much stricter. Car makers are now required to produce vehicles with a certain

percentage of recyclable parts, and this affects materials choice as well.

It is probably evident, but still worth of mentioning, the role played by price:

if carbon fiber cost as steel, there is no doubt we would driving much safer and

faster cars; this could seem like an unintelligent remark, but the history of cars

is sparkling with huge errors driven by a inappropriate consideration of money.

75

B. INFLUENCE OF MATERIAL SELECTION ON CHASSISPERFORMANCE AND MANUFACTURING

76

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78

determination of the paris curve for a twip steel for body-in-white applications

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