ON THE RELATIONSHIP BETWEEN FRACTURE TOUGHNESS AND FRACTAL DIMENSION= IN AISI 4340 STEEL

ON THE RELATIONSHIP BETWEEN FRACTURE TOUGHNESS AND FRACTURE SURFACE FRACTAL DIMENSION IN AISI 4340 STEEL

LUIS RAMOS = CARNEY

A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL

OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF

DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORI= DA

2006

Luis Ramos = Carney

This docume= nt is dedicated to Robert F. and Maria J. Carney

TABLE OF CO= NTENTS

Page<= /o:p>

ACKNOWLE= DGEMENTS...................................................................= ..................... &= nbsp; iv

LIST OF TABLES.....................................................................= ................................. = vii

LIST OF FIGURES....................................................................= ................................ = viii

&nb= sp;

ABSTRACT= ...........................................................................= .................................... &= nbsp; xi

CHAPTER

1 INTRO= DUCTION....................................................................= ...................….. &= nbsp; 1

&nb= sp;

General Comments on Fracture Toughness………………………= 230;………… &= nbsp; 2

Significance of Fracture Surfaces………………………R= 30;…………………... &= nbsp; 3

2 LITER= ATURE REVIEW..............................……………R= 30;………………… &= nbsp; 5

&nb= sp;

Fractal Geometry………………………R= 30;…………………………&#= 8230;……… &= nbsp; 5

Application of Fractals to Material Surfaces………………&= #8230;……………….. &= nbsp; 12

3 MATER= IAL AND METHODS....................................................................= …. = 30

&nb= sp;

Material……………………= …………………………̷= 0;…………………….. = 30

Fracture Toughness Specimens………………………= 230;……………………. &= nbsp; 31

Tensile Testing………………………̷= 0;…………………………= 230;………... &= nbsp; 40

Fracture Toughness Testing………………………̷= 0;………………………… &= nbsp; 40

Fractal Dimension Measurements………………………= …………………….. &= nbsp; 42

4 RESUL= TS AND DISCUSSION.................................................................= ....... = 49

Results……………………&= #8230;…………………………= ;……………………… 49

Discussion…………………̷= 0;…………………………= 230;………………….. = 50

5 CONCL= USIONS.....................................................................= .......................... 97

TABLE OF CO= NTENTS

(Continued)=

Page<= /o:p>

APPENDIX

A SAMPLE OF L= OAD DISPLACEMENT DIAGRAMS......................................... 100

B TEXBO= OK IMAGES USED TO VERIFY FRACTAL MEASUREMENTS.. 102

C AISI 4340 SPECIMEN; SLIT-ISLAND FRACTAL IMAGES….................... 103

REFERENC= ES………....................................................= ............................................ 131

LIST OF TABLES

Table &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; Page

3-1= Chemical Composition of Raw Material........................................…………= ;……... 30

&nb= sp;

3-2= Tempering Schedule for CT Samples..............................................………&= #8230;…….. 32

3-3= Test Images and Resulting D*.........................................................……&#= 8230;………... 46

3-4 Fractal Dimension Results............................……………̷= 0;……………………….. 48

4-1 Fracture toughness - Fractal Dimen= sion Results............................……………̷= 0;…. 49

4-2 Percentage of Area Covered by Micr= ovoids v. Sample group.……………………..<= span style=3D'mso-tab-count:1'> 53

4-3 Summary of Principal Findings Deri= ved from Mathematical Models…………….. &= nbsp; 65

LIST OF FIGURES

Figure &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; Page

2-1= Representation of roughness average….........................................………= ;………. 6

&nb= sp;

2-2=    Lines of similar R_{a but
different fractal
dimension.........................………………=
;        &=
nbsp;   6}

2-3 Variation of boundary length= with ruler length...................................……………= ; &= nbsp; 7

2-4 Typical Richardson plot…………………………&= #8230;…………………………= ;…… 8

2-5 Typical cross section result= ing in slit-islands………………………= …………….. &= nbsp; 9

2-6 Material class and correspon= ding interpretation of a₀………………&#= 8230;…………... 14

2-7 Fracture toughness-fractal dimension relationships in ceramics……………̷= 0;….. = 15

2-8 Graphical representation of = C/r₁ and D*…………………………= 230;……………... 16

2-9 D* v Charpy impact for sever= al steels…………………………= ;………………….. 20

2-10 & 2-11<= span style=3D'mso-spacerun:yes'> Fracture toughness v. fractal dime= nsion under two conditions………….. 22

2-12 Fractal plot of AISI 4340 at diffe= rent tempering temperatures…………………R= 30; 25

2-13 Linearized fractal plot…………………………&= #8230;…………………………= ;……. 25

2-14 Fractal dimension v. tempering temperature………………………&= #8230;…………… 25

2-15 & 2-16<= span style=3D'mso-spacerun:yes'> Interpretation of fractal plots…………………………= …………………... 26

2-17 Step length v. energy of fracture………………………R= 30;………………………… 27

2-18 Fractal dimension v. surface energy…………………………= ;…………………….. 28

3-1 to 3-4 Typical specimen microstructures a= t high and low hardness values…………. 34

3-5 Microhardness traverse………………………R= 30;…………………………&#= 8230;……… 35

3-6 to 3-9 Images of specimen prior-Austenite grains…………………………= ;……….. 37

LIST OF FIGURES

(Continued)

Figure &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; Page

3-10 EDS spectrum of overall intergranu= lar region…………………………= ;………….. 38

3-11 EDS spectrum of overall intergranu= lar region (expanded)……………………&#= 8230;.. 38

3-12 EDS spectrum from fracture surface fissures………………………R= 30;………….. 39

3-13 EDS spectrum comparing overall fra= cture surface v. fissure region……………… 39

3-14 Hardness v. yield and ultimate ten= sile strengths………………………= 230;……….. 40

3-15 & 3-16<= span style=3D'mso-spacerun:yes'> CT specimens of high and low fract= ure toughness………………………. 41

3-17 CT specimen cut and polished to re= veal slit-islands………………………= …….. = 43

4-1 Fracture toughness v. D*…………………………= 230;…………………………&= #8230;. 50

4-2 to 4-7 Fractographic images of CT fracture surfaces………………………R= 30;….. 51

4-8 Fracture toughness v. D*…………………………= 230;…………………………&= #8230;. 54

4-9 Graph of Ln(K_IC) = v. D*…………………………= 230;…………………………&= #8230;…. 58

4-10 Graph of K_IC v. (= D*)^1/2 …………………………̷= 0;…………………………= 230;… 59

4-11 Graph of a₀^{1/2 v.
K_IC……………………̷=
0;…………………………=
230;…………. 61}

4-12 Graph a₀ v. r_{p
…………………………̷=
0;…………………………=
230;………….. 63}

4-13 Stress field ahead of a crack tip…………………………&#= 8230;……………………… 66

4-14 Strain field ahead of a crack tip…………………………&#= 8230;……………………… 66

4-15 Fracture toughness v. temperature = for ASME SA533 steel………………………. 69

4-16 Generation scheme for Koch fractal curve…………………………= ……………… 72

LIST OF FIGURES

(Continued)

Figure &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; &= nbsp; &nbs= p; Page

4-17 Idealized fractal surface initiator patterns………………………R= 30;………………. 73

4-18 Fractal void nucleation model, X-Y plane…………………………= ……………. 76

4-19 Fractal void nucleation model, X-Z plane…………………………= ……………. 77

4-20 Optical micrograph of A36 steel at= the crack tip…………………………&#= 8230;…... 82

4-21 SEM fractograph of A36 steel showi= ng microvoids………………………&#= 8230;….. 82

4-22 to 4-27a Fractographs of high fracture toug= hness specimen…………………...83-88

4-28 to 4-33a Fractographs of low fracture tough= ness specimen……………………89-94

4-34 Cross section of AISI 4340 steel a= t 40 HRC…………………………&#= 8230;………… 95

4-35 Cross section of AISI 4340 steel a= t 40 HRC…………………………&#= 8230;………… 95

4-36 Cross section of AISI 4340 steel a= t 40 HRC…………………………&#= 8230;………… 96

CHAPTER 1

Introduction

Since the inception of the science of fractography, fracture surfaces have been an invaluable source of information regarding t= he fracture event. Investigation= s into the various modes of failure have produced significant advances in the qualitative and quantitative understanding of how and why fractures develop= ^1-3. The application of this knowledge = has increased the safety and efficiency with which new structures and mechanical system can be created and used. New investigative methods and techniques promise to add more information to thi= s important field of materials science.

In recent years a great deal of interest has develope= d in the area of quantitative fractography.&nbs= p; That is, measuring the size of fracture features or material propert= ies directly from a fracture surface. One of the concepts that has emerged as a potentially useful tool in these efforts is that of fractal geometry.=

Fractal geometric analysis, when applied to a fracture surface, provides a measure of its irregularity. To date, a large number of studies= using these techniques have been performed on a variety of fracture surfaces. A subject of particular interest h= as been the potential relationship that exists between plane-strain fracture toughness, usually expressed as K_IC, and the degree of fracture surface tortuosity. Results h= ave been mixed. Some studies have= shown a positive correlation, some have shown a negative correlation and others s= till report no correlation at all⁴.

A review of the literature shows that the efforts mad= e to date in correlating K_IC to the fractal dimension have employed a wide variety of materials, fracture modes, and measurement methods. In most of these studies the mater= ial is unique, the fracture process is not well understood and the preferred measurement method is debatable. The variety and complexity of the information obtained has made the assessment of material properties-surface feature relationships, and the drawing of general conclusions, a difficult task.

Despite inconsistencies in the literature, some gener= al consensus is now emerging among researchers⁴. The separate elements employed in = the construction of a fractal dimension-fracture toughness relationship are becoming more formalized and accepted as new studies demonstrate their validity. The present study is intended to become part of one of the building blocks which demonstrates how fractal analysis might be applied and what material property information mi= ght be gained from fracture surfaces. In particular, this study focuses on the fractal dimension-fracture toughness relationship in heat treated AISI 4340 steels. The significance of any relationsh= ips that emerge will also be considered.

General Comments on Fracture Toughness=

It is the principal goal of every structural or mechanical engineer to prevent the sudden failure of their particular component or assembly by des= ign. The consequences of ignoring the conditions that lead to fracture are disastrous. It is, therefore, of extreme inter= est to understand the factors that may control and predict this potential outcome.=

Unstable fracture in metals typically begins from some preexisting fl= aw or crack. Cracks may develop = in a component for a number of reasons; fatigue, stress-corrosion and hydrogen embrittlement being but a few. Regardless of how the flaw initiates and grows, unstable crack propagation is, in gene= ral, the final event that causes structural failure. A knowledge of when a component is approaching this end condition is vital in original design and failure prevention. Furthermore, a knowledge of what material features assist in delaying or preventing this occurrence are, for obvious reasons, crucial.

The parameter most commonly used to describe a material’s resistance to unstable fracture is fracture toughness. It is generally represented by val= ues of the plane-strain fracture toughness factor K_IC. In practice, a structure containin= g a flaw can assume any value of stress intensity at a crack tip up to this critical quantity. When it is reached, the component fails catastrophically. Stress intensity depends on applied stresses, flaw size and flaw geometry.&nbs= p; The critical stress intensity is, in general, a material related property.

The value = of fracture toughness (critical stress intensity factor, K_IC) for a particular material is commonly found through standardized testing methods. It is now well known= that a number of geometric and microstructural features such as specimen size, loading geometry, crack geometry and orientation, grain size, second phase particles etc., will affect the measurement and apparent value of this parameter. It is, therefore of practical and scientific interest to determine how material characteristics influence fracture toughness, the various methods through which this proper= ty may be properly measured and how the fracture mechanism itself is related to the critical stress intensity factor⁵.

Significance of Fracture Surfaces

Just as re= search on the relationship between fracture toughness and microstructure has added= new knowledge about the mechanism of fracture, investigations into the relation= ship between fracture surface features and fracture toughness offer to do the same. There is reason to thin= k that these two factors are related.⁶

An examina= tion and comparison of metallic fracture surfaces created by overload reveals that t= here is an apparent connection between fracture surface roughness and the associ= ated fracture toughness. That is, materials with relatively smooth fracture surfaces appear to have a lower resistance to fracture. This = notion seems to be intuitively correct. It appears logical for relatively smooth surfaces to consume little energy in their formation and, thus, reflect this fact in low values of fracture toughness. The mathematical t= ools are now available to determine, in a quantitative way, if this notion is in= deed correct and what that relationship might be. The connection between microstruct= ure, fracture surface texture and fracture toughness is expected to lead to a be= tter understanding of the fracture process, the development of tougher alloys, a= nd the identification of pre-fracture material properties after a component has failed.

In order t= o study the relationship between fracture toughness and the resulting fracture surf= ace topography, fractal geometry will be used.= Fractal geometry is a non-Euclidean geometry that can be employed to= quantitatively describe the tortuosity of fracture surfaces, which are characterized by th= eir fractal dimension. A 4340 ste= el will be heat treated to vary the fracture toughness and correspondingly cha= nge the fracture surface topography. These heat treatments will cause the behavior to vary from fracture = in a brittle manner to fracture in a ductile mode. The results will be compared to ex= isting and new theories to explain the observed relationship between toughness and= the fractal dimension of the fracture surface.

= ;

CHAPTER 2

LITERATURE REVIEW

Fractal Geome= try

Concepts

The concep= t of fractal geometry is relatively young. Fractal geometry expands the concept of dimension and recognizes that there are an infinite number of dimensions in between the topological dimensions of 1, 2 & 3. I= n the particular case of irregular surfaces, for instance, it is possible to obta= in fractal dimension values between 2.0 and 3.0. This number, to a large extent, represents the degree of surface irregularity. A planar surface would be expected= to produce values near 2.0 while more tortuous surfaces would be expected to be closer to 3.0.⁷

Although r= ougher surfaces are thought to exhibit a greater fractal dimension, this statement= is not strictly true. The indust= ry standards that cover roughness usually define it as a number which represen= ts an average deviation from a mean line.&nbs= p; For example; the International Standards Organization (ISO 4287) def= ines the most commonly used roughness parameter R_a (roughness average= ) as⁸:

(2-1) R_a =3D 1/L _<=
![endif]>

This expre= ssion is illustrated graphically in Figure 2-1.

X direction

L

<= /span>

0

<= ![endif]>

Figure 2-1.= Representation of roughness averag= e (R_a) per ISO 4287.

= ;

Line 2

Line 1

<= ![endif]>

Figure 2-2.= Lines of similar R_a but different fractal dimension.

Consider t= he two lines in Figure 2-2. If the traditional engineering definition of roughness is applied to these lines t= hen R_a would be found to be very similar. Both profiles deviate about the sa= me amount from a mean line. If t= heir fractal dimensions were measured, however, these values would be expected to differ significantly. Line 2 = is considerably more tortuous (irregular) than line 1.

An accurate definition of fractal dimension, then, is not based strictly on roughness.<= span style=3D'mso-spacerun:yes'> Fractal objects possess at least t= hese two important qualities:

1) Scale invariant self-similarity: Fractal objects display identical = levels of irregularity no matter what the magnification of the object. In fact, given two images of the s= ame object at vastly different magnifications, the viewer would not be able to determine the actual scale of the object without a reference scale.

2) The geometric features of length, = area or volume change at a predictable linear rate on a log-log graph when the measurement device changes. F= or example, if an irregular line is measured with a step length of 1 cm (using dividers, for instance) the total length will be longer than if a step leng= th of 2 cm is used.

= ;

Measurement Through Richardson Plots<= o:p>

The scienc= e of measuring fractal curves made an important advance when Richardson attempted to correlate the l= ength of national boundaries with their respective military and economic conflicts. The boundary measurements were made from various maps, however, he quickly realized that these perimeter values varied widely depending on the scale of the map. He further learned that boundary dimensions also varied if the scale was kept constant and the measuring unit was altered.⁹

Figure 2-3= shows Richardson’s measurement process. Each of = the four identical images produces a different boundary length depending on the ruler length.

<= ![endif]>

Figure 2-3. Variation of boundary length with = ruler length. Smaller rulers produce larger perimeters [Ref. 4, pg 27, with kind permission of Springer Science & Business Media].

The variat= ion of perimeter with measurement scale can be demonstrated with a log-log graph.<= span style=3D'mso-spacerun:yes'> The typical result is a linear relationship as shown in Figure 2-4. Plots of this type are often used today in the fractal analysis of irregular lines or perimeters and are referred to as = Richardson plots.

Stride length too large for object size

<= /span><= /span>

Stride length near or less than boundary width.

<= /span><= ![endif]>

Figure 2-4. Richardson plot produces a straight line on a log-log graph. The slope gives the fractal dimension. Note that the devi= ations from linearity at the ends are due to measuring unit effects. [Ref 4, pg 29, with kind permission of Springer Science & Business Media].

The mathem= atical representation of a Richardson= graph (for a boundary line), and the one which is characteristic of fractal relationships, is given by:

(2-2) = L =3D Ks^(1-D)

The linearized version is:

(2-3) = Log(L) =3D Log (K) + (1-D)Log(s)

where in both (2-2) and (2-3):

L =3D Measured length. = K =3D A constant.

S =3D The measurement scale. D =3D The fractal dimension.

Other Measurement Techni= ques

Dimensional analysis

Mandelbrot= and co-workers¹⁰were the first to examine fracture surfaces using techniques derived from Richar= dson’s observations. For the case of fracture surfaces, the Richard= son relationship was modified to read as follows:

(2-4) A =3D Kp^2/D

The linearized version is:

(2-5) Log(A) =3D Log(K) + 2/D_=
s Log(p)

where in both (2-4) and (2-5):

A =3D Measured area. = K =3D A constant.

p =3D The perimeter length. D_{s =3D
The surface fractal dimension.}

2/D_s =3D The slope obtained from the Log(A) - = Log(p) plot.

In practic= e, this technique is applied by mounting and polishing a fracture surface in such a= way as to allow only the cross sections of "high spots" or "islands" of the fracture surface to show on a sectioning plane.<= span style=3D'mso-spacerun:yes'> Figure 2-5 illustrates an example = of a group of such islands.

<= /span>

Low spot

<= ![endif]>

Figure 2-5. T= ypical cross section through a fracture surface revealing several islands and low spots.

Once a set= of islands is obtained, the surface may be digitized and a computer program ma= y be applied to obtain data measuring areas and perimeters for all the islands available. The data is then u= sed to obtain graphs of log area v log perimeter.

The Minkowsky dimension<= sup>11

The Minkow= sky dimension is determined for a line or boundary by sweeping out the feature = from beginning to end with circles of various radii. A graph of log area of the circle = v log radii of the circle produces a slope which gives the Minkowsky fractal dimension.

The Kolmogorov dimension= ¹²

The Kolmog= orov dimension is determined for a line or boundary by covering the feature with grids of different sizes. A g= raph of log grid size v log number of grids through which the feature passes produces a slope which gives the Kolmogorov fractal dimension.

Fourier analysis^13<=
o:p>

A log-log = graph is constructed of the magnitude v. frequency for the Fourier transform. This graph produces a straight line whose slope is related to the fractal dimension.

Selection of Measurement Technique and Inherent Differences.