Size Effect on Fatigue Crack Growth of a Quasibrittle
Material
A THESIS
SUBMITTED TO THE FACULTY OF THE GRADUATE SCHOOL
OF THE UNIVERSITY OF MINNESOTA
BY
Jonathan Manning
IN PARTIAL FULFILLMENT OF THE REQUIREMENTS
FOR THE DEGREE OF
MASTER OF SCIENCE
Professors Joseph F. Labuz and Jia-Liang Le
January 2013
i
Abstract
The Paris-Erdogan law describes the rate of fatigue crack growth as a function
of the amplitude of the applied stress intensity factor. This equation, however, does not
include a dependence of the crack growth on the structure size, which has been
observed experimentally for concrete. The size effect on the fatigue crack growth is
derived based on two hypotheses: (1) the scaling of the critical energy dissipation for
fatigue crack growth has the same form as that of fracture energy for monotonic
loading; (2) the difference in transitional sizes between the fatigue and monotonic
loading is purely due to the difference in the fracture process zone (FPZ) size. The size-
dependent fatigue crack growth law is verified experimentally through size effect tests
on Berea sandstone. Using digital image correlation, it is shown that the FPZ length is
approximately 7 mm and 11 mm for monotonic and cyclic loading, respectively.
Optimal fitting resulted in transitional sizes of 34 mm and 54 mm for monotonic
loading and cyclic loading, respectively, which shows a proportional relationship
between the FPZ length and the transitional size.
ii
Table of Contents
List of Tables……………………………………………………………………. iii
List of Figures…………………………………………………………………… iv
1.0 Introduction…………………………………………………………………. 1
1.1 Motivation………………………………………………………………….. 1
1.2 Objective……………………………………………………………………. 2
1.3 Scope and Organization…………………………………………………….. 2
2.0 Background ………………….……………………………………………… 3
2.1 Fracture and Fatigue………………………………………………………… 3
2.2 Size Effect on Nominal Strength……….…………………………………… 4
2.3 Fracture Process Zone………………………………………………………. 6
2.4 Digital Image Correlation….………………..……………………..………... 8
3.0 Fatigue Crack Growth – Theoretical Framework………………………….. 12
3.1 Fatigue Crack Growth at the Large-size Limit…...…………………………. 10
3.2 Scaling of Fatigue Crack Growth Law………..…………………..………… 15
4.0 Monotonic Size Effect Test…..……………………………………………... 17
4.1 Experimental Setup…………………………………………………………. 17
4.2 Nominal Strength Calculation………………………………………………. 18
4.3 Fracture Properties from Size Effect Analysis……………………………… 22
4.4 Fracture Process Zone Length – Monotonic Loading………………………. 25
4.5 Displacement Gradient Method for Tracking FPZ…………………………. 35
5.0 Fatigue Crack Growth Size Effect Test………...……………………………… 40
5.1 Experimental Setup…………………..……………………………………... 41
5.2 Compliance Calibration Method……………………………………………. 43
5.3 Paris-Erdogan Law calculated from Compliance Method………………….. 41
5.4 Fracture Process Zone Length – Cyclic Loading …………………………... 51
5.5 Paris-Erdogan Law Calculated from DIC Method………………………….. 56
6.0 Conclusions……………………………………………………...…………………. 62
7.0 References………………………………………………………………………….. 64
iii
List of Tables
4.1 Specimen properties for monotonic loading……………………………………. 19
4.2 Average nominal strength………………………………………………………. 19
4.3 Fracture parameter for Berea sandstone………………………………………… 23
4.4 Displacement gradient at tip of FPZ for a medium sized specimen…………….. 37
5.1 Specimen properties for cyclic loading…………………………………………. 40
5.2 Young’s Modulus used to calibrate CMOD compliance curves………………… 43
5.3 Cycles analyzed for Paris-Erdogan Law………………………………………… 47
5.4 Fracture process zone lengths for all four specimens……………………... 56
5.5 Cycles where the FPZ tip increases by 1 mm for specimen 2-4……………….... 57
iv
List of Figures
2.1 Size effect results for various rocks and ceramics……………………………….. 5
2.2 Fracture process zone schematic…………………………………………………. 6
2.3 Locations of acoustic emission at peak load…………………………………...... 7
2.4 Digital image of a painted specimen…………….……………………………..... 9
2.5 Schematic of DIC process……………………….……………………………..... 11
3.1 Cyclic FPZ composed of many micro-cracks…….……………………………… 13
4.1 Specimen geometry for three-point bending experiments...……………………… 17
4.2 Experimental setup ………………………………………………………………. 18
4.3 Load vs. CMOD for small specimens……………………………………………. 19
4.4 Load vs. CMOD for medium specimens…………………………………………. 20
4.5 Load vs. CMOD for large specimens….…………………………………………. 20
4.6 Nominal strength vs. CMOD for one specimen of each size…………………….. 21
4.7 Linear regression of experimental results………………………………………… 23
4.8 Size effect of nominal strength…………………………………………………… 24
4.9 Horizontal displ. contours – medium specimen – 30% to 40% peak load………. 25
4.10 Horizontal displ. measurements along horizontal lines – 30% to 40% peak……. 26
4.11 Horizontal displ. contours – medium specimen – 60% to 70% peak load………. 27
4.12 Horizontal displ. measurements along horizontal lines – 60% to 70% peak …… 28
4.13 Horizontal displ. contours – medium specimen – 90% to 100% peak load……… 29
4.14 Horizontal displ. contours – medium specimen – 100% to 90% post-peak load… 30
4.15 Post peak fracture process zone vs. traction free surface – medium specimens…. 31
4.16 Horizontal displ. contours – large specimen – 95% to 100% peak load………… 32
4.17 Horizontal displ. contours – large specimen – 100% to 95% post-peak load…… 28
4.18 Post peak fracture process zone vs. traction free surface – large specimens…….. 33
4.19 Post peak fracture process zone vs. traction free surface – small specimens……. 34
4.20 Displacement gradient vs. distance from the notch tip…………………………... 36
4.21 Displacement gradient calculation……………………………………………….. 37
4.22 Displacement gradient at tip vs. incremental CMOD – medium specimen……... 38
4.23 Displacement gradient contours for a medium specimen………………………... 38
4.24 Displacement gradient at tip vs. incremental CMOD – large specimen………… 39
4.25 Displacement gradient contours for a large specimen…………………………… 39
v
5.1 Load vs. time for cyclic 3-point bending tests…………………………………… 40
5.2 Zoomed in view of CMOD gage, Numerical analysis…………………………… 41
5.3 CMOD vs. Effective crack length ratio……………………….…………………. 42
5.4 Load vs. CMOD for different cycles…………………………………………….. 44
5.5 Peak and valley CMOD measurements vs. cycles………………………………. 44
5.6 Effective crack evolution vs. cycles……………………………………………... 45
5.7 da/dN vs. ΔK for medium specimen…………………………………………….. 46
5.8 Effective crack vs. cycles for specimen 2-5, showing stages of crack growth….. 47
5.9 Paris-Erdogan law for all four specimens plotted individually…………………. 48
5.10 Paris-Erdogan law for all three sizes with a fixed slope……………………….... 48
5.11 Size-dependent Paris-Erdogan law using D0 = 34mm…………………………... 50
5.12 Size-dependent Paris-Erdogan law using D0 = 54mm………………………….. 50
5.13 Horizontal displacement contours – medium specimen – cycle 6,000………….. 52
5.14 Horizontal displacement contours – medium specimen – cycle 13,900………… 53
5.15 Horizontal displacement contours – large specimen – cycle 12,000……………. 54
5.16 Horizontal displacement contours – large specimen – cycle 20,500……………. 54
5.17 Horizontal displacement contours – large specimen – cycle 21,600……………. 55
5.18 Horizontal displacement contours – medium specimen – cycle 2………………. 57
5.19 Effective crack ratio vs. cycles computed using DIC method…………………... 58
5.20 da/dN vs. ΔK calculated form DIC method for medium specimen 2-5…………. 59
5.21 da/dN vs. ΔK calculated form DIC method for large specimen 3-4…………….. 60
5.22 da/dN vs. ΔK calculated form DIC for all four specimens……………………… 60
5.23 Size-dependent Paris-Erdogan law applied to DIC results with D0 = 47mm…… 61
1
Chapter 1: Introduction
1.1 Motivation
In many fields of engineering, the designed products will be exposed to repeated
loading, i.e., cars driving over a bridge, gas expanding and contracting inside a vessel, or
weight being applied to a prosthetic leg when walking. Understanding how these
structures behave through time is of great interest in determining the safety of an aging
structure.
Geoengineering is a challenging field because the natural material within the earth
has a high degree of variability. However difficult, the expectations for safety and
reliability remain the same for structures built within the earth, i.e., tunnels, underground
buildings or oil pipe lines. Great works have been constructed in the field of
geoengineering in the past century, but many phenomena are still not completely
understood, and one of such fields is fatigue. Very few studies have focused on fatigue in
rock materials, and very little is published to explain the lifespan predictions of
geological structures that are exposed to cyclic loading.
This research is focused on investigating fatigue, specifically, subcritical crack
growth under cyclic loading in a rock. Understanding this phenomenon in greater detail
will provide a better means to design geological structures and predict a more accurate
lifespan.
2
1.2 Objective
The main objectives of the research are to:
Develop a size-dependent Paris-Erdogan equation
Determine the size effect on nominal strength and fracture properties from
monotonic three-point bending experiments
Investigate the validity of the size dependent Paris-Erdogan law by performing
cyclic three-point bending experiments
Compare the fracture process zone length under monotonic loading and cyclic
loading using digital image correlation
1.3 Scope and Organization
This thesis considers the classical Paris-Erdogan law and proposes a size-
dependent version. The thesis is organized into five chapters. Chapter 2 presents a brief
literature review of fatigue, size effect and the experimental technique called digital
image correlation. Chapter 3 discusses the size effect for the Paris-Erdogan law and
proposes a size dependent equation. Chapter 4 presents results from monotonic three-
point bending experiments that are used to determine the size effect of the nominal
strength and the fracture properties. The fracture process zone during monotonic loading
is also investigated using digital image correlation. Chapter 5 presents the results of
cyclic three-point bending experiments. The Paris-Erdogan law is computed as well as
the size-dependent version that is proposed in Chapter 3. The fracture process zone
under cyclic loading is also investigated. Chapter 6 presents a summary of the findings
and conclusions.
3
Chapter 2: Background
2.1 Fracture and Fatigue
Fatigue has been of interest for many years and the expression dates back to the
1800’s when it was used to explain the straining of masts on large sea vessels. The first
known study of fatigue was performed on metals around 1829 by W.A.J Albert, a
German mining engineer, who performed experiments on chains made of iron [1].
During the 1840’s, as larger infrastructure (bridges, railways, etc.) became more popular,
the study of fatigue began to expand [2, 3] and continued into the 20th
century, with many
studies focused on iron railways [4, 5].
Fracture mechanics began with the ground-breaking work of Inglis (1913) and
Griffith (1921), who provided a mathematical approach based on stress analysis and
energy balance [6, 7]. The work laid the foundation for the modern field of linear elastic
fracture mechanics (LEFM) that is largely responsible for the reliability and safety of
bridges, buildings, aircraft, etc. Irwin (1957) expanded on the work and introduced the
term stress intensity factor to quantify the near tip stress field for a crack in a linear
elastic body [8]. The initiation of a “linear elastic” crack, under monotonic loading
conditions, is characterized by a critical value of the stress intensity factor, Kc. This
value is typically classified further by the opening mode where, KIc is the critical stress
intensity factor under mode I separation (tensile opening). This value, KIc, is also
referred to as the fracture toughness of the material.
This approach to fracture was adapted for fatigue by Paris et al. [9] and Paris and
Erdogan [10] by relating the crack growth rate to the change in stress intensity factor
from maximum to minimum load. In the most general form, the Paris-Erdogan law is
( )
, (2.1)
where a = crack length, N = number of cycles, ΔKI = amplitude of stress intensity factor
calculated from linear fracture mechanics, and C and m are both empirical constants.
Paris & Erdogan performed experiments on aluminum alloys with different geometries to
test the law and found that it was consistent for many different combinations of stress
4
range and crack length [10]. It is important to note that Eqn. 1.1 holds only for fixed
environmental conditions and fixed load ratio, R.
(2.2)
Many researchers have expanded on the general form of the Paris-Erdogan law to include
the influence of the load ratio, R [11, 15], mixed-mode fatigue [13], and high temperature
fatigue [14, 15]. This law was originally developed for metals [10] and has since been
applied to ceramics [16] and concrete [17, 18].
Time-dependent crack growth under subcritical conditions (K < KIc) has also been
observed under non-cyclic loading [19]. This phenomenon can cause fractures and faults
in stressed rock masses, which is of interest when dealing with subsurface structures. It
has been reported that time-dependent fracturing of rock is also related to the increase in
seismicity prior to a volcanic eruption or earthquake [20]. The relationship between the
subcritical crack growth rate (da/dt) and the applied stress intensity factor is similar to the
Paris-Erdogan law and is expressed as a power law. The environmental conditions, such
as temperature, relative humidity and pressure, impact the growth rate. Water saturated
rock exhibits crack velocities that are 2-4 orders of magnitude greater than air saturated
rock [21, 22]. It has been reported that the growth rate for air saturated rock increases
with increasing relative humidity [23, 24].
2.2 Size effect on Nominal Strength
The theory of plasticity is commonly used to predict the “strength” of a structure,
where the strength of geometrically similar specimens is independent of structure size.
This theory does not hold true when a structure is made of brittle or quasibrittle materials
such as concrete, ceramics, or rock. Instead, there is a well pronounced size effect that
explains the change in strength as a function of structure size.
The sources of size effect include boundary layer effect, diffusion phenomena,
hydration heat, statistical size effect, fracture mechanics, and the fractal nature of crack
surfaces. Bazant and Planas [25] define the strength of a structure as the value of the
5
nominal stress at peak load, which is a parameter that is proportional to the load divided
by a cross-sectional area
(2.3)
where P = applied load, b = thickness of a two-dimensional structure and D =
characteristic dimension of the structure. For quasibrittle structures, the nominal strength
varies from a plastic limit for very small structures to a linear elastic fracture mechanics
(LEFM) limit characterized by a slope of -1/2. Through analysis of the energy required
for crack growth, Bažant introduced a size effect function that bridges the two limits,
plasticity and LEFM, and only depends on two parameters:
√ ⁄ (2.4)
where ft’ = tensile strength of the material, B = dimensionless constant and D0 = a
constant with the dimension of length [26]. Figure 2.1 shows the size effect equation
plotted alongside the two limits.
Figure 2.1 – Size Effect Results for various kinds of rocks and ceramics
(Fathy 1992; Bazant, Gettu and Kazemi 1991; McKinney and Rice 1981) [25]
6
2.3 Fracture Process Zone
LEFM predicts infinite stress at a crack tip but in real materials, stresses must
remain finite. As a result, a non-linear zone develops that is composed of a region
characterized by progressive softening and a region characterized by plastic hardening
[25]. For quasibrittle materials, the region of plastic hardening is often negligible and the
majority of the nonlinear zone undergoes softening due to microcracking, crack
deflection, interface breakage, and other phenomena. This zone, commonly called the
fracture process zone (FPZ), is of interest in predicting failure and deciding the
applicability of LEFM.
Figure 2.2 displays a conceptual representation of the FPZ where cohesive
stresses are present within the process zone [27, 28]. The traction inside the process zone
is due to ligaments that remain unbroken as the surfaces separate. Ultimately, the
opening displacement between the two surfaces reaches a critical value, wc, and the
traction becomes zero.
Figure 2.2 – Schematic of the fracture process zone (FPZ) showing the traction free crack and
cohesive tractions that restrict the opening of the FPZ
Cohesive tractions
𝜎
lp
Traction-free crack
7
The length of the FPZ can be written as:
(
)
(2.5)
where η = a dimensionless constant, KIc = mode I fracture toughness and ft = tensile
strength. Irwin estimated the constant, η, by cutting off the LEFM stress field and
placing a limit equal to the yield strength, σY. This concept yields η = 1/π, but stress
redistribution of the plastic (ductile) material is not considered [25].
The FPZ length was observed experimentally by Labuz and Biolzi [29] through
acoustic emission (AE). Transducers, attached to a granite beam, detect acoustic
emissions, which are caused by the development of microcracks. At peak load, the
events localize and provide an estimate for the length and width of the fracture process
zone, as shown in Fig. 2.3. Optical methods, such as speckle interferometry and digital
image correlation, have also been used to measure opening displacements and process
zone lengths in sandstone [30, 31].
Figure 2.3 – Locations of acoustic emission at peak load in a granite beam [47]
8
2.3 Digital Image Correlation
Digital Image Correlation (DIC) is a particle-tracking technique that uses digital
images to generate displacement fields. The technique stemmed from earlier work during
the 1960’s that introduced laser speckles as a tool for observing displacements. The laser
speckle, generated by shining a laser on the surface of a specimen, has a unique size,
shape and intensity that are products of the local microscopic imperfections. When the
specimen deforms, the speckles move and the surface displacement can be tracked by
evaluating the movement of the speckles [32].
Extensive numerical analysis is required to estimate the speckle movement from
the reference image (undeformed surface) to the current image (deformed surface). The
process attempts to track small regions, called subsets, by performing correlation analysis
between the two images [33]. The location of maximum correlation between the
reference image and the current image coincides with the location of the displaced subset.
Many algorithms, with slightly different correlation functions, have been introduced that
use methods such as coarse-fine search [34], Newton-Raphson [35], and Fast Fourier
Transform [36]. Studies have also been focused on analyzing the system error, which can
arise during reconstruction of the intensity patterns [37] and during interpolation in the
approximation of sub-pixel intensity values [38].
The correlation methods have since been applied to many experimental studies.
Peters et al. analyzed rigid body movement using digital correlation methods to calculate
angular velocity and linear velocity of discrete points [39]. Another study, by Chu et al.
deemed the correlation method accurate in the determination of rigid body translations
and rotations [40]. The study also tested the method in a uniform finite-strain test and
found errors to be minimal.
9
2.3.1 DIC Basics
One of the great advantages for using DIC is the ease of setup, which only
requires a camera, lens, image acquisition system, and a software package or code that
can perform the numerical correlation. The camera is setup perpendicular to the
specimen and the lens is focused to create a clear image. The specimen is typically
painted with black and white paint to create speckles. During an experiment, images of
the specimen are taken before and after deformation. The camera converts the light
energy into a digital image that consists of many small squares called pixels. The total
number of pixels per image is commonly called the resolution, which varies depending
on the camera. During the image acquisition process, each pixel is assigned a digital
value, called a grayscale value, which is related to the amount of light energy at that
location. Grayscale values for an eight bit digital image range from 0-255, where
extremely bright (white) pixels have higher grayscale values than dark (black) pixels.
Figure 2.4 – (Left) Digital image of a painted specimen in 3-point bending. (right) Zoomed-in
view of a region of interest (ROI) and a subset used for DIC analysis
Subset
Region of interest
10
Fig. 2.4 displays an image of a specimen with painted speckles that was taken
during a three-point bending experiment. The black region near the middle of the bottom
edge is the notch that is cut into the specimen. The resolution of the image is 1600x1200
pixels and each pixel has a grayscale value from 0-255. There are nearly 2 million pixels
in this image and only 255 different grayscale values, which means there are many pixels
with the same grayscale value. This makes it impossible to track individual pixels.
Instead, groups of pixels, called subsets, are tracked that have a unique pattern of
grayscale values. The size of the subset is set by the user and typically ranges from
20x20 pixels to 128x128 pixels.
The objective is to scan the current image to find the new location of the
displaced subset. Many times the displacements are very small and it is not necessary to
scan the entire image to locate the subset. Instead, the user specifies another parameter,
called the region of interest (ROI), which is a region, many times smaller than the entire
image, where the algorithm scans for the displaced subset. Restricting the scanning area
substantially increases the speed of the correlation process. Fig. 2.4 also shows a
zoomed-in view of a region of interest that is almost four times larger than the subset but
still very small in relation to the entire image.
The DIC software package used for this research is DaVis, which is produced by
LaVision Inc. (Ypsilanti, MI). The package combines the most advanced Fast Fourier
Transform algorithms with the highest quality software to create an easy to use interface
for analysis [41]. Fig. 2.5 shows a schematic of the analysis procedure. The software
recognizes two photos at different times, t1 and t2, and discretizes the image into smaller
cells (subsets). A correlation field between t1 and t2 is computed in each cell using Fast
Fourier Transformation and the maximum correlation corresponds to the displacement.
The software uses multi-pass processing and Whittaker reconstruction to maximize the
sub-pixel accuracy to 0.01 pixels.
11
Figure 2.5 – Schematic of Digital Image Correlation process performed by DaVis [31]
The user has control of many different parameters that impact the analysis.
Depending on the desired accuracy, speed or expected deformation, the parameters can
be adjusted to meet the specific needs. For more accurate analysis, the software can
perform a multi-pass method where multiple iterations are performed. The resulting
vector field from the first iteration is passed into the second iteration as a reference field.
The software also allows for overlap between subsets so that a higher resolution
(vectors/area) is achieved without decreasing the size of the subsets. Larger subsets are
advantageous because they have a pattern with more information (more pixels) and thus
the measured displacement is more precise. Post-processing or filtering of the vector
field can also be performed to smooth and create a continuous vector field.
The analysis throughout this thesis uses the multi-pass method with square
subsets of 64x64 pixels. The overlap between subsets is 75% creating a square grid with
vectors spaced every 16 pixels. The vector field is also post-processed using a median
filter and a smoothing function.
12
Chapter 3: Fatigue Crack Growth – Theoretical
Framework
3.1 Fatigue Crack Growth at the Large-size Limit
Suppose a quasibrittle structure with a notch is subjected to cyclic tensile loading,
shown in Fig. 3.1. Due to the stress concentration, a fracture process zone develops
ahead of the crack. This zone is characterized by progressive softening where the stress
decreases at increasing deformation. In a brittle or quasibrittle material, the zone is
composed of many small microcracks. At the tip of each microcrack, there is a micro-
scale FPZ, which is composed of sub-microcracks, etc., all the way down to the nano-
scale. The energy dissipation during crack growth at the macro-scale is equivalent to the
total energy dissipated at the nano-scale level:
∑
(3.1)
where = critical energy dissipation per unit crack growth, a = length of crack, N =
cycle, = critical energy dissipation per unit growth of nano-crack, ai = length of nano-
crack and n = number of nano-cracks.
Figure 3.1 – Cyclic FPZ composed of many micro-cracks
Cyclic FPZ
Micro-cracks
𝜎(𝑡)
13
Le and Bazant explained that a crack in a nano-scale element does not advance
smoothly but rather advances in discrete jumps, which corresponds to the jump over the
activation energy barriers of the nano-particle connectors [42]. It was shown that the
nano-crack growth rate under constant stress is related to the frequency of crack jumps in
a nano-element:
⁄ (3.2)
where Q0 = the dominant activation energy barrier on the free energy potential surface, k
= the Boltzmann constant, T = absolute temperature, Ka = the stress intensity factor of the
nano-element, ( ) ⁄ , = the spacing of nano-particles, = the
perimeter of the radial crack front, h = 6.626 x 10-34
and E1 = the Young’s modulus of
the nano-element. If the stress is cyclic, the equation can be written as:
⁄ (3.3)
where ΔK = amplitude of the stress intensity factor. This equation resembles the Paris-
Erdogan law with an exponent of 2. Nevertheless, the classical Paris-Erdogan law for
quasibrittle materials usually has an exponent, m, greater than 10. Substituting Eqn. 3.3
into Eqn. 3.1:
∑
(3.4)
Since there are numerous nano-cracks in the FPZ, the summation can be
expressed as:
∑
(3.5)
where = the average stress intensity factor of the nano-cracks. Within the LEFM
framework, the stress intensity factor amplitude of a nano-crack is proportional to the
stress intensity factor amplitude of the macro-crack.
14
Therefore, Eqn. 3.4 yields:
( ) (3.6)
The number of nano-cracks, n, can be expressed as a function of the stress
intensity factor, Young’s Modulus (E) and critical energy dissipation. Using dimensional
analysis, n depends on the dimensionless quantity ⁄ . Further assuming self-
similarity (i.e., power law):
(
) (
)
(3.7)
Substituting Eqn. 3.7 into Equation 3.6 yields:
( ) (3.8)
Assuming A/Em can be rewritten as a new constant, C, and p = 4m:
( ) ⁄ (3.9)
Eqn. 3.9 represents a fatigue crack growth law for the large-size limit. It agrees
well with the original Paris-Erdogan law, which has been applied to quasibrittle
structures [43, 44, 45]. The power law relation stems from the hypothesis of self-
similarity of function f, presented in Eqn. 3.7. This hypothesis holds well for the
medium stress amplitudes but deviates for very small or very large amplitudes. Le and
Bazant [42] explain that at small amplitudes (ΔK → 0), the distinction between static and
cyclic FPZs may become blurred causing the deviation from self-similarity. For large
amplitudes (ΔK → KIc), the failure may be dynamic because the stress amplitude
approaches the fracture toughness. Many experiments have shown the power-law
deviation at both limits [1, 25, 46].
15
3.2 Scaling of Fatigue Crack Growth Law
The forgoing analysis is limited to structures that are much larger than the FPZ
size (i.e., the large-size limit). In order to apply Eqn. 3.9 for smaller sizes, a scaling
relationship will be introduced for the critical energy dissipation, U*. It is assumed that
this relationship takes a form similar to the scaling of fracture energy, Gf, which is
derived from the size effect law on nominal monotonic strength:
√ ⁄ (3.10)
where ft’ = tensile strength of the material, B = dimensionless constant and D0m = the
transitional size for monotonic loading. The nominal strength is reached when the stress
intensity factor is equal to the critical stress intensity factor KIc, which can be written in
the following form:
√ ( ) (3.11)
Substituting the relationship between fracture energy and fracture toughness,
( ) into Eqn. 3.11 and solving for the nominal strength yields:
( ) (3.12)
where ( ) ( ) . Equation 3.12 is equal to Eqn. 3.10 squared:
( )
( )
( ⁄ ) (3.13)
Simplifying Eqn. 3.13, the size effect on fracture energy can be written as:
[
] (3.14)
where ( )
( ) ⁄ , which is the fracture energy for an infinitely sized
specimen [26].
16
The first hypothesis is that the size effect on the critical energy dissipation, U*,
takes the form of Eqn. 3.13 and can be expressed as:
[
] (3.15)
where is the critical energy dissipation per unit crack growth for a infinitely sized
specimen and D0c = the transitional size under cyclic loading. The equation holds for
infinitely sized specimens, ( ) and as D → 0, the energy dissipation goes to zero.
Combining Eqn. 3.9 into Eqn. 3.7, the final equation for the size-dependent Paris-
Erdogan law is
[ √
]
(3.16)
Since is a very difficult parameter to determine, it can be combined into the constant
C, which is fit to experimental data. This equation is very similar to the equation
proposed by Bažant and Xu [43], which was scaled by normalizing the amplitude of the
stress intensity factor by the size-dependent critical stress intensity factor, KIc. Bažant
and Xu found that the optimum fit was achieved when the transitional size used for cyclic
loading, D0c, was much larger than the transitional size computed during monotonic
loading, D0m.
The second hypothesis is that the transitional size for both monotonic and cyclic
loading can be expressed as:
( ) (3.17)
where γ = a non-dimensional constant and lpm and lpc = the length of the fracture process
zone under monotonic and cyclic loading, respectively. Eqn. 3.17 states that the change
in D0i from monotonic loading to cyclic loading is purely due to the change in FPZ size
from monotonic to cyclic loading. Experimental verification is essential to test the
hypotheses.
17
Chapter 4: Monotonic Size Effect Test
4.1 Experimental Setup
Prior to conducting fatigue tests, monotonic three-point tests were performed to
determine the size effect on nominal strength and investigate the fracture process zone
size. A total of nine specimens, three of each size, were tested using a closed-loop, servo
hydraulic 1 MN load frame (MTS Systems, Eden Prairie, MN). The nominal dimensions
of the specimens are shown in Fig. 4.1. The setup for monotonic loading is fairly simple
and is shown in Fig 4.2. The specimen is centered between two supports and the load is
applied at the center of the beam. A charge-coupled device (CCD) camera is attached to
a fixed frame and takes pictures for digital image correlation. A clip gage measures the
crack mouth opening displacement (CMOD).
To increase the accuracy of the digital image correlation process, the surface of
the specimen is painted with black and white paint. The surface is first coated with white
paint and then sprayed lightly with black spray paint. The result is a surface that has
unique patterns of speckles.
Figure 4.1 – Specimen Geometry for three-point bending tests
d
S = 2.5d
b = 20mm
a0 = 0.2d
Small: d = 25.4mm
Medium: d = 50.8mm
Large: d = 101.6mm
18
Figure 4.2 – Experimental Setup
4.2 Nominal Strength Calculation
The final dimensions for all nine specimens are shown in Table 4.1 alongside the
measured peak load. The load vs. CMOD curves for all specimens are shown in Figures
4.3-4.5. CMOD control was used so that post-peak crack growth could be observed. The
nominal strength is calculated with the following equation:
(4.1)
where Pu = ultimate load, b = thickness and d = depth of the beam. Nominal stress vs.
CMOD is plotted in Fig. 4.6 for a small, medium and large specimen. Table 4.2 shows
the final nominal strength used for each size.
CCD Camera
CMOD gage
Specimen
19
Table 4.1: Specimen properties for monotonic loading
Specimen S[mm] a0
[mm] b [mm] d [mm] Pu [N]
σn [MPa]
Sma
ll Spec. 1-1 63.50 5.38 20.01 25.40 381.8 0.751
Spec. 1-2 63.50 5.51 19.99 25.38 404.5 0.797
Spec. 1-3 63.50 5.21 20.02 25.45 464.9 0.912
Med
ium
Spec. 2-1 127.00 11.20 20.14 50.83 626.2 0.612
Spec. 2-2 127.00 10.80 20.08 50.58 662.0 0.652
Spec. 2-3 127.00 10.49 20.08 50.66 698.7 0.687
Larg
e Spec. 3-1 254.00 20.43 20.04 101.62 1029 0.505
Spec. 3-2 254.00 20.57 20.02 101.70 1190 0.584
Spec. 3-3 254.00 20.72 20.19 101.73 1178 0.573
Table 4.2: Average nominal strength
Figure 4.3 – Load vs. CMOD for small specimens (d=25.4mm)
0
50
100
150
200
250
300
350
400
450
500
0 50 100 150 200
Load
[N
]
CMOD [microns]
Spec 1-1
Spec 1-2
Spec 1-3
Small (d=25.4mm)
Size Average nominal strength
Small 0.820 MPa
Medium 0.650 MPa
Large 0.554 MPa
20
Figure 4.4 – Load vs. CMOD for medium specimens (d=50.8mm)
Figure 4.5 – Load vs. CMOD for large specimens (d=101.8mm)
0
100
200
300
400
500
600
700
800
0 50 100 150 200 250 300 350
Load
[N
]
CMOD [microns]
Spec 2-1
Spec 2-2
Spec 2-3
Medium (d=50.8mm)
0
200
400
600
800
1000
1200
1400
0 100 200 300 400
Load
[N
]
CMOD [microns]
Spec 3-1
Spec 3-2
Spec 3-3
Large (d=101.8mm)
21
Figure 4.6 – Nominal strength vs. CMOD for one specimen of each size
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
0 100 200 300 400
σn
[MP
a]
CMOD [microns]
Small (Spec 1-2)Medium (spec 2-2)Large (Spec 3-2)
22
4.3 Fracture Properties from Size Effect Method
The size effect of the ultimate nominal stress (σNu), shown in Table 4.2 and Fig.
4.6, can be characterized by Bažant’s size effect equation [25]:
√ ⁄ (4.2)
where = tensile strength of the material, D = size of specimen, B = dimensionless
constant and D0 = constant with dimension of length. This equation can be algebraically
rearranged to a linear equation in the following form:
(4.3)
where
,
,
√ ,
(4.4)
Fig. 4.7 shows the regression plot used to compute the constants A and C. D0 and
B are both functions of these constants, as shown in Eqn. 4.4, and are also related to
the classical fracture parameters by the following equations,
√ (4.5)
(4.6)
( )
(4.7)
where KIf = fracture toughness, cf = critical effective crack extension for a semi-infinite
crack in an infinite body, Gf = critical energy release rate, k0 = the dimensionless stress
intensity factor computed from LEFM.
23
For this particular geometry,
( ) √ ( )
( ) ⁄ (4.8)
where α = a/d. Table 4.3 summarizes the fracture parameters for the Berea sandstone
specimens and the size effect curve (Eqn 4.4) is plotted along with the experimental data
in Fig. 4.8.
Table 4.3: Fracture parameters for Berea sandstone
Figure 4.7 – Linear regression of experimental results (A = 0.885MPa-2, C=0.026MPa-2 mm-1)
Y = 0.885X + 0.026
0.00
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.00 0.01 0.02 0.03 0.04 0.05 0.06
(1/σ
n)2
[m
m]-1
[MP
a]-2
1/d [mm]-1
C
A
Parameter Value
KIf 0.53 MPa m1/2
Gf 23.3 J/m2
cf 6.5 mm
D0m 34.0 mm
Bft’ 1.06 MPa
24
Figure 4.8 – Size effect curve for monotonic strength of Berea sandstone
0.1
1.0
0.1 1.0 10.0
σ/B
f t'
D/D0
Plasticity Limit
25
4.4 Fracture Process Zone Size
Analyzing the displacement fields from digital image correlation (DIC) provides a
way to estimate the size of the fracture process zone. This zone, in front of a crack tip, is
considered a material property and travels with a propagating crack. In a three-point
bending test, a stress concentration appears near the notch tip causing this non-linear
fracture process zone to develop.
Fig. 4.9 shows the horizontal displacement contours for an interval from 30-40%
peak load for a medium-sized specimen. The notch-tip is at point (0, 0) and the notch is
shown by a white box. The contours show that the material is moving to the right
(positive displacement) on the right side of the notch and to the left (negative
displacement) on the left side of the notch. The contours merge at the notch tip.
Figure 4.9 – Incremental horizontal displacement contours for a medium sized specimen from 30-40% peak
load
26
To better understand the behavior near the tip, it is helpful to look at the
displacements along a horizontal cross section. For instance, Fig. 4.10a shows the
horizontal displacement values along a horizontal line 0.3 mm above the notch tip. The
influence of the notch is shown by an increase in the gradient (∂ux/∂x). If a cross section
is taken farther away from the notch, the gradient decreases, as shown in Fig. 4.10b for a
horizontal line taken at y = 2.5 mm. The gradient will become zero at the neutral axis.
(a) (b)
Figure 4.10 –Horizontal displacement measurements from 30% to 40% peak load along horizontal lines
at (a) y = 0.3mm. (b) y = 2.5mm
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 0 × 0
y = 0.3mm
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 0 × 0
y = 2.5mm
27
As the load increases, a damage zone develops. Lin and Labuz [31] showed that
the tip of this zone is represented by the location where horizontal displacement contours
merge. Fig. 4.11 shows the incremental horizontal displacement contours from 60%-70%
peak load. The contours merge at a location y = 4 mm. The horizontal cross section
directly above the notch tip (y = 0.3 mm) displays an even larger gradient, shown in Fig.
4.12a. Moving farther from the notch tip causes the gradient to decrease as shown in
Figures 4.12b-d.
Figure 4.11 – Incremental horizontal displacement contours for a medium sized specimen
from 60-70% peak load
Tip
28
(a) (b)
(c) (d)
Figure 4.12 –Horizontal displacement measurements from 60% to 70% peak load along the line
(a) y = 0.3 mm. (b) y = 2.5 mm (c) y = 5.0 mm (d) y=9.0 mm
The fracture process zone is assumed to be fully developed once the specimen
reaches the peak load. Figure 4.13 shows the horizontal displacement contours for 90-
100% peak load. The tip of the damage zone is at approximately y = 9 mm and thus the
length of the fully-developed fracture process zone is 9 mm.
As the specimen continues into the post-peak regime, unstable crack propagation
begins. At peak load, the traction-free crack is assumed to form and as it propagates, the
load capacity of the specimen decreases. To estimate the length of this traction-free
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 × 0
y=0.3 mm
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 0 × 0
y=2.5 mm
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-15 -5 5 15ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 0 × 0
y=5.0 mm
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
4.0
-15 -5 5 15ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 0 × 0
y=9.0 mm
29
surface, another measurement, called the critical opening displacement, ωc, must be
identified. Because it is impossible to pinpoint the tortuous crack path and measure
exactly at the crack face, the opening displacement will instead be measured a few
millimeters away. Fig. 4.13b shows the total displacement measurements from 70% pre-
peak to 100% peak along two vertical lines located at x = 1.7 mm and x = -1.6 mm. The
critical opening displacement, which represents the threshold for a traction free surface, is
the distance between the two lines at y = 0 mm. For this particular specimen ωc = 51 μm.
(a) (b)
Figure 4.13 – (a) Incremental horizontal displacement contours for a medium sized specimen from
90-100% peak load (b) Total horizontal displacement measurements along vertical lines at x = -1.6mm
(left line) and x = 1.7mm (right line)
-5
0
5
10
15
20
25
30
35
-40 -20 0 20 40
y (m
m)
ux (micron)
wc
Tip
30
(a) (b)
Figure 4.14 – (a) Incremental horizontal displacement contours for a medium sized specimen from 100%
peak -90% post-peak (b) Total horizontal displacement measurements along vertical lines at x = -1.6mm
(left line) and x = 1.7mm (right line)
Using the critical opening displacement, it is now possible to locate the tip of the
traction free surface post-peak. Fig. 4.14a shows the displacement contours from 100%
peak to 90% post-peak and the tip of the fracture process zone is at about y = 15 mm.
Fig. 4.14b shows that the traction free crack extends up to y = 8.5 mm, and thus the
process zone is about 6.5 mm in length. Fig. 4.15 summarizes the fracture process zone
farther into post-peak and includes the two other medium specimens that were tested.
The average FPZ size for the medium specimens was 7 mm.
-5
0
5
10
15
20
25
30
35
-60 -30 0 30 60
y (m
m)
ux (micron)
wc
a
lp
wc
a
lp
Tip
31
Figure 4.15 – Post-peak fracture process zone vs. traction free crack for medium sized specimens
The same analysis is performed on a large specimen (d = 101.6 mm). Fig. 4.16a
shows the horizontal displacement contours from 90% to 100% peak load. The critical
opening displacement, ωc, is calculated again for the large specimen by determining the
total opening displacement between two vertical lines from the localization load (70%
peak load) to the peak load. Fig. 4.16b shows the horizontal displacement measurements
along vertical lines at x = -1.8 mm and x = 1.5 mm, which are used to calculate ωc = 58
μm. This value is then used in the post-peak regime, as shown in Fig. 4.17, to track the
traction free surface. Fig. 4.18 shows the FPZ size vs. traction free crack for two large
specimens and the average FPZ size was 7 mm. The third large specimen exhibited a
very tortuous crack and it became impossible to estimate the FPZ with the current
method.
0
1
2
3
4
5
6
7
8
9
10
0 2 4 6 8 10 12 14
Frac
ture
Pro
cess
Zo
ne
Len
gth
(m
m)
Traction Free Crack (mm)
Specimen 2-1
Specimen 2-2
Specimen 2-3
32
(a) (b)
Figure 4.16 – (a) Incremental horizontal displacement contours for a large-size specimen from 95% pre
peak -100% post-peak (b) Total horizontal displacement measurements from 70% pre-peak to 100%peak
along vertical lines at x = -1.8mm (left line) and x = 1.5mm (right line)
The same procedure was used on the small-size specimens, shown in Fig. 4.19,
and it was found that the average FPZ size was 3.5 mm. This contradicts the assumption
that the FPZ size is independent of size, but there are other factors that may be at play.
Boundary effects may be causing a decrease in size because the small sized specimens
only have a ligament approximately 20mm in length. With some of the ligament in
compression, there may not be enough “room” for the FPZ to develop to its full size.
In conclusion, the length of the fully developed fracture process zone, lp, under
monotonic loading is 7 mm. The transitional size, D0m, is also known and thus the non-
dimensional parameter, γ, can be computed from Eqn. 3.17.
(4.8)
-5
0
5
10
15
20
25
30
35
-40 -20 0 20 40
y (m
m)
ux (micron)
wc
lp
Tip
33
. (a) (b)
Figure 4.17 – (a) Incremental horizontal displacement contours for a large-size specimen from 100% peak
-95% post-peak (b) Total horizontal displacement measurements from 70% pre-peak to 95%post-peak
along vertical lines at x = -1.8mm (left line) and x = 1.5mm (right line)
Figure 4.18 – Post-peak fracture process zone vs. traction free crack for large-sized specimens
-5
0
5
10
15
20
25
30
35
-60 -30 0 30 60
y (m
m)
ux (micron)
wc
a
lp
0
1
2
3
4
5
6
7
8
9
10
0 5 10 15
Frac
ture
Pro
cess
Zo
ne
Len
gth
(m
m)
Traction Free Crack (mm)
Specimen 3-2
Specimen 3-3
Tip
34
Figure 4.19 – Post-peak fracture process zone vs. traction free crack for small-sized specimens
0
1
2
3
4
5
6
7
0 1 2 3 4 5 6 7 8
Frac
ture
Pro
cess
Zo
ne
Len
gth
(m
m)
Traction Free Crack (mm)
Specimen 1-1
Specimen 1-2
Specimen 1-3
35
4.4 Displacement Gradient Method for Tracking Fracture Process Zone
In Sec. 4.3, a method was introduced to determine the length of the fracture
process zone that was developed by Lin & Labuz [31]. This method was based on
experiments that used both digital image correlation (DIC) and acoustic emission (AE) to
analyze the growth and propagation of the FPZ and it was found that the tip coincided
with the merging of horizontal displacement contours. In this section, another method
will be introduced that attempts to quantitatively characterize the behavior of the fracture
process zone by analyzing the displacement gradient.
The displacement gradient describes the change of displacement along a
continuous line segment:
𝜕𝑢
𝜕𝑥 (4.9)
where ux is the displacement in the x-direction. Fig. 4.11 shows the horizontal
displacement measurements along horizontal lines at different distances from the notch
tip. The stress concentration due to the notch results in an increase in the slope of the
line. The effect of the stress concentration is well pronounced when y = 0.3 mm (Fig.
4.12a) and becomes less pronounced as the distance away from the notch tip increases,
e.g. y = 9.0 mm (Fig. 4.12d). The slope of this line is the displacement gradient, g, and it
is possible to compute this value. Fig. 4.20 shows how the displacement gradient
(calculated from x = -1 mm to x = 1 mm) decreases as the distance from the notch tip
increases.
36
Figure 4.20 – Displacement gradient (from x=-1mm to x=1mm) vs. distance from the notch tip for a
medium sized specimen from 60% to 70% pre peak
This calculation is used to quantify the fracture process zone tip. Using the
assumption that the tip of the fracture process zone coincides with the location where
horizontal contours merge, it is possible to calculate the displacement gradient at this
location. For instance, in Fig. 4.13a, the incremental horizontal contours from 90% peak
load to 100%, for a medium sized specimen, merge at a location of about y = 9 mm. The
displacement gradient calculated at a cross section y = 9 mm yields a value of 7.3x10-3
,
shown in Fig. 4.21a. From 100% to 90% post peak, the contours merge at about y = 15
mm and the gradient is equal to 6.2x10-3
, shown in Fig. 4.21b.
The two values for the displacement gradient at the FPZ tip are slightly different,
even though the load increment is the same (10%). This suggests that using load as the
increment is not appropriate. A better correlation is developed if one analyzes the
incremental CMOD between the two images rather than the load. Table 4.4 shows the
incremental CMOD listed alongside the displacement gradient calculated at the location
where the contours meet. As shown in Fig. 4.22, there is a linear relationship between
the displacement gradient at the tip and the incremental CMOD.
0
0.5
1
1.5
2
2.5
3
3.5
4
0 2 4 6 8 10
Dis
pla
cem
ent
grad
ien
t (x
10
-3)
Distance from notch tip (mm)
37
(a) (b)
Figure 4.21 – (a) Displacement gradient calculation at the tip (y=9mm) for 90% to 100% peak.
(b) Displacement gradient calculation at the tip (y=15mm) for 100% to 90% post-peak.
Table 4.4: Displacement gradient at tip of fracture process zone for a medium sized specimen
Load Increment FPZ
Location (mm)
CMOD Increment
(μm)
Displacement Gradient at
Tip (10-3)
70-80% pre-peak 3 12.6 2.6
80-90% pre-peak 5 17.4 3.0
90-100% pre-peak 9 46.5 7.3
100-90% post-peak 15 45.5 6.6
90-80% post-peak 19 30.9 4.6
80-70% post-peak 21 25 3.5
70-60% post-peak 24 35.3 4.8
Using this equation, one can calculate a limit where, if the displacement gradient
is above the limit, it is considered inside the FPZ and if it is below the limit, it is
considered outside the FPZ. For example, the incremental CMOD for a medium
specimen from 90% to 100% peak-load is 46.5 μm. The equation shown in Fig. 4.22
estimates the displacement gradient at the tip of the FPZ to be 6.8 x 10-3
. Figure 4.23a
shows the displacement gradient contour plot where the FPZ is gray and corresponds to
the region with displacement gradient is greater than 6.8 x 10-3
. Figure 4.23b shows the
FPZ during post-peak loading. This procedure may also be helpful to determine the
width of the process zone.
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 × 0
-10.0
-8.0
-6.0
-4.0
-2.0
0.0
2.0
4.0
6.0
8.0
10.0
-15 -5 5 15
ux
(mic
ron
)
x (mm)
𝜕𝑢
𝜕𝑥 × 0
38
This approach was also implemented for a large specimen. Figure 4.24 shows the
linear relationship between the displacement gradient and the incremental CMOD, and
Figure 4.25 shows two displacement gradient plots that outline the fracture process zone.
Figure 4.22 – Displacement gradient calculation at the tip vs. Incremental CMOD between photos for a
medium-size specimen
(a) (b)
Figure 4.23 – Displacement gradient contour for (a) 90-100% peak load (b) 80-70% post-peak load
y = 0.1332x + 0.5642 R² = 0.9547
0
1
2
3
4
5
6
7
8
0 10 20 30 40 50
Dis
pla
cem
ent
Gra
die
nt
(10-3
)
at F
PZ
tip
Incremental CMOD (microns)
0.0068 𝒅𝒖𝒙 𝒅𝒙⁄
0.0039 𝒅𝒖𝒙 𝒅𝒙⁄
39
Figure 4.24 – Displacement gradient calculation at the tip vs. Incremental CMOD between photos
for a large-size specimen
(a) (b)
Figure 4.25 – Displacement gradient contour for (a) 90-95% pre- peak load (b) 80-70% post-peak load
for a large-size specimen
y = 0.2397x + 0.2277 R² = 0.91
0
2
4
6
8
10
12
14
0.0 10.0 20.0 30.0 40.0 50.0 60.0
Dis
pla
cem
ent
Gra
die
nt
(10
-3)
at
FP
Z ti
p
Incremental CMOD
0.0032 𝒅𝒖𝒙 𝒅𝒙⁄
0.0060 𝒅𝒖𝒙 𝒅𝒙⁄
40
Chapter 5: Size Effect Test on Fatigue Crack growth
5.1 Experimental Setup
Cyclic three-point bending experiments were performed to investigate size effect
on fatigue crack growth. The physical setup is identical to the monotonic test (Fig. 4.2),
the load, however, cycles between a maximum load, Pmax, and a minimum load, Pmin,
with a frequency of 1 Hz (Fig. 5.1). Crack mouth opening displacement (CMOD) is
recorded at the peak and valley of each cycle. The maximum load, Pmax, is set at
approximately 75% of the monotonic strength and Pmin is approximately 3%. One small
specimen, two medium specimens and one large specimen were prepared for cyclic
testing; dimensions are shown in Table 5.1.
Figure 5.1 – Load vs. time for cyclic three-point bending tests
Table 5.1: Specimen properties for cyclic loading
Specimen S [mm] a0 [mm] b [mm] d [mm] Pmax [N] Pmin [N]
Small Spec 1-4 63.50 5.37 19.99 25.47 310 15
Medium Spec 2-4 127.00 10.70 20.05 50.79 495 26
Medium Spec 2-5 127.00 11.45 20.04 50.89 480 28
Large Spec 3-4 254.00 20.05 20.16 100.55 930 33
Pmax
P
Pmin
t
. . .
P
a0
S
d
b = thickness
41
5.2 Compliance Calibration Method
To track crack growth through time, the compliance calibration method is used.
The displacement divided by the load is called the compliance and it is uniquely related
to geometry and crack length. Several investigators have used the compliance method to
determine crack length [47].
For this research, the compliance curve relates the CMOD compliance to an
effective crack ratio, α. This relationship was developed by modeling beams of different
notch lengths and computing the compliance using a finite element software, Abaqus ®.
The CMOD gauge does not measure the exact opening displacement of the notch, but
instead measures the displacement between two points where the CMOD clips are glued,
as shown in Fig. 5.2. Consideration was also taken to account for the thickness of the
CMOD clips. Fig 5.3 shows the final CMOD vs. effective crack length curves for the
three sizes with a Young’s modulus, E = 12 GPa.
Figure 5.2 – (Left) Zoomed-in view of CMOD gage, (Right) Numerical analysis performed with Abaqus
Gage
Length
1/2 Gage
Length
42
Figure 5.3 – CMOD vs. effective crack ratio for Young’s modulus = 12GPa
The measured compliance during the first cycle of loading should equal the
calculated compliance (from Fig. 5.3) for an effective crack ratio of α = a0/d where a0 is
the initial notch length d is the beam depth.. Any difference between the measured
compliance and the calculated compliance is assumed to be due to an effective Young’s
modulus in bending. The Young’s modulus used within the model is calibrated so that
the calculated compliance is equal to the measured compliance from the experiment.
Table 5.2 shows the Young’s modulus that was used for each beam. Once calibrated, a
polynomial is then fit to compute the effective crack length for a given CMOD
compliance.
0.000
0.050
0.100
0.150
0.200
0.250
0.300
0.00 0.10 0.20 0.30 0.40 0.50 0.60
CM
OD
Co
mp
lian
ce (
mic
ron
/N)
Effective crack ratio (a/d)
Small Beam (d=25.4mm)
Medium Beam (d=50.8mm)
Large Beam (d=101.8mm)
43
Table 5.2: Young’s modulus used to calibrate the CMOD compliance vs. effective crack curve for the first
cycle of loading
5.3 Paris-Erdogan Law Calculated from Compliance Method
Crack mouth opening displacement (CMOD) measurements were recorded at the
maximum (peak) and minimum (valley) load during each cycle. Figure 5.4 shows load
vs. CMOD measurements at different cycles. It is observed that the compliance increases
through time, which is a result of crack growth. Ultimately, the compliance begins to
increase at a faster rate and the specimen fails (Fig. 5.5). Using these measurements, the
compliance is computed:
(5.1)
where ΔP = the change in load and ΔCMOD = the change in crack mouth opening from
the minimum load to the maximum load. The effective crack length is then determined
using the calibrated curves from Fig. 5.3. Figure 5.6 shows the effective crack evolution
vs cycles for all four specimens.
Specimen α0
Cycle 1 CMOD
Compliance
[micron/N]
Young’s Modulus
[GPa]
Small – Specimen 1-4 0.21 0.063 6.25
Medium – Specimen 2-4 0.21 0.056 5.15
Medium – Specimen 2-5 0.23 0.053 5.80
Large – Specimen 2-4 0.20 0.044 5.35
44
Figure 5.4 – Load vs CMOD for different cycles
Figure 5.5 – Peak and valley CMOD measurements vs number of cycles
0
50
100
150
200
250
300
350
400
450
0.0 10.0 20.0 30.0 40.0 50.0 60.0 70.0
Load
(N
)
CMOD (microns)
0
10
20
30
40
50
60
70
80
90
100
0 1000 2000 3000 4000 5000 6000
CM
OD
(m
icro
ns)
Cycles
Failure
45
Figure 5.6 – Effective crack evolution
The Paris-Erdogan law can be constructed for each specimen by computing the
crack growth rate (da/dN) using a forward difference equation:
( ) ( )
(5.2)
where a(N1) is the effective crack length at cycle number N1 and a(N2) is the effective
crack length at cycle number N2. The change in stress intensity factor (ΔK) is computed
using
( )
√ ( ) (5.3)
where Pmax and Pmin are the maximum and minimum applied loads, b = thickness, d =
depth and k(α) is the dimensionless stress intensity factor, given in Eqn. 4.8. The value
used for alpha (α) is the average of a(N1) and a(N2) divided by the depth, d.
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
10 100 1000 10000 100000
Effe
ctiv
e C
rack
Len
gth
Rat
io (
a e/d
)
Cycles
Small Spec 1-4
Medium Spec 2-5
Medium Spec 2-4
Large Spec 3-4
46
Figure 5.7 shows da/dN vs ΔK in a log-log scale for medium specimen 2-5. Note
that the classical Paris-Erdogan law will not explain the entire behavior of the specimen.
Instead, there appears to be four unique stages of crack growth. At the beginning of the
test, the specimen exhibits crack growth that decreases in speed, shown by stage I. This
phenomenon has been observed previously and is defined as small flaw growth [1, 48-
50]. The rate then reaches a somewhat constant value during cycles 3,000 to 8,000,
shown by stage II. During stage III, The specimen exhibits an increasing growth rate that
can be explained by the Paris-Erdogan law. The last stage is associated with fast crack
growth. Although stages I and II were present for all four specimens, this portion of each
experiment was removed so the focus was on the region that can be explained by the
Paris-Erdogan Law. For specimen 2-5, which is shown in Fig. 5.7 and Fig. 5.8, the
analysis is performed from cycle 8,000 to failure. Table 5.3 lists the first cycle that is
included in the analysis of the Paris-Erdogan law for the other three fatigue tests.
Figure 5.7 – da/dN for medium specimen 2-5 plotted on log-log scale
0.000
0.001
0.010
0.250
da/
dN
ΔK (MPa-m1/2)
Stage I
Stage II
Stage III
Stage IV
0.300 0.400 0.500
47
Figure 5.8 – Effective crack length ratio for specimen 2-5 showing
the four apparent stages of crack growth
Table 5.3: Cycles analyzed for Paris-Erdogan law
Specimen First cycle analyzed for
Paris-Erdogan Law
Total Cycles To
Failure
Small 1-4 2,000 5,827
Medium 2-4 10,000 13,911
Medium 2-5 8,000 10,701
Large 3-5 13,000 21,276
The Paris-Erdogan law for all four specimens is shown in Fig. 5.9. The interval
(N2 – N1) used to calculate the crack growth rate (Eqn. 5.2) varies for each specimen in an
attempt to capture approximately 20-40 points per specimen. Figure 5.10 displays the
best-fit lines using a fixed slope for all three sizes and combining the two medium sized
specimens.
0.20
0.25
0.30
0.35
0.40
0.45
0 2000 4000 6000 8000 10000 12000
Effe
ctiv
e C
rack
Len
gth
Rat
io (
a e/d
)
Cycles
I II III IV
48
Figure 5.9 – Paris-Erdogan law for all four specimens plotted individually
Figure 5.10 – Paris-Erdogan law for all three sizes with a fixed slope, m = 17.6
5E-05
0.0005
0.005
0.250
da/
dN
(m
m/c
ycle
)
ΔK (MPa-m1/2)
Spec 1-4Spec 2-4Spec 2-5Spec 3-4
0.300 0.400 0.500
m= 19.0 logC= 6.0 m= 14.8 logC= 2.8 m= 19.0 logC= 4.5 m= 21.7 logC= 3.9
5E-05
0.0005
0.005
0.250
da/
dN
(m
m/c
ycle
)
ΔK (MPa-m1/2)
Small (d=25.4mm)Medium (d=50.8mm)Large (d=101.6mm)
m = 17.6
log C = 5.28
log C = 3.91 log C = 2.61
0.300 0.400 0.500 0.550
49
It is obvious from Fig. 5.10 that there exists a size effect in the Paris-Erdogan law
for Berea sandstone. In Chapter 3, it is assumed that the size effect can be introduced
into the Paris-Erdogan law by scaling the critical energy dissipation per unit crack
growth, U*. The hypothesis is that the size effect takes the same form as the scaling of
the fracture energy, Gf. This assumption yields the following size-dependent version of
the Paris-Erdogan law:
( √ ⁄ )
(5.4)
where C and p = empirical constants that are fit, ΔK = the change in stress intensity factor
from minimum to maximum load, and D0c = the transitional size under cyclic loading.
The transitional size calculated from the monotonic tests, D0m, is 34 mm. Assuming,
D0c = D0m, the optimum fit for the size-dependent Paris-Erdogan law (Eqn. 5.5) is shown
in Fig. 5.11. The data from the small and medium-size specimens nearly converge to a
single line but the data from the large specimen is still clearly separate. This suggests
that the transitional size, D0c, may not be equal to D0m. Assuming D0c ≠ D0m, the optimal
fit occurs when D0c = 54 mm, shown in Fig. 5.12. The data for all three sizes converge
fairly well to a single line. Thus, the assumption of size effect on critical energy
dissipation is a reasonable concept for scaling.
The second hypothesis from Chapter 3 is that the monotonic and cyclic
transitional sizes, D0m and D0c, are functions of the FPZ length, lp, and a non-dimensional
constant, γ (Eqn. 3.16). The foregoing analysis shows that the transitional size for cyclic
loading, D0c = 54 mm, is 58% larger than the transitional size for monotonic loading, D0m
= 34 mm. Eqn. 3.16 suggests that the increase in transitional size is purely due to an
increase in the FPZ size. Digital image correlation is used to investigate this hypothesis.
50
Figure 5.11 – Size-dependent Paris-Erdogan law using D0 = 34 mm
Figure 5.12 – Size-dependent Paris-Erdogan law using D0 = 54 mm
0.0001
0.001
0.40
da/
dN
ΔK·(1+D0/D)1/2 (MPa m1/2)
SmallMediumLarge
p = 17.6 log C = 2.37 logC = 2.23 log C= 1.66
D0 = 34 mm
0.500 0.600
0.0001
0.001
0.50
da/
dN
ΔK·(1+D0/D)1/2 (MPa m1/2)
SmallMediumLarge
p = 17.6 logC = 0.99
D0 = 54 mm
0.600 0.700
51
5.4 Fracture Process Zone Length – Cyclic Loading
From digital image correlation (DIC), the length of the fracture process zone
(FPZ) under monotonic loading was estimated to be 7mm (Chapter 4). The procedure is
replicated for the fatigue tests to investigate the growth and size of the FPZ during cyclic
loading.
In order to analyze the images in a consistent manner, it is important to consider
the digital images at a known load level. In the monotonic tests, photographs were taken
every two seconds, and it was possible to match the time of the photograph to the load
value that is recorded in the data file. This is much more difficult for a fatigue test that
can run for hours, with cycles between maximum and minimum load in 1 Hz. A system
was developed that is capable of triggering the CCD camera at a user-specified load
level. For all the cyclic tests, the camera was triggered at the maximum load and at the
minimum load once every 20 cycles. It is then possible to analyze the displacement
contour plots at different cycles to estimate the location of the FPZ.
As mentioned previously in Chapter 4, the tip of the FPZ coincides with the
location where the horizontal displacement contours merge. The region from the notch
tip to the FPZ tip is comprised of process zone and traction free length, although the
traction free surface exists only if the opening displacement between two vertical lines is
greater than the critical opening displacement, wc. The average critical opening
displacement from the monotonic experiments was approximately 40 μm for two lines
placed 3 mm apart.
Figure 5.13a shows the horizontal displacement contours from minimum to
maximum load for medium specimen 2-4 at cycle 6,000. The tip of the FPZ is estimated
to be at y = 5 mm. To determine if part of the damage zone is traction free surface, the
opening displacement between two vertical lines is analyzed. Figure 5.13b shows the
horizontal displacement measurements along lines at x = -1.6 mm and x = 1.6 mm. At
the notch tip (y = 0 mm), the total opening displacement between the two lines is 22 μm,
which is less than the critical opening displacement (wc = 40 μm). Thus, the entire region
is FPZ and the length is 5 mm.
52
. (a) (b)
Figure 5.13 – (a) horizontal displacement contours for medium specimen 2-4 – cycle 6,000
(b) Horizontal displacement measurements along vertical lines at x = -1.6 mm (left line) and x = 1.6 mm
(right line)
To reduce the amount of images recorded, photos are only taken once every 20
cycles and as a result, the final cycle may not be photographed. For this particular
specimen, the final cycle imaged is 13,900, which is 11 cycles before failure. Fig. 5.14a
shows the horizontal displacement contours for cycle 13,900 with the tip of the FPZ at y
= 10 mm. At the notch tip, the total opening displacement is 35 μm, which is still less
than the critical opening displacement (wc = 40 μm). Thus, the FPZ length is 10 mm. It
is impossible to determine if the FPZ grows any larger during the next 11 cycles or if
traction free surface begins to develop because no images were taken after cycle 13,900.
From the available data, it can be concluded that the cyclic FPZ length for this specimen
is at least 10 mm.
-5
0
5
10
15
20
25
30
-25 -13 0 13 25
y (m
m)
ux (micron)
lp
Tip
53
. (a) (b)
Figure 5.14 – (a) horizontal displacement contours for medium specimen 2-4 – cycle 13,900
(b) Horizontal displacement measurements along vertical lines at x = -1.6mm (left line) and x = 1.6mm
(right line)
The analysis is performed on a large specimen (specimen 3-4) and the horizontal
displacement contours for cycle 12,000 are shown in Fig. 5.15a. The tip of the process
zone is at approximately y = 8 mm and the opening displacement at the notch is 30μm,
meaning there is no traction free surface. For the medium specimen, there was no
traction free surface created until the very end. The large specimen, on the other hand,
displayed traction free surface growth around cycle 20,000, which is about 1,500 cycles
before failure.
Fig. 5.16a shows the displacement contours for cycle 20,500, where the tip of the
FPZ is at 13 mm. Fig. 5.16b shows that the opening displacement between two vertical
lines at x = -1.4 mm and x = 1.8 mm exceeds the critical opening displacement and thus
some of this region is considered traction free. The opening displacement is 43 μm at the
notch tip and is 40 μm at y = 1.2 mm, meaning the traction free surface is 1.2 mm and the
FPZ length is 11.8 mm.
-5
0
5
10
15
20
25
30
-25 -13 0 13 25
y (m
m)
ux (micron)
lp
Tip
54
. (a) (b)
Figure 5.15 – (a) horizontal displacement contours for large specimen 3-4 – cycle 12,000
(b) Horizontal displacement measurements along vertical lines at x = -1.4mm (left line) and x = 1.8mm
(right line)
. (a) (b)
Figure 5.16 – (a) horizontal displacement contours for large specimen 3-4 – cycle 20,500
(b) Horizontal displacement measurements along vertical lines at x = -1.4mm (left line) and x = 1.8mm
(right line)
-5
5
15
25
35
45
-20 -10 0 10 20 30
y (m
m)
Δu (micron)
lp
-5
5
15
25
35
45
-40 -20 0 20 40
y (m
m)
Δu (micron)
lp
a
wc
Tip
Tip
55
. (a) (b)
Figure 5.17 – (a) horizontal displacement contours for large specimen 3-4 – cycle 21,260
(b) Horizontal displacement measurements along vertical lines at x = -1.4mm (left line) and x = 1.8mm
(right line)
Fig. 5.17a shows the displacement contours for cycle 21,260, which is 16 cycles
before failure. The tip of the FPZ is at y = 17 mm. The location where the opening
displacement is less than the critical opening displacement is at y = 7.0 mm, thus the FPZ
length for the large specimen is 10 mm. The average length of the FPZ calculated after
growth of traction free surface was 11 mm.
The same procedure is performed for the small specimen and the length of the
process zone during fatigue is 5 μm. This substantial decrease in size, compared to the
medium and large sized specimens, is similar to what was found during monotonic
loading. This is most likely caused by boundary effects that restrict the region where the
FPZ can develop.
-5
5
15
25
35
45
-40 -20 0 20 40
y (m
m)
Δu (micron)
lp
a
wc
Tip
56
Table 5.4 summarizes the FPZ sizes for all four fatigue tests. Also shown is the
monotonic FPZ length for each size. All four specimens generate a larger FPZ during
cyclic loading and the average increase is 43%.
Three of the four specimens did not exhibit any traction free surface until after the
last image was taken, and thus the length of the FPZ cannot be considered “fully-
developed.” This could explain why there was less increase in the FPZ size for the small
and medium size specimens. Only the large specimen displayed traction free growth and
thus the length of the fully-developed FPZ during cyclic loading is 11 mm. This value
can be used to compute the non-dimensional parameter, γ, from Eqn. 3.17:
(5.5)
The value for γ computed from monotonic loading was also 4.9, which supports the
hypothesis that the increase in transitional size is due to the increase in the fracture
process zone.
Table 5.4: Fracture process zone lengths for all four specimens
Specimen Cyclic FPZ length
(mm)
Monotonic FPZ length
(mm)
Percent increase in FPZ
size during cyclic
loading
Small 1-4 5 3.5 43%
Medium 2-4 10 7.0 43%
Medium 2-5 9 7.0 29%
Large 3-5 11 7.0 57%
5.5 Paris-Erdogan Law calculated from DIC Method
Digital image correlation can also be implemented to look at the growth of the
damage zone through time. The difficulty in using the proposed method is that the
location where the fringes meet cannot be pinpointed more accurately than within a
millimeter. Fig. 5.18 shows the horizontal contours for cycle 2 with the tip at
approximately y = 1 mm. The final length of the process zone is 10 mm, and thus there
can only be 9 points to represent each time the FPZ length increases by approximately 1
mm, shown in Table 5.5.
57
For example, at cycle 100, the tip is estimated at y = 2 mm and at cycle 1,500, the tip is
estimated at y = 3 mm. In between cycles 100 and 1,499, growth can be observed but the
tip is still estimated at y = 2 mm. Cycle 1,500 represents the first time when the tip is
estimated at y = 3 mm.
Figure 5.18 – Horizontal displacement contours for cycle 2 for a medium specimen (2-4)
Table 5.5: Cycles where the FPZ tip increases by 1 mm for medium specimen 2-4
Cycle FPZ tip [mm]
2 1
100 2
1500 3
4000 4
6000 5
8500 6
10380 7
12880 8
13700 9
13900 10
Tip
58
An effective crack ratio that incorporates the damage zone can be represented by
the following equation:
(5.6)
where a0 = the initial crack length, Δae = the length of the damage zone determined from
DIC (FPZ + traction free surface) and d = depth of the beam. Figure 5.19 plots the
effective crack ratio from DIC (Eqn. 5.6) alongside the effective crack ratio estimated
from the compliance method. The results are fairly close throughout and this suggests
that the FPZ tip may be a good estimate for the length of the effective crack.
Figure 5.19 – Effective crack ratio vs. cycles computed using the compliance method and DIC method
The effective crack growth rate can be calculated using a forward difference
function (Eqn. 5.2) and the change in stress intensity factor can be calculated using the
alpha computed from DIC (Eqn. 5.6). Fig. 5.20 plots da/dN vs. ΔK from DIC method
alongside the data calculated from the compliance method for specimen 2-5. Despite the
limitations of the DIC method, the results lie close to the data from the compliance
method. The DIC method also observes a crack growth that decreases at the beginning of
the test that was shown in Fig. 5.7. As mentioned previously, the focus is to analyze the
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
0.50
10 100 1000 10000 100000
Effe
ctiv
e C
rack
Len
gth
Rat
io (
a e/d
)
Cycles
DIC Method
Compliance Method
59
region where the Paris-Erdogan law holds, which for this specimen is only from cycle
8,000 until failure. During this period, the FPZ tip grows from 5mm in length to 9 mm in
length and thus only four points that can be plotted.
The Paris-Erdogan law, for the large specimen, holds from cycles beyond 13,000.
During this period, the crack grows from 8 mm to 17 mm. Traction free surface is also
included in this growth, which reaches 7.8 mm at cycle 21,260 (16 cycles before failure).
A total of nine data points can be computed, which are shown in Fig. 5.20. Again, the
data points are fairly close to the results obtained using the compliance method.
The DIC results for all four tests are shown in Fig. 5.22. The size-dependent
Paris-Erdogan law (Eqn. 5.4) can again be fit, shown in Fig. 5.23. The D0c value that
produces the optimum fit is 47 mm, which is a 38% increase from the monotonic value.
This result compares well with the transitional size from the compliance method, D0c =
58 mm.
Figure 5.20– da/dN vs. ΔK calculated from the compliance method and DIC method
for medium-size specimen 2-5
0.000
0.001
0.010
0.250
da/
dN
(m
m/c
ycle
)
ΔK (MPa-m1/2)
Compliance Method
DIC Method
0.300 0.350 0.400 0.450 0.500
60
Figure 5.21 – da/dN vs. ΔK calculated from the compliance method and DIC method
for large-size specimen 3-4
Figure 5.22 – da/dN vs. ΔK calculated from DIC method for all four specimens
0.000
0.001
0.010
0.100
0.300
da/
dN
(m
m/c
ycle
)
ΔK (MPa-m1/2)
Compliance Method
DIC Method
0.400 0.500 0.600 0.700 0.800
0.0001
0.001
0.01
0.1
0.250
da/
dN
(m
m/c
ycle
)
ΔK (MPa-m1/2)
Small
Medium
Large
m = 11.9 log C = 2.51 m = 12.8 log C = 2.12 m = 19.5 log C = 3.00
0.300 0.400 0.500 0.500
61
Figure 5.23 – Size dependent Paris-Erdogan law fit to results from DIC method
0.0001
0.001
0.01
0.1
0.4
da/
dN
ΔK·(1+D0/D)1/2 (MPa-m1/2)
Small
Medium
Large
m = 19.0 log C = 1.39
D0 = 47 mm
0.500 0.600 0.700 0.800
62
Chapter 6: Conclusions
In order to ensure the safety of subsurface structures, an understanding of crack
initiation and growth in geological materials is essential. Subcritical crack growth (SCG)
occurs when stress intensity values are less than the fracture toughness, and can result in
time-dependent fracturing and faulting of rock masses. Many studies have focused on
SCG under constant loading and investigated environmental effects. Other studies have
also observed SCG under cyclic loading, called fatigue, which is described by the Paris-
Erdogan law. This law, however, lacks scaling parameters to take into account structural
size. Theoretical framework and experimental results are presented to investigate size
effect of the Paris-Erdogan law in a quasibrittle material.
When loaded, a structure with a notch composed of a quasibrittle material will
develop a fracture process zone (FPZ) that is composed of microcracks. At the tip of
each microcrack, there exists an even smaller FPZ composed of nano-cracks. It has been
shown that the growth rate of the nano-cracks can be expressed as a power law with an
exponent of two. Using this expression and assuming self-similarity, a fatigue crack
growth law is developed that agrees well with the original Paris-Erdogan law. The self-
similarity assumption holds for mid-range stress amplitudes but breaks down when the
stress amplitude is very low (ΔK →0) or very high (ΔK → KIc), where K = stress intensity
factor and KIc = fracture toughness. The law also lacks a size effect that has been
observed for quasibrittle materials.
The scaling relationship is introduced into the law through the critical energy
dissipation per unit crack length, U*. The size effect on this parameter was assumed to be
of the same form introduced by Bažant to scale the fracture energy, Gf. The final size-
dependent Paris-Erdogan law is:
( √ ⁄ )
where C and p = empirical constants, ΔK = amplitude of stress intensity factor, D = the
specimen size, D0c = the transitional size under cyclic loading, a = crack length and N =
number of cycles.
63
This law is verified experimentally through cyclic three-point bending experiments on
three different size specimens. An effective crack growth was computed using the
compliance calibration method that relates the compliance of the crack mouth opening
displacement to a unique crack length. The compliance curves were constructed from
numerical simulations of beams with different notch lengths. A strong size effect was
observed when the classical Paris-Erdogan law was plotted (da/dN vs ΔK). The size-
dependent version caused the data to converge to a single line with constants p = 17.6 and
C = 9.85. The results suggest that the scaling assumption is reasonable. However, an
optimal fit occurs when the cyclic transitional size, D0c, is equal to 58 mm, which is 58%
greater than the monotonic transitional size, D0m = 34 mm.
The second hypothesis is that this increase in transitional size was purely due to
an increase in FPZ length from monotonic loading to cyclic loading. To investigate this
assumption, digital image correlation (DIC) is used to estimate the length of the FPZ for
both monotonic and cyclic loading. The procedure involves analyzing the horizontal
displacement contours; the location where the contours merge coincides with the tip of
the FPZ. Traction free length is determined by analyzing the opening displacement at
peak load to obtain the critical crack opening displacement. The fracture process zone
was found to be 7 mm and 11 mm for monotonic and cyclic loading, respectively. The
increase in fracture process zone size was 57%, which supports the hypothesis that the
increase in transitional size is proportional to the increase in FPZ size.
Future work is recommended to further verify the size effect on Paris-Erdogan
law for other quasibrittle materials. Digital image correlation is a valuable tool, but it
would be advantageous to perform experiments that combine two techniques for
estimating FPZ length. For example, using acoustic emission as well as DIC, could help
to increase confidence in the estimation of FPZ length.
64
Chapter 7: References
[1] Suresh, S. Fatigue of Materials. Cambridge: Cambridge UP, 1998. Print.
[2] Hodgkinson, E.A. (1849) Report of commissioners appointed to enquire into the
application of iron to railway structures, Command Paper No. 1123. London: His
Majesty’s Stationery Office.
[3] Rankine, W.J.M. (1843) On the causes of unexpected breakage of the journals of
railway axles and the means of preventing such accidents by observing the law of
continuity in their construction. Proceeding of the Institute of Civil Engineering,
London 2, 105-108.
[4] Wöhler, A. (1860) Versuche über die festigkeit der eisenbahnwagenachsen.
Zeritschrift für Bauwesn 10; English Summary (1867). Engineering 4, 160-161.
[5] Fairburn, W. (1864) Experiments to determine the effect of impact, vibratory action,
and long continued changes of load on wrought iron girders. Philisophical
Transactions of the Royal Society, London 154, 311.
[6] Inglis, C.E. (1913) Stresses in a plate due to presence of cracks and sharp corners.
Transactions of the Institute of Naval Achitects 55, 219-241.
[7] Griffith, A.A. (1921) The phenomenon of rupture and flow in solids. Philisophical
Transaction of the Royal Society, London A221, 163-197.
[8] Irwin G.R. (1957) Analysis of stresses and strains near the end of a crack traversing a
plate. Journal of Applied Mechanics 24, 361-364.
[9] Paris, P.C., Gomez, M.P. & Anderson, W.P. (1961) A rational analytic theory of
fatigue. The Trend in Engineering 13, 9-14.
[10] Paris, P.C., Erdogan, F. (1963) A critical analysis of crack propagation laws.
Journal of Basic Engineering 85, 528-534.
[11] Docker, H., Marci, G. (1983) Threshold range and opening stress intensity factor in
fatigue. International Journal of Fatigue 5, 187-191.
[12] Kishimoto, H. Ueno, A. & Kawamoto, H. (1987) Fatigue crack propagation in
ceramics. Society of Materials Science Japan, 36, 1122-1127.
[13] Gao, Hua, Alagok, N., Brown, M.W. & Miller, K.J. (1985) Growth of fatigue cracks
under combined mode I and mode II loads. Multiaxial Fatigue, Special Technical
Publication 853, pp. 184-202. Philadelphia: American Society for testing and
Materials
65
[14] Ewart, L., Suresh, S. (1992) Elevated-temperature crack growth in polycrystalline
alumina under static and cyclic loads. Journal of Materials Science Letters 5,
774-778.
[15] El-Shabasy, A.B., Lewandowski, J.J. (2004) Effects of load ratio, R, and test
temperature on fatigue crack growth of fully pearlitic eutectoid steel.
International Journal of Fatigue 26, 305-309.
[16] Ritchie, R.O. (1988) Mechanisms of fatigue crack-propagation in metals, ceramics
and composites- role of crack tip shielding. Materials Science and Engineering
103, 15-28.
[17] Cook, R.F., Fairbanks, C.J. & Lawn, B.R. (1985) Microstucture-Strength Properties
in Ceramics .1. Effect of crack size on Toughness, and .2. Fatigue Relations.
Journal American Ceramic Society 68. 604-623.
[18] Hilsdorf, H.K. & Kesler, C.E. (1966) Fatigue Strength of Concrete under Varying
Flexural Stresses. ACI Journal Proceedings 63. 1059-1075.
[19] Atkinson, B.K. (1982). Subcritical crack-propagation in rocks – theory, experimental
results and applications. Journal of Structural Geology. 4, 41–56.
[20] Kilburn, C.R.J., Voight, B. (1998) Slow rock fracture as eruption precursor at
Soufriere Hills volcano. Geophysics Research Letters. 25, 3665–3668.
[21] Waza, T., Kurita, K., Mizutani, H., (1980) The effect of water on the subcritical
crack growth in silicate rocks. Tectonophysics 67, 25–34.
[22] Nara, Y., Takada, M., Igarashi, T., Hiroyoshi, N., Kaneko, K., (2009) Subcritical
crack growth in rocks in an aqueous environment. Exploration Geophysics. 40,
163–171.
[23] Nara, Y., Hiroyoshi, N., Yoneda, T., Kaneko, K., (2010) Effect of temperature and
relative humidity on subcritical crack growth in igneous rock. International
Journal of Rock Mechanics and Mining Sciences, 47, 640–646.
[24] Nara, Y., Morimoto, K., Hiroyoshi, N., Yoneda, T., Kaneko, K., Benson, P.M.
(2012) Influence of relative humidity on fracture toughness of rock:
Implicationsfor subcritical crack growth. International Journal of Solids and
Structures, 49, 2471-2481.
[25] Bažant, Z.P., Planas, J., (1998). Fracture and size effect in concrete and other
quasibrittle materials. CRCPress.
[26] Bažant, Z. P. (1984). Size effect in blunt fracture: concrete rock, metal. Journal of
Engineering Mechanics, Vol. 110, 518-535.
66
[27] Barenblatt G.I. (1959) The formation of equilibriujm crack during britlle fracture –
general ideas and hypotheses, azially-symmetric cracks. Journal of Applied
Mathematics and Mechanics, 23, 434-444.
[28] Dugdale D.W. (1960) Yielding of steel sheets containing slits. Journal of the
Mechanics and Physics of Solids, 8, 59-68.
[29] Labuz, J.F., Biolzi, L., (1998) Characteristic strength of quasi-brittle materials.
International Journal of Solids and Structures, 35, 4191-4203.
[30] Lin, Q., Fakhimi, A., Haggerty, M., Labuz, J.F. (2009) Initiation of tensile and
mixed-mode facture in sadstone. International Journal of Rock Mechanics and
Mining Sciences,46, 489-497.
[31] Lin, Q., Labuz, J. F. (2013) Fracture of sandstone characterized by digital image
correlation. International Journal of Rock Mechanics and Mining Sciences.
[32] Yamaguchi,I. (1981) Speckle Displacement and Decorrelation in the Diffraction and
Image Fields for Small Object Deformation. International Journal of Optics. Vol.
28. Issue 10. 1359-1376.
[33] Peters, W.H., Ranson, W.F. (1982) Digital imaging techniques in experimental stress
analysis. Optical Engineering 21 (3), 427-31.
[34] Sutton, M.A., Wolters, W.J., Peters, W.H., Ranson, W.A, McNeill, S.R. (1983)
Determination of displacements using an improved digital correlation method.
Image and Vision Computing 1 (3), 133-139.
[35] Bruck, H.A., McNeill, S.R., Sutton, M.A., Peters, W.H. (1989) Digital image
correlation using Newton-Raphson method of partial differential correction.
Experimental Mechanics. Volume 29. Issue 3. 261-267.
[36] Chen, D.J., Chiang, Y.S. Tan, Don, D.S, (1993) Digital speckle-displacement
measurement using a complex spectrum method. Applied Optics, Vol. 32, Issue
11. 1839-1849.
[37] Sutton, M.A., McNeill, S.R., Jang, J., Babai, M. (1988) Effects Of Subpixel Image
Restoration On Digital Correlation Error Estimates. Optical Engineering 27(10)
271070.
[38] Schreier, H.W., Braasch, J.R., Sutton, M.A. (2000) Systematic errors in digital
image correlation caused by intensity interpolation. Optical Engineering 39(11)
2915-2921.
[39] Peters, W.H., Ranson, W.F., Sutton, M.A., Chu, T.C., Anderson, J. (1983)
Application of digital correlation methods to rigid body mechanics. Optical
Engineering 22(6) 226738.
67
[40] Chu, T.C., Ranson, W.F., Sutton, M.A., (1985) Applications of digital-image-
correlation techniques to experimental mechanics. Experimental Mechanics,
Vol25, Issue 3, 232-244.
[41] StrainMaster. (2007) Advanced image based Tools for Optical Defromation and
Strain Measurement [Brochure]. Ypsilanti, MI: LaVision Inc.
[42] Le, J.L., Bažant, Z.P., Bažant, M.Z., (2011) Unified nano-mechanics based
probabilistic theory of quasibrittle and brittle structures: I. Strength, static crack
growth, lifetime and scaling. Journal of the Mechanics and Physics of Solids. Vol
59, Issue 7, 1291-1321.
[43] Bažant, Z. P., Xu, K. (1991) Size effect in fatigue fracture of concrete. ACI
Materials Journal, 88(4), 390-99.
[44] Ogawa, T., (1995) Fatigue crack growth of monolithic and composite ceramics.
Cyclic Fatigue in Ceramics. Elsevier Science B. V. and The society of Materials
Science, Japan. 167-188.
[45] Studarta, A.R., Filser, F., Kochera, P., Gaukler, L.J., (2007) Fatigue of zirconia
under cyclic loadin in water and its implications for the design of dental bridges.
Dental Materials. 23, 106-114.
[46] Andersons, J., Hojo, M., Ochiai, S., (2004) Emperical model for stress ratio effect on
fatigue delamination growth rate in composite laminates. International Journal of
Fatigue 26,567-604.
[47] Swartz, S. E., Hu, K. K. and Jones, G. L. (1978) Compliance monitoring of crack
growth in concrete. Journal of Engineering Mechanics Division –ASCE 104. 789-
800.
[48] Pearson, S. (1975) Initiation of fatigue cracks in commercial aluminum alloys and
the subsequent propagation of very short cracks. Engineering Fracture Mechanics
7, 235-247.
[49] Lankford, J. (1983) The effect of environment on the growth of small fatigue cracks.
Fatigue of Engineering Materials and Structures 5, 233-248.
[50] Tanaka, K. Hojo, M., Nakai, Y. (1983) Crack initiation and early propagation in 3%
silicon iron. Fatigue Mechanisms: Advances in Quantitative Measurement of
Fatigue Damage, Special Technical Publication 881, 207-232. Philidelphia:
American society for Testing and Materials.