FRACTURE MECHANICS : OVERVIEW
CONTENTS
Failure modes
Historical Development
Research summary
The Energy Release Rate criterion
The Stress Intensity Factor criterion
Time dependent crack growth
Fracture Mechanics approach to design Vs traditional approach
Fracture Mechanics approach to Fatigue design
Safe - Life and Fail - Safe approaches to Fatigue design
Fracture Mechanics Family Tree
FRACTURE MECHANICS: OVERVIEW
FAILURE MODES Our understanding of how materials fail and our ability to prevent such failures in service has increased
considerably since World War II
Catastrophic service failures are determental to the economy of a nation.
Commonly observed modes of failure are
- Yielding
- Excessive deformation
- Buckling
- Fatigue
- Fracture
- Creep
- Environmental degradation of stiffness and strength
- Vibration and Noise
- Wear, etc.,
Designing components / structures to avoid these failure modes is not a new idea.
Design against FRACTURE (Failure due to Crack Propagation) is a relatively new approach. So also Fatigue
Analysis based on Fracture Mechanics concepts.
The use of Fracture Mechanics has undoubtedly prevented a substantial number of component / structural failures in
service.
FRACTURE MECHANICS : OVERVIEW
HISTORICAL DEVELOPMENT
Land marks
- Griffith (1920’s) Energy Balance approach
- George Irwin (1948) Stress Intensity Factor approach
- Wells (1961) Crack Tip Opening Displacement concept
- Rice (1968) Path – Independent Integral
Griffith applied the results from stress concentration around an elliptical hole to predict FRACTURE (unstable
Propagation of a Crack)
Griffith’s Theory : A Crack becomes unstable and thus FRACTURE occurs when the strain energy change that results
from an Incremental crack growth is sufficient to overcome the surface energy of the material
Griffith’s theory accurately predicted the relationship between fracture strength and crack length in glass. Subsequent efforts to apply the same to metals was unsuccessful.why?
The Griffith’s theory only applies to ideally brittle solids.
A modification to Griffith’s theory that made it applicable to metals did not come till 1948.
A group of researchers directed by George Irwin at the Naval Research Laboratory in Washington D.C. studied the FRACTURE problem in detail. The subject we know as Fracture Mechanics was born in this lab during the decade following World War II. Fracture Mechanics progressed from being a scientific curiosity to an Engineering Discipline primarily because of this groups investigation of the structural failure of Liberty ships during World War II.
Investigations revealed that the Liberty ship failures were caused by a combination of
three factors
1. The welds, which were produced by semi-skilled workforce, contained crack
like flaws.
2. Most of the FRACTURES initiated on the deck, at square hatch corners,
where there was a local Stress Concentration.
3. The steel from which the Liberty ships were build had Poor Toughness, as
measured by Charpy Impact tests
In the longer term, structural steels were developed with vastly improved toughness
as measured by Fracture Toughness Tests. Weld Quality Control Standards were
developed and implemented and Engineering Analysis reduced the Stress
Concentration effects. Consequently, catastrophic failures of ship structures did not
reoccur.
FRACTURE MECHANICS : OVERVIEW
Research Summary
A group of researchers at the Naval Research Laboratory, Washington, D.C. led by
Dr George R. Irwin created the basic tools for the Analysis and Prediction of
FRACTURE (Failure due to Crack Propagation).
Irwin’s first major contribution was to extend the Griffith’s theory to metals by
including the energy dissipated by local plastic deformation.
Orowan independently proposed a similar modification to Griffith’s theory.
Mott extended the Griffith theory to a rapidly propagating crack (Dynamic Fracture).
Irwin in 1956 developed the energy release rate concept, which is related to the
Griffith Theory, but in a form useful for Engineering Analysis. He used the
Westergaard approach (a semi inverse technique for analysis of stress and
displacements around a crack tip) to show that the stresses and displacements in the
immediate vicinity of the crack tip could be described by a single parameter that was
related to the energy release rate. This crack-tip characterizing parameter later became
known as the Stress Intensity Factor (SIF) denoted by K
During the same period of time, M.L. Williams derived crack tip solutions that were
identical to Irwin’s.
In 1956, Wells applied Fracture Mechanics to show that the fuselage structural failure
in several Comet Jet aircraft resulted from fatigue cracks growing to a critical size.
These cracks initiated at windows and were caused by insufficient local reinforcement,
combined with square corners which produced severe stress concentrations.
Another early application of Fracture Mechanics occurred in General Electric in 1957.
Winnie and Wundt used Irwin’s energy release rate approach to investigate the failure
large steam turbine rotors. They were able to predict the bursting behavior of large
disks extracted from rotor forgings, and applied this knowledge to the prevention of
FRACTURE in actual rotors.
In 1960, Paris and coworkers failed to find a receptive audience for their ideas on the
Fracture Mechanics approach to Fatigue Crack growth Analysis.
Linear Elastic Fracture Mechanics (LEFM) is not valid when significant plastic
deformation precedes FRACTURE. In 1960 – 61, several researchers developed
analysis to correct for yielding at the crack tip. Irwin’s plastic zone correction was
simple extension of LEFM. Dugdale and Barenblaat developed elaborate models based
on a narrow strip of yielded material at the crack tip.
Wells proposed in 1961, Crack Tip Opening Displacement (CTOD) as an alternative
fracture parameter when significant plastic deformation at the crack tip precedes
FRACTURE.
In 1968, Rice developed another parameter to account for nonlinear material behavior
around the crack tip. By idealizing plastic deformation as nonlinear elastic, Rice was
able to generalize the energy release rate to nonlinear material behavior. He showed
that this nonlinear energy release rate can be expressed as a line integral, which he
called the J-integral, evaluated along an arbitrary contour around the crack tip.
The same year, Hutchinson, Rice and Rosengren related the J-integral to crack tip
stress fields in nonlinear materials. This showed that J- integral can also be viewed as
non linear Stress Intensity Factor as well as a non linear energy release rate.
Fracture Mechanics analysis is widely applied in the design of Nuclear Reactor
components. One major difficulty in applying Fracture Mechanics in this case was
that most nuclear pressure vessel steels were too tough to be characterized with
LEFM without resorting to very large test specimens for Fracture Toughness Testing
to measure KIC
Begley and Landers at Westinghouse, decided to characterize fracture toughness of
Nuclear Pressure vessel steels with the J - integral. Their experiments were successful
and led to the publication of a Standard Test procedure to measure JIC of materials .
Ten years later ! JIC is also a measure of Fracture Toughness of materials.
Material Toughness characterization is only one aspect of Fracture Mechanics. In
order to apply Fracture Mechanics concepts to modern design one must have a
mathematical relation between Toughness, applied stress and flaw (crack) size. This
is provided by Phenomenological Fracture criteria.
Shih and Hutchinson provided the theoretical frame work for Elastic – Plastic
Fracture Mechanics Analysis based on the J – integral. An engineering approach for
EPFM analysis was then developed at EPRI (1981).
In the UK, Well’s CTOD parameter was applied extensively to Fracture Mechanics
Analysis of welded structures.
Shih in 1981 demonstrated a relationship between the J – integral and CTOD
implying that both parameters are equally valid for EPFM analysis.
Much of the theoretical foundations of dynamic fracture mechanics was also
developed during 1960 – 1980.
Recent trends in Fracture Mechanics research
More sophisticated material models are being included in Fracture Mechanics
Analysis.
To incorporate time – dependent non linear material behavior into Fracture
Mechanics Analysis, Viscoplasticity or Viscoelasticity is employed.
Vicoplasticity is motivated by the use of tough, creep – resistant high
temperature materials.
Viscoelasticity reflects the increasing proportion of Polymeric materials in
engineering applications.
Fracture Mechanics has also been used (and sometimes abused ) in the
characterization of laminated composite materials.
Development of micro structural models and models to relate local and global
fracture behavior of materials. A related topic is the effort to characterize and
predict geometry dependence of fracture toughness.
New approaches where traditional single – parameter fracture mechanics breaks
down.
FRACTURE MECHANICS : OVERVIEW
The Energy Release Rate Criterion
Crack extension ( FRACTURE) occurs when the energy available for crack growth is
sufficient to overcome the resistance of material to crack growth. The resistance may
include the Surface energy, Plastic work, or other type of energy dissipation associated
with a propagating crack.
The energy release rate, G , is defined as the rate of change in potential energy with
crack area for a linear elastic material. At the moment of fracture G = G c the critical
energy release rate, is a measure of the material fracture toughness.
For a through crack of length 2a in an infinite plate subjected to a remote tensile stress
σ, the energy release rate is
G = σ2 a / E
where, E is the Youngs modulus of Elasticity of the material.
At fracture G = Gc and
Gc = f2 ac / E
where, σf is the fracture stress and ac is the measured crack length at
the onset of Fracture.
The energy release rate is a driving force, while Gc is the material resistance to crack
propagation.
FRACTURE MECHANICS : OVERVIEW
The Stress Intensity Factor Criterion
The singular stress field around a crack tip
X
KI is the Mode I Stress Intensity Factor. It is the AMPLITUDE of stress singularity at
the crack tip. The singularity of the type γ-1/2 .
Fracture occurs when KI = KIC. KIC is a measure of the fracture toughness of the
material.
For an infinite plate with a central crack of length 2a, the SIF is
KI = σ
KI is the driving force and KIC is the resistance of the material to crack propagation.
KIC is assumed to be a size independent material property.
Relation between KI and G
G = KI2 / E
The energy release rate and stress intensity factor approaches to predict fracture ( as
failure due to crack propagation) are equivalent for linear elastic material behavior.
FRACTURE MECHANICS : OVERVIEW
Time Dependent Crack Growth
Fracture Mechanics plays a key role in Life prediction of component that are subjected
to time – dependent crack growth mechanisms such as fatigue or stress – corrosion
cracking.
The fatigue crack growth rate in metals is described by the Paris law
is the crack growth per cycle, is the SIF range
C and m are material dependent constants.
Damage Tolerance Approach Design is illustrated in this figure. The initial crack size a0
is inferred from NDT, and the critical crack size ac is computed using applied stress and
fracture toughness. An allowable crack size is then defined by dividing the critical size
by a safety factor. The service life of the component can then be inferred by calculating
the time required for the flaw to grow from initial size to the maximum allowable size.
ac
The Fracture Mechanics Approach to Design Vs Traditional Approach
In the traditional approach to design and material selection a material is assumed to be
adequate , if its strength (yield or ultimate) is greater than the maximum allowed stress.
This approach may guard against brittle fracture by imposing a safety factor on stress,
combined with minimum tensile elongation requirements of material.
The Fracture Mechanics approach has three important variables as seen in the following fig.
FRACTURE TOUGHNESS
FLAW SIZE
APPLIED STRESS
Fracture Mechanics quantifies the critical combinations of these three variables
There are two alternative approaches to Fracture Analysis: The energy release rate
criterion and the Stress Intensity Factor criterion. These two are equivalent in certain
circumstances.
FRACTURE MECHANICS APPROACH TO FATIGUE DESIGN
Invokes a “defect – tolerant” philosophy based on the premise that all engineering
components are inherently flawed. The size, shape and location of a pre-existing
flaw(s) is determined by NDT.
If no flaw is found in the component, Proof tests are conducted at a stress level
slightly higher than the service stress. If no cracks are detected by the NDT and if
catastrophic failure does not occur during the proof test, the largest (undetected)
initial crack size is estimated from the resolution of the NDT.
The fatigue life is then defined as the number of cycles (or time) to propagate the
dominant cracks from the initial size to some critical size. The critical size based on
the Fracture Toughness of the material, the LIMIT load for the component, the
design allowable strain or the permissible change in compliance of the component.
The prediction of crack propagation life using the defect – tolerant approach involves
empirical Fatigue Crack Growth Laws based on Fracture Mechanics.
Various methods are available to include the effect of mean stress, stress
concentrations, environments, variable amplitude loading spectra and multiaxial
stress state in the estimation of Fatigue Crack Growth.
This intrinsically conservative approach to fatigue is widely used in fatigue – critical
applications. Examples, Aerospace and Nuclear Power Engineering.
Optimization of materials microstructure to improve resistance to both crack initiation
and crack growth would require a trade-off.
SAFE – LIFE AND FAIL – SAFE APPROACHES TO FATIGUE DESIGN
Developed by Aerospace Engineers
In the safe – life approach to fatigue design, the typical cyclic load spectra, which are
imposed on a structure / component in service are first determined. The components are
either analyzed or tested in the laboratory under load conditions which are typical of
service load spectra, and a useful fatigue life is estimated for the components.
The estimated fatigue life is suitably modified with a factor of safety (or a factor of
ignorance) then provides a prediction of safe - life for the component.
At the end of ‘safe - life’, the component is automatically retired from service, even if
no failure has occurred during service and the component has considerable residual life.
Although an estimate of fatigue life may be obtained from tests on the actual component,
the safe – life method is intrinsically theoretical in nature. This procedure has to account
for several unknowns; unexpected changes in loading conditions; errors in the
estimation of typical service load spectra; scatter in test results; variability in properties
among different batches of the same material; existence of initial defects in the
production process; corrosion of the parts; and human errors in the operation.
By selecting a large margin of safety a safe operating life can be guaranteed.
The approach is conservative and may not be desirable from the view point of economy
and performance. However, if fatigue cracks are nucleated in the component in service,
the component may fail catastrophically. In the safe – life approach the emphasis is
therefore on the prevention of crack initiation!
The fail – safe approach to fatigue design, by contrast, is based on the argument that,
even if an individual member of a structure fails, there should be sufficient structural
integrity in the remaining parts to enable the structure to operate safely until the crack
is detected. Components with multiple load paths are generally fail – safe because of
redundancy. In addition, the component may contain crack arresters to prevent
undesirable levels of crack growth.
The fail – safe approach therefore mandates PERIODIC INSPECTION along with a
requirement that the NDT techniques be capable of identifying flaws to enable prompt
REPAIRS or REPLACEMENTS.
Whatever philosophy is employed in fatigue design, it is preferable that the critical
components of a structure be inspected periodically. This step eliminates dangerous
consequences arising from false estimates and errors in the design stage, especially
with the safe – life approach
FRACTURE MECHANICS : FAMILY TREE
Linear Elastic Fracture Mechanics
LEFM
Linear elastic
time – independent material behavior
Elastic – Plastic Fracture Mechanics
EPFM
Non linear
time – independent material behavior
Dynamic Fracture Mechanics
Non linear time – dependent material behavior
Viscoelastic Fracture Mechanics
Viscoplastic Fracture Mechanics
The specific branch of Fracture Mechanics, one should use in a particular problem
that obviously depends on material behavior, component geometry, applied loads,
operating environment, etc.,
It is unlikely that all these topics can be covered in a single module. This module is
limited in scope to the study of Linear Elastic Fracture Mechanics. However, it
should form the foundation for the study of EPFM, DFM, VEFM, etc., in future
modules.
PRACTICAL USES OF FRACTURE MECHANICS
Provide a conceptually different approach to Engineering Design Practice; Namely The Damage Tolerance Design Methodology
Enables to quantify toughness of the materials as Resistance to Fracture (a failure mode due to crack propagation)
Enables stress analysis of components/structures with cracks
Helps to evaluate Fracture Mechanics parameters
Crack-tip Stress Intensity Factors (Ki) (i= 1,2,3)
Strain Energy Release Rates (Gi) (i=1,2,3)
Path-Independent Integral (J)
Crack Tip Opening Displacement (CTOD)
Identifies Fracture Criteria to predict residual strength of the cracked materials, components and structures as well as the direction of crack propagation.
Defines Fatigue Crack Growth Laws that enable life estimation of cracked components/structures
Helps to fix Non destructive Inspection Intervals
Supports Service failure Investigations involving fatigue and Fracture.
PRACTICAL USES OF FRACTURE MECHANICS
ASSIGNMENTS
1. FE modeling and SIF evaluation of different cracked bodies (with known target solutions)
2. Prediction of Residual strength of cracked bodies (comparison of different fracture criteria)
3. Prediction of fatigue crack growth using different FCG laws (for a given cracked body with known SIF solutions)
4. Fatigue Analysis – fatigue life of notched components using ANSYS (low cycle fatigue)
5. Fatigue Analysis – fatigue life of components using ANSYS (High cycle fatigue)
6. G evaluation (Penny shaped crack at interface)
7. J evaluation (DCB test)
8. CTOD evaluation (Compact Tension Test)
9. Critical study of standards for K, J, CTOD testing
10. Commentary on Analytical, experimental, Phenomenological and Computational aspects of Fracture Mechanics
11. Material Information System for Fracture Mechanics analysis and Fatigue Analysis: Surey & Assessment
12. Compendium of SIF solutions: 3D cracks.
13. Atlas of Fatigue Curves:Study and Update
14. Prediction of Crack Tip Plastic Zone(Shape or Size)
