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Fracture mechanicsME 524
University of Tennesee Knoxville (UTK) /
University of Tennessee Space Institute (UTSI)Reza Abedi
Fall 2014
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Disclaimer The content of this presentation are produced by R. Abedi and also
taken from several sources, particularly from the fracture mechanicspresentation by Dr. Nguyen Vinh [email protected]
University of Adelaide (formerly at Ton Duc Thang University)
Other sources
Books:
T. L. Anderson, Fracture Mechanics: Fundamentals and Applications, 3rd Edition, CRC Press, USA, 2004 S. Murakami, Continuum Damage Mechanics, Springer Netherlands, Dordrecht, 2012. L.B. Freund, Dynamic Fracture Mechanics, Cambridge University Press, 1998. B. Lawn, Fracture of Brittle Solids, Cambridge University Press, 1993. M.F. Kanninen and C.H. Popelar, Advanced Fracture Mechanics, Oxford Press, 1985. R.W. Hertzberg, Deformation & Fracture Mechanics of Engineering Materials. John Wiley & Sons, 2012.
Course notes:
V.E. Saouma, Fracture Mechanics lecture notes, University of Colorado, Boulder. P.J.G. Schreurs, Fracture Mechanics lecture notes, Eindhoven University of Technology (2012). A.T. Zender, Fracture Mechanics lecture notes, Cornell University. L. Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.html
MSE 2090: Introduction to Materials Science Chapter 8, Failure
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Outline Brief recall on mechanics of materials
- stress/strain curves of metals, concrete
Introduction
Linear Elastic Fracture Mechanics (LEFM)- Energy approach (Griffith, 1921,
Orowan and Irwin 1948)
- Stress intensity factors (Irwin, 1960s)
LEFM with small crack tip plasticity
- Irwins model (1960) and strip yield model
- Plastic zone size and shape
Elastic-Plastic Fracture Mechanics- Crack tip opening displacement(CTOD), Wells 1963
- J-integral (Rice, 1958)
Dynamic Fracture Mechanics
3
Source: Sheiba, Olson, UT Austin
Source: Farkas, Virgina
Source: Y.Q. Zhang, H.
Hao,
J. Appl Mech (2003)
Wadley Research Group,
UVA3
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Outline (cont.)Fatigue
- Fatigue crack propagation & life prediction
- Paris law
Ductile versus Brittle Fracture
- Stochastic fracture mechanics
- Microcracking and crack branching in brittle
fracture
RolledAlloys.com
DynamicFractureBifurcation PMMA
Murphy 2006
quippy documentation (www.jrkermode.co.uk)
wikipedia
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Outline (cont.)Computational fracture mechanics
- FEM aspects:
- Isoparametric singular elements
- Calculation of LEFM/EPFM Integrals
- Adaptive meshing, XFEM
- Cohesive crack model (Hillerborg, 1976)
- Continuum Damage Mechanics
- size effect (Bazant)
http://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htm
Singular Element
Cracks in FEM Adaptive mesh XFEM bulk damage
P. Clarke UTSI
underwood.faculty.asu.edu
P. Leevers
Imperical College
Cohesive Zone
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Introduction
Cracks: ubiquitous !!!
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Definitions Crack, Crack growth/propagation
A fracture is the (local) separation of anobject or material into two, or more, pieces
under the action of stress.
Fracture mechanics is the field ofmechanicsconcerned with the study of thepropagation of cracks in materials. It uses
methods of analytical solid mechanics to
calculate the driving force on a crack andthose of experimental solid mechanics to
characterize the material's resistance to
fracture(Wiki).
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http://en.wikipedia.org/wiki/Mechanicshttp://en.wikipedia.org/wiki/Solid_mechanicshttp://en.wikipedia.org/wiki/Fracturehttp://en.wikipedia.org/wiki/Fracturehttp://en.wikipedia.org/wiki/Solid_mechanicshttp://en.wikipedia.org/wiki/Mechanics -
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Objectives of FM
What is the residual strength as a function of cracksize?
What is the critical crack size? How long does it take for a crack to grow from a
certain initial size to the critical size?88
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Approaches to
fracture Stress analysis
Energy methods
Computational fracture mechanics Micromechanisms of fracture (eg. atomic
level)
Experiments Applications of Fracture Mechanics
covered in
the course
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New Failure analysis
Stresses
Flaw size a Fracturetoughness
Fracture Mechanics (FM)
1970s
- FM plays a vital role in the design of every critical structural or machine component
in which durability and reliability are important issues (aircraft components, nuclear
pressure vessels, microelectronic devices).
- has also become a valuable tool for material scientists and engineers to guide their
efforts in developing materials with improved mechanical properties.1010
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Design philosophies
Safe life
The component is considered to be free of defects after
fabrication and is designed to remain defect-free during
service and withstand the maximum static or dynamic
working stresses for a certain period of time. If flaws,cracks, or similar damages are visited during service,
the component should be discarded immediately.
Damage tolerance
The component is designed to withstand the maximum
static or dynamic working stresses for a certain period
of time even in presence of flaws, cracks, or similar
damages of certain geometry and size.
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New Failure analysis
Stresses
Flaw size a Fracturetoughness
Fracture Mechanics (FM)
1970s
- FM plays a vital role in the design of every critical structural or machine component
in which durability and reliability are important issues (aircraft components, nuclear
pressure vessels, microelectronic devices).
- has also become a valuable tool for material scientists and engineers to guide their
efforts in developing materials with improved mechanical properties.1212
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1. Preliminaries
2. History
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U it
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Units
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I di i l t ti
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Indicial notationa 3D vector
two times repeated index=sum,
summation/dummy index
tensor notation
i: free index(appears precisely
once in each side of an equation)
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Engineering/matrix notation
Voigt notation
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S / i
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Stress/strain curve
Wikipedia
necking=decrease of cross-sectional
area due to plastic deformation
fracture
1: ultimate tensile strength1717
St t i
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Stress-strain curvesTrue stress and true (logarithmic) strain:
Plastic deformation:volume does not change
(extension ratio)
elationship between engineering and true stress/strain
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Principal stresses
Principal stresses are those stresses that act on principal surface.Principal surface here means the surface where components of shear-
stress is zero.
Principal direction
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Material classification /
Tensile test
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F t T
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Fracture TypesShearing
Plastic deformation (ductile material)
- Applied stress =>
- Dislocation generation and motion =>- Dislocations coalesce at grain boundaries =>
- Forming voids =>
- Voids grow to form macroscopic cracks
- Macroscropic crack growth lead to fracture
Dough-like orconical features
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F t T
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Fracture TypesCleavage
Fatigue
- Cracks grow a very short distance every time
(or transgranular)
split atom bonds
between grain boundaries
mostly brittle
Clam shell structures mark the location of cracktip after each individual cyclic loading
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F t T
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Fracture TypesDelamination (De-adhesion)
Crazing
- Common for polymers
- sub-micormeter voids initiate
stress whitening because
of light reflection from
crazes
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3.1 Fracture modes
3.2 Ductile fracture
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure27
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure
After crack
commencement
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure29
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure30
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure31
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure32
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure34
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure35
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3.3 Ductile to brittle transition
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Lecture source:
Prof. Leonid Zhigilei, http://people.virginia.edu/~lz2n/mse209/index.htmlMSE 2090: Introduction to Materials Science Chapter 8, Failure37
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Charpy v-notch test
Influence of temperature on Cv
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1. Temperature Effects
Temperature decrease => Ductile material can become brittle
BCC metals: Limited dislocation slip systems at low T =>
Impact energy drops suddenly over a relatively narrow temperature range around DBTT.
Ductile to brittle transition temperature (DBTT)or
Nil ductility transition temperature (T0)
FCC and HCP metalsremain ductile down to very low temperatures
Ceramics, the transition occurs at much higher temperatures than for metals
FCC and HCP metals (e.g. Cu, Ni, stainless steal)
strain
stress
- Titanic in the icy water of Atlantic (BCC)- Steel structures are every likely to fail in winter39
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Source: Tapany Udomphol, Suranaree University of Technology
http://eng.sut.ac.th/metal/images/stories/pdf/14_Brittle_fracture_and_impact_testing_1-6.pdf40
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2. Impurities and alloying effect on DBTT
Alloying usually increases DBTT by inhibiting dislocation motion.
They are generally added to increase strength or are (an unwanted)outcome of the processing
For steal P, S, Si, Mo, Oincrease DBTT while Ni, Mgdecease it.
Decrease of DBTT by Mg: formation of
manganese-sulfide (MnS) and consumption of
some S. It has some side effects
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3. Radiation embrittlement through DBTT
Medianfractureto
ughness(KJC
)
T0 T0Temperature
Irradiation effect:
1. Strengthening
2. More brittle
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3. Radiation embrittlement through DBTT
Wallins Master Curve
Irradiation inceases T0
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4. Hydrogen embrittlement through DBTT
Hydrogen in alloys drastically reduces
ductility in most important alloys: nickel-based alloys and, of course, both
ferritic and austenitic steel
Steel with an ultimate tensile strength of
less than 1000 Mpa is almost insensitive
A very common mechanism in
Environmentally assisted cracking (EAC):
High strength steel, aluminum, & titanium
alloys in aqueous solutions is usually
driven by hydrogen production at the
crack tip (i.e., the cathodic reaction)
Different from previously thought anodic
stress corrosion cracking(SCC)
Reason (most accepted) Reduces the bond strength between
metal atoms => easier fracture.
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Grains
Polycrystalline material:
Composed of many small crystals or grains
Grain boundary barrier to dislocation motion:High angle grain boundaries block slip and
harden the material
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5. Grain size
dcr dd
Small grain size =>
1. Lower DBTT (more duct ile)
2. Increases thoughness
DUCTILITY & STRENGH INCREASE
SIMULTANEOUSLY!!!!
ONLY STRENGTHING MECHANISM
THAT IMPROVES DUCTILITY
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Lowering Grain size
dcr dd
Small grain size =>
1. Lower DBTT (more duct ile)
2. Increases thoughness
DUCTILITY & STRENGH INCREASE SIMULTANEOUSLY!!!!
ONLY STRENGTHING MECHANISM THAT IMPROVES DUCTILITY
Grain boundaries have higher energies (surface energy)Grains tend to dif fuse and get
larger to lowe the energy
Heat treatments that provide
grain refinement such as air
cooling, recrystallisationduring hot working help to
lower transition temperature.
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DBTT relation to grain size analysis
dcr(T)
d
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6 Size effect and embrittlement
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6.Size effect and embrittlement
Experiment tests: scaled versionsof real structures
The result, however, depends on the size of thespecimen that was tested
From experiment result to engineering design:knowledge of size effect required
The size effect is defined by comparing the nominalstrength (nominal stress at failure) N of geometrically
similar structures of different sizes.
Classical theories (elastic analysis with allowablestress): cannot take size effect into account
LEFM: strong size effect ( )
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Size effect is crucial in concrete structures
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Size effect is crucial in concrete structures(dam, bridges), geomechanics (tunnels):
laboratory tests are small
Size effect is less pronounced in mechanicaland aerospace engineering the structures or
structural components can usually be tested
at full size. geometrically similarstructures of different sizes
b is thickness
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Structures and tests
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Structures and tests
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Size effect (cont.)
1. Large structures are softer than small structures.2. A large structure is more brittle and has a lower
strength than a small structure.
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Size effect
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Size effect
For very small structures the curve approaches the horizontal line and, therefore, the failure
of these structures can be predicted by a strength theory. On the other hand, for large
structures the curve approaches the inclined line and, therefore, the failure of these
structures can be predicted by LEFM.
Larger sizeMore brittle
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Bazants size effect law
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7 R t ff t d tilit
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7. Rate effects on ductility Same materials that show temperature toughness
sensitivity (BCC metals) show high rate effect
Polymers are highly sensitive to strain rate(especially for T > glass transition temperature)
Strain rate
1. Strength
2. Ductility
Strain rate similar to T 56
St i t ff t I t t h
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Strain rate effects on Impact toughness
Ki Computed using
quasistatic relations
(Anderson p.178)
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Ductile to brittle transition
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Ductile to brittle transition
Often hardening (increasing strength) reduces ductility
Phenomena affecting ductile/brittle response1. T (especially for BCC metals and ceramics)
2. Impurities and alloying
3. Radiation
4. Hydrogen embrittlement
5. Grain size6. Size effect
7. Rate effect
8. Confinement and triaxial stress state
Decreasing grain size is the only mechanismthat hardens and promotes toughness
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4. Linear Elastic Fracture Mechanics (LEFM)
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4.1Griffith energy approach
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Atomistic view of fracture
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r
xx0 Pc
Atomistic view of fracture
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Atomistic view of fracture
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Atomistic view of fracture
r
xx0 Pc
Summary
Cause: Stress Concentration!66
Stress concentration
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Stress concentration
Geometry discontinuities: holes, corners, notches,cracks etc: stress concentrators/risers
load lines
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Stress concentration (cont )
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Stress concentration (cont.)
uniaxial
biaxial
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Elliptic hole
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Elliptic holeInglis, 1913, theory of elasticity
radius of curvature
stress concentration factor [-]0 crac
!!!
2a
b
2a
b
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Griffiths work (brittle materials)
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Griffith s work (brittle materials)
FM was developed during WWI by English aeronautical
engineer A. A. Griffith to explain the following observations:
The stress needed to fracture bulk glassis around100 MPa
The theoretical stress needed for breaking atomic bondsis approximately 10,000 MPa
experiments on glass fibers that Griffith himselfconducted: the fracture stress increases as the fiber
diameter decreases => Hence the uniaxial tensile
strength, which had been used extensively to predict
material failure before Griffith, could not be a specimen-independent material property.
Griffith suggested that the low fracture strength observed
in experiments, as well as the size-dependence of
strength, was due to the presence of microscopic flaws
in the bulk material. 70
Griffiths size effect experiment
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Griffith s size effect experiment
the weakness of isotropic solids... is due to the presence of
discontinuities or flaws... The effective strength of technical materials
could be increased 10 or 20 times at least if these flaws could be
eliminated.'' 71
Fracture stress: discrepancy
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Fracture stress: discrepancybetween theory and experiment
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ause o screpancy:
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2a
b
m
p y1. Stress approach
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Griffiths verification experiment
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Griffith s verification experiment Glass fibers with artificial cracks (much
larger than natural crack-like flaws), tension
tests
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4.1.3. Cause of discrepancy:
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2. Energy approach
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Strain energy density
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Strain energy density
This work will be completely stored in the structure
in the form of strain energy. Therefore, the external work and strain energy
are equal to one another
Strain energy density
In terms of stress/strain
Consider a linear elastic bar of stiffness k, length L, area A, subjected to a
force F, the work is
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Strain energy density
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Strain energy density
Plane problems
Kolosov coefficient
Poissons ratio
shear modulus
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Energy balance during
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Energy balance duringcrack growth
external worksurface energy
kinetic energy
internal strain energyAll changes with respect to time are caused by changes in
crack size:
Energy equation is rewritten:
It indicates that the work rate supplied to the continuum by the applied loads is equal to
the rate of the elastic strain energy and plastic strain work plus the energy dissipated in
crack propagation
slow process
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Potential energy
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Brittle materials: no plastic deformation
Potential energy
sis energy required to form a unit of new surface
(two new material surfaces)
(linear plane stress, constant load)
Inglis solution
Griffiths through-thickness
crack
[J/m2=N/m]
7979
Comparison of stress & energy
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Comparison of stress & energy
approaches
8080
Energy equation for
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Griffith (1921), ideally brittle solids
Irwin, Orowan (1948), metals
plastic work per unit area of surface created
Energy equation forductile materials
Plane stress
(metals)
Griffiths work was ignored for almost 20 years
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Generalization of Energy
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Generalization of Energy
equation
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Energy release rate
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Energy release rateIrwin 1956
G: Energy released during fracture per unit of newly
created fracture surface area
energy available for crack growth (crack driving force)
fracture energy, considered to be a material property (independent of the
applied loads and the geometry of the body).
Energy release rate failure criterion
Crack extension force
Crack driving forcea.k.a
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Evaluation of G
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Evaluation of G
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Evaluation of G
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Evaluation of G
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G from experiments
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G from experiments
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G from experiment
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G from experiment
Fixed grips
a2: OB, triangle OBC=U
a1: OA, triangle OAC=U
90
B: thickness
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G from experiment
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G from experiment
Dead loads
B: thicknessa2: OB, triangle OBC=U
a1: OA, triangle OAC=U
OAB=ABCD-(OBD-OAC)
9191
G in terms of compliance
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G in terms of compliance
Fixed grips
inverse of stiffness
P
u
92
G in terms of compliance
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G in terms of compliance
Fixed load
inverse of stiffness
P
u
93
G in terms of compliance
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G in terms of complianceFixed grips Fixed loads
Strain energy release rate is identical for fixed
grips and fixed loads.
Strain energy release rate is proportional to the
differentiation of the compliance with respect to
the crack length.
94
Crack extension resistance curve (R-curve)
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R-curve
Resistance to fracture increases with growing
crack size in elastic-plastic materials.
Irwin
crack driving
force curve
SLOW
Irwin
Stable crack growth:
fracture resistance of thin
specimens is represented
by a curve not a single
parameter.9595
R-curve shapes
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pflat R-curve
(ideally brittle materials)
rising R-curve
(ductile metals)
stable crack growthcrack grows then stops,
only grows further if
there is an increase of
applied load
slope
96
Double cantilever beam (DCB)
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example
97
Double cantilever beam (DCB)
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example: load control
98
Double cantilever beam (DCB)
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example: displacement control
99
Example
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p
100
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Gc for different crack lengths are almost the same: flat
R-curve. 101
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102
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103103
Examples
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p
Double cantilever beam (DCB)
load control
disp. control
load control
a increases -> G decreases!!!
a increases -> G increase
104
4.2. Stress solutions, Stress Intensity FactorK (SIF)
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K (SIF)
105
Elastodynamics Boundary value problem
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106
Kinematics
displacement uvelocity v
acceleration a
strain
Kinetics
stress
traction
body force b
linear momentum p
106
Elastostatics Boundary value problem
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107
Constitutive equation
Hooks lawIsotropic3D
2D (plane strain)
2D (plane stress)
Balance of linear momentum
107
Displacement approach
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108108
Stress function approach
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109
What are Airy stress function approach?Use of stress function
Balance of linear momentum is automatically satisfied (no body force, static)
How do we obtain strains and displacements?
Strain: compliance e.g.
Displacements: Integration of
Can we always obtain u by integration? No
3 displacements (unknowns)6 strains (equations)
Need to satisfy strain
compatibility condition(s)
109
Stress function approach
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Generally no need to solve biharmonic function:
Extensive set of functions from complex analysis
No need to solve any PDE
Only working with the scalar stress function
110
Complex numbers
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111111
Stress function approach
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112112
Crack modes
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Crack modes
ar
113
Crack modes
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114
Westergaards complex stressf i f d I
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function for mode I1937 Constructing appropriate stress function (ignoring const. parts):
Using
116
Westergaards complex stressf ti f d I
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function for mode I
shear modulus
Kolosov coef.
117
Griffiths crack (mode I)
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boundary conditions
infinite plateis imaginary
1I
I
118
Griffiths crack (mode I)
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boundary conditions
infinite plate 119
Griffiths crack (mode I)
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120
Recall
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singularity
Crack tip stress field
inverse square root
121
Plane strain problems
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Plane strain
Hookes law
122
Stresses on the crack plane
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On the crack plane
crack plane is a principal
plane with the followingprincipal stresses
123
Stress Intensity Factor (SIF)
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Stresses-K: linearly proportional
K uniquely defines the crack tip stress field
modes I, II and III:
LEFM: single-parameter
SIMILITUDE
124
Singular dominated zone
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crack tip
K-dominated zone
(crack plane)
125
Mode I: displacement fieldRecall
-
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Displacement field
Recall
126
Crack face displacement
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Crack Opening Displacement
ellipse
127
Crack tip stress field in polardi t d I
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coordinates-mode I
stress transformation
128
Principal crack tip stresses
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129
Mode II problem
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Stress function
Boundary conditions
Check BCs
130
Mode II problem
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Stress function
mode II SIF
Boundary conditions
131131
S f
Mode II problem (cont.)
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Stress function
mode II SIF132
Universal nature of theasymptotic stress field
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asymptotic stress field
Irwin
Westergaards, Sneddon etc.
(mode I) (mode II)
133
Crack solution using V-Notch
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Airy stress function assumed to be
Compatibility condition
Final form of stress function
135
Crack solution using V-Notch
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Stress values
Boundary conditions
Eigenvalues and eigenvectors for nontrivial solutions
Mode I
Mode II136
Crack solution using V-Notch
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Sharp crack
After having eigenvalues, eigenvectors are obtained from
First termof stress expansion
(n = 2 constant stress)
Mode I
Mode II
137
Inclined crack in tension
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Recall
+
Final result
12
138
Cylindrical pressure vessel with an inclinedthrough-thickness crack
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thin-walled pressureclosed-ends
139
Cylindrical pressure vessel with an inclinedthrough-thickness crack
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?
This is why an overcooked hotdog usuallycracks along the longitudinal direction first(i.e. its skin fails from hoop stress, generatedby internal steam pressure).
Equilibrium
140
Computation of SIFs
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Analytical methods (limitation: simplegeometry)
- superposition methods
- weight/Green functions
Numerical methods (FEM, BEM, XFEM)
numerical solutions -> data fit -> SIF
handbooks
Experimental methods- photoelasticity
141
SIF for finite size samples
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Exact (closed-form) solution for SIFs: simple crack
geometries in an infiniteplate.
Cracks in finite plate: influence of external boundariescannot be neglected -> generally, no exact solution
142
SIF for finite size samples>
higher stress concentration
geometry/correction
factor [-]
dimensional
analysis
mode contributions are additive
Assumption: self-similar crack growth!!!
Self-similar crack growth: planar crack remains planar (
same direction as )
157157
SIF in terms of complianceB: thickness
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A series of specimens with different crack lengths: measurethe compliance C for each specimen -> dC/da -> K and G
B: thickness
158158
K as a failure criterionFailure criterion
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Failure criterion
Problem 1: given crack length a, compute the maximumallowable applied stress
Problem 2: for a specific applied stress, compute themaximum permissible crack length (critical crack length)
Problem 3: compute provided crack length and stressat fracture
fracture toughness
161
5. Elastoplastic fracture mechanics5.1 Introduction to plasticity
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5.2. Plastic zone models
5.3. J Integral5.4. Crack tip opening displacement (CTOD)
168
5.2. Plastic zone models- 1D Models: Irwin, Dugdale, and Barenbolt models
2D d l
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- 2D models:
- Plastic zone shape
- Plane strain vs. plane stress
169
Introduction Griffith's theory provides excellent agreement with experimental data
for brittle materials such as glass For ductile materials such as steel
http://en.wikipedia.org/wiki/Brittlehttp://en.wikipedia.org/wiki/Ductilehttp://en.wikipedia.org/wiki/Steelhttp://en.wikipedia.org/wiki/Steelhttp://en.wikipedia.org/wiki/Ductilehttp://en.wikipedia.org/wiki/Brittle -
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for brittlematerials such as glass. For ductilematerials such as steel,
the surface energy () predicted by Griffith's theory is usually
unrealistically high. A group working under G. R. Irwinat the U.S.Naval Research Laboratory(NRL) during World War II realized that
plasticity must play a significant role in the fracture of ductile
materials.
crack tipSmall-scale yielding:
LEFM still applies with
minor modifications done
by G. R. Irwin
(SSY)
170
Validity of K in presence of aplastic zone
http://en.wikipedia.org/wiki/Brittlehttp://en.wikipedia.org/wiki/Ductilehttp://en.wikipedia.org/wiki/Steelhttp://en.wikipedia.org/wiki/G._R._Irwinhttp://en.wikipedia.org/wiki/U.S._Naval_Research_Laboratoryhttp://en.wikipedia.org/wiki/U.S._Naval_Research_Laboratoryhttp://en.wikipedia.org/wiki/U.S._Naval_Research_Laboratoryhttp://en.wikipedia.org/wiki/U.S._Naval_Research_Laboratoryhttp://en.wikipedia.org/wiki/G._R._Irwinhttp://en.wikipedia.org/wiki/Steelhttp://en.wikipedia.org/wiki/Ductilehttp://en.wikipedia.org/wiki/Brittle -
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plastic zone
crack tip Fracture process usually occursin the inelastic region not the K-
dominant zone.
is SIF a valid failure criterion for materials
that exhibit inelastic deformation at the tip ?171
Validity of K in presence of aplastic zone
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plastic zone
same K->same stresses applied on the disk
stress fields in the plastic zone: the same
K still uniquely characterizes the crack
tip conditions in the presence of asmall plastic zone.
[Anderson]
LEFM solution
172
Paradox of a sharp crack
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At crack tip:
An infinitely sharp crack is merely a mathematical
abstraction.
Crack tip stresses are finitebecause
(i) crack tip radius is finite (materials
are made of atoms) and (ii) plasticdeformation makes the crack blunt.
173
5.2.1 Plastic zone shape: 1D models- 1storder approximation
2nd order Irwin model
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- 2ndorder Irwin model
- Strip yield models (Dugdale, and Barenbolt models)
- Effective crack length
174
Plastic correction:1storder approximation
A k d b d i l t diti
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A cracked body in a plane stresscondition
Material: elastic perfectly plasticwith yield stressOn the crack plane
(yield occurs)
first order approximation of plastic zone size: equilibrium
is not satisfied
stress singularity is truncated b
yielding at crack tip
175175
2.Irwins plastic correction
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176
2.Irwins plastic correction
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stress redistribution:
Plane strain
plastic zone: a CIRCLE !!!
177
Von Mises Yield Criterion
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Plane stress
Plane strain
178
proposed by Dugdale and Barrenblatt3. Strip Yield Model
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Infinite plate with though thickness crack 2a
Plane stress condition Elastic perfectly plastic material
Hypotheses:
All plastic deformation concentrates ina line in front of the crack.
The crack has an effective length whichexceeds that of the physical crack bythe length of the plastic zone.
: chosen such that stress singularityat the tip disappears.
179179
SIF for plate with normalforce at crack
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force at crack
Anderson p64180
3. Strip Yield Model (cont.)Superposition principle
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Irwins result
(derivation follows)
close to
0.318
0.392
181
3. Strip Yield Model:Dugdale vs Barenblatt model
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g
Dugdale: Uniform stress
More appropriate for
polymers
Barenblatt: Linear stress
More appropriate for
metals
182
5.2.1 Plastic zone shape: 1D models- 1storder approximation
- 2nd order Irwin model
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2 order Irwin model
- Strip yield models (Dugdale, and Barenbolt models)
- Effective crack length
183
Effective crack lengthIrwin Dugdale
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Irwin Dugdale
Crack tip blunting
184
Effective crack length
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187
Validity of LEFM solutioncrack tip
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crack tip
LEFM is better applicable to materials of high yield
strength and low fracture toughness
From Irwin,
Dugdale, etc
188
5.2.2 Plastic zone shape: 2D models- 2D models
- plane stress versus plane strain plastic zones
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p p p
189
Plastic yield criteriavon-Mises criterion Tresca criterion
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Maximum shear stress
s is stress deviator tensor
3D 2D
190
Plastic zone shapevon-Mises criterion
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Mode I, principal stresses
plane stress
Principal stresses:
plane strain191
Plastic zone shapevon-Mises criterion Tresca criterion
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plane stress
192
Plastic zone shape:Mode I-III
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193
Plastic zone sizes:Summary
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Source: Schreurs (2012)194
What is the problem
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with these 2D shapeestimates?
195
Stress not redistributed
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stress redistribution
(approximately) in 1D
196
Stress redistributed for 2D
Dodds 1991 FEM solutions
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Dodds, 1991, FEM solutions
Ramberg-Osgood material model
Effect of definition of yield
(some level of ambiguity)
Effect of strain-hardening:Higher hardening (lower n) =>
smaller zone197
Plane stress/plane strain
constrained by the
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dog-bone shape
constrained by the
surrounding material
Plane stress failure: more ductile Plane strain failure: mode brittle
198
Plane stress/plane strain
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Higher percentage of plate thickness is in plane
strain mode for thicker plates
Plane stress
Plane strain
As the thickness increases more through
the thickness bahaves as plane strain
199
ane stress p ane stra n:Fracture loci
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Loci of maximum shear stress for plane stress and strain
200
Plane stress/plane strain:What thicknesses are plane stress?
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Change of plastic loci to plane stress
mode as relative B decreases.Nakamura & Park, ASME 1988
Plane strain
201
Plane stress/plane strainToughness vs. thickness
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Plane strain fracture toughness
lowest K (safe)(Irwin)
202
Fracture toughness tests Prediction of failure in real-world applications: need the
value of fracture toughness
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value of fracture toughness
Tests on cracked samples: PLANE STRAINcondition!!!Compact Tension
Test
ASTM (based on Irwins model)
for plane strain condition:
203
Fracture toughness test
ASTM E399
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plane strain
ASTM E399
Linear fracture mechanics is only useful when the plasticzone size is much smaller than the crack size
205
5.3. J Integral (Rice 1958)
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206
IntroductionIdea
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deformation theory of plasticity can
be utilized
Monotonic loading: an elastic-plastic
material is equivalent to a nonlinear
elastic material
Idea
Replace complicated plastic model with nonlinear elasticity(no unloading)
207
J-integralEshelby, Cherepanov, 1967, Rice, 1968
C t f J i t l t
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Components of Jintegral vector
J integral in fracture
strain energy density
Surface traction
208
J-integral WikipediaRice used J1 for fracture characterization
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(1) J=0 for a closed path
(2) is path-independent
notch:traction-free209
5.3. 1. Path independence of J
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210
J Integral zero for a closed loop
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Gauss theorem
and using chain rule:
211
J Integral zero for a closed loop
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212
Path independence of J-integral
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J is zero over a closed path
AB, CD: traction-free crack faces
(crack faces: parallel to x-axis)
which path BC or AD should be used t
compute J?
213
5.3. 2. Relation between J and G(energy release property of J)
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214
Energy release rate of J integral:Assumptions
-
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215
Energy release rate of J-integralcrack grows, coord. axis move
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Self-similar crack growth
nonlinear elastic
,
216
J-integral symmetric skew-symmetric
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J-integral is equivalent to the
energy release rate for a
nonlinear elastic material
under quasi-static condition.
Gauss theorem,
Gauss theorem
217
Generalization of J integral
Dynamic loading
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Dynamic loading
Surface tractions on crack surfaces Body force
Initial strains (e.g. thermal loading)
Initial stress from pore pressures
cf. Saouma 13.11 & 13.12 for details
218
5.3. 3. Relation between J and K
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219
J-K relationshipBy idealizing plastic deformation
as nonlinear elastic, Rice was able
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(previous slide)
J-integral: very useful in numerical computation of
SIFs
to generalize the energy release rateto nonlinear materials.
220
J-K relationshipIn fact both J1 (J) and J2 are related to SIFs:
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Hellen and Blackburn (1975)
J1 & J2: crack advance for (= 0, 90) degrees
221
5.3. 4. Energy Release Rate,crack growth and R curves
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222
Nonlinear energy release rateGoal: Obtain J from P-Curve
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223223
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Nonlinear energy release rateGoal: Analytical equation for J
-
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225225
Nonlinear energy release rateGoal: Experimental evaluation of J
Landes and Begley, ASTM 1972
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226
g y
P-curves for differentcrack lengths a
Jas a function of
Rice proposes a
method to obtain J withonly one test for certain
geometries
cf. Anderson 3.2.5 for details
226
Crack growth resistance curve
A: R curve is nearly vertical:
ll t f t
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228
small amount of apparent
crack growth from blunting
: measure of ductile
fracture toughness
Tearing modulus
is a measure of crack
stability
A
If the crack propagates longer we even observe a flag R value
Rare in experiments because it
requires large geometries!228
Crack growth and stability
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229
The JRand J are similar to R and G curves for LEFM: Crack growth can happen when J = JR
Crack growth is unstable when
229
5.3. 5. Plastic crack tip fields; Hutchinson,Rice and Rosengren (HRR) solution
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230
RambergOsgood model
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231
Elastic model:
Unlike plasticity unloading
in on the same line
Higher n closer to
elastic perfectly plastic
231
Hutchinson, Rice andRosengren(HRR) solution Near crack tip plastic strains dominate:
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232
Assume the following r dependence forand
*
1. Bounded energy:
2. - relation *
232
HRR solution:Local stress field based on J
Final form of HRR solution: HRR solution
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234
LEFM solution
J plays the role of Kfor local , , u fields!
234
HRR solution:Angular functions
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235235
HRR solution:Stress singularity
HRR solution
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236
LEFM solution
Stress is still singular but with a weaker power of singularity!
236
5.3. 6. Small scale yielding (SSY) versuslarge scale yielding (LSY)
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237
Limitations of HRR solution
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238
McMeeking and Parks, ASTM STP
668, ASTM 1979
238
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From SSY to LSYLEFM: SSY satisfied and
generally have Generally
have
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240
have
PFM (or NFM): SSY is
gradually violated andgenerally
LSY condition:
Relevant parameters:
G (energy) K (stress)
Relevant parameters:
J (energy & used for stress)
No single parameter can
characterize fracutre!
J + other parameters
(e.g. T stress, Q-J, etc)240
LSY: When a single parameter (G, K,J, CTOD) is not enough?
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241
Crack growing out ofJ-dominant zone
Nonlinear vs plastic models
241
LSY: When a single parameter (G, K,J, CTOD) is not enough? T stress
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242
plane strain
Plastic analysis: is redistributed!Kirk, Dodds, Anderson
High negative T stress:
- Decreases
- Decreases triaxiality
Positive T stress:
- Slightly Increases
and reduce
triaxiality
242
LSY: When a single parameter (G, K,J, CTOD) is not enough? J-Q theory
QCrack tip
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243
n: strain hardening in HRR analysis
243
5.3. 7. Fracture mechanics versus material(plastic) strength
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244
Governing fracture mechanismand fracture toughness
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245
Fracture vs. Plastic collapseP
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a
W
P
(cracked section)
Yield:
short crack: fracture by plastic collapse!!!
high toughness materials:yielding
before fracture
LEFM applies when
unit thickness
246
Example
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250
5.4.Crack tip opening displacement(CTOD), relations with J and G
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252
Crack Tip OpeningDisplacement
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COD is zero at the crack
tips. 253
Crack Tip Opening Displacement:First order aproximation
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(Irwins plastic correction, plane stress)
CTOD
Wells 1961
COD is taken as the separation of the faces of the effective crack at the tip of the physical crack
254
Crack Tip Opening Displacement:Strip yield model
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Stresses that yielded K = 0
CTOD
K = 0
For
255
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CTOD-G-K relationFracture occurs
The degree of crack blunting increases
in proportion to the toughness
of the material
Wells observed:
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Under conditions of SSY, the fracture criteria
based on the stress intensity factor, the strainenergy release rate and the crack tip opening
displacement are equivalent.
of the material
material property
independent of
specimen and crack
length (confirmed by
experiments)
257
CTOD-J relation When SSY is satisfied G = J so we expect:
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In fact this equation is valid well beyond validity of LEFM and SSY
E.g. for HRR solution Shih showed that:
is obtained by 90 degree method:Deformed position corresponding to r* = r
and = -forms 45 degree w.r.t crack tip)
for
258
CTOD experimentaldetermination
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similarity of triangles: rotational factor [-], between 0 and 1
Plastic Hinge
Rigid
For high elastic deformation contribution, elastic corrections should be added
261
6. Computational fracture mechanics6.1. Fracture mechanics in Finite Element Methods
6.2. Traction Separation Relations (TSRs)
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262
6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
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6.1.3. Extraction of K (SIF), G6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
263
Numerical methods to solve PDEs Finite Difference (FD) & Finite Volume (FV) methods FEM (Finite Element Method) FD
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BEM (Boundary Element Method) MMs (Meshless/Meshfree methods)
FEMBEM
MMs
264
Fracture models Discrete crack models (discontinuousmodels): Cracks are explicitly modeled
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- LEFM- EPFM
- Cohesive zone models
Continuous models: Effect of (micro)cracksand voids are incorporated in bulk damage- Continuum damage models
- Phase field models
Peridynamic models: Material is modeledas a set of particles
265
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6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
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6.1.3. Extraction of K (SIF), G6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
267
Isoparametric Elements Geometry is mapped from a parent
element to the actual element
The same interpolation is used for
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geometry mapping and FEM solution(in the figure 2ndorder shape
functions are used for solution and
geometry)
Geometry map and solution are
expressed in terms of parent element Actual element
(Same number of nodes
based on the order)
Order of element
268
Singular crack tip solutions
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269
Finite Elements for singularcrack tip solutions
Direct incorporation of singular
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terms: e.g. enriched elements byBenzley (1974), shape functions
are enriched by KI, KIIsingular
terms
XFEMmethod falls into thisgroup (discussed later)
Quarter point (LEFM) and
Collapsed half point (Elastic-
perfectly plastic) elements: By
appropriate positioning ofisoparametric element nodes
create strain singularities
More accurate
Can be easily used in FEM software
270
Isoparametric singular elements
LEFM
Quarter point
Quad elementQuarter point collapsed
Quad elementQuarter point
Tri element
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singular form only along these lines
NOT recommended
Improvement:
- from inside all element
Problem
- Solution inaccuracy and sensitivity
when opposite edge 3-6-2 is curved
Improvement:
- Better accuracy and
less mesh sensitivity
Elastic-perfectly
plastic
1storder
2ndorder
Collapsed Quadelements
271
Motivation: 1D quadrature elementFind that yields singularity at x1
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Parent element
272
Motivation: 1D quadrature element
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Parent element
273
Moving singular position
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Transition elements:
According to this analysis
mid nodes of next layers
move to point from point
Lynn and Ingraffea 1977)
274
6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
6.1.3. Extraction of K (SIF), G
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6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
275
1. K from local fields1. Displacement
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or alternatively from the first quarter point element:
Recall for 1D
Mixed mode generalization:276
1. K from local fields
2. Stress
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or can be done for arbitrary angle () taking
angular dependence f() into account
Stress based method is less accurate because:
Stress is a derivative field and generally is one order less accurate than displacement
Stress is singular as opposed to displacement
Stress method is much more sensitive to where loads are applied (crack surface or
far field)
277
2. K from energy approaches1. Elementary crack advance (two FEM solutions for a and a+ a)
2. Virtual Crack Extension: Stiffness derivative approach
3. J-integral based approaches (next section)
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After obtaining G(or J=Gfor LEFM) K can be obtained from
278
2.1 Elementary crack advanceFor fixed grip boundary condition perform two simulations(1, a) and (2, a+a):
All FEM packages can compute strain (internal) energy Ui
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1 2
Drawback:
1. Requires two solutions
2. Prone to Finite Difference (FD) errors279
2.2 Virtual crack extension Potential energy is given by
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Furthermore wen the loads are constant:
This method is equivalent to J integral method (Park 1974)280
2.2 Virtual crack extension: Mixed mode For LEFM energy release rates G1and G2are given by
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Using Virtual crack extension (or elementary crack advance) computeG1and
G2for crack lengths a, a + a
Obtain KIand KIIfrom: Note that there are two
sets of solutions!
281
4
4
6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
6.1.3. Extraction of K (SIF), G
6 1 4 J i t l
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6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
282
J integralUses of J integral:1. LEFM: Can obtain KIand KIIfrom J integrals (G = J for LEFM)
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2. Still valid for nonlinear (NLFM) and plastic (PFM) fracture mechanics
Methods to evaluate J integral:
1. Contour integral:
2. Equlvalent (Energy) domainintegral (EDI):
Gauss theorem: line/surface (2D/3D) integral surface/volume integral
Much simpler to evaluate computationally
Easy to incorporate plasticity, crack surface tractions, thermal strains,etc.
Prevalent method for computing J-integral
Line integralVolume integral
283
J integral: 1.Contour integral Stresses are available and also more accurate at Gauss points
Integral path goes through Gauss points
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Cumbersome to formulate
the integrand, evaluate
normal vector, andintegrate over lines (2D)
and surfaces (3D)
Not commonly used
284
J integral: 2.Equivalent DomainIntegral (EDI)
General form of J integral
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Inelastic stressCan include (visco-) plasticity,
and thermal stresses
Elastic
Plastic Thermal (
temperature)
Kinetic energy density
: J contour approaches Crack tip (CT)
Accuracy of the solution deteriorates at CT
Inaccurate/Impractical evaluation of J using contour integral
285
J integral: 2. EDI: DerivationDivergence theorem: Line/Surface (2D/3D) integral Surface/Volume Integral
Application in FEM meshes
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2D mesh coverscrack tip
Original J integral contour
Surface integral after
using divergence theorem
Contour integral added to create closed surface
By using q = 0this integral in effect is zero
Zero integral on 1(q= 0)
Divergence theorem
286
J integral: 2. EDI
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Plasticity effectsThermal effects
Body force Nonzero cracksurface traction
General form of J
Simplified Case:
(Nonlinear) elastic, no thermal strain, no body force, traction free crack surfaces
This is the same as deLorenzis approach where
finds a physical interpretation (virtual crack extension)
287
J integral: 2. EDI FEM Aspects Since J0 0 the inner J0collapses to the crack tip (CT)
J1will be formed by element edges
By using spider web (rozet) meshesany
reasonable number of layers can be used
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reasonable number of layers can be usedto compute J:
Spider web (rozet) mesh:
One layer of triangular elements (preferably singular, quadrature
point elements) Surrounded by quad elements
1 layer 2 layer3 layer
288
J integral: 2. EDI FEM Aspects Shape of decreasing function q:
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Pyramid qfunction Plateau qfunction
Plateau q function useful when inner elements are not very accurate:
e.g. when singular/quarter point elements are not used
These elements do not contribute to J
289
J integral: Mixed mode loading For LEFM we can obtain KI, KII, KIIIfrom J1, J2, GIII:
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290
6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
6 1 3 Extraction of K (SIF) G
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6.1.3. Extraction of K (SIF), G6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
291
Different elements/methods to compute K
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J integral EDI method is by far the most accurate method
Interpolation of K from uis more accurate from : 1) higher
convergence rate, 2) nonsingular field. Unlike it is almost insensitive
to surface crack or far field loading
Except the first contour (J integral) or very small rthe choice of
element has little effect
J integral EDI K from stress K from displacement u
292
Different elements/methods to compute K:Effect of adaptivity on local field methods
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Even element h-refinement cannot improve K values by much
particularly for stress based method
K from stress K from displacement u
293
General Recommendations1. Crack surface meshing:
Nodes in general should be duplicated:
Modern FEM can easily handle duplicate nodes
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If not, small initial separation is initially introduced
When large strain analysis is required,initial mesh has finite crack tip radius.
The opening should be smaller than
5-10 times smaller than CTOD. Why?
294
General Recommendations2. Quarter point vs mid point elements / Collapsed elements around the
crack tip
For LEFM singular elements triangle quarter
elements are better than normal tri/quad and
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2ndorder
elements are better than normal tri/quad, andquarter point quad elements (collapsed or not)
Perturbation of quarter point by eresults in O(ge2) error in K(g=
h/a)
For elastic perfectly plastic material collapsed quad elements (1st
/2ndorder) are recommended
Use of crack tip singular elements are more important for local field
interpolation methods (uand ). EDI J integral method is less
sensitive to accuracy of the solution except the 1stcontour is used.
1storder
295
General Recommendations3. Shape of the mesh around a crack tip
a) Around the crack tip triangular singular elements are
recommended (little effect for EDI J integral method)
b) Use quad elements (2nd
order or higher) around the first contour
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b) Use quad elements (2 order or higher) around the first contourc) Element size: Enough number of elements should be used in
region of interest: rs, rp, large strain zone, etc.
d) Use of transition elements away from the crack tip although
increases the accuracy has little effect
Spider-web mesh (rozet)
a
b
d
296
General Recommendations4. Method for computing K
Energy methods such as J integral and G virtual crack extension
(virtual stiffness derivative) are more reliable
J integral EDI is the most accurate and versatile method
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J integral EDI is the most accurate and versatile method Least sensitive method to accuracy
of FEM solution at CT particularly if
plateau q is used
K based on local fields is the least accurate and most sensitive to
CT solution accuracy.
Particularly stress based method is not recommended.
Singular/ quarter point elements are recommended for these
methods especially when Kis obtained at very small r
297
6.1Fracture mechanics in Finite
Element Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
6.1.3. Extraction of K (SIF), G
6 1 4 J integral
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6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
298
Whats wrong with FEM forcrack problems
Element edges must conform to the crack
geometr make s ch a mesh is time cons ming
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geometry: make such a mesh is time-consuming,especially for 3D problems.
Remeshing as crack advances: difficult. Example:Bouchard et al.
CMAME 2003
299
Capturing/tracking cracks
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Fixed mesh Crack tracking XFEM enrichedelements
Crack/void capturing bybulk damage models
Brief overview in
the next sectionBrief overview in
continuum damage models
300
Fixed meshes Nodal release method (typically done on fixed meshes)
Crack advances one element edge at a time by releasing FEM nodes
Crack path is restricted by discrete geometry
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Also for cohesive elements they can be used for both extrinsic and intrinsic
schemes. For intrinsic ones, cohesive surfaces between all elements induces anartificial compliance (will be explained later)
301
Adaptive meshes Adaptive operations align element boundaries with crack direction
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Abedi:2010
Cracks generated by refinementoptions Element edges move to desireddirection
Element splitting:
Smoother crack path by element splitting:
cracks split through and propagate
between newly generated elements
302
6.1Fracture mechanics in Finite
Element Methods (FEM)6.1.1. Introduction to Finite Element method
6.1.2. Singular stress finite elements
6.1.3. Extraction of K (SIF), G
6.1.4. J integral
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6.1.4. J integral
6.1.5. Finite Element mesh design for fracture mechanics6.1.6. Computational crack growth
6.1.7. Extended Finite Element Method (XFEM)
303
Extended Finite ElementMethod (XFEM)
Belytschko et al 1999 set of enriched nodes
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standard part enrichment part
Partition of Unity (PUM) enrichment function
known characteristics of the problem (crack tip singularity,
displacement jump etc.) into the approximate space.
304
XFEM: enriched nodes
nodal support
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nodal support
enriched nodes = nodes whose support is cut by the
item to be enrichedenriched node I: standard degrees of freedoms
(dofs) and additional dofs305
XFEM for LEFMcrack tip with known
displacement
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crack edgedisplacement: discontinuous
across crack edge
306
XFEM for LEFM (cont.)Crack tip enrichment functions:
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blue nodes
red nodes
Crack edge enrichment functions:
307
XFEM for cohesive cracksWells, Sluys, 2001
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No crack tip solution is known, no tip
enrichment!!!
not enriched to ensure zero
crack tip opening!!!
308
XFEM: examples
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CENAERO, M. Duflot
Northwestern Univ.
309
6.2. Traction Separation Relations (TSRs)
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312
Cohesive models
Cohesive models remove stress singularity predicted by Linear
Elastic Fracture Mechanics (LEFM)
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: Stress scale
: Displacement scale
: Length scaleTraction is related to displacement jump across fracture surface
313
Sample applications of cohesive models
fracture of polycrystalline material
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delamination of compositesfragmentation
Microcracking
and branching
314
Cohesive models
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315
Process zone (Cohesive zone)
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316
Shape of TSR
smoothed trapezoidaltrapezoidalcubic polynomial
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bilinear softeninglinear softeningexponential
317
Cohesive model TableSource: Namas Chandra, Theoretical and Computational Aspects of Cohesive Zone Modeling, Department
of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University
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318
Cohesive model TableSource: Namas Chandra, Theoretical and Computational Aspects of Cohesive Zone Modeling, Department
of Mechanical Engineering, FAMU-FSU College of Engineering, Florida State University
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319
Scales of cohesive model
2 out of the three are independent
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Process zone size
Influences time step for time
marching methods
320
Why process zone size is important?
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321
When LEFM results match cohesive solutions?
crack speed process zone size
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p process zone size
322
Artificial compliance for intrinsic cohesive models
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323
4.3 Mixed mode fracture4.3.1 Crack propagation criteriaa) Maximum Circumferential Tensile Stress
b) Maximum Energy Release Rate
c) Minimum Strain Energy Density
4.3.2 Crack Nucleation criteria
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324
Motivation: Mixed mode crackpropagation criteria
Pure Mode I fracture:
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Mixed mode fracture (in-plane)
Note the similarity with yield surface plasticity model:
Example: von Mises yield criterion
325
Motivation: Experiment verification ofthe mixed-mode failure criterion
a circle inKI KII l
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Data points do not fall exactly on the circle.
KI, KII plane
self-similar growth326
Mixed-mode crack growth
Combination of mode-I, mode-II and mode-III
loadings: mixed-mode loading.
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Cracks will generally propagate along a curved
surface as the crack seeks out its path ofleast
resistance.
g g
Only a 2D mixed-mode loading (mode-I and
mode-II) is discussed.
327
4.3 Mixed mode fracture4.3.1 Crack propagation criteriaa) Maximum Circumferential Tensile Stress
b) Maximum Energy Release Rate
c) Minimum Strain Energy Density
4.3.2 Crack Nucleation criteria
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328
Maximum circumferentialstress criterionErdogan and Sih
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(from M. Jirasek) principal stress329
Maximum circumferentialstress criterion
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330
Maximum circumferentialstress criterionFracture criterion
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332
Experiment
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XFEM
333
Modifications to maximumcircumferential stress criterion
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334
4.3 Mixed mode fracture4.3.1 Crack propagation criteriaa) Maximum Circumferential Tensile Stress
b) Maximum Energy Release Rate
c) Minimum Strain Energy Density
4.3.2 Crack Nucleation criteria
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335
Maximum Energy ReleaseRate
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G: crack driving force -> crack will grow in the
direction that G is maximum
336
Maximum Energy ReleaseRate
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Stress intensity factors for kinked crack extension:
Hussain, Pu and Underwood (Hussain et al. 1974)
337
Maximum Energy ReleaseRate
Maximizationcondition
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338
4.3 Mixed mode fracture4.3.1 Crack propagation criteriaa) Maximum Circumferential Tensile Stress
b) Maximum Energy Release Rate
c) Minimum Strain Energy Density
4.3.2 Crack Nucleation criteria
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339
Strain Energy Density(SED) criterion Sih 1973
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340
Strain Energy Density(SED) criterion
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Minimizationcondition
Pure mode I (0 degree has smallest S)
341
Strain Energy Density(SED) criterion
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342
Comparison:a) Crack Extension angle
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Zoom view (low KIIcomponent)
Good agreement for low KII
~ 70 degree angle for mode II !
343
Comparison:b) Locus of crack propagation
Least conservative
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Most conservative
344
Mixed mode criteria:Observations
1. First crack extension 0is obtained followed by on whether crack extends in0
direction or not.
2. Strain Energy Density (SED) and Maximum Circumferential Tensile Stressrequire an r0but the final crack propagation locus is independent of r0.
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3. SED theory depends on Poisson ratio .
4. All three theories give identicalresults for small ratios of KII/KIand diverge
slightly as this ratio increases
5. Crack will always extend in the direction which attempts to minimize KII/KI
minimize6. For practical purposes during crack propagation all three theories yield very
similar paths as from 4 and 5 cracks extend mostly in mode I where the there is
a better agreement between different criteria
345
4.3 Mixed mode fracture4.3.1 Crack propagation criteriaa) Maximum Circumferential Tensile Stress
b) Maximum Energy Release Rate
c) Minimum Strain Energy Density
4.3.2 Crack Nucleation criteria
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346
Crack nucleation criterion
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347
Crack nucleation criterion
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8. Fatigue8.1. Fatigue regimes8.2. S-N, P-S-N curves
8.3. Fatigue crack growth models (Paris law)
- Fatigue life prediction8 4 Variable and random load
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8.4. Variable and random load
- Crack retardation due to overload
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Fatigue examples
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(source Course presentation S. Suresh MIT)
Fatigue fracture is prevalent! Deliberately applied load reversals (e.g. rotating systems)
Vibrations (machine parts)
Repeated pressurization and depressurization (airplanes) Thermal cycling (switching of electronic devices)
R d f ( hi hi l l )
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Random forces (ships, vehicles, planes)(source: Schreurs fracture notes 2012)
Fatigue occurs occur always and everywhere and ismajor source of mechanical failure
Fatigue Fatigue occurs when a material is subjected to repeated loading and unloading (cyclicloading). Under cyclic loadings, materials can fail (due to fatigue) at stress levels well below
their yield strength or crack propagation limit-> fatigue failure.
ASTMdefines fatigue life, Nf, as the number of stress cycles of a specified characterthat a specimen sustains before failureof a specified nature occurs.
http://en.wikipedia.org/wiki/ASTM_Internationalhttp://en.wikipedia.org/wiki/Structural_failurehttp://en.wikipedia.org/wiki/Structural_failurehttp://en.wikipedia.org/wiki/ASTM_International -
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blunting
resharpening
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Fatigue striationsFatigue crack growth:
Microcrack formation in accumulated
slip bands due to repeated loading
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