fracture me chanc is me 524

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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

2
mailto:[email protected]:[email protected]


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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

4
http://www.jrkermode.co.uk/http://www.jrkermode.co.uk/


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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

5
http://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htmhttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htmhttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htmhttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htmhttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htmhttp://www.fgg.uni-lj.si/~/pmoze/ESDEP/master/toc.htm


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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).

77
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

13

U it


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Units

14

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
http://people.virginia.edu/~lz2n/mse209/index.htmlhttp://people.virginia.edu/~lz2n/mse209/index.html


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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:



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3.3 Ductile to brittle transition

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Lecture source:



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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
http://eng.sut.ac.th/metal/images/stories/pdf/14_Brittle_fracture_and_impact_testing_1-6.pdfhttp://eng.sut.ac.th/metal/images/stories/pdf/14_Brittle_fracture_and_impact_testing_1-6.pdfhttp://eng.sut.ac.th/metal/images/stories/pdf/14_Brittle_fracture_and_impact_testing_1-6.pdfhttp://eng.sut.ac.th/metal/images/stories/pdf/14_Brittle_fracture_and_impact_testing_1-6.pdf


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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

55

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


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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
http://en.wikipedia.org/wiki/Glasshttp://en.wikipedia.org/wiki/Glass


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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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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


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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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

7878

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

83

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

84

Evaluation of G
http://en.wikipedia.org/wiki/Failure_theory_(material)http://en.wikipedia.org/wiki/Failure_theory_(material)


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85

Evaluation of G


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86

Evaluation of G


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87

Evaluation of G


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88

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

90

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)

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G in terms of compliance


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Fixed grips

inverse of stiffness

P

u

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Fixed load

inverse of stiffness

P

u

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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.

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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

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Double cantilever beam (DCB)


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example

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example: load control

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example: displacement control

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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


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)

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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



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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



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boundary conditions

infinite plate 119



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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



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Stress values

Boundary conditions

Eigenvalues and eigenvectors for nontrivial solutions

Mode I

Mode II136



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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


150/384

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


151/384

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


152/384

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


153/384

- 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


154/384

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


155/384

176

2.Irwins plastic correction


156/384

stress redistribution:

Plane strain

plastic zone: a CIRCLE !!!

177

Von Mises Yield Criterion


157/384

Plane stress

Plane strain

178

proposed by Dugdale and Barrenblatt3. Strip Yield Model


158/384

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


159/384

force at crack

Anderson p64180

3. Strip Yield Model (cont.)Superposition principle


160/384

Irwins result

(derivation follows)

close to

0.318

0.392

181

3. Strip Yield Model:Dugdale vs Barenblatt model


161/384

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


162/384

2 order Irwin model

- Strip yield models (Dugdale, and Barenbolt models)

- Effective crack length

183

Effective crack lengthIrwin Dugdale


163/384

Irwin Dugdale

Crack tip blunting

184

Effective crack length


164/384

187

Validity of LEFM solutioncrack tip


165/384

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


166/384

p p p

189

Plastic yield criteriavon-Mises criterion Tresca criterion


167/384

Maximum shear stress

s is stress deviator tensor

3D 2D

190

Plastic zone shapevon-Mises criterion


168/384

Mode I, principal stresses

plane stress

Principal stresses:

plane strain191

Plastic zone shapevon-Mises criterion Tresca criterion


169/384

plane stress

192

Plastic zone shape:Mode I-III


170/384

193

Plastic zone sizes:Summary


171/384

Source: Schreurs (2012)194

What is the problem


172/384

with these 2D shapeestimates?

195

Stress not redistributed


173/384

stress redistribution

(approximately) in 1D

196

Stress redistributed for 2D

Dodds 1991 FEM solutions


174/384

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


175/384

dog-bone shape

constrained by the

surrounding material

Plane stress failure: more ductile Plane strain failure: mode brittle

198

Plane stress/plane strain


176/384

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


177/384

Loci of maximum shear stress for plane stress and strain

200

Plane stress/plane strain:What thicknesses are plane stress?


178/384

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


179/384

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


180/384

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


181/384

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)


182/384

206

IntroductionIdea


183/384

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


184/384

Components of Jintegral vector

J integral in fracture

strain energy density

Surface traction

208

J-integral WikipediaRice used J1 for fracture characterization


185/384

(1) J=0 for a closed path

(2) is path-independent

notch:traction-free209

5.3. 1. Path independence of J


186/384

210

J Integral zero for a closed loop


187/384

Gauss theorem

and using chain rule:

211

J Integral zero for a closed loop


188/384

212

Path independence of J-integral


189/384

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)


190/384

214

Energy release rate of J integral:Assumptions


191/384

215

Energy release rate of J-integralcrack grows, coord. axis move


192/384

Self-similar crack growth

nonlinear elastic

,

216

J-integral symmetric skew-symmetric


193/384

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


194/384

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


195/384

219

J-K relationshipBy idealizing plastic deformation

as nonlinear elastic, Rice was able


196/384

(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:


197/384

Hellen and Blackburn (1975)

J1 & J2: crack advance for (= 0, 90) degrees

221

5.3. 4. Energy Release Rate,crack growth and R curves


198/384

222

Nonlinear energy release rateGoal: Obtain J from P-Curve


199/384

223223


200/384

Nonlinear energy release rateGoal: Analytical equation for J


201/384

225225

Nonlinear energy release rateGoal: Experimental evaluation of J

Landes and Begley, ASTM 1972


202/384

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


203/384

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


204/384

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


205/384

230

RambergOsgood model


206/384

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:


207/384

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


208/384

234

LEFM solution

J plays the role of Kfor local , , u fields!

234

HRR solution:Angular functions


209/384

235235

HRR solution:Stress singularity

HRR solution


210/384

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)


211/384

237

Limitations of HRR solution


212/384

238

McMeeking and Parks, ASTM STP

668, ASTM 1979

238


213/384

From SSY to LSYLEFM: SSY satisfied and

generally have Generally

have


214/384

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?


215/384

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


216/384

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


217/384

243

n: strain hardening in HRR analysis

243

5.3. 7. Fracture mechanics versus material(plastic) strength


218/384

244

Governing fracture mechanismand fracture toughness


219/384

245

Fracture vs. Plastic collapseP


220/384

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


221/384

250

5.4.Crack tip opening displacement(CTOD), relations with J and G


222/384

252

Crack Tip OpeningDisplacement


223/384

COD is zero at the crack

tips. 253

Crack Tip Opening Displacement:First order aproximation


224/384

(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


225/384

Stresses that yielded K = 0

CTOD

K = 0

For

255


226/384

CTOD-G-K relationFracture occurs

The degree of crack blunting increases

in proportion to the toughness

of the material

Wells observed:


227/384

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:


228/384

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


229/384

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)


230/384

262

6.1Fracture mechanics in FiniteElement Methods (FEM)6.1.1. Introduction to Finite Element method

6.1.2. Singular stress finite elements


231/384

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


232/384

BEM (Boundary Element Method) MMs (Meshless/Meshfree methods)

FEMBEM

MMs

264

Fracture models Discrete crack models (discontinuousmodels): Cracks are explicitly modeled


233/384

- 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


234/384




235/384




267

Isoparametric Elements Geometry is mapped from a parent

element to the actual element

The same interpolation is used for


236/384

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


237/384

269

Finite Elements for singularcrack tip solutions

Direct incorporation of singular


238/384

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


239/384

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


240/384

Parent element

272

Motivation: 1D quadrature element


241/384

Parent element

273

Moving singular position


242/384

Transition elements:

According to this analysis

mid nodes of next layers

move to point from point

Lynn and Ingraffea 1977)

274



6.1.3. Extraction of K (SIF), G


243/384

6.1.4. J integral



275

1. K from local fields1. Displacement


244/384

or alternatively from the first quarter point element:

Recall for 1D

Mixed mode generalization:276

1. K from local fields

2. Stress


245/384

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)


246/384

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


247/384

1 2

Drawback:

1. Requires two solutions

2. Prone to Finite Difference (FD) errors279

2.2 Virtual crack extension Potential energy is given by


248/384

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


249/384

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 1 4 J i t l


250/384

6.1.4. J integral



282

J integralUses of J integral:1. LEFM: Can obtain KIand KIIfrom J integrals (G = J for LEFM)


251/384

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


252/384

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


253/384

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


254/384

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


255/384

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


256/384

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:


257/384

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:


258/384

290



6 1 3 Extraction of K (SIF) G


259/384




291

Different elements/methods to compute K


260/384

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


261/384

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


262/384

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


263/384

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


264/384

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


265/384

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 4 J integral


266/384

6.1.4. J integral



298

Whats wrong with FEM forcrack problems

Element edges must conform to the crack

geometr make s ch a mesh is time cons ming


267/384

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


268/384

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


269/384

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


270/384

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



6.1.4. J integral


271/384

6.1.4. J integral



303

Extended Finite ElementMethod (XFEM)

Belytschko et al 1999 set of enriched nodes


272/384

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


273/384

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


274/384

crack edgedisplacement: discontinuous

across crack edge

306

XFEM for LEFM (cont.)Crack tip enrichment functions:


275/384

blue nodes

red nodes

Crack edge enrichment functions:

307

XFEM for cohesive cracksWells, Sluys, 2001


276/384

No crack tip solution is known, no tip

enrichment!!!

not enriched to ensure zero

crack tip opening!!!

308

XFEM: examples


277/384

CENAERO, M. Duflot

Northwestern Univ.

309

6.2. Traction Separation Relations (TSRs)


278/384

312

Cohesive models

Cohesive models remove stress singularity predicted by Linear

Elastic Fracture Mechanics (LEFM)


279/384

: Stress scale

: Displacement scale

: Length scaleTraction is related to displacement jump across fracture surface

313

Sample applications of cohesive models

fracture of polycrystalline material


280/384

delamination of compositesfragmentation

Microcracking

and branching

314

Cohesive models


281/384

315

Process zone (Cohesive zone)


282/384

316

Shape of TSR

smoothed trapezoidaltrapezoidalcubic polynomial


283/384

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


284/384

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


285/384

319

Scales of cohesive model

2 out of the three are independent


286/384

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

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Artificial compliance for intrinsic cohesive models


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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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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

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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.

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Maximum circumferentialstress criterionErdogan and Sih


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(from M. Jirasek) principal stress329

Maximum circumferentialstress criterion


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Maximum circumferentialstress criterionFracture criterion


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Experiment


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XFEM

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Modifications to maximumcircumferential stress criterion


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Maximum Energy ReleaseRate


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G: crack driving force -> crack will grow in the

direction that G is maximum

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Stress intensity factors for kinked crack extension:

Hussain, Pu and Underwood (Hussain et al. 1974)

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Maximizationcondition


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Strain Energy Density(SED) criterion Sih 1973


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Strain Energy Density(SED) criterion


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Minimizationcondition

Pure mode I (0 degree has smallest S)

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Strain Energy Density(SED) criterion


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Comparison:a) Crack Extension angle


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Zoom view (low KIIcomponent)

Good agreement for low KII

~ 70 degree angle for mode II !

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Comparison:b) Locus of crack propagation

Least conservative


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Most conservative

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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

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Crack nucleation criterion


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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

7/26/2019 Fracture Me Cha

fracture me chanc is me 524

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