numerical simulation of fracture of materials · 2017. 1. 3. · numerical simulation of fracture...
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Numerical Simulation of Fracture of Materials
Allgemeines Wahlfach / Optional Course WS 2009/2010
Dr. Steffen Brinckmann, Dr. Rebecca Janisch
Contact information: [email protected],[email protected]
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Why should we care?
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What will you learn here?
Classical fracture mechanics
Fracture & FEM
Density Functional Theory
Quantum mechanics
Application of DFT: ’abinit’
Application of FEM: ’Abaqus’
Apply to two fracture cases
Location normally: IA 1/21; CIP: UHW 1219
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FAQ??
Q: Do I have enough computer knowledge?A: Programming part will be very easy.
Q: Why are the notes not “complete”?A: To increase active thinking and participation,
certain parts are left blank.
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The formal things:
Time of classes: Tuesday 8AM - 9:30AM
and??
QUESTIONS ???
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Classical fracture mechanics
H.L. Ewalds and R.J.H. Wanhill: Fracture Mechanics(hard to get)
D. Gross and T. Seelig: Bruchmechanik, Springer
T.L. Anderson: Fracture Mechanics: Fundamentals andApplications, CRC PR Inc.
or search for Fracture Mechanics at Amazon, Google, ...
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Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
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Overview of Fracture
Let’s look at it!
What is fracture?
Are there different types?
What are we doing here?
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Fracture of concrete
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Fracture of metals
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Fracture of steel
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Fracture of wood
www.ecometry.biz/PicturesPatterns
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Fracture of a polymere
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Fracture of ???
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Fracture of explosive
Rae et al. (2002) Proc. R. Soc. Lond.
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Overview of Fracture
Let’s look at it!
What is fracture?
Are there different types?
What are we doing here?
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What is failure?
Yielding dominated Fracture dominated- plasticity - localized- ductile - brittle- material dominates - crack dominated
Defects: Defects:
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Some definitions ...
crack tip
crack process zone
crac
k fro
nt
crack wake
crack surface
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Tangential and normal cracks
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Different modes
mode I mode II mode III
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Intergranular and Transgranularfracture cannot always be differentiated. Why?
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Fracture of materials with precipitates
void nucleation void growth void coallescenceinitial state
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Especially in polymers ...
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Some metals ...
slow crackfast crack
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At temperature ...
stress to fracture
1
10
meltingtemperature
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Question: Why?
energy to fracture
meltingtemperature
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In these lectures we concentrate on
mode I loading
brittle cracks
single cracks
metals
without precipitates
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Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
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Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and Plasticity
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The stress is a tensor.The traction is a vector.
SV
appliedt
nt1
2
traction: ti = σijnj
normal stress: σ11, σ22
shear stress: σ12, σ23
When do traction & stress have the same value?
What are the units of traction & stress?
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The strain is a tensor.
If Xi is the original position,
and xi is the current position of a particle
then it has moved by ui = xi − Xi.
The strain is εij = 12 (ui,j + uj,i)
What is the unit of strain?
ui,j: ith displacement component differentiated in jth direction.ui,j = ∂ui
∂Xj
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Plane stress and strain are 2D approximations.
PLANE STRESS PLANE STRAIN
picture: sheet of paper levee / dike
tickness: t→ 0 t→∞σ33 = 0 ε33 = 0
What other stress and strain components arezero?
What constrains can lead to plane strain?
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Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and plasticity
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Equilibrium: Tug-of-war
traction:∫
S ti dS = 0
stress: σij,j = 0
divergence of stress: divσ = 0
gradient: ∂∂xσ = 0
What is the difference?
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Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and plasticity
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Elastic behavior:σ
E
ε
Hooke’s law: σij = Cijklεkl
linear elastic: Cijkl = const
How does the material unloaded?
What are the units?
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Elastic constants:
Young’s modulus: E = 2µ(1 + ν)
Poisson’s ratio: ν = E2µ − 1
Shear modulus: µ = E2(1+ν)
What are the units?
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Plastic behavior:
y
σ
E
ε
σH
Yield: σmises = σflow
Flow stress: σflow = σy + Hε
Mises stress: σmises = 12
(σij − σkk
3 δij) (σij − σkk
3 δij)
How does the material unloaded?
How much energy is dissipated?
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Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
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Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
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Pre-Griffith models
Principal Stress model
Principal strain model
Mohr-Coulomb model
Drucker-Prager model
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Stress space
σI
σII
Principal stress = largest Eigenvalue of stress tensor.
2D: σI = 12(σx + σy) +
√[12(σx − σy)
]2 + σ2xy
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Principal Stress modelRankine, Lamé, Navier (∼1800)
Fracture occurs,if principal stress reaches strength.σcompression ≤ σI ≤ σtension
Tσ
Tσ
σC
σC
σI
σII
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Principal Stress model (continuation)
Advantages: Disadvantages:
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Principal Strain modelSaint-Venant, Bach (∼1880)
Fracture occurs,if principal strain reaches critical value.
εcompression ≤ εI ≤ εtension
σC
σI
σII
σC
Tσ
Tσ
Assume: σIII = 0
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Principal Strain model (continuation)
Advantages: Disadvantages:
Principal strain = largest Eigenvalue of strain tensor.EεI = σI − ν(σII + σIII)
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Mohr-Coulomb modelMohr, Coulomb (end 1800s)
Fracture occurs,if Mohr-circle touches boundary.
σI
σII
Tσ
Tσ
σC
σCσIσIII
σII
σ
τ
Assume: σIII = 0
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Mohr-Coulomb model (continuation)
Advantages: Disadvantages:
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Drucker-Prager modelDrucker, Prager (∼1920)
Fracture occurs,if shear stress reaches value whichdepends on hydrostatic pressure.
σI
σII
σC
Tσ
TσσC
Assume: σIII = 0
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Drucker-Prager model (continuation)
Advantages: Disadvantages:
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Disadvantage of Pre-Griffith models
Only maximum stress important (no fatigue)
Stress concentrations always lead to failure
Plasticity not included
Not for complex loading conditions
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Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
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Griffith Energy Balance ApproachGriffith (1920)
w
h
2a
Utotal = U0 + Ua + Uγ − UF
U0: elast. energy of loadeduncracked plate
Ua: change in elast. energydue to crack formation πσ2a2
E
Uγ: cleavage energy
UF: work by external forces
a� h ∼ w; unit thickness
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What are the units?
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Griffith Energy Balance Approach (continuation)
if γs is the surface formation energy per area:
U =
equilibrium: dUda = 0 =
4aγs
πσ2a2
E
a
U
instable equilibrium
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Griffith Energy Balance Approach (continuation)
σ√
a =√
2γsEπ
Material properties:
Configuration:
Advantages: Disadvantages:
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Irwin’s extensionIrwin (1948)
Fracture occurs,if πσ
2aE ≥ πσ2
c aE = Gc = R
≥ 2(γs + γp)
σc:
Gc: critical energy release rate
R:
γp: plastic strain work per surface area
γs:
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Irwin’s extension (continuation)
Advantages: Disadvantages:
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What are the units of R and γs?
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Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
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A convenient coordinate system
σ11
σ22
rϕ
2
1
Plane strain: κ = 3− 4ν, σ33 = ν(σ11 + σ22)Plane stress: κ = 3−ν
1+ν , σ33 = 0
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Mode I elastic fields
σ11
σ22
σ12
=KI√2πr
cosϕ1− sin ϕ
2 sin 3ϕ2
1 + sin ϕ2 sin 3ϕ
2sin ϕ
2 cos 3ϕ2
u1
u2=
KI
2G
√r
2π(κ− cosϕ)
cos φ2
sin φ2
What is stress at the crack tip?
What is the shape at the crack tip?
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Irwin’s Stress Intensity Factor ModelIrwin (1950s)
Fracture occurs,if KI = σ
√πa ≥ KIc
KI: MPa√
m
KIc:
plane stress: G = K2I
E
plane strain: G = K2I
E (1− ν2)
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Stress Intensity Factors
Gross, Seelig; Bruchmechanik
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Convenient: Stress Intensity Factors are added up..
KI shear stress + KI point load + KI external load ≥ KIc
However, crack interaction is not possible:KI center−cracked + KI edge−cracked
However, different modes add up:KI + KII≥ KIc
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Irwin’s Stress Intensity Factor ModelIrwin (1950s)
Fracture occurs,if KI = σ
√πa ≥ KIc
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The critical values are determined experimentally.
Ewalds, Wanhill; Fracture mechanics
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The critical values is a material parameter.
Courtney; Mechanical behavior of Materials
Why are not all on the diagonal?
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The critical values is temperature dependent.
30CrNiMo8 (20◦): KIc = 3650MPa√
mm
30CrNiMo8 (−20◦): KIc = 2000MPa√
mm
Why??
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Stress Intensity Factors (continuation)
Advantages: Disadvantages:
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The J-Integral
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The J-Integral
ds ds2
U
ti
U =∫ εij
0 σkl dεkl
J =∫
S
[U ds2 − ti ui,1 dS
]ui,1: strain
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The J-Integral
Fracture occurs, if J ≥ Jc
J = G =K2
I
E(1− ν2)
contour-independent !
applicable also for plastic cases
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The J-Integral (continuation)
Advantages: Disadvantages:
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Plastic zone
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Plasticity at the crack tip
yσ
22σ
r
Plane strain
2rp = 13π
(KIσy
)2
Plane stress
2rp = 1π
(KIσy
)2
only for: small-scale plasticity!
What is the effective crack-length?
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Plasticity at the crack tip
thickn
ess
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Test your understanding
The R-curve
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Definition of R-curve
Griffith and Irwin had said:πσ2a
E = G(Fapplied, a) ≥ Gc(a) = R(a)
1G(F ,a)
2G(F ,a)
R,G
a
R(a)G(F ,a)c
a0
Stable crack growth:∂G∂a
∣∣F=const. ≤
dRda
Why does R-curve increase? Why that criterium?Why ∂a?
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Application of R-curve
a
2h
b
applied force
dGda = +24F2a
EBh3
applied displacement
dGda = −48F2a
EBh3
What is the difference? Why? Explain!
Why is it an analytical expression?
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