Numerical Simulation of Fracture of Materials · 2017. 1. 3. · Numerical Simulation of Fracture...
Transcript of Numerical Simulation of Fracture of Materials · 2017. 1. 3. · Numerical Simulation of Fracture...
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]
Why should we care?
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
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.
The formal things:
Time of classes: Tuesday 8AM - 9:30AM
and??
QUESTIONS ???
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, ...
Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
Overview of Fracture
Let’s look at it!
What is fracture?
Are there different types?
What are we doing here?
Fracture of concrete
Fracture of metals
Fracture of steel
Fracture of wood
www.ecometry.biz/PicturesPatterns
Fracture of a polymere
Fracture of ???
Fracture of explosive
Rae et al. (2002) Proc. R. Soc. Lond.
Overview of Fracture
Let’s look at it!
What is fracture?
Are there different types?
What are we doing here?
What is failure?
Yielding dominated Fracture dominated- plasticity - localized- ductile - brittle- material dominates - crack dominated
Defects: Defects:
Some definitions ...
crack tip
crack process zone
crac
k fro
nt
crack wake
crack surface
Tangential and normal cracks
Different modes
mode I mode II mode III
Intergranular and Transgranularfracture cannot always be differentiated. Why?
Fracture of materials with precipitates
void nucleation void growth void coallescenceinitial state
Especially in polymers ...
Some metals ...
slow crackfast crack
At temperature ...
stress to fracture
1
10
meltingtemperature
Question: Why?
energy to fracture
meltingtemperature
In these lectures we concentrate on
mode I loading
brittle cracks
single cracks
metals
without precipitates
Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and Plasticity
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?
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
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?
Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and plasticity
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?
Overview of Continuum Mechanics
Stress & Strain
Equilibrium
Elasticity and plasticity
Elastic behavior:σ
E
ε
Hooke’s law: σij = Cijklεkl
linear elastic: Cijkl = const
How does the material unloaded?
What are the units?
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?
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?
Classical fracture mechanics
Overview of Fracture
Overview Continuum Mechanics
Classical Fracture Mechanics
Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
Pre-Griffith models
Principal Stress model
Principal strain model
Mohr-Coulomb model
Drucker-Prager model
Stress space
σI
σII
Principal stress = largest Eigenvalue of stress tensor.
2D: σI = 12(σx + σy) +
√[12(σx − σy)
]2 + σ2xy
Principal Stress modelRankine, Lamé, Navier (∼1800)
Fracture occurs,if principal stress reaches strength.σcompression ≤ σI ≤ σtension
Tσ
Tσ
σC
σC
σI
σII
Principal Stress model (continuation)
Advantages: Disadvantages:
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
Principal Strain model (continuation)
Advantages: Disadvantages:
Principal strain = largest Eigenvalue of strain tensor.EεI = σI − ν(σII + σIII)
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
Mohr-Coulomb model (continuation)
Advantages: Disadvantages:
Drucker-Prager modelDrucker, Prager (∼1920)
Fracture occurs,if shear stress reaches value whichdepends on hydrostatic pressure.
σI
σII
σC
Tσ
TσσC
Assume: σIII = 0
Drucker-Prager model (continuation)
Advantages: Disadvantages:
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
Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
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
What are the units?
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
Griffith Energy Balance Approach (continuation)
σ√
a =√
2γsEπ
Material properties:
Configuration:
Advantages: Disadvantages:
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:
Irwin’s extension (continuation)
Advantages: Disadvantages:
What are the units of R and γs?
Overview of Classical Fracture Mechanics
Pre-Griffith models
Energy Balance models
Current engineering models
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
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?
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)
Stress Intensity Factors
Gross, Seelig; Bruchmechanik
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
Irwin’s Stress Intensity Factor ModelIrwin (1950s)
Fracture occurs,if KI = σ
√πa ≥ KIc
The critical values are determined experimentally.
Ewalds, Wanhill; Fracture mechanics
The critical values is a material parameter.
Courtney; Mechanical behavior of Materials
Why are not all on the diagonal?
The critical values is temperature dependent.
30CrNiMo8 (20◦): KIc = 3650MPa√
mm
30CrNiMo8 (−20◦): KIc = 2000MPa√
mm
Why??
Stress Intensity Factors (continuation)
Advantages: Disadvantages:
The J-Integral
The J-Integral
ds ds2
U
ti
U =∫ εij
0 σkl dεkl
J =∫
S
[U ds2 − ti ui,1 dS
]ui,1: strain
The J-Integral
Fracture occurs, if J ≥ Jc
J = G =K2
I
E(1− ν2)
contour-independent !
applicable also for plastic cases
The J-Integral (continuation)
Advantages: Disadvantages:
Plastic zone
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?
Plasticity at the crack tip
thickn
ess
Test your understanding
The R-curve
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?
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?