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2/15/2016 1 1Lecture #7 – Fall 2015 1D. Mohr
151-0735: Dynamic behavior of materials and structures
by Dirk Mohr
ETH Zurich, Department of Mechanical and Process Engineering,
Chair of Computational Modeling of Materials in Manufacturing
Lecture #7: • Basic Notions of Fracture Mechanics
• Ductile Fracture
© 2015
2/15/2016 2 2Lecture #7 – Fall 2015 2D. Mohr
151-0735: Dynamic behavior of materials and structures
Basic Notions of Fracture Mechanics
2/15/2016 3 3Lecture #7 – Fall 2015 3D. Mohr
151-0735: Dynamic behavior of materials and structures
Fracture Mechanics
Fracture mechanics is a branch of mechanics that is concerned withthe study of the propagation of cracks and growth of flaws. Thestarting point of a fracture mechanics analysis therefore is astructure with a pre-existing crack or flaw. Central questions infracture mechanics are for example:
• Under which mechanical loads does a pre-existing crack propagate?
• What is the maximum size of a crack that canbe tolerated in a structure that is subject to aknown mechanical load such that the crackdoes not propagate?
crack
2/15/2016 4 4Lecture #7 – Fall 2015 4D. Mohr
151-0735: Dynamic behavior of materials and structures
Griffith theory – an energy approach
Griffith‘s (1921) made an attempt to come up with a fracturecriterion for brittle solids by writing down the energy balanceduring the growth of a crack.
E
ta 22
t
a2 dada
Consider an isotropic linear elastic plate subject to uniaxial tension and let W0denote the elastic strain energy stored in that system. According to Inglis (1913) analysis, the introduction of a through thickness crack of length 2a reduces the elastic strain energy (energy release) by
2/15/2016 5 5Lecture #7 – Fall 2015 5D. Mohr
151-0735: Dynamic behavior of materials and structures
Griffith theory
stot atE
taUU
4
22
0
t
a2 dada
Introducing the free surface energy per unit area, s, the total energy of the system then reads
After differentiating with respect to a, weobtain the rate of change of the totalenergy as a function of the crack lengthand the applied stress
stot t
E
ta
da
dU
42
2
2/15/2016 6 6Lecture #7 – Fall 2015 6D. Mohr
151-0735: Dynamic behavior of materials and structures
Griffith theory
t
a2 dada
Note that for small flaws and for lowapplied stresses, the surface energy termdominates, i.e. the total energy of thesystem would increase as the crackadvances. However, when the criticalcondition dUtot/da=0 is met, the change intotal energy becomes negative (energy rel
stot t
E
ta
da
dU
42
2
0422
st
E
ta
According to Griffith, this condition defines the onset of unstablecrack growth. The critical far field stress for fracture initiationtherefore reads
a
Esc
2
ease).
2/15/2016 7 7Lecture #7 – Fall 2015 7D. Mohr
151-0735: Dynamic behavior of materials and structures
Linear Elastic Fracture Mechanics (LEFM)
The stress fields in the vicinity of cracks in elastic media can becalculated using linear elastic stress analysis. In particular, theanalytical solutions have been developed for three basic modes offracture:
Mode I“opening mode”
Mode II“in-plane shear mode”
Mode III“out-of-plane shear mode”
2/15/2016 8 8Lecture #7 – Fall 2015 8D. Mohr
151-0735: Dynamic behavior of materials and structures
Linear elastic fracture mechanics
The leading terms of analytical solutions for the crack tip stressfields are typically expressed through the product of a radial andcircumferential term. For example, the stress field for Mode I planestress loading reads
2
322
sinsin1cos2
r
ax
2
322
sinsin1cos2
r
ay
23
22cossincos
2
r
axy
http://www.fracturemechanics.org
2/15/2016 9 9Lecture #7 – Fall 2015 9D. Mohr
151-0735: Dynamic behavior of materials and structures
Note that the governingterms all exhibit a singularityat the crack tip,
ijr
0
lim
Linear elastic fracture mechanics
ar /
crack
2
322sinsin1cos
2
r
ay
2/15/2016 10 10Lecture #7 – Fall 2015 10D. Mohr
151-0735: Dynamic behavior of materials and structures
Stress Intensity Factor
r
ay
2]0[
The limit
00
2lim:
yr
I rK
and thus aK I
is called stress intensity factor. It characterizes the magnitude of thestresses at the crack tip. An analog factor can be defined for Mode IIand Mode III fracture. In the above example, we have
The expressions of the stress intensity factor for different crack shapesand loading conditions can be found in many textbooks.
2/15/2016 11 11Lecture #7 – Fall 2015 11D. Mohr
151-0735: Dynamic behavior of materials and structures
In linear elastic fracture mechanics, it is assumed that a crackpropagates if the stress intensity factor reaches a critical value,
cI KK
Fracture toughness
The critical value KC for Mode I fracture under plane strainconditions is called fracture toughness. Its units are .mMPa
It is considered as a materialproperty which is measured usingSingle Edge Notch Bend (SENB) orCompact Tension (CT) specimens,see ASTM E-399 standard.
2/15/2016 12 12Lecture #7 – Fall 2015 12D. Mohr
151-0735: Dynamic behavior of materials and structures
The stress intensity factor characterizes only the leading term ofthe stress field near the crack tip. The exact solution for the near-tip fields includes also non-singular higher order terms
K-dominance
The annular region withinwhich the singular terms (so-called K-fields) dominate isdescribed by the radius rK ofthe zone of K-dominance.
sorder termhigher ][2
],[
ijij fr
Kr
ar /
crack
exact
K-field
2/15/2016 13 13Lecture #7 – Fall 2015 13D. Mohr
151-0735: Dynamic behavior of materials and structures
The K-fields can still be meaningful even if the “real” mechanicalsystem is different from that assumed in the theoretical analysis.Examples include situations where
Small scale yielding condition
• the crack is not sharp;• the material deforms plastically• micro-cracks are present near the crack tip
The condition of applicability of linear elastic fracture mechanics isthat the radius rp of the zone of inelastic deformation at the cracktip must be well confined inside the region of K-dominance.
Kp rr
2/15/2016 14 14Lecture #7 – Fall 2015 14D. Mohr
151-0735: Dynamic behavior of materials and structures
The classical fracture mechanics approach is based on theassumption that failure is the outcome of the growth of a pre-existing crack (and that all materials contain flaws).
An alternative to fracture mechanics
An alternative approach consists of assuming that a solid isinitially crack-free. Phenomenological fracture criteria in terms ofmacroscopic stresses and strains are then often employed topredict the onset of fracture.
A simple example of a phenomenological criterion for brittlesolids is to assume that fracture initiates when the maximumprincipal stress exceeds a critical value,
critI
2/15/2016 15 15Lecture #7 – Fall 2015 15D. Mohr
151-0735: Dynamic behavior of materials and structures
Ductile Fracture
2/15/2016 16 16Lecture #7 – Fall 2015 16D. Mohr
151-0735: Dynamic behavior of materials and structures
• Bending under tension
• Shear induced fracture
Fracture in Automotive Applications
• Fractures on tight radii during stamping cannot be predicted by Forming Limit Diagram (FLD)
FLD
• Usually termed as shear fracture, presents little necking, shows slant fracture
(Courtesy of ThyssenKrupp)
(Courtesy of US Steel)
2/15/2016 17 17Lecture #7 – Fall 2015 17D. Mohr
151-0735: Dynamic behavior of materials and structures
Interrupted tension experiments
• Flat notched tensile specimens(1.4mm initial thickness)
500mm
20mm
2/15/2016 18 18Lecture #7 – Fall 2015 18D. Mohr
151-0735: Dynamic behavior of materials and structures
1850mm
RD
Th
RD
Th
2/15/2016 19 19Lecture #7 – Fall 2015 19D. Mohr
151-0735: Dynamic behavior of materials and structures
191950mm
RD
Th
2/15/2016 20 20Lecture #7 – Fall 2015 20D. Mohr
151-0735: Dynamic behavior of materials and structures
… BUT: Almost no voids just below fracture surface!
Signature of voids on fracture surface!
Surface versus Cross-section View
2/15/2016 21 21Lecture #7 – Fall 2015 21D. Mohr
151-0735: Dynamic behavior of materials and structures
Failure Mechanism Summary
Define “strain to fracture” as strain at the onset of shear localization
2
2. Void volume fraction increases (more nucleation)
1
1. Onset of necking
e=0.2
3
3. Shear localization
e~0.9
44. Void sheet failure
e>1.0
2/15/2016 22 22Lecture #7 – Fall 2015 22D. Mohr
151-0735: Dynamic behavior of materials and structures
A Think Model of the Ductile Fracture Process
initial porosity
growth & nucleation
primary localization
growth & nucleation
secondary localization
nucleation & growth
final fracture
① ② ③ ④ ⑤ ⑥ ⑦
2/15/2016 23 23Lecture #7 – Fall 2015 23D. Mohr
151-0735: Dynamic behavior of materials and structures
Definition of “strain to fracture”
initial porosity
growth & nucleation
primary localization
growth & nucleation
secondary localization
nucleation & growth
final fracture
① ② ③ ④ ⑤ ⑥ ⑦
RVE
macroscopic equiv. plastic strain at instant of first localization
Strain to fracture =
2/15/2016 24 24Lecture #7 – Fall 2015 24D. Mohr
151-0735: Dynamic behavior of materials and structures
Tomographic observations
2xxx aluminum
2/15/2016 25 25Lecture #7 – Fall 2015 25D. Mohr
151-0735: Dynamic behavior of materials and structures
Void evolution in a plastic solid
…. void evolution depends on stress state
2/15/2016 26 26Lecture #7 – Fall 2015 26D. Mohr
151-0735: Dynamic behavior of materials and structures
Decomposition of the Stress Tensor
= +
HYDROSTATIC PART(average stress)
DEVIATORIC PART(differences among stresses)
𝜎𝑚 =𝜎𝐼 + 𝜎𝐼𝐼 + 𝜎𝐼𝐼𝐼
3
𝛔𝐈
𝛔𝐈𝐈
𝛔𝐈𝐈𝐈 𝛔𝒎
𝛔𝒎
𝛔𝒎
(𝛔𝐈𝐈𝐈−𝛔𝐦)
(𝛔𝐈 − 𝛔𝐦)
(𝛔𝐈𝐈−𝛔𝐦)
2/15/2016 27 27Lecture #7 – Fall 2015 27D. Mohr
151-0735: Dynamic behavior of materials and structures
Effect of Stress State on Void Evolution
HYDROSTATIC PART
DEVIATORIC PART
… controls void growth
STRESS TRIAXIALITY
𝜂 =𝜎𝑚ത𝜎
LODE PARAMETER
… is characterized by:
… controls shape change … is characterized by:
𝛔𝒎
2/15/2016 28 28Lecture #7 – Fall 2015 28D. Mohr
151-0735: Dynamic behavior of materials and structures
Definition of Lode Parameter
• Maximum shear stress (radius of biggest circle):
• Normal stress on plane of max. shear (center of biggest circle)
𝜏 =𝜎𝐼 − 𝜎𝐼𝐼𝐼
2
σI
𝜎𝑁 =𝜎𝐼 + 𝜎𝐼𝐼𝐼
2
𝜎𝑁
𝜎𝐼𝐼 − 𝜎𝑁
σIII σII
𝐿 =𝜎𝐼𝐼 − 𝜎𝑁
𝜏LODE PARAMETER
𝜏
• Position of the intermediate principal stress:
2/15/2016 29 29Lecture #7 – Fall 2015 29D. Mohr
151-0735: Dynamic behavior of materials and structures
Which states have the same Lode parameter?
• Uniaxial tension
• Plane strain tension
• Pure shear
𝜎𝜎
𝜎
𝜎/2
𝜎/2
ത𝜎 = 𝜎𝜎𝑚 = 𝜎/3
⇒ 𝜂 = 1/3
𝜎𝐼𝐼 = 𝜎𝐼𝐼𝐼 = 0 ⇒ 𝐿 = −1
𝜎/2ത𝜎 = 3/2𝜎
𝜎𝑚 = 𝜎/2⇒ 𝜂 = 1/ 3
ത𝜎 = 3/2𝜎𝜎𝑚 = 0 ⇒ 𝜂 = 0
𝜎𝐼𝐼 = (𝜎𝐼 + 𝜎𝐼𝐼𝐼)/2 ⇒ 𝐿 = 0
𝜎𝐼𝐼 = (𝜎𝐼 + 𝜎𝐼𝐼𝐼)/2 ⇒ 𝐿 = 0
2/15/2016 30 30Lecture #7 – Fall 2015 30D. Mohr
151-0735: Dynamic behavior of materials and structures
Lode angle parameter
I
III
II
IIIs
IIs
Is
plane
-10
+1
III
II
I
• Stress triaxiality:
m
• Lode angle parameter
)arccos(2
1
3
3
2
27
J
• Normalized third stress invariant
2/15/2016 31 31Lecture #7 – Fall 2015 31D. Mohr
151-0735: Dynamic behavior of materials and structures
Lode angle parameter
• Lode angle parameter
L )arccos(2
1
III III
1
axisymmetric compression
IIIIII
generalized shear0
IIIIII
axisymmetric tension
1
I
III
II
IIIs
IIs
Is
plane
-10
+1
III
II
I
• Lode parameter (Lode, 1926)
IIII
IIIIIIL
2
2/15/2016 32 32Lecture #7 – Fall 2015 32D. Mohr
151-0735: Dynamic behavior of materials and structures
Plane stress states
For isotropic materials, the stress tensor is fully characterized bythree stress tensor invariants,
},,{ 321 JJI or
while the stress state is characterized by the two dimensionlessratios of the invariants, e.g.
}/ ,/{2/3
2321 JJJI or
2
1
33 J
I
},,{ IIIIII
or }/ ,/{ IIIIIII } ,{
with
2/3
2
3
2
33arccos
21
J
J
and
2/15/2016 33 33Lecture #7 – Fall 2015 33D. Mohr
151-0735: Dynamic behavior of materials and structures
Plane stress states
Biaxial tension (III0
Tension-compression
(II0
Biaxial comp. (I0
generalized shear (0
axisymmetric compression (1
axisymmetric tension (1
Under plane stress conditions, one principal stress is zero. Thestress state may thus be characterized by the ratio of the two non-zero principal stresses.
As a result, the stresstriaxiality and the Lode angleparameter are no longerindependent for planestress, i.e. we have afunctional relationship
][
2/15/2016 34 34Lecture #7 – Fall 2015 34D. Mohr
151-0735: Dynamic behavior of materials and structures
Linear Mohr-Coulomb approximation
Unit Cell with Central Void
Stresses onPlane of Localization
Results from Localization Analysis
2/15/2016 35 35Lecture #7 – Fall 2015 35D. Mohr
151-0735: Dynamic behavior of materials and structures
fe
Principal stress space },,{ IIIIII
Haigh-Westergaardspace },,{ },,{ pe
Mixed strain-stress space
],[1 e ff k ],[ ff
Isotropic hardening lawCoordinate
transformation
𝜏 + 𝑐(𝜎𝐼 + 𝜎𝐼𝐼𝐼) = 𝑏
Mohr-CoulombHosford-
ത𝜎𝐻𝑓
][ pk e
Hosford-Coulomb Ductile Fracture Model
f
2/15/2016 36 36Lecture #7 – Fall 2015 36D. Mohr
151-0735: Dynamic behavior of materials and structures
n
HC
fg
cb
1
],[
1
e
fe
],,,,[ cbaff ee
Stress triaxiality
3 material parameters
von Mises equivalent plastic strain to fracture
Lode angle parameter
• General form
• Detailed expressions
IIIIaa
IIII
a
IIIII
a
IIIHC ffcffffffg 2||||||1
21
21
21
)1(
6cos
3
2][
If
)3(
6cos
3
2][
IIf
)1(
6cos
3
2][3
f
Hosford-Coulomb Ductile Fracture Model
2/15/2016 37 37Lecture #7 – Fall 2015 37D. Mohr
151-0735: Dynamic behavior of materials and structures
• Influence of parameter b
b = strain to fracture for uniaxial tension (or equi-biaxial tension)
b=0.2
b=0.3
b=0.4
b=0.5
a=1.3c=0.05
Hosford-Coulomb Ductile Fracture Model
2/15/2016 38 38Lecture #7 – Fall 2015 38D. Mohr
151-0735: Dynamic behavior of materials and structures
8.0a
2a
1.0c
1a
2.1a
5.1a
0c
1.0c
2.0c
35.0c
1a
Compare: Mohr-Coulomb
Can easily adjust the depth of the “plane strain valley”
• Influence of parameter a
Hosford-Coulomb Ductile Fracture Model
2/15/2016 39 39Lecture #7 – Fall 2015 39D. Mohr
151-0735: Dynamic behavior of materials and structures
• Influence of parameter c
c=0
c=0.1
c=0.2a=1.3n=0.1
c=0.05
Hosford-Coulomb Ductile Fracture Model
2/15/2016 40 40Lecture #7 – Fall 2015 40D. Mohr
151-0735: Dynamic behavior of materials and structures
Rapid calibration guideline
Step I:Identify b
Step II:Identify a
Step III:Identify c
a=2c=0
b=0.4c=0
b=0.4a=1.15
b=0.4 a=1.15 c=0.14
Hosford-Coulomb Ductile Fracture Model
2/15/2016 41 41Lecture #7 – Fall 2015 41D. Mohr
151-0735: Dynamic behavior of materials and structures
Excel program for calibration
Hosford-Coulomb Ductile Fracture Model
2/15/2016 42 42Lecture #7 – Fall 2015 42D. Mohr
151-0735: Dynamic behavior of materials and structures
fe
fe
plane stress
fe
3D View 2D View
plane stress
],[ ee ff
“heart” of the model:
Hosford-Coulomb Ductile Fracture Model
2/15/2016 43 43Lecture #7 – Fall 2015 43D. Mohr
151-0735: Dynamic behavior of materials and structures
Damage Accumulation
],[ e
e
f
pdD
0D (initial)
1D (fracture)
• Example: uniaxial tension
],[ ee ff
Define “damage indicator”
VIDEO
2/15/2016 44 44Lecture #7 – Fall 2015 44D. Mohr
151-0735: Dynamic behavior of materials and structures
Damage Accumulation
• Example: uniaxial compression followed by tension
],[ e
e
f
pdD
0D (initial)
1D (fracture)],[ ee ff
Define “damage indicator”
VIDEO
2/15/2016 45 45Lecture #7 – Fall 2015 45D. Mohr
151-0735: Dynamic behavior of materials and structures
Damage Accumulation
Non-linear loading path effect!
],[ e
e
f
pdD
0D (initial)
1D (fracture)],[ ee ff
Define “damage indicator”
• Example: uniaxial compression followed by tension
2/15/2016 46 46Lecture #7 – Fall 2015 46D. Mohr
151-0735: Dynamic behavior of materials and structures
Reading Materials for Lecture #7
• S. Suresh, “Fatigue of Materials”, Cambridge University Press, 1998.
• D. Mohr and S.J. Marcadet, http://www.sciencedirect.com/science/article/pii/S0020768315000700
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