Lecture #8: Ductile Fracture (Theory & Experiments)...D. Mohr 2/15/2016 Lecture #8 –Fall 2015 2 2 2 151-0735: Dynamic behavior of materials and structures Ductile Fracture (continuation

2/15/2016 1 1Lecture #8 – 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 #8: • Ductile Fracture (Theory & Experiments)

© 2015



Ductile Fracture (continuation from previous lecture)



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




• 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



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



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

][



Linear Mohr-Coulomb approximation

t

Unit Cell with Central Void

Stresses onPlane of Localization

Results from Localization Analysis



Mohr-Coulomb Failure Criterion

21 ][max cc n tn

According to the Mohr-Coulomb model, failure occurs along a plane of normal vector n for which the linear combination of the shear stress t and the normal stress n stresses acting on that plane reaches a critical value c2:

n

t

n

with the dimensionless “friction coefficient” c1.



Mohr-Coulomb criterion

21 ][max cc n tn

The maximization problem

has an analytical solution which is given by the solution of the equality for the ordered principal stresses

) ) bc IIIIIIII

with the coefficients2

1

1

1 c

cc

2

1

2

1

2

c

cb

and

222][ maxmax

bc

n

IIIIIIII

tt

Observe that the first term corresponds to the maximum shear stress, while the second term is the normal stress acting on the plane of maximum shear:



Mohr-Coulomb criterionfor plane stress

after Bai (2008)



Hosford-Coulomb criterion

)a

a

IIII

a

IIIII

a

IIIHF

1

)()()(2

1

The Mohr-Coulomb model can be seen as a linear combination of the Trescaequivalent stress and the normal stress,

bc IIII

stressTresca

IIII )(

As a generalization, the Tresca stress is substituted by the Hosford equivalent stress,

bc IIIIHf )(

which results in the so-called Hosford-Coulomb criterion:



Hosford equivalent stress

) ka

a

IIII

a

IIIII

a

IIIHF

1

)()()(2

1

The Hosford-Coulomb stress may be considered as an interpolation between the Tresca and von Mises envelopes. The limiting cases are obtained for:

• a=1 (Tresca):

IIIIaHF 1

• a=2 (von Mises):

2aHF

1a

2a

21 a



Hosford-Coulomb criterion

First in-plane stress

Seco

nd

in-p

lan

e st

ress

EMC

Yield

67.0

58.0

33.0

0

Hosford-Coulomb

Mohr-Coulomb

Mises



Principal stress space },,{ IIIIII

Haigh-Westergaardspace },,{

Coordinate Transformation

I

III

II

IIIs

IIs

Is

plane

-10

+1

III

II

I

)1fI

)2fII

)3fIII

)1(

6cos

3

2][1

f

)3(

6cos

3

2][2

f

)1(

6cos

3

2][3

f

with



f

Principal stress space },,{ IIIIII

Haigh-Westergaardspace },,{ },,{ p

Mixed strain-stress space

],[1 ff k ],[ ff

Isotropic hardening lawCoordinate

transformation

𝜏 + 𝑐(𝜎𝐼 + 𝜎𝐼𝐼𝐼) = 𝑏

Mohr-CoulombHosford-

ത𝜎𝐻𝑓

][ pk

Hosford-Coulomb Ductile Fracture Model

f



n

HC

fg

cb

1

],[

1

f

],,,,[ cbaff

Stress triaxiality

3 material parameters

von Mises equivalent plastic strain to fracture


• 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




f

f

plane stress

f

3D View2D View

plane stress

Hosford-Coulomb Fracture Initiation Model- for proportional loading -



• 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




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




• Influence of parameter c

c=0

c=0.1

c=0.2a=1.3n=0.1

c=0.05




f

f

f

a=1.89b=522.2c=0.001

SH PU NT20

NT6

CH

a=1.47b=1020.8c=0.008

SH

CHPU

NT20

NT6

SH

CHPU

a=1.29b=1371.5c=0.096

NT20

NT6

DP590DP780 TRIP780

Application of the Hosford-Coulomb Model



CP1000DP1000 CP1200

Application of the Hosford-Coulomb Model



Common feature for most metals:

Biaxial Tension Valley

/2

/2



f

f

f


Biaxial tension valley

Biaxial tension valley is due to Lode effect!

plane stress

plane stress



f


Biaxial tension valley

Biaxial tension valley is due to Lode effect!



f

f

plane stress

f

3D View 2D View

plane stress

],[ ff

“heart” of the model:




Damage Accumulation

],[

f

pdD

0D (initial)

1D (fracture)

• Example: uniaxial tension

],[ ff

Define “damage indicator”

VIDEO



Damage Accumulation

• Example: uniaxial compression followed by tension

],[

f

pdD

0D (initial)

1D (fracture)],[ ff


VIDEO



Damage Accumulation

Non-linear loading path effect!

],[

f

pdD

0D (initial)

1D (fracture)],[ ff


• Example: uniaxial compression followed by tension



Calibration Experiments

I. Shear test II. Plate bending III. Mini-Punch

20mm

60mm

• All experiments can be performed in a uniaxial testing machine

• Strains to fracture can be directly measured on specimen surface (no FEA needed)

Focus on simplicity and robustness of experimental technique:



Plate bending



Punch test



Model Calibration

SHEAR

PUNCH

BENDING

• Non-linearity in loading paths negligible

SHEAR

BENDING

PUNCH



Many different flat shear specimen designs exist for use in uniaxial loading frames …

Shear Fracture Specimen Design

… but we nonetheless developed a new geometry …

20mm



Shear Fracture Specimen Design

Stre

ss T

riax

ialit

y

0.33

0.167

0.00

0.50

0.667

Major challenge: Fracture prone to initiate prematurely at nearly plane strain tension conditions near boundaries!



Shear Specimen Optimization Problem



Shear Specimen - Optimization



Smiley Shear Specimen

• Basic geometry

• Optimized geometry

apparent shear fracture strain:

0.74

apparent shear fracture strain:

0.86



Typical smiley-shear experiment

• Average equivalent plastic strain rate: ~0.001 /s• Camera resolution: 4 mm/pixel



f

Other fracture experiments

PunchButterfly shear

Central hole tension

Notched tension



Flat Notched Tensile Specimens



Hybrid experimental-numerical determination of the loading history

Boundary displacement up to the onset of fracture (first surface crack)

Loading history

Surface strain field

Front view: Whole specimen

1pix=50µm

Back view: Gage section1pix<10µm

ExperimentFEA

Location of onset of fracture: Not known experimentallyElement with highest plastic strain

42



Approach

1. Identification of plasticity model for large strains based on

multi-axial experiments on specimens with homogeneous stress

and strain fields (“material test”)

2. Validation of the plasticity model for very large strains and

multi-axial loading based on experiments on specimens with

heterogeneous stress and strain fields (“structural test”)

3. Determination of loading path to fracture and assessment of

errors



Strain hardening

np

sA )(

44



Discretization errors

Fine mesh gives a converged result

Coarse

Fine

Medium

Very fine

2 elements through half thickness



16 elements through half thicknessEq

. pla

stic

str

ain

[-]

45



Notched tension: Exp. & FEA



Side view of FEA (R=20mm)

Side view of FEA (R=10mm)

Side view of FEA (R=6.67mm)

Notched tension: Exp. & FEA



Experimental detection of the onset of fracture

Crack propagation unstable in most experiments

t = 617s t = 618s

Instant of onset of fracture: appearance of the first surface crack

Location of onset of fracture: unknown experimentally

48



Loading path to fracture

49



Summary plots



Tensile specimen with central hole



Tensile specimen w/ central hole



Punch experiments



Punch experiments



Reading Materials for Lecture #8

• C. Roth and D. Mohr (2015), “Ductile fracture experiments with locally proportional loading histories”, Int. J. Plasticity, http://www.sciencedirect.com/science/article/pii/S0749641915001412

Documents

Lecture #8: Ductile Fracture (Theory & Experiments)...D. Mohr 2/15/2016 Lecture #8 –Fall 2015 2 2 2 151-0735: Dynamic behavior of materials and structures Ductile Fracture (continuation