lecture #7: basic notions of fracture mechanics ductile ... · linear elastic fracture mechanics...

$: Lecture #7: Basic Notions of Fracture Mechanics Ductile ... · Linear elastic fracture mechanics lim crack r/a 3 2 cos 2 1 sin 2 sin 2 ... lim 2 o T S V y r K I r ... The classical$
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



Basic Notions of Fracture Mechanics



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



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



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



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).



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”

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&uact=8&ved=0CAcQjRxqFQoTCMr9zY-k8MgCFUmCPgodb_oILQ&url=http://rsta.royalsocietypublishing.org/content/370/1965/1827&psig=AFQjCNGe8mHcQayXIjffRlmXuTj-QcXSZg&ust=1446503582131228








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

http://www.fracturemechanics.org/



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



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.



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.



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



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



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



Ductile Fracture



• 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)



Interrupted tension experiments

• Flat notched tensile specimens(1.4mm initial thickness)

500mm

20mm



1850mm

RD

Th

RD

Th



191950mm

RD

Th



… BUT: Almost no voids just below fracture surface!

Signature of voids on fracture surface!

Surface versus Cross-section View



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



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



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 =



Tomographic observations

2xxx aluminum



Void evolution in a plastic solid

…. void evolution depends on stress state



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)



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



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:



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



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

Unit Cell with Central Void

Stresses onPlane of Localization

Results from Localization Analysis



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



n

HC

fg

cb

1

],[

1

e

fe

],,,,[ cbaff ee

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




• 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




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




Excel program for calibration




fe

fe

plane stress

fe

3D View 2D View

plane stress

],[ ee ff

“heart” of the model:




Damage Accumulation

],[ e

e

f

pdD

0D (initial)

1D (fracture)

• Example: uniaxial tension

],[ ee ff

Define “damage indicator”

VIDEO



Damage Accumulation

• Example: uniaxial compression followed by tension

],[ e

e

f

pdD

0D (initial)

1D (fracture)],[ ee ff


VIDEO



Damage Accumulation

Non-linear loading path effect!

],[ e

e

f

pdD

0D (initial)

1D (fracture)],[ ee ff


• Example: uniaxial compression followed by tension



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

http://www.sciencedirect.com/science/article/pii/S0020768315000700

lecture #7: basic notions of fracture mechanics ductile ... · linear elastic fracture mechanics...

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