Lecture 7 Energy Release Rate

7/28/2019 Lecture 7 Energy Release Rate

1/20

CEE 770 Meeting 7

Objectives of This Meeting

Learn Generalized Stiffness Derivative Technique (akaGeneralized Virtual Crack Extension techniques) for:

Computing G, and its derivatives

Using these values to predict single crack tip stability

Using these values to predict multiple crack tip stability

Using these values to predict shape evolution of a 3D

crack

136


2/20

Review Pages 72-74, VCE/Stiffness

Derivative Methods

Recall from your studies of LEFM theory that energy release rate (aka crackdriving force) for a single 2D crack tip, in FEM context, is defined as:

fK uuuT =2

1 (43, p.72)

aaaG

+

=

fK TT

uuu21 (44, page 73, from Parks, 1975)

Where a is the length of crack, u is the nodal displacement vector,Kis the global

stiffness matrix,fthe global force vector, and nonzero contributions to

and occur only over elements adjacent to the crack front.aK / af /

137


3/20

Typical Near-Tip Meshing for a

Virtual Crack Extension, a

138


4/20

Energy Release Rate and Its Rates

aaaG

+

=

fK TT

uuu21 (44, page 73, from Parks, 1975)

This is OK for a single, 2D crack tip, BUT for more general, multiple crack2D case, and always for 3D:

i

T

i

T

i

i

aaa

G

fKuuu +=

2

1

Ti

Sd

Tip 1

1

2

3

4

3

4

5

6

5

6

2D Multiple Crack System

2

3D crack front

1

2

3

45

6

(69)

139


5/20

Energy Release Rate and Its Rates: Key Issues

How to compute the derivatives in

accurately?

What is the meaning ofa for 3D crack?

How to compute higher order derivatives? And whycompute them?

i

T

i

T

i

iaaaG

fKuuu +=

2

1

140


6/20

When Can a 2D Crack Propagate?

When Can a 3D Crack Change Shape?

For a 2D system with a single crack, the necessary and sufficientconditions for crack extension are:

G = Gc, and

G/a 0

where,G = - /a =Potential energy of the system

G/a = - 2 /a2

Therefore, one needs to be able to calculate both G and G/aaccurately.

Moreover, in many 2D situations it is possible that the system willhave multiple cracks!

Why????(70)

141


7/20

The Sufficiency Condition of LEFM

GIc

G

aa0

a0

P0

P0

P0

P0

P1

P1

P1

P2

P2

P2

P3

P3

P3

142


8/20

The Generalized Virtual Crack Extension MethodHwang, Wawrzynek, Tayebi, Ingraffea, "On the Virtual Crack Extension Method for Calculation of the Rates of Energy Release Rate,

Engineering Fracture Mechanics, 59, 521-542, 1998

Provides Gi and Gi/ai for multiple crack systems.

subjected to arbitrary thermal loading, crack-face loading,

body forces, in 2D/3D/axisymmetric problems, from a single FEA.

ji

T

i

T

jji

T

ji

T

j

i

aaaaaaaaa

G

ffKK 22

2

1u

uuu

uu ++=

Non-Zero for these

loadings

Null for 2D

Non-Zero for 3Dwhen i = j

i

T

i

T

i

i aaaG

fK

uuu +=

21

(71)

143


9/20

How to Compute The Derivatives

Analytically and Accurately

Gi

aj = uT K

aiK

1 f

aj K

aju

+f

aj K

aju

T

K1T f

ai

i j

i = j

For a multiply cracked system, the following expressions are the rates of energyrelease rate for crack tip i. They are the analytical and generalized version of thestiffness derivative technique (Parks, 1975). Even more general expressions whichaccount for mixed-mode are also available in Hwang et al.,1998:

Gi

ai= uT

K

aiK1

f

ai

K

aiu

1

2 uT

2K

ai2 u +

f

ai

K

aiu

T

K1T f

ai+ uT

2f

ai2

(72)

(73)

144


10/20

How to Compute The DerivativesAnalytically and Accurately

Note that:1. These operations need only be performed on a small number of

elements around the crack tip.

2. The element stiffness derivatives can be computed analytically via:

k= BTDB + BTDB + Tr( )BTDB[ ]v

dV (74)

2k= 2BTDB + 2BTDB+ BTD2B + 2 BTDB + 2Tr( ) BTDB + BTDB( )[ ]vdV

where is the virtual strain-like matrix,

= J1N

1

N

2

n1 n

2

=11 12

21 22

where 's are the geometry changes of the meshes due to virtual crack extension.

(75)

(76)

145

H C Th D i i


11/20



2Gi

ai2 =

1

2uT

3K

ai3 u 2u

T 2K

ai2

u

ai uT

K

ai

2u

ai2

(77)

uai

T

Kai

uai

+ 2

uai

2

T

fai

+ 2 uai

T

2

fai

2 + uT 3

fai

3

u

ai= K1

f

ai

K

aiu

and

2u

ai 2= K1

2f

ai2

2K

ai2u 2

K

ai

u

ai

3k= 3BTDB + 32BTDB +3BTD2B+ BTD3B[ ]

v

dV

+ 2 2B

TDB + 6 B TDB + B TDB( )+ 2 Tr()B TDB[ ]

v

dV

+ 3 2B

TDB + 2BTDB + B TD2B[ ]

v

Tr( )dV

3

B = 6 ( )3

B (78)146


12/20



You also need the force derivatives when relevant, assembled from elementalderivatives from the virtually perturbed elements:

fa =

feae

2f

a

2=

2fe

a

2

e

(79)

For example, if there is crack face pressure, p

fe = NT

p ds =s N

T

p + Tr()NT

p[ ]dss (80)

2fe =

2N

Tpds

s

= NT2p+ 2Tr()NTp + 2 NTp[ ]dss

147


13/20

If You Want Stress Intensity Rather ThanEnergy Release Rate Derivatives

Then, where H = E for plane stress and H = E/(1 - 2) for plane strain:

KI( )i = GiH

KI( )i

aj= 1

2Gi

aj

HGi

(81)

2 KI( )iai

2 =1

2

2Gi

ai2

H

Gi

1

4Gi

Gi

ai

2 H

Gi(82)

148


14/20

Example: Stability of Multiple 2D Crack Systems

Key issue: a/d ratio and its effect on crack tip interaction

Applied Uniform Displacement = 0.5 in.

Applied Uniform Displacement

a

d =2 in.

Think:When a/d is very small

and as a/d grows

.and finally?

E=29000ksi Deformed shape

149

C t d E R l R t d


15/20

Computed Energy Release Rates and

Rates of Energy Release Rates

150

a Top

Crack

Middle

Crack

Bottom

Crack

0.5 119 42 117

1 94 21 94

2 16 12 19

3 21 -3 16

4 -27 -16 -18

Total Rates ofG

0

50

100

150

200

250

300

0 1 2 3 4 5

Crack Length, a

Ene

rgy

Release

Ra

te,

G

Top, Bottom Cracks

Middle Crack

Observe: one can use the generalized VCE technique to predictsensitivity coefficients, strength of crack interaction, and

propagation stability of multiple 2D crack systems.


16/20

2D, Mixed-Mode Case

As previously discussed, we can decompose energy release rate into

components from symmetric and anti-symmetric fields (p.86-88)

,

( ) ( )iIIiIi

GGG +=

( ) ( ) ( )i

IT

II

i

T

IiIa

fuu

a

KuG

+=2

1

GI( )ia

j

= uI( )T K

ai

K1 fI

aj

K

aj

uI

+

fI

aj

K

aj

uI

T

K1T fI

ai

GII( )iaj

= uII( )T K

aiK

1 fII

aj

K

ajuII

+

fII

aj

K

ajuII

T

K1TfII

ai

For i j

GII( )i = 1

2uII( )

T K

aiuII + uII( )

TfII

ai(83)

Then,

(84)

151


17/20

2D, Mixed-Mode Case

For i = j

GI( )iai = u

I

( )T K

aiK

1 fI

ai

K

aiuI

1

2 uI

( )T2K

ai2 uI

+fI

ai

K

ai

uI

T

K1TfI

ai

+ uT2fI

ai2

,

(85)

GII( )iai = uII( )

T K

ai K

1 fII

ai

K

ai uII

1

2 uII( )

T 2K

ai2 uII

+fII

ai

K

ai

uII

T

K1T fII

ai

+ uT2fII

ai

2

(86)

152


18/20

Crack Growth Model:Crack Growth Model:

Principle of Minimum Potential EnergyPrinciple of Minimum Potential Energy

Maximize = (a0) (a0 +a)w.r.t. Next Crack Extension a

a

(a)

aa0 = G0 a + 1/2 G0 a

WE NEED TO KNOW G and dG

for Mult iple Planar/Non-Planar Crack Systemsin 2D/3D Under Arbitrary Loading

G0

G0

G(a)

a

aa0

153


19/20

Three Dimensional Problem:Find New Crack Front Shape, a(s)

( ) ( ) ( ) ( )

+=

FrontCrack

dssasGsasG 2

1 Maximize

(87)( )=FrontCrack

dssaA w.r.t a(s) subjected to

Piecewise Linearly Approximate G(s), a(s), G(s) along thecrack front.

s

Crack front

GiGj Gk

ai ajak

G(s)

a(s)154


20/20

Three Dimensional Problem:

Find New Crack Front Shape, a(s)

This problem is at the current state-of-the-art. For someinteresting insights and examples, see

Hwang, Wawrzynek, Ingraffea, " On the virtual crack extension method for calculating thederivatives of energy release rates for a 3D planar crack of arbitrary shape under mode-I

loading", Engineering Fracture Mechanics, 68, 7, 925-947, 2001.

155

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Lecture 7 Energy Release Rate