Configurational Equilibrium of Cracks Affected by Surface Stress and - - PowerPoint PPT Presentation

Configurational Equilibrium of Cracks Affected by Surface Stress and Crack-Tip Point Load

Chien H. Wu

University of Illinois at Chicago

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Configurational Equilibrium

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∂ ∂ = Potential Energy Configuration

b g b g

Effect of Surface Stress and Crack-Tip Load

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Stretch unit cube along x, then split Split unit cube, then stretch along x Work A = Work B

Σ = ∂Γ ∂ε /

2 2 Γ Σε ( ) ( ) + + = W ε

b g

Work B

W( ) ( ) ε ε + = 2Γ Work A

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Stretch Then Split Split Then Stretch 1 1 1 1+ε 1+ε 1+ε

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Surface Stress

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1 1+ε

Γ ε

a f: Surface Energy per Unit Referential Area

γ ε

b g: Surface Energy per Unit Spatial Area

γ ε ε ε ε ε γ ε

a f a f a f a f a f

= + = − = ∂Γ ∂ε = + ∂γ ∂ε 1 1 1 Γ Γ Σ , Linear Density: Γ Γ Σ Γ Σ Γ ε ε γ ε ε

a f a f

= + = + −

• ,

( ) Γo: Surface Tension Coefficient

Σ o: Surface Stress Coefficient

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Referential and Spatial Forms

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Behaves like a pre-stressed membrane that is perfectly fitted on the bounding surface of a bulk material body. Interacts with the deformation of the bulk material through the curvature of the surface. Becomes increasingly important at reduced scales. Effect on linear and curvilinear cracks.

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Surface Stress Coefficient

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Griffith Crack (1921 & 1924)

σ11 σ22 R → ∞ Configurational Energy: = Π Π ∆Π

0 +

Stress Intensity Factor:

1

K = σ π

22

a ∆Π Γ = − + + κ µ π σ 1 8 4

2 22 2

a a ∂∆Π ∂ = ⇒ = ≡ + 2 16 1

1 2 1 2

a K K C µΓ κ For Configurational Equilibrium:

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Configurational Energy

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Griffith Crack with Surface Stress

σ22 σ11 R → ∞ Configurational Energy: = Π Π ∆Π

0 +

Stress Intensity Factor:

1

K (same) = σ π

22

a ∆Π Γ Σ = − + + − + − κ µ π σ κ µ σ σ 1 8 4 1 8 4

2 22 2 22 11

a a a a

f

∂∆Π ∂ = ⇒ + −

F H G I K J

− = 2 2 1

1 2 11 22 1 1 2

a a K Σ π σ σ K K C For Configurational Equilibrium: Σ0

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Configurational Energy

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Griffith Crack with Surface Stress and Crack-Tip Point Load P

Configurational Energy: = Π Π ∆Π

0 +

Stress Intensity Factor:

1

K (still the same) = σ π

22

a ∆Π Γ Σ = − + + − − + − − κ µ π σ κ µ σ σ 1 8 4 2 1 8 4 2

2 22 2 22 11

a a a ( ) ( ) P P a

f

∂∆Π ∂ = ⇒ + − −

F H G I K J

− −

F H G I K J

= 2 2 1 1 2

1 2 11 22 1 1 2

a a K Σ Γ P K P K C π σ σ For Configurational Equilibrium: Σ0

σ11

σ 22

R → ∞

P P

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Configurational Energy

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Surface Stress, Internal Pressure p,and Crack-Tip Load P

Configurational Energy: = Π Π ∆Π

0 +

Stress Intensity Factor:

1

K = p πa ∆Π Γ Σ = − + + − + − − κ µ π κ µ 1 8 4 2 1 2 2

2 2

a a a p P P p ( ) ( ) ∂∆Π ∂ = ⇒ + − − + − −

F H G I K J

= 2 2 2 1 1 1 2

1 2 1 1 2

a a K ( ) ( ) Σ Γ P K P K C π κ κ

a f

For Configurational Equilibrium:

R → ∞

P P

Pressure p

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Configurational Energy

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Circular-Arc Cracks, Remote Tension and Surface Stress

σ

R → ∞

ρ

2φ

σ

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Configurational Energy

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Circular-Arc Cracks, Remote Tension and Surface Stress

Remote Tension:

(o)

K K K

I

• II
• 2

2 2 2 2 2

1 2 ≡ + = +

F H I K

( ) ( )

sin / sin σ πρ φ φ Remote Tension and Surface Stress: K K K

I II 2 2 2 2 2 2

1 2 ≡ + = −

F H G I K J

+

F H I K

σ φ πρ πρ φ φ Σ sin / sin For Configurational Equilibrium: ∆Π Σ = − + − − −

F H I K

4 1 2 1 3

2

ρφΓ κ µ φ φ πσρ φ π

a fa f a f

cos cos ∂∆Π = ⇒ −L

N M O Q P

− = 2 4 2

2 2

ρ∂φ φ πρ φ K 2 Σ sin / sin K KIC

σ

ρ

2φ

Remote Tension

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Configurational Energy

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Remote Loading, Surface Stress and Crack-Tip Load

σ11

σ 22

2 Σ − P

R → ∞ P The Bulk-Surface System Free-Body Of The “Surface” P P P

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Elasticity Solution

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Circular-Arc Crack,Remote Loading and Surface Stress The Bulk-Surface System Free-Body Of The “Surface”

σ

R → ∞

ρ

2φ

2 Σ 2 Σ

Σ 0 ρ

Σ 0 ρ

σ

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Elasticity Solution

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Remote Loading, Surface Stress and Crack-Tip Load

u u u

• s

s s α αβ αβ α αβ αβ α αβ αβ

ε τ ε τ ε τ , , , , , ,

c h d i d i

a f a f a f a f a f a f

= + Total Load Surface Stress Strain Energies: U W dA R u R u R d P u u

e rr r r

≡ = + + − − −

z z

ε σ θ θ σ θ θ θ

αβ θ θ π

c h a f a f a f a f a f a f a f

2 2

2 1 1

, , 1 2 Σ a, a, U W dA R u R u R d

e

• rr

r

• r
• a f

a f a f a f

d i a f a f a f a f

≡ = +

z z

ε σ θ θ σ θ θ θ

αβ θ θ π

2

2

, , U W dA P u u

e s s s s

a f a f a f a f

d i a f a f a f

≡ = − − −

z

εαβ 1 2 2

1 1

Σ a, a,

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Elasticity Solution - Griffith Crack

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U R u R u R d P u u U U P u u U U

e rr r s r s

• e
• e

s

• e
• e

s

= + + − − − + + = − − − + +

z

2 2 2

2 1 1 1 1

σ θ θ σ θ θ θ

θ θ π a f a f

a f a f a f a f a f a f a f a f

a f a f a f a f a f a f a f a f a f a f

, , 1 2 Σ Σ a, a, a, a, Interaction Energy: U P u u

e

• s
• a f

a f a f

a f a f a f

= − − − 2

1 1

Σ a, a, Total Elastic Strain Energy: U U U U

e e

• e

s e

• s

= + +

a f a f a f

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Total Elastic Energy

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Potential of External Loads: V R u R u R d P u u

rr r r

= + + − −

z

σ θ θ σ θ θ θ

θ θ π a f a f

a f a f a f a f

, ,

2 1 1

2 a + a, a,

Elastic Energy of the Bulk: U U U U

e e

• e

s e

• s

= + +

a f a f a f

Total Potential Energy: Π Γ = + − = − − − + − U U V U U U P

s e e

• e

s e

• s

a f a f a f a

f

4 2 a Surface Energy: U dx u u

s =

= + − −

−

z

2 4 2

1 1 1

Γ Γ Σ ε

a f a f a f

a +a

a a, a,

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Energy of the Bulk-Surface System

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∆Π Γ Σ = − + + − − + − − κ µ π σ κ µ σ σ 1 8 4 2 1 8 4 2

2 22 2 22 11

a a a ( ) ( ) P P a

f

∂∆Π ∂ = ⇒ + − −

F H G I K J

− −

F H G I K J

= 2 2 1 1 2

1 2 11 22 1 1 2

a a K Σ Γ P K P K C π σ σ

σ11

σ 22

P P

U P

e

• s

a f a

fa f a f

= − + − 2 1 4

11 22

Σ κ µ σ σ a U z z

e s

a f

a f

= − + − + → κ πµ 1 2

2

Σ ln a a a z U R

e

• a f

a fa f a f

= − + + − + + + ⋅⋅⋅ π µ κ σ σ σ σ κ µ π σ

2 11 22 2 11 22 2 2 22 2

16 1 2 1 8 a

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Final Results For σ

σ

11 22

, , P, and R Σ → ∞

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Then and ∆Π Γ = − ( ) 4 2 P a ∂∆Π ∂ = ⇒ = 2 2 a P Γ

σ11

σ 22

P P

Let

11 22

σ σ = = 0

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The Role of P

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a

α

∆

P P P / 2 F

a

b g

b

b g

a A Smooth Wedge of Angle

a f

2α b A Smooth Edge of Thickness

a f

2∆ Π Γ Γ

α α

α = +

L N M O Q P

∂Π ∂ = ⇒ 2 2 3

2

EI a a a , P = 6 Π ∆ Γ Γ

∆ ∆

= +

L N M O Q P

∂Π ∂ = ⇒ 2 3 2

2 3

EI a a a , P = 2

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Splitting A Double-Cantilever Beam