Fracture Mechanics 2

8/3/2019 Fracture Mechanics 2

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Elements of Fracture Mechanics


Jonah H. Lee

Department of Mechanical Engineering

University of Alaska Fairbanks

Jonah H. Lee Fracture Mechanics

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Griffith Crack Theory

Figure: 8.1: Through-thickness crack in a large plate.

The potential energy (strain energy - work done) of a plate with a crack(γ s = energy/area, J /m 2):

PE with crack = PE w/o crack− Dec. PE + Inc. surface energy

U = U 0 −πσ2a 2t

E + 4a · t · γ s

surface energy = area · γ s = 2 · (2a · t ) · γ s

strain energy ≈ σ · · V /2 ≈ σ · σ/E · πa 2t


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Griffith Crack Theory - continued

Equilibrium condition:

∂ U

∂ a = 4t γ s − 2πσ2at

E = 0.

Equilibrium but unstable, crack will grow (negative second derivative)

For plane stress

σ =

2E γ s

πa

for plane strain

σ =

2E γ s

πa (1 − ν 2)

σ is the applied stress.


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Effect of Plastic Deformation

Materials with limited plastic deformation, fracture energy ≈ surface energy(glasses).

Materials with extensive plastic deformation, fracture energy surface energy(metals, polymers):

σ =

2E (γ s + γ P )

πa =

2E γ s

πa (1 +

γ P

γ s )

γ P = plastic deformation energy

γ s , so

σ ≈

2E γ s

πa (γ P

γ s )

(Eq. 7.27): σa =applied stress, σmax ≈ 2σa a /ρ; (Eq. 7.9): σc = E γ

a 0

Let σc (theoretical cohesion) = σmax (maximum stress due to stressconcentration):

σa =1

2

E γ s

a (ρ

a 0) =

2E γ s

πa (πρ

8a 0)

plastic deformation γ P

→increase of ρ: crack tip blunting.


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El t f F t M h i




Critical Energy Release Rate (Fig. 8.2)



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Determination of Critical Energy Release Rate

Let da =increase in crack length, d δ=displacement increment due to P , work done

(by P )=dW = Pd δ, V =stored elastic strain energy=

1

2 P δ or

1

2

P 2

M (M =stiffness);Surface energy is needed for that crack to extend and comes from:

G =dU

da = P

d δ

da − dV

da

= work done by external force - release of strain energy

Fixed grip (displacement control) condition, d δ = 0, load drops from P 1 to P 2,

stiffness drops from M 1 to M 2 but δ1 = δ2 =P 1

M 1=

P 2

M 2

(∂ U

∂ a )δ = −1

2P 2

∂ (1/M )

∂ a

Critical energy release rate

Gc =1

2P 2max

∂ (1/M )

∂ a

where P max =load at fracture.



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Stress Analysis of Cracks

Figure: 8.3

Mode I - opening or tensile mode.Mode II - sliding or in-plane shear mode.

Mode III - tearing or antiplane shear mode.



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Figure: 8.5

Example of crack-tip stress in y -direction obtained using theory of elasticity (Airystress functions):

σy = K √ 2πr

cos θ2

1 + sin θ

2sin 3θ

2

Singular stress at r = 0.

K =f (σ,a ) →stress-intensity factor , with dimensions Pa · m 1/2.



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Figure: 8.7 - Stress-intensity-factor (K ) solutions for a central (a) and edge (b) crack.

K I = Y Pa 1/2

tW



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Design philosophy, e.g., Fig. 8.7a:

K = K c = σ√ πa

K c → Material selection σ → Design stress

a → Allowable flaw size (NDT)



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Crack-Tip Plastic-Zone Size Estimation

At θ = 0, σy = K /√

2πr , at yielding, σy = σys , plastic-zone size=r y ≈ K 2

2πσ2ys

(K

is a function of a .)

Plane stress: r y

≈

1

2π

K 2

σ2

ys

; Plane strain: r y

≈

1

6π

K 2

σ2

ys Effective K for an infinite plate with a small central notch (Eq. 8-45)

K eff =σ√ πa

1− 1

2

σ

σys

21/2



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Fracture-Mode Transition: Plane Stress Versus Plane Strain

Figure: 8.14 - Variation in fracture

toughness with plate thickness

Figure: 8.15 - Effect of relative plasticzone size to plate thickness (r y /t )

K c =fracture toughness varies with plate thickness (higher at plane stress, lower atplane strain) (Fig. 8.14).

K IC =plane-strain fracture toughness → remains constant after thickness t 2, or,small r y /t (Figs. 8.14 and 8.15).

Plane strain fracture (flat, smooth), plane stress fracture (slant, rough).



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Plane-Strain Fracture-Toughness (K IC ) Testing - ASTM E3990-90

Figure: 8.7(e) - K I for 3-point bending.

Figure: 8.18 - Example of load-displacementcurves during K IC testing.



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Test sample initially fatigue loaded to extend the machined notch to a prescribedamount a ; e.g., three-point bend bar, Fig. 8.7(e):

K I = Y 6Ma 1/2

tW 2

Test done using displacement control.

Measure specimen load P and (crack-opening) displacement δ until fracture.Applied stress is found via (maximum) load P using P − δ curve (Fig. 8.18)

formula for K for a particular geometry is then used.

Restrictions:

t and a ≥ 2.5K IC

σys 2



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Plane Stress and Related Models

Figure: 8.13(b) - Dugdale plastic zone strip model where plastic zones R extend as thin strips fromeach end of the crack. 2c is the initial crack length.

Cohesive-zone model (Fig. 8.13)

R /c =

π2

8 (

σ

σys )

2

J-integral based on ’simple’ elastic-plastic analysis using a deformation plasticitytheory (path-independent).


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Fracture Mechanics 2