Crack Shape Evolution

William T. Riddell

Civil and Environmental Engineering

Outline

• Introduction• Work by Mahmoud –

the PPP.• Beyond the PPP

– Complex crack shapes– Non-planar cracks.

• Summary and Conclusions.

Key References

• Mahmoud [EFM 1988a]• Mahmoud [EFM 1988b]• Mahmoud [EFM 1990]• Mahmoud [EFM 1992]• Gera and Mahmoud [EFM 1992]

• Newman and Raju [NASA TM 1979, EFM 1988]

Simple Question

• How can we quantify the evolution of a crack shape?

Cracked Configuration

• Crack size and shape defined by a and c• Remaining crack front interpolated as elliptical quadrants• Bending and uniform tension loading considered

a

c

Simulating Crack GrowthAssume a cyclic

load history

Assume initial a, c

Calculate K()

Calculate Ka and Kc

Calculate a and c

Update a and c

Newman and Raju

Paris model

4 different approaches

time

0.0 0.2 0.4 0.6 0.8 1.0

cra

ck le

ngt

h/d

ep

th

0

1

2

3

4

crack depthcrack width

Crack Lengths vs Time

time

0.0 0.2 0.4 0.6 0.8 1.0

crac

k le

ngth

/dep

th

0

1

2

3

4

crack depthcrack width

Crack Lengths vs Time

time

0.0 0.2 0.4 0.6 0.8 1.0

crac

k le

ngth

/dep

th

0

1

2

3

4

crack depthcrack width

Crack Lengths vs Time

• a and c define size and shape for semi-elliptical cracks

Outline

• Introduction

• Work by Mahmoud – the PPP.

• Beyond the PPP– Complex crack shapes– Non-planar cracks.

• Summary and Conclusions.

Simulating Crack GrowthAssume a cyclic

load history

Assume initial a, c

Calculate K()

Calculate Ka and Kc

Calculate a and c

Update a and c

Newman and Raju

Paris model

4 different approaches

• Mahmoud reformulated so time was not explicitly in formulation• a/t (size) and a/c (shape) remaining variables.

Resulting Differential Equation

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

Resulting Differential Equation

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

da

dc=

F(a,c)

G(a,c)

a(to) = ao

a/c(to) = ao/co

Evolution of Crack Shape

a/t

0.0 0.2 0.4 0.6 0.8 1.0

a/c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Other initial sizes and shapes

Col 1 vs Col 2

a/t

0.0 0.2 0.4 0.6 0.8 1.0

a/c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Col 1 vs Col 2

a/t

0.0 0.2 0.4 0.6 0.8 1.0

a/c

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

Preferred Propagation Pattern

• Cracks evolve toward same path, regardless of initial size and shape

Mahmoud’s Idea Mathematicized• System is autonomous,

i.e., equations do not explicitly contain t.

• The a-c plane is the phase plane

• Curve developed by solution to equations is a trajectory

• Mahmoud’s equations are asymptotically stable.

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

Mahmoud’s Idea Mathematicized• System is autonomous,

i.e., equations do not explicitly contain t.

• The a-c plane is the phase plane

• Curve developed by solution to equations is a trajectory

• Mahmoud’s equations are asymptotically stable.

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

Mahmoud’s Idea Mathematicized• System is autonomous,

i.e., equations do not explicitly contain t.

• The a-c plane is the phase plane

• Curve developed by solution to equations is a trajectory

• Mahmoud’s equations are asymptotically stable.

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

Mahmoud’s Idea Mathematicized• System is autonomous,

i.e., equations do not explicitly contain t.

• The a-c plane is the phase plane

• Curve developed by solution to equations is a trajectory

• Mahmoud’s equations are asymptotically stable.

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

Mahmoud’s Idea Mathematicized• System is autonomous,

i.e., equations do not explicitly contain t.

• The a-c plane is the phase plane

• Curve developed by solution to equations is a trajectory

• Mahmoud’s equations are asymptotically stable.

da

dt= F(a,c)

dc

dt= G(a,c)

a(to) = ao

c(to) = co

So What?• You do not have a chance of predicting

crack growth rate if you do not know crack shape.

• There is significantly less variation in the way that crack shape evolves, compared to crack growth rate.

• Comparison of crack shape evolution is one way to evaluate the accuracy of a crack growth simulation.

Quantification of Error• Defined residual Ri

• Compare predicted and observed evolution of shape.

• Evaluate 4 different ways to find KA and KC

• Evaluated empirical relationships from Portch [1979], Kawahara and Kurihara [1975], and Iida [17].

Ri = pred. (a/c)i – obs. (a/c)i

Extract Values of K

• Value at specific

a

c

a

c

• Weighted average over

• Reduce Kc by a factor

Refined Question

• How can crack shape be quantified?– Graphically describe evolution as 2D problem

in phase plane– Quantify differences between predicted and

observed shapes.

Mahmoud’s Conclusions• PPP’s depend on Paris exponent, m.

• PPP change as loading ranges from pure tension to pure bending.

• Point values with factor best overall prediction.

• Kawahara and Kurihara’s equation best of three empirical equations considered.

Outline

• Introduction

• Work by Mahmoud – the PPP.

• Beyond the PPP– Complex crack shapes– Non-planar cracks.

• Summary and Conclusions.

More Complex Planar Cracks

• 4 point bend specimens• Off-center initial notch

76.2 mm

61.3 mm50.8 mm

304.8 mm

C

C

p = 51.1 kNR = 0.214

bottom view

side view

Edge AEdge B

Section C-C

cracktip 2

cracktip 1

12.9mm

a1 a2

38.1 mm

Crack Shape Evolution

• Crack size and shape not easily characterized by two values

B

cracktip 1 crack

tip 2

A

Crack Lengths vs Cycles

• Crack lengths defined as distance traveled by crack tip

cracktip 1 crack

tip 2

a1 a2

Crack Shape Evolution

• Slight divergence in crack shape

• Small variation compared to rates

cracktip 1 crack

tip 2

a1 a2

Extension to Complex Shapes

• Concept of PPP applies to the shape of planar cracks, not just 2 crack lengths.

• Concept of phase plane is useful, even when 2 DOF do not completely characterize crack shape.

Non-Planar Crack Growth

• Tests by Pook and Greenan [1984]

Side View

Bottom View25 mm

360 mm

75 mm25 mm

300 mm

P = 18 to 79 kNR = 0.1

• 3D Simulations compared to Pook’s data

Experiments and Simulations

3D concept of PPP

• L.P. Pook “On Fatigue Crack Paths,” IJF 1995– There can exist a surface that attracts crack

growth

Non-Planar Crack Growth

• Initial notch at angle to principal stresses

76.2 mm

61.3 mm50.8 mm

304.8 mm

C

C

p = 51.1 kNR = 0.214

bottom view

side view

Edge AEdge B Section C-C

cracktip 2

cracktip 1

12.9mm

a1 a2

38.1 mm

Photo of Resulting Face

• Factory roof facets near top of initial notch

A B

Path of Crack Tips

• Good agreement for crack tip path.• Not much progress toward attracting surface

until crack tips pass edges and crack becomes a through crack.

Application to Non-Planar Cracks

• Attracting surface and phase plane both useful concepts.

• Even though its very hard to quantify non-planar crack shape, 2 DOF can help evaluate simulations.

• Both PPP and Attracting Surface describe tendency for cracks to evolve certain ways, despite initial crack size and shape.

Outline

• Introduction

• Work by Mahmoud – the PPP.

• Beyond the PPP– Complex crack shapes– Non-planar cracks.

• Summary and Conclusions.

Summary and Conclusions• Mahmoud presented concept of PPP for

evolution of aspect ratio for a cracked configuration.

• This is an example of an autonomous system that is asymptotically stable. – there are rules for stability based on eigenvalues.

• PPP can be used to evaluate fidelity of simulations when compared to experiments.

• Concept of a-c phase plane is useful for more complex geometries as well.

Summary and Conclusions

• Pook presented concept of attracting surface for non-planar crack growth.

• Attracting surface is straightforward to visualize under symmetric load cases.

• Under some real cases, crack growth is non-planar. The attracting surface is less obvious here.

Parting Question

• There is a mathematical way to show Mahmoud’s PPP are asymptotically stable.

• Is there an analogous way to identify, define, or show an attracting surface exists?