Introduction to Fracture Mechanics phần 5 docx

Check: We note that the Plane Stress case reduces to our first order estimate for θ = 0.. Also note that KI/ σys2 has dimensions of length.. Next we will compare the extent of the plast

Substitute the known expression for σ xx, σ yy and σ xy

in the Mode I crack problem (derived last time) and

find that:











 



2

sin

1 2

cos 2

1









r

K I





K

(plane strain); σ3= 0 (plane stress)

2

cos 2

2 3









r

K I













 



2

sin

1 2

cos 2

2









r

K I

Substitute in to the Mises yield condition:

Plane strain:

2

2

2 cos

1 2

1

sin 2

3

I

r

K









Plane stress:

2 2

2

2 cos

sin 2

3 1

I

r

K













These expressions can be used to solve for the radius

of the plastic zone rp as a function of θ:

Plane strain:

          

















2

3 4

2

I p

K r











 

p

Plane stress:

      













2

I p

K r

Check: We note that the Plane Stress case reduces to

our first order estimate for θ = 0.

Also note that (KI/ σys)2 has dimensions of length

Next we will compare the extent of the plastic zone

Next we will compare the extent of the plastic zone

in the two situations, plane stress and plane strain,

for two cases, θ = 0 and θ = 45˚.

For θ = 0, ,

3

1





1



stress plane

r

strain plane

r p p

For θ = 45˚, ,

3

1





1 381

.



stress plane

r

strain plane

r

p p

3

Extent of the plastic zone is significantly larger for the

plane stress case.

Plastic Zone Shape Plane stress/plane strain

Plastic Zone Shape Plane stress/plane strain

Plastic Zone Size Engineering Formulae

1



















ys

I p

K r





For Plane Strain:

For Plane Strain:

2

3

1



















ys

I p

K r





Similar analyses can be done to determine the plastic zone size and shape for Mode II and Mode III loading

Specimen Thickness Effects

Plane stress/plane strain

Meaning of ς

Recall the Strain Energy Release Rate ς

What does it physically represent? It is the rate of

decrease of the total potential energy with respect to crack length (per unit thickness of crack front),i.e

 

a

PE











What is the connection between ς and K?