Introduction to Fracture Mechanics phần 2 pptx

   d dWP WP, and thus G, can be evaluated for different loading G WP, and thus G, can be evaluated for different loading conditions.. Modes of FractureThe three basic modes of separ

Irwin introduced, for this purpose, the energy release

rate G which is defined as

.







d

dWP

WP, and thus G, can be evaluated for different loading

G

WP, and thus G, can be evaluated for different loading

conditions This definition is valid for both linear and

nonlinear elastic deformation of the body G is a function

of the load (or displacement) and crack length It is

independent of the boundary conditions, in particular

whether the loading is fixed-displacement or fixed-load

The Griffith criterion for fracture initiation in an ideally

brittle solid can be re-phrased in terms of G such that

.

2 '

2

S

a



 





We define the compliance C (inverse of the stiffness) of

G

a cracked solid as C=u / F It can be shown that

2

2

da

dC B

F



Thus measurements of compliance as a function of crack length allow the energy release rate to be evaluated

G

Modes of Fracture

The three basic modes of separation of the crack

surfaces (modes of fracture) are depicted below:

Combinations of modes (mixed-mode loading) are

also possible

Modes of Fracture Definitions

separate in a direction normal to the plane of the crack The displacements are symmetric with respect to

the x – z and x – y planes.

are mutually sheared in a direction normal to the

crack front The displacements are symmetric with

respect to the x – y plane and anti-symmetric with

respect to the x – z plane.

Modes of Fracture

Definitions

crack faces are sheared parallel to the crack front The displacements are antisymmetric with respect

to the x – y and x – z planes

to the x – y and x – z planes

The crack face displacements in modes II and III find an analogy to the motion of edge dislocations and screw dislocations, respectively

Plane Crack Problem

The preceding analysis considered fracture from an energy standpoint We now carry out a linear elastic stress analysis of the cracked body, which will

allow us to formulate critical conditions for the

allow us to formulate critical conditions for the

growth of flaws more precisely An analysis of this

type falls within the field of Linear Elastic

Fracture Mechanics (LEFM).

We consider a semi-infinite crack in an infinite plate

of an isotropic and homogeneous solid as shown below:

Our goal is to develop expressions for the stresses,

strains and displacements around the crack tip

Plane Crack Problem

Equilibrium Equations

The equilibrium equations (no body forces) are

, 0

1

















r r

r

rr r

rr    







, 0

2











r r

r

r

r   







where and are the polar coordinates as shown previously

Plane Crack Problem Strain-Displacement

The strain-displacement relations for polar coordinates are:

.

1

1









,

1 ,















r r

u r

rr

.

1 2

1























r

u r

u

u r

r



The strain compatibility equation in polar coordinates is:

0

1 1

1 1

2

2

2 2

2

2 2

2







































r r

r r

r r

r r

r

rr rr

r















   

Plane Crack Problem Hooke’s Law

Hooke’s Law (for plane stress, ): zz  0

,

rr

E     

,







 rr  rr 

E

.

2 G   G   

For the case of plane strain ( ): zz  0

2Grr    rr 

.

2 G r  G  r   r

2G     rr