Introduction to Fracture Mechanics pdf

Introduction to Fracture Mechanics Introduction to Fracture MechanicsFrom Suresh: Fatigue of Materials Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com... Key Ide

Trang 1

Introduction to Fracture Mechanics Introduction to Fracture Mechanics

From Suresh: Fatigue of Materials

Simpo PDF Merge and Split Unregistered Version - http://www.simpopdf.com

Trang 2

Importance of Fracture Mechanics :

 All real materials contain defects: understand

the influence of these defects on the strength ofthe material Defect-tolerant design philosophy

the material Defect-tolerant design philosophy

 Relevance for Fatigue: understand the initiation

and growth of fatigue cracks

We will use two approaches, an energy-based

approach and a more rigorous mechanics approach

Trang 3

Key Idea : Griffith (1921) postulated that for unitcrack extension to occur under the influence of the

applied stress, the decrease in potential energy of

Griffith Fracture Theory

Introduction

applied stress, the decrease in potential energy of

the system, by virtue of the displacement of the

outer boundaries and the change in the stored elastic

energy, must equal the increase in surface energy

due to crack extension

Trang 4

Consider the center-cracked plate shown below.The in-plane dimensions of the plate are large compared to the crack length.

Trang 5

Using the results of Inglis (1913) Griffith found that

the net change in potential energy of the plate caused

by the introduction of the crack is:

'

2 2

Trang 6

The total system energy is then given by

.

4 '

2 2

S S

E

B

a W

W

Griffith noted that the critical condition for the onset

of crack growth is:

, 0

2 '

dW dA

Trang 7

where A=2aB is the crack area and dA denotes an

incremental increase in the crack area

Thus the stress required to initiate fracture is:

As the second derivative, d2U/da2 is negative, the

above equilibrium condition gives rise to unstable

crack propagation This applies for brittle materials;

Trang 8

Orowan (1952) extended Griffith’s brittle fracture

concept to metals by simply adding a term representing

plastic energy dissipation The resultant expression

for fracture initiation is

,

) (

where is the plastic work per unit area of surface

created Generally is much larger than p

Trang 9

Energy Release Rate Crack Driving Force

Consider an elastic plate with an edge crack of length

Trang 10

The total mechanical potential energy of a cracked elastic body is given by the general expression

F

W   

where is the stored elastic strain energy and is  wF

the work done by the external forces. F

Irwin (1956) proposed an approach for the

characterization of the driving force for fracture in cracked bodies, which is conceptually equivalent to

Trang 11

Irwin introduced, for this purpose, the energy release

rate G which is defined as

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

Trang 12

The Griffith criterion for fracture initiation in an ideally

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

.

2 '

2

da

dC B

Trang 13

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

Trang 14

Modes of Fracture Definitions

Mode I (tensile opening mode): The crack faces

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.

Mode II (in-plane sliding mode): The crack faces

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.

Trang 15

Modes of Fracture

Definitions

Mode III (tearing or anti-plane shear mode): The crack faces are sheared parallel to the crack front The displacements are antisymmetric with respect

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

Trang 16

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

growth of flaws more precisely An analysis of this

type falls within the field of Linear Elastic

Fracture Mechanics (LEFM).

Trang 17

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

Trang 18

Plane Crack Problem

Equilibrium Equations

The equilibrium equations (no body forces) are

, 0

r

rr r

Trang 19

Plane Crack Problem Strain-Displacement

The strain-displacement relations for polar coordinates are:

u r

u r r

rr

.

1 2

u

u r

1 1

Trang 20

Plane Crack Problem Hooke’s Law

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

Trang 21

Plane Crack Problem Airy Stress Function

For the plane problem, the equations of equilibrium are satisfied when the stress components are expressed by

the Airy stress function throughx

,,

r r

r

Using these definitions for the stresses and Hooke’s

Trang 22

It can be shown that the compatibility equation, when expressed in terms of the Airy stress function, satisfies the biharmonic equation:

2 2

2

2 2

r

The boundary conditions for this plane crack problem

The boundary conditions for this plane crack problem are: for    r  0     .

These conditions express the fact that the crack is

traction-free (no loads applied to crack face)



Trang 23

A choice of the Airy stress function for the present

crack problem should be such that x has a singularity

at the crack tip, and is single-valued We try a solution

Trang 24

, sin

Note that we have expressed x as a symmetric part and

an anti-symmetric part The symmetric part provides

the Mode I solution while the anti-symmetric part

provides the Mode II solution We will derive the

Trang 26

The admissible cases are: (i) cos λ π = 0, hence where Z is an integer including zero, and thus

B1= A1λ /(λ + 2) or (ii) sin λπ = 0 and hence λ = Z and

B1= A1 Since the governing equations are linear, any

linear combination of the admissible solutions provides

a solution, hence λ can have any satisfying:

Z





Where Z is a positive or negative integer, including zero

Trang 27

Out of all the possible values of λ, how do we decide

the appropriate value of λ?

The value of λ cannot be found from any mathematical argument We need to use a physical, based on the total strain energy around the crack tip From the expressions for the stresses, and Therefore the  ~ r   ~ r  .

for the stresses, and Therefore the strain energy density is given by



 ij ~ r ij ~ r  .

.

~ 2

Trang 28

The total strain energy within an annular region,with

inner and outer radii r0 and R, respectively, centered at

the crack tip, with unit thickness is

~ 2

1

0 0

1 2

2 0

2

0    rdrd   R r  drd 

r ij

We assert that the strain energy should be bounded

(Φ < ∞) as r00 Using this physical argument, we

see that λ > -1 (λ = 1 gives ) If λ < 1, the

strain energy will not be bounded

Trang 29

Thus the physically admissible values of λ are

, 2

) (

, ,

2

, 2

3 ,

1

, 2

1 ,

0

, 2

where Z is –1, 0, or a positive integer Taking the most

dominant singular term (λ = 1/2 and thus B1=A1/3)

3

2

3 cos 3

1 2

Trang 30

The higher order terms, with exponents greater than

zero, vanish as r  0 We write where

K I is the stress intensity factor.

Thus we have that:

I ij

The first term is the leading singular term for linear elastic

mode I crack problems is a function of θ alone (no r

dependence) The second term, generally referred to as the

‘T term’, is a non-singular term which can be important in

some situations involving fatigue is the Kronecker

I ij

~



Trang 31

From Hooke’s law, the strains are linearly related to the stresses so that

Since the strains are calculated from the

K r

Trang 32

Plane Crack Problem Stress Intensity Factors

The stress intensity factors for Modes I, II and III are

The stress intensity factor K depends on loading and geometry

Many different geometries have been evaluated, either

analytically or numerically, and are available in the literature,

Trang 33

Plane Crack Problem Similitude

For a crack of length 2a1 in an infinite plate, subjected

to an applied stress σ1 the stress intensity factor is

known to be Consider two large plates, one with a center crack of length 2a1, the other with a center crack of length 2a2 A stress σ1 is applied to the first

1

1 a

K I   

crack of length 2a2 A stress σ1 is applied to the first

plate, and a stress σ2 is applied to the second plate If

we choose σ1, σ1, σ2 and σ2 so that then the

fields at the crack tip are identical in both cases This

is the principle of similitude, which is very important in fracture mechanics as it allows results from laboratory

) 2 ( )

Trang 34

Plane Crack Problem Stress Intensity Factors

How do we apply this analysis to the failure of actual materials? It has been found experimentally that when

the stress intensity factor K (which depends on the

geometry and loading) attains a critical value K C (a

material property) the crack begins to grow, i.e., the

critical condition for the onset of fracture is

K → K c

The condition can also be expressed in terms of the

energy release rate, i.e., ς → ςc

What are some typical values for K ?

Trang 35

Si3 N4

Al2O3Glass

2 5

Trang 36

Fracture Mechanics #2:

Role of Crack Tip Plasticity

Trang 37

Plastic Zone Size Estimate

Consider inelastic and permanent deformation at the crack tip (stresses are too high for the material to

remain elastic)

First order estimate of plastic zone size:

Assume: plane stress, and the material behavior is elastic-perfectly plastic Set the stress σyy= σys

(along the line θ = 0)

Trang 38

2 2

2 1

What about the details of the plastic zone shape? The

shape of the plastic zone is obtained by examining the yield condition, in conjunction with asymptotic K-field

results, for all angles θ around the crack tip Either the

Mises or the Tresca criterion can be applied

Trang 39

Plastic Zone Shape

Recall that for the Tresca yield condition yielding

occurs when  max   ys / 2

We will use the Mises yield condition The Mises

condition in terms of principal stresses is given as

1 3

2 3 2

2 2

Trang 40

On the plane θ = 0, σxy= 0 and thus σxx and σyy are the principal stresses σ1 and σ2 The stresses σ z ≡ σ3; σz =0 for plane stress, σ z = ν(σ xx + σ yy) for plane strain

However, in general the shear stress σ xy is not zero and the principal stresses σ1 and σ2 cannot be determined so easily

The principal stresses σ1 and σ2 are evaluated as follows (can use Mohr’s circle, for example):

2

2 1

22

Trang 41

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

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

cos 2

Trang 42

Substitute in to the Mises yield condition:

1 2

1

sin 2

2

2 cos

sin 2

3 1

Trang 43

These expressions can be used to solve for the radius

of the plastic zone rp as a function of θ:

2

I p

K r

Trang 44

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

in the two situations, plane stress and plane strain,

for two cases, θ = 0 and θ = 45˚.

Trang 45

strain plane

r

p p

.



stress plane

r

strain plane

r

p p

3

Extent of the plastic zone is significantly larger for the

Trang 46

Plastic Zone Shape Plane stress/plane strain

Trang 47

Plastic Zone ShapeSimpo PDF Merge and Split Unregistered Version - http://www.simpopdf.comPlane stress/plane strain

Trang 48

Plastic Zone Size Engineering Formulae

K r





For Plane Strain:

K r





Similar analyses can be done to determine the plastic

Trang 49

Specimen Thickness Effects

Plane stress/plane strain

Trang 50

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

What is the connection between ς and K?

Trang 52

For the general 3-D case, plane strain and anti-plane strain loading:

Trang 53

K Dominance Domain of validity

There exists an annular zone where the K solution

is valid:

Trang 54

The outer radius, r 0, is the radial distance at which the approximate, asymptotic singular solutions

deviate significantly (say, by more than 10%)

from full elasticity solutions which include higher order terms It is found that,

a

where is the crack length.a

Trang 55

Example Problem Size Requirements

Consider a low strength steel with σys=350 MPa,

KIc=250 MPa and E = 210 GPa What are the

Minimum specimen size requirements for a valid

KIc measurement?

m

Tiêu đề	Introduction to Fracture Mechanics
Tác giả	Suresh
Trường học	Unknown University
Chuyên ngành	Fracture Mechanics
Thể loại	Bài báo
Năm xuất bản	2025
Thành phố	Unknown

Định dạng
Số trang	56
Dung lượng	1,45 MB