Lecture 1 - Introduzione · Lecture 10 –Fracture mechanics Mechanical Engineering. Fracture Mechanics Introduction • Until 1950, catastrophic events such as the failure of Liberty

Fracture Mechanics

Fracture mechanicsLecture 10 – Fracture mechanics

Mechanical Engineering

Fracture Mechanics

Introduction

• Until 1950, catastrophic events such as the failure of Liberty ships were unpredictable and unexplained in the light of a design based on the assumption that the material is free from defects.

• In classical design, also called the "mechanics of materials", the material is assumed:

• Homogeneous → continuity of effort and deformation

• Isotropic → description of the response of the material with scalars

• Free from defects → absence of local concentrations, validity of theoretical solutions for stress fields


Fracture Mechanics

Introduction (cont.)

• In the mechanics of the materials, a state of stress is admissible if the calculated stress value (DESIGN STRESS) in each point of the component is lower than a characteristic limit value of the material (ALLOWABLE)


2. Definizione dei carichi

3. Determinazione del campo di sforzo

3. Determinazione della natura della

sollecitazione (statico, fatica, creep, etc.)

4. Confronto con gli ammissibili

1. Definizione della geometria

𝜎𝑑 ≤ 𝜎𝑎𝑙𝑙. = 𝑛𝜎𝑌

Fracture Mechanics

Introduction (cont.)

• The allowable value of the stress is determined by experimental tests (traction, compression, buckling, fatigue, etc.)

• Identification of the possible «failure modes»

• Es.: LINEPIPE DESIGN

• Definizione degli «stati limite» (limit state criteria)

• Spessore minimo e maggiore rapporto D/t

• Bursting under combined load

• Local buckling/collapse

• Fracture (flow stress: averagebetween yield and ultimate stress)

• Low-cyce fatigue

• Ratcheting


Fracture Mechanics

Fracture Mechanics

• In fracture mechanics it is assumed that the materials are not free from defects

• Defects exist in the material:

• due to the production process

• they form during operation due to normal operation loads.

• The defects are in the form of "cracks": two-dimensional defects that cause the reduction of the nominal resistant section and generate a high concentration of tensions


Fracture Mechanics

Cleavage fracture

• Some metallic materials show a significant variation of the breaking mechanism as a function of temperature

• Brittle fracture vs. ductile breakage

• The fragile fracture occurs mainly in BCC and HCP-structured metals

• It is promoted by:

• Temperature

• Triaxlity of the stress state

• Deformation speed

• FCC metals are ductile even at very low temperatures


Fracture Mechanics

Cleavage

• Characteristics of brittle fracture:

• Absence of plastic deformation (macroscopic): breakage in the elastic field!

• The fracture plane is perpendicular to the applied stress

• Low deformation energy

• The fracture propagates with the speed of sound


Fracture Mechanics

Cleavage

• Characteristics of brittle fracture:

• Absence of plastic deformation (macroscopic): breakage in the elastic field!

• The fracture plane is perpendicular to the applied stress

• Low deformation energy

• The fracture propagates with the speed of sound

Mechanical EngineeringGranbury, TX, USA (2010)

Fracture Mechanics

Cleavage

• Transgranular fracture (Cleavage)

The fracture plane passes through the grains. The fracture surface has a "multifaceted" appearance precisely because of the different orientation of the cleavage planes of the grains

• Intergranular fracture

• The fracture plane is along the grain edges (the grain edges are weakened or brittle due to the presence of impurities or precipitates)


Fracture Mechanics

Cleavage

• Macroscopic features

• "Flat" surface

• Absence of a streak

• «Crystalline» appearance (bright faceted)

•


• Microscopic features

• «Chevron marks»

• «River pattern»

Fracture Mechanics

Cleavage


T T1 2 f0.2 T

rv

s

r

0

RA

1

2

3

30

60

Du

ttili

ta'

(%)

Temperatura (K)

90

Tensio

ne

Fracture Mechanics

Fracture stress

• The value of stress (normal or shear) at rupture for a defect-free crystal lattice can be estimated based on the binding energy.

• Taking into account that the force necessary to remove from the equilibrium position two planes of atoms varies with distance:

• The value of the fracture stress can be estimated from the work necessary for separation:

• No real material shows these resistance values. Explanation: presence of defects in the lattice


App

lied

Forc

e (F

) →

r →x0

Cohesive force

W

0

c

E

x

E cohesive

Fracture Mechanics

Fracture stress

• During fracture two new free fracture surfaces are formed.

• Equating the value of the energy required for the formation of the two surfaces to the bonding energy:

• By replacing the / ratio in the expression of the theoretical failure stress:


App

lied

Forc

e (F

) →

r →x0

Cohesive force

W

0

1sin

2s c c

xdx

0 0

sc

EE

x x

Fracture Mechanics

Inglis solution (Ref: C. E. Inglis, “Stresses in a plate due to the presence of cracks and sharp corners”, Trans. Inst.

Naval Architects, 55, 1913, pp 219-241. )

• Inglis determined the solution for a generic elliptical hole subject to remote stress in infinite plate:


Fattore di concentrazione delle tensioni:

Kt = peak stress/nominal stress

Kt = A/0

0 01 2 1 2A

a a

b

– curvature radius2b

a

Fracture Mechanics



• For → 0, the stress at point A becomes infinite

• This is unrealistic because no material is able to sustain an inflexible effort

• The solution provides that for any stress applied to the remote, the stress at the tip becomes infinite!

• In real materials the presence of plasticity redistributes stresses and promotes the rounding of the tip (blunting)

• In the absence of plasticity, the minimum physical radius at the tip is not zero but approaches the atomic equilibrium distance of the crystalline lattice: ≈ x0


0 01 2 1 2A

a a

b

0

0

2A

a

x

a/ >> 1

x0

Fracture Mechanics



• By equating the theoretical value of the stress at the most critical point with the theoretical resistance value of the material we can obtain the expression of the stress at remote fracture for a defect of size a:

• Therefore: the stress at which a plate with a crack loaded with remote stress fractures depends on:

• material: E e s

• geometry: flaw size

Nota: in this solution the hypothesis that continuous mechanics applys at an atomic scale is made!! (not true)


0

0 0

2 sEa

x x

0

0

0

1

2 4

s sf

E x E

x a a

Fracture Mechanics

Griffith solution (1920)

• In 1927 A.A. Griffith published a work in which he demonstrates the intimate connection between the applied stress and the size of the defect for the estimation of the resistance of structures with defects

• He used Inglis' analysis for elliptical hole in the case of unstable propagation of a defect

• Griffith referred to the first law of thermodynamics (energy conservation)

• GRIFFITH CRITERION

A defect becomes unstable (and therefore hasfracture) when the variation of the deformationenergy resulting from an increase in its size is largeenough to overcome the surface energy of thematerial


Fracture Mechanics

Soluzione di A.A. Griffith (1920)

• Griffith's solution correctly predicts fracture in intrinsically fragile materials: glass.

• Attempts to adapt the solution to the case of metals failed

• The criterion presupposes that the fracture work is exclusively that of formation of the fracture surfaces true only for brittle materials (glass and ceramics)

• A modification of the model for its application to metals was not presented until 1948


Fracture Mechanics

Griffith (1920)

• Energy conservation equation (powers):

• E total energy

• P potential energy (external forces and internal strain energy)

• Ws work to create new surfaces

• Substituting time derivative with that respect to the variable size of the defect


P, D

P, D

2a

B

s EE W K P

d AA

dt t A A

Fracture Mechanics

Griffith (1920)

• Minimization: derive with respect to a and impose equality to zero

• Deformation energy released: Saint Venanthypothesis, elliptical volume of dimensions 2a x 4a.

• Fracture surface energy:


P, D

P, D

2a

B

0sWE

A A A

P

2 2

0

a B

E

P P

2 4s s sW A aB

2

2 0ss

W a

A A E

P

1

22 sf

E

a

Fracture Mechanics

Griffith (1920): comparison

• Comparison with previous solutions

• The prediction of fracture stress with local stress criterion and Griffith differs by about 40%

• However the two approaches are consistent

• Solutions differ when the radius of curvature at the tip is much larger than the atomic distance

• Griffith's model implies independence from the radius of curvature


2 sc

E

a

0

sc

E

x

4

sc

E

a

Legame atomico Sforzo locale Rilascio di energia di deformazione

Fracture Mechanics

Griffith (1920): strain energy release rate

• Definition

• Infinite plate

• U deformation energy stored in the body.


dG

dA

P sdW

RdA

Crack driving force Resistance

2

2c s

aG

E

G

1 1

P

dU dUG

B da B da D

Fracture Mechanics

Griffith (1920): flaw stability

• The stability condition of a crack is obtained by minimizing the derivative of G


0dG

da

Fracture Mechanics


• stability condition of a crack is obtained by minimizing the derivative of G


0dG

da

Fracture Mechanics


• The definition of G according to Griffith is independent of the load condition (imposed load vs. imposed displacement) for infinitesimal advances of the defect.


P

P

D

dD

1

2dU Pd D

a+da

IMPOSED LOAD

a

Fracture Mechanics


• The definition of G according to Griffith is independent of the load condition (imposed load vs. imposed displacement) for infinitesimal advances of the defect.


D

P

D

-dPdU

a+da

IMPOSED

DISPLACEMENT

a

Fracture Mechanics


• NON LINEAR ELASTICITY


Fracture Mechanics


• NON LINEAR-ELASTICITY

• It is an approach that has been proposed to try to solve the elasto-plastic problem and is based on the observation that during a tensile loading it is not possible to distinguish between an elastic-plastic behavior and a non-linear elastic

• Only by performing unloading it is possible to recognize an elastic-plastic behavior

• The elastic-non-linear material is described by the same constitutive equations of the linear elastic one with the only difference that the modulus of elasticity varies according to the deformation


Fracture Mechanics

Crack resistance curve

• The crack resistance curve or R-curve represents the resistance of the material to the advance a defect

• It is a function of the material (even if it suffers from the geometric effect of the sample used for its determination, see below)

• Two types of curves:

• Fragile materials: stepped

• Ductile materials: power law


Fracture Mechanics

G in practical applications

• The Griffith criterion establishes that a size defect assigns to become unstable under the action of a remote stress when the strain energy rate released by infinitesimal advance exceeds the fracture surface energy

• The strain energy release rate G is a function of load and geometry,

• The released energy can be measured by the change in compliance


1

2

dUG

da

Fracture Mechanics


• In the case of an infinite plate defect, the expression of the crack driving force is given by

• G increases with the crack length and with the square of the remote stress

• Unit of measurement: Nmm or kJ/m2


2aG

E

G

a

G

a

Fracture Mechanics


• The nature of the dG/da sign depends on the load condition, while the G value is independent

• In quite general terms it is possible to state that the systems solicited in conditions of imposed displacement result in stable propagation conditions for a defect, while the imposed load condition always implies an unstable propagation situation.

• In real applications it is not possible to know a priori whether the system behaves with imposed load or displacement. Moreover, geometrical variations (for example the growth of the defect) can make it possible to move from one condition (eg imposed displacement) to another (eg imposed load)


Fracture Mechanics

Double cantilever beam (DCB)

• Consider the double cantilever beam geometry configuration (DCB)

• Analytical relationships

• From the theory of the beam theory, the compliance is given by:


3

1

2 ; =C P

3 f

aC

E I

Fracture Mechanics

G estimation: DCB geometry

• Crack driving force in load control condition is:

• For the displacement control condition is:

• The two expressions are numerically equivalent


2 2 2

1

1/ 21 1

2I

P f

d PdU P dC P aG

w da w da w da wE I

2 2 2

3 2

/ 21 1 3 9

2 2I

d CdU d EI EIG

w da w da da a a

Fracture Mechanics


• Defect’s stability condition:

• Imposed load UNSTABLE!

• Imposed displacement STABLE!


0da

dGI

IwE

aP

da

dG

f

I

1

22

IwEC

a

da

dG

f

I

1

2

24

Fracture Mechanics



dU

• Experimental estimation of critical G

• Typical behaviour of a displacement controlled condition

Fracture Mechanics


• Experimental estimation of critical G

• R-curve example for a graphite/PEEK laminate: GC independent respect to a


Fracture Mechanics

G estimation: ENF geometry

• End Notch Flexure (ENF) specimen: mode II (sliding mode)

• Analytical relationships

• From the theory of the beam theory, the compliance is given by :

• The crack driving force is:


)22(2

933

22

aLw

CaPGII

3

1

33

8

32

whE

aLC

f

Fracture Mechanics


• Defect stability condition :

• Imposed load UNSTABLE!

• Imposed displacement

• It becomes negative (STABLE) if:


0IIdG

da

1

32

2

8

9

f

II

Ehw

aP

da

dG

32232

1

2

32

91

8

9

aL

a

ChwE

a

da

dG

f

II

LL

a 7.03

3

Fracture Mechanics


• Typical response


dU

Fracture Mechanics


• Example


0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5 4,0 4,5 5,0 5,5 6,0 6,5 7,0 7,5

0

50

100

150

200

250

300

350

400

450

500

FEM a=55 mm

FEM a=47 mmFEM a=39 mm

a=56 mm

a=48 mm

a=38 mm

WOVEN CARBON/EPOXY COMPOSITE 95250-4 RTM/IM7 4 HS Fabric

experimental crack length

a = 33 mmlo

ad (

N)

displacement (mm)

Documents

Lecture 1 - Introduzione · Lecture 10 –Fracture mechanics Mechanical Engineering. Fracture Mechanics Introduction • Until 1950, catastrophic events such as the failure of Liberty