Fracture Mechanics

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Fracture mechanics

Lecture 10 – Fracture mechanics

Mechanical Engineering

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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

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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

𝜎

_𝑑

≤ 𝜎

_{𝑎𝑙𝑙.}

= 𝑛𝜎

_𝑌

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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: average

between yield and ultimate stress)

• Low-cyce fatigue

• Ratcheting

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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

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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

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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

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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 Engineering Granbury, TX, USA (2010)

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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)

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Cleavage

• Macroscopic features

• "Flat" surface

• Absence of a streak

• «Crystalline» appearance (bright faceted)

•

• Microscopic features

• «Chevron marks»

• «River pattern»

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Cleavage

T₁ T₂ 0.2 T_f



rv

s r

0

 ^RA

1 2

3

30 60

Duttilita' (%)

Temperatura (K)

90

Tensione

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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

Applied Force (F) →

x₀ r →

Cohesive force W

0 c

E x

 

  E

cohesive



 

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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:

Applied Force (F) →

x₀ r →

Cohesive force W

0

1 sin

s

2

c c

x dx



 

  

 

 

      

0 0

s c

E E

x x



 

  



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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:

K

_t

= peak stress/nominal stress K

_t

= 

_A

/

₀

0

1 2

0

1 2

A

a a

  b 



 

           



– curvature radius

b

2

  a

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Inglis solution (

. )

• 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

1 2

0

1 2

A

a a

  b 



 

           

0

0 A

2 a

   x

a/ >> 1

x₀

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Inglis solution (

. )

• 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 a E

^s

x x

  

₀ ⁰

0

1 2 4

s s

f

E x E

x a a

 

    

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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 has fracture) when the variation of the deformation energy resulting from an increase in its size is large enough to overcome the surface energy of the material

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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

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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

2a

B

s E

E  P  W  K

d A

dt t A A A

  

 

  

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Griffith (1920)

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

• Deformation energy released: Saint Venant

hypothesis, elliptical volume of dimensions 2a x 4a.

• Fracture surface energy:

P, D

2a

B

s

0 W E

A A A

 P 

  

  

2 2 0

a B E P  P  

2 4

s s s

W  A



 aB



2

2 0

s

W a

A A E

 

P      

 

1

2

_s 2 f

E a

 



 

    

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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

_s

c

E a

 



 

    

0 s c

E x

  

4

s c

E a

  

Legame atomico Sforzo locale Rilascio di energia di deformazione

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Griffith (1920): strain energy release rate

• Definition

• Infinite plate

• U deformation energy stored in the body.

G d

dA

  P dW

^s

R  dA

Crack driving force Resistance

2

c

2

s

G a

E G







1 1

P

dU dU

G B da B da

_D

   

         

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Griffith (1920): flaw stability

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

dG 0

da 

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Griffith (1920): flaw stability

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

dG 0

da 

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Griffith (1920): flaw stability

• 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

dU  2 PdD

a+da

IMPOSED LOAD

a

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Griffith (1920): flaw stability

• 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 dU -dP



a+da

IMPOSED DISPLACEMENT

a

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Griffith (1920): flaw stability

• NON LINEAR ELASTICITY

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Griffith (1920): flaw stability

• 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









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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

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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 G dU

 da

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G in practical applications

• 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

2

a

G E

 

G

a



G

a



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G in practical applications

• 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)

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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:

Mechanical Engineering 3

1

2 ; =C P 3

_f

C a

E I 

 

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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 1 1/ 2

I

2

P f

d P

dU P dC P a

G w da w da w da wE I



  

       



²



² ²

3 2

1 1 / 2 3 9

2 2

I

d C

dU d EI EI

G w da

_

w da da a a

    

 

              

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G estimation: DCB geometry

• Defect’s stability condition:

• Imposed load UNSTABLE!

• Imposed displacement STABLE!

 0 da dG

_I

I wE

a P da

dG

f I

1

2



I wE C

a da

dG

f I

1 2

4 

2





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G estimation: DCB geometry

dU

• Experimental estimation of critical G

• Typical behaviour of a displacement controlled condition

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G estimation: DCB geometry

• Experimental estimation of critical G

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

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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:

) 2 2

( 2

9

3 3

2 2

a L

w

Ca G

_II

P

 

3 1

3 3

8 3 2

wh E

a C L

f

 

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G estimation: ENF geometry

• Defect stability condition :

• Imposed load UNSTABLE!

• Imposed displacement

• It becomes negative (STABLE) if:

Mechanical Engineering II

0 dG da 

1 3 2

2

8 9

f II

E h w

aP dG da





 

 ₂ ₃ ₂ ₂ ₃

1 2

3 2

1 9 8

9

a L

a C

h w E

a da

dG

f

II



L L

a 0.7 3

3 



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G estimation: ENF geometry

• Typical response

dU

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G estimation: ENF geometry

• 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 mm FEM 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 mm

load (N)

displacement (mm)