introduction to fracture

$: Introduction to fracture$
FRACTURE MECHANICS

1 Fracture Mechanics| SMiE| AAiT| AAU 2016

Introduction

Why Structures Fail? Most Eng’g structures are fails owing to one and more of the listed phenomena:

Excessive Elastic Deformation Unstable Elastic Deformation (Buckling) Plastic Deformation Fracture Fatigue Creep

In general, various failure mechanisms may be classified into the two broad fields Deformation and Fracture.

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Fracture Mechanics| SMiE| AAiT| AAU 2016

Introduction, cont’d

Causes for failure(s) can be Negligence and/or improper design of the product or component, Inefficient manufacturing and/or construction of the structure Improper operation of the structure

Further, Application of a new design or material for new product design and

manufacturing

Build with Ethylene tetrafluoroethylene, ETFE

Fig. 1.1 Birds nest Stadium, China

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Design Philosophies

Safe Life: defect-free manufacturing and service condition, and serve for a certain period of time with limited either static or dynamic load.

Fail Safe: its probable failure would not be catastrophic.

Damage Tolerance: withstand the maximum working stresses for a certain period of time even in presence of flaws, cracks, or similar damages of certain geometry and size.

Remark: The consequences of fracture can be minor or they can be costly, deadly or both.

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Potential Area/Industries Failure Due Fracture

Probably encountered in any industry dealing with structures Automotive Electronics Healthcare Aviation Civil Nuclear Defense Maritime

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Notable Fracture

The DeHavilland comet, a catastrophic crash on 1954, Evidence of fatigue cracking was found that originated from the aft lower

corner of the forward escape hatch and also from the right-hand aft corner.

Fig. 1.2 Description of point of crack initiation on DeHavilland comet

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Notable Fracture, Cont’d

Liberty ship, were cargo ships built in the United States during World War II. there were nearly 1,500 instances of significant brittle fractures among 2710, and

Twelve ships broke in half without warning. ,[http://en.wikipedia.org/wiki/Liberty_ship]

Fig. 1.3 Photo of a Brittle fracture on Liberty Ship

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Notable Fracture, Cont’d

A Missouri Air National Guard F-15C, broke in to two in flight, November 2007. a manufacturing defect in which a fuselage longeron was machined to below

its design thickness.

Fig. 1.5 Crashed A Missouri Air National Guard F-15C Ref: http://www.hoax-slayer.com/F15-crash.html

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Notable Fracture, Cont’d 9

German Train ICE Accident in 1998:


Fracture Mechanics

Fracture mechanics is a field of solid mechanics that deals with the mechanical behavior of cracked bodies.

Fracture mechanics pose and finds answers to questions related to designing components and processes against fracture.

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Fracture Mechanics, Cont’d

Consider a structure in which a crack develops due to the application of repeated loads or a combination of loads and environmental attack the crack will grow with time…result for higher stress concentration, and

the strength of the structure is decreased.

a. Crack growth curve b. Residual strength curve Fig. 1.6. The engineering problem

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The objects of fracture mechanics would be to predict and develop prediction methods are how fast cracks will grow and how fast the residual strength will decrease

Hence, it answers question to a. What is the residual strength as a function of crack size? b. What size of crack can be tolerated at the expected service load; i.e. what is the critical

crack size? c. How long does it take for a crack to grow from a certain initial size to the critical size? d. What size of pre-existing flaw can be permitted at the moment the structure starts its

service life? e. How often should the structure be inspected for cracks?

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Wide range of application of Engineering fracture mechanics

Fig. 1.7. Application of Engineering Fracture Mechanics

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C. E. Inglis, 1913, is the first person for major step in the direction of quantification of the effects of crack like defects.

He published a stress analysis for an elliptical hole in an infinite linear elastic plate loaded at its outer boundaries. He modeled the crack like discontinuity by making the minor axis very much less than the

major,

…Eq. 1.1

Fig. 1.8. An infinite plate with Elliptical hole

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A.A. Griffith ,transformed the Inglis’s analysis by calculating the effect of the crack on the strain energy stored in an infinite cracked plate. He proposed that this energy, which is a finite quantity, should be taken as a measure of the

tendency of the crack to propagate.

But his proposal is limited to surface energy and elastic behavior of crack deformation.

In 1956, Irwin developed the energy release rate concept, which is related to the Griffith theory but is in a form that is more useful for solving engineering problems.

stored elastic strain energy which is released as a crack grows

dissipated energy which includes plastic deformation and the surface energy

…Eq. 1.2

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Method of Crack/Crack Like Defect Analysis

A. Linear Elastic Fracture Mechanics(LEFM)

In linear and elastic , the elastic energy release rate, G, and the stress intensity factor K can be used

B. Elastic Plastic Fracture Mechanics (EPFM) In the elastic-plastic region or yielding fracture mechanics

(YFM), the fracture characterizing parameters are the J- integral and the crack-tip-opening displacement, CTOD.

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Method of Crack Analysis, Cont’d

C. Fatigue fracture on a component subjected to fluctuating stresses fail at stress levels much lower than its monotonic fracture strength. Fatigue is an dangerous time-dependent type of failure which can occur without any obvious

warning.

Three distinct stages in the fatigue failure of a component: Crack Initiation, Incremental Crack Growth, and the Final Fracture.

Fig. 1.9.

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Method of Crack Analysis, Cont’d

D. Creep can be defined as a time-dependent deformation of materials under constant load (stress). The resulting progressive deformation and the final rupture, can be considered as

two distinct, yet related, modes of failure. For metals, creep becomes important at relatively high temperatures, i.e., above 0.3 of their

melting point in Kelvin scale.

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Fracture Criteria 19


Introduction

The art of mechanical component design, and material selection can be categorized under Classical mechanical design approach; strength assessment in relation to induced stress in component

Stiffness assessment in relation to induced deformation in component

Remark: not effective for all type of mechanical design

Modern design approach: include fracture mechanics

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Fundamental fracture Concepts: 21

Fracture is the separation of a component into, at least, two parts This separation can occur locally due to formation and growth of

cracks. Steps : crack formation, and crack propagation

The formation of cracks may be a complex fracture process, which strongly depends on the microstructure of a particular crystalline or amorphous solid, applied loading, and environment.


Ductile Vs Brittle 22

Ductile Fracture is a high-energy process in which a large amount of energy dissipation is associated with a large plastic deformation before crack instability occurs. …(Dislocation Mediated)

(a) Necking, (b) Cavity Formation, (c) Cavities coalesce form crack (d) Crack propagation, (e) Fracture

Fig. 1.11.

Crac

k gro

ws 9

0o to

ap

plied

stre

ss

45O -

max

imum

shea

r st

ress


Ductile Vs Brittle, Cont’d

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Example on Cup-and-cone fracture in Al)

Scanning Electron Microscopy. Spherical “dimples” micro-cavities that initiate crack formation.


Ductile Vs Brittle, Cont’d 24

Brittle Fracture is a low-energy process (low energy dissipation), which may lead to catastrophic failure since the crack velocity is normally high….. (Low Dislocation Mobility)

Crack propagation is fast

Propagates nearly perpendicular to direction of applied stress

Often propagates by cleavage - breaking of atomic bonds along specific crystallographic planes

Little or no appreciable plastic deformation


Ductile Vs Brittle, Cont’d 25

Brittle fracture in a mild steel

A. Transgranular fracture: Cracks pass through grains. Fracture surface: faceted texture because of different orientation of cleavage planes in grains.

B. Intergranular fracture: Crack propagation is along grain boundaries

(grain boundaries are weakened/ embrittled by impurity. Fracture Mechanics| SMiE| AAiT| AAU 2016

Fig 1.12

THEORETICAL STRENGTH 26

A theoretical strength can be approximated to determine the stress required for fracture of atomic bonding between atom. i.e. owing to a consideration that fracture occurs when sufficient stress are

applied on the atomic level to break the bond.


The bond energy is given by:

Where: x0 is the equilibrium spacing and P is the applied force.

Eq. 1-3

Cont’d

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A reasonable estimate of the cohesive strength at the atomic level can be obtained by idealizing the interatomic force-displacement relationship as one half the period of a sine wave, so we may write:

Where: λ is defined in Fig.2.1. For small displacements, we may consider further simplification by assuming:

Eq. 1-4

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Eq. 1-5

Cont’d


Hence, the bond stiffness (i.e., the spring constant) can be defined by:

Multiply (1-4) by the number of bonds per unit area and the equilibrium

spacing x0 (gage length), Then k can be converted to Young’s modulus E and P to the cohesive stress σc

. The eq. 1-4 rearrange to

` 29

Eq. 1-6

Eq. 1-7

Cont’d


Assuming λ ≈ x0 , we may write above equation as

E.g. For Young’s modulus of 210 GPa, a fracture stress of 70000MPa obtained, which is almost 25 times the strength of the most strong steels!!

Remark: it is observed that components fails due fracture besides inherent material fracture

strength is high as that of stress due external load, this is the existence of numerous defects in ordinary materials.

Eq. 1-8

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Cont’d


STRESS-CONCENTRATION FACTOR

Let’s take a problem of an elliptical hole subjected to uniform remote stress ….

Figure 1.14 An infinite plate with an elliptical hole subjected to remote uniform normal stress.

Major diameter = 2a and Minor diameter = 2b

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Cont’d

And let’s take elliptical—hyperbolic coordinate system

Where α and β are coordinate directions

X = c coshα cosβ Y = c Sinhα sinβ

c is the position of the common foci for both the ellipses and the hyperbolae.

Let αo denote the boundary of an elliptical hole

a = c coshαo (β = 0) b = c Sinhαo (β = π/2)

…Eq. 1-9

…Eq. 1-10

Figure 1.15 Elliptical—hyperbolic coordinate system

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Cont’d

The in-plane strain can be written as, “ A Treatise on the Mathematical Theory of Elasticity, by Love, 1944”

Where are the displacements along coordinate directions,

The quantities, h1 and h2 ,which are formally defined as

…Eq. 1.11

…Eq. 1.6a …Eq. 1.6b

1

2111h

uhhuh

uandu

2

2121h

uhhu

h

uhhhuh

hh

11

22

2

1

2

+ +

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Cont’d

For this particular problem the boundary conditions can be satisfied identically by retaining only three terms in the series solution and the exact solution for the stresses have relatively simple forms given by

…Eq. 1.11a sinh 2 cosh 2 cosh 2

cosh2 cos 2

sinh 2 cosh2 cosh 2 2 cos 2cosh 2 cos 2

sin 2 cosh 2 cosh2cosh2 cos 2

…Eq. 1.11b

…Eq. 1.11c

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And, as per Inglis(1913) finding, stress for the case of uniaxial remote stress perpendicular to the major axis of the ellipse is, when α = αo

The stress normal to boundary, σαα = 0, and above equation provides the distribution of the tangential stress at the hole:

Thus as per the above equation, the stress concentration factor for an elliptical hole in uniaxial tension is

…Eq. 1-13

…Eq. 1-12

…Eq. 1-14

+ = 1

| = 1

Stress Concentration factor(k) 1 2 tanh 1 2 ⁄

Cont’d

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Stress Distribution For Elliptical Hole

Distribution of the tangential stress around an elliptical hole for a/b =3

On the other hand, if b→0 then → ∞ , i.e. stress is singular and it is meaningless.

Concluding, k is used to analyze the a stress at a point in the vicinity of a notch having a radius ≫0. However, if a crack is formed having ≅0the stress field at the crack tip is defined

in terms of the stress-intensity actor(KI) instead of the stress-concentration factor.

Fig. 1.16

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GRIFFITH CRACK THEORY

Griffith (1921) derived a criterion for crack growth using an energy approach. It is based on the concept that energy must be conserved in all processes.

He proposed that when a crack grows the change (decrease) in the potential energy stored in the system, U, is balanced by the change (increase) in surface energy, S, due to the creation of new crack faces.

For fracture to occur energy must be conserved so, ∆ ∆ 0

The change in surface energy, ∆ where is the new surface area created and

is the surface energy per unit area,

The change in area 2 ∆ ,

Fig. 1.17

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Griffith Crack Theory, Cont’d

Inserting these values and dividing across by ∆ and rewriting as a partial derivative, we get Griffith’s relationyields

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If this equation is satisfied then crack growth will occur.

And, the energy release rate, G is defined as 1

is a measure of the energy provided by the system to grow the crack and depends on the material, the geometry and the loading of the system.

Hence, as result a crack will extend when

2

…Eq. 1-15

…Eq. 1-16

…Eq. 1-17

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In 1948 Irwin andOrowanindependently proposed an extension to the Griffith theory, +

where is the plastic work dissipated in the material per unit crack surface area created (in general >> ). Then the criterion for fracture become

12

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The Griffith and Irwin/Orowan approaches are mathematically equivalent, the only difference is in the interpretation of the material toughness .

In general is obtained directly from fracture tests which will be discussed later and not from values of and .

…Eq. 1-19

…Eq. 1-18

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Further, considering Elastic energy due to the presence of the crack as

Where 1 for plane stress and 1 for plane strain Substituting eq. 1-20 in eq. 1-16 and compute the derivation, result yields

Crack size

Surface energy 2

Further, rearranging eq. …. gives a significant expression in LEFM 2

…Eq. 1-21

…Eq. 1-20

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Fracture Primary Factor

In fracture mechanics, the three primary factors that control the susceptibility of a structure to brittle failure.

1. Material Fracture Toughness: 2. Crack size 3. Stress level

Fig. 1.18 modern design approach

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Cont’d

1. Material Fracture Toughness: may defined as the ability to

carry loads or deform plastically in the presence of a notch.

It may be described in terms of the critical stress intensity factor under

conditions of plane stress (KC) or plane strain (KIC ).

Or it may be considered in terms of critical energy release rate, .

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Cont’d

2. Crack/flaw/ size. Fractures initiate from discontinuities that can vary from extremely small cracks to much larger fatigue cracks.

Remark: Although good fabrication practice and inspection can minimize the size and number of cracks, most complex mechanical components cannot be fabricated without discontinuities of one type or another.

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Cont’d

3. Stress Level. For the most part, tensile stresses are necessary for brittle fracture to occur. These stresses are determined by a stress analysis of the particular component.

Remark: Other factors such as temperature, loading rate, stress concentrations, residual stresses, etc., influence these three primary factors.

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Relationship between K, a - σ

Figure 1.19 Fracture-locus curve for material with differing fracture resistance

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Thank You! ?

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