Fracture 1

Fracture Mechanics for Modern Engineering Design:Overview

FRACTURE MECHANICS:

Latest addition to Modern Engineering Designers Arsenal

FRACTURE is failure due to UNSTABLE CRACK

PROPAGATION in solids and structures under STRESS

The new branch of SOLID MECHANICS dealing with cracked

solids/ structures is known as Fracture Mechanics

Engineers in general,and designers in particular,should appreciate the

role of Fracture Mechanics

Prediction,Prevention,Control and Treatment of fracture represents a

bulk of engineering practice today

We now weave together the essential concepts underlying Fracture Mechanics to help design engineers

Elements of Solid Mechanics (Theory of Elasticity)

Fracture Criteria

Stress Analysis of Cracks

Stress Intensity Factors

Plasticity Effects

Mixed Mode Fracture

Fatigue: FM approach

Fracture Testing

Dynamic Effects

Fracture Control

Discussions of modern engineering design is incomplete without Reliability Analysis.Reliability Analysis of machine components can be enhanced using Fracture Mechanics Approach to predict crack growth

SOURCE

[1] K.R.Y Simha

Fracture Mechanics:For Modern Engg.Design

Universities Press, 2001

[2] Sih,G.C.,(Editor)

Mechanics of Fracture Vols. 1 to 7

Noordhoff, Leyden, (1973 to 1981)

[3] Liebowitz,H (Editor)

FRACURE – an Advanced Treaties. Vols. 1 to 7

Academic Press, (1968 to 1972)

FRACTURE MECHANICS AND FATIGUE

CONTENTS

Fracture Mechanics Overview

Linear Elastic Fracture Mechanics

Fracture Toughness Testing of Materials

Computational Fracture Mechanics

Fatigue of Materials

Fatigue Crack Propagation

Topics for Advanced Study

• Mathematical foundations of LEFM

• Elastic – Plastic Fracture Mechanics

• Mathematical foundations of EPFM

• Dynamic and Time dependent fracture

• Fracture Mechanisms – Metals

• Fracture Mechanisms – Non Metals

• Fracture tests of Non Metals

Assignments

TEXT BOOKS

1. R.J.Sanford, Principles of Fracture Mechanics,Prentice Hall,2003

2. T. L. Anderson, Fracture Mechanics: Fundamentals and Applications, 2/e, CRC Press,1995.

3. S. Suresh, Fatigue of Materials, 2/e, Cambridge University Press,2003.

Principles of Fracture Mechanics

R.J.Sanford, Professor Emeritus, Mechanical Engineering, University of Maryland, College Park

Prentice Hall

2003

Latest Text Book

About the Author

• Prof. Emeritus R.J.Sanford has had two careers involving fracture

mechanics. He spent 22 years at the Naval Research Laboratory as a

research engineer during a period of intense fracture mechanics discovery

at NLR under the direction of George R.Irwin.(1960-1982)

• He left NLR in 1982 to join the faculty at the university of Maryland.At

the college park campus his focus has been on graduate education in

SOLID MECHANICS and FRACTURE MECHANICS!

• He is fellow in the Society for Experimental Mechanics and has received

both their Hetenyi Award (for research) and the Frocht Award (for

teaching excellence).

• He is a member of committee E 08 of the American Society for Testing

and Materials (ASTM).

About the Book

• The selection of topics and order of presentation in the book evolved

from a graduate course in fracture mechanics developed by the author over

last two decades.

• The focus is on the mathematical basis of Linear Elastic Fracture

Mechanics and their applications in Engineering Design.

• The presentation is a conversational, yet rigorous in manner with the

focus on the general formulation of the theory.

• The origins and limitations of simplified theories presented in other

introductory text books is thereby apparent.

• Unified mathematical treatment based on the Westergard Formulation

provides a coherent basis for the analytical,numerical, and experimental

treatment of crack problems in two dimensions.

About the Book

• Introductory chapter on Linear Theory of Elasticity with pivotal results

for the circular hole, the elliptical hole, and the wedge leading up to the

general problem of bodies with cracks.

• Thorough treatment of Fatigue Crack Growth behaviour and introduction

to NASGRO 3.0 and AFGROW 4.0 computer programs for life time

prediction analysis using complex empirical FCG models.

• Extensive TABLES of fracture properties for a wide variety of metallic

materials in both English and S.I units from the NASA data base.

• Broad spectrum of exercises at the end of each chapter ranging from

basic analytical derivations to parametric numerical analysis.

• A selection of open-ended design problems suitable for capstone project

assignments or take-home examinations.

Preface

• In the current state of development, the discipline of LEFM is a mature science that can be and, indeed is being introduced into the basic UG/PG education programs in mechanical, civil, aerospace and engineering mechanics departments.

• The quantitative prediction of FCG in a wide range of Engineering materials is of major importance in contemporary engineering design since over 80 percent of all brittle fractures have their origins in FCG.

• Definable within the context of the Linear Theory of Elasticity, the principles of fracture mechanics have a wide range of engineering design applications including the analysis of Brittle Fracture of low-toughness structural materials and many nonmetallics.

• The focus is on the mathematical principles of LEFM and their application to Engineering Design.

Preface

• The end-of-chapter exercises and comprehensive design problems are taken from examinations given at the end of each course.

• Included in this text are two appendices – listing (a) Strength and Fracture Properties and (b) Fatigue data for a wide variety of metallic materials, adapted from the NASA/ nasgro database.

• An appendix contains an extensive tabulation of Westergard stress functions and the corresponding SIF (K) solutions.(MODE I ONLY)

• The subject matter is based on the publications of pioneers in Fracture Mechanics over the last 40 years.

FRACTURE MECHANICS : OVERVIEW

CONTENTS

Failure modes

Historical Development

Research summary

The Energy Release Rate criterion

The Stress Intensity Factor criterion

Time dependent crack growth

Fracture Mechanics approach to design Vs traditional approach

Fracture Mechanics approach to Fatigue design

Safe - Life and Fail - Safe approaches to Fatigue design

Fracture Mechanics Family Tree

FRACTURE MECHANICS: OVERVIEW

FAILURE MODES Our understanding of how materials fail and our ability to prevent such failures in service has increased

considerably since World War II

Catastrophic service failures are determental to the economy of a nation.

Commonly observed modes of failure are

- Yielding

- Excessive deformation

- Buckling

- Fatigue

- Fracture

- Creep

- Environmental degradation of stiffness and strength

- Vibration and Noise

- Wear, etc.,

Designing components / structures to avoid these failure modes is not a new idea.

Design against FRACTURE (Failure due to Crack Propagation) is a relatively new approach. So also Fatigue

Analysis based on Fracture Mechanics concepts.

The use of Fracture Mechanics has undoubtedly prevented a substantial number of component / structural failures in

service.


HISTORICAL DEVELOPMENT

Land marks

- Griffith (1920’s) Energy Balance approach

- George Irwin (1948) Stress Intensity Factor approach

- Wells (1961) Crack Tip Opening Displacement concept

- Rice (1968) Path – Independent Integral

Griffith applied the results from stress concentration around an elliptical hole to predict FRACTURE (unstable

Propagation of a Crack)

Griffith’s Theory : A Crack becomes unstable and thus FRACTURE occurs when the strain energy change that results

from an Incremental crack growth is sufficient to overcome the surface energy of the material

Griffith’s theory accurately predicted the relationship between fracture strength and crack length in glass. Subsequent efforts to apply the same to metals was unsuccessful.why?

The Griffith’s theory only applies to ideally brittle solids.

A modification to Griffith’s theory that made it applicable to metals did not come till 1948.

A group of researchers directed by George Irwin at the Naval Research Laboratory in Washington D.C. studied the FRACTURE problem in detail. The subject we know as Fracture Mechanics was born in this lab during the decade following World War II. Fracture Mechanics progressed from being a scientific curiosity to an Engineering Discipline primarily because of this groups investigation of the structural failure of Liberty ships during World War II.

Investigations revealed that the Liberty ship failures were caused by a combination of

three factors

1. The welds, which were produced by semi-skilled workforce, contained crack

like flaws.

2. Most of the FRACTURES initiated on the deck, at square hatch corners,

where there was a local Stress Concentration.

3. The steel from which the Liberty ships were build had Poor Toughness, as

measured by Charpy Impact tests

In the longer term, structural steels were developed with vastly improved toughness

as measured by Fracture Toughness Tests. Weld Quality Control Standards were

developed and implemented and Engineering Analysis reduced the Stress

Concentration effects. Consequently, catastrophic failures of ship structures did not

reoccur.


Research Summary

A group of researchers at the Naval Research Laboratory, Washington, D.C. led by

Dr George R. Irwin created the basic tools for the Analysis and Prediction of

FRACTURE (Failure due to Crack Propagation).

Irwin’s first major contribution was to extend the Griffith’s theory to metals by

including the energy dissipated by local plastic deformation.

Orowan independently proposed a similar modification to Griffith’s theory.

Mott extended the Griffith theory to a rapidly propagating crack (Dynamic Fracture).

Irwin in 1956 developed the energy release rate concept, which is related to the

Griffith Theory, but in a form useful for Engineering Analysis. He used the

Westergaard approach (a semi inverse technique for analysis of stress and

displacements around a crack tip) to show that the stresses and displacements in the

immediate vicinity of the crack tip could be described by a single parameter that was

related to the energy release rate. This crack-tip characterizing parameter later became

known as the Stress Intensity Factor (SIF) denoted by K

During the same period of time, M.L. Williams derived crack tip solutions that were

identical to Irwin’s.

In 1956, Wells applied Fracture Mechanics to show that the fuselage structural failure

in several Comet Jet aircraft resulted from fatigue cracks growing to a critical size.

These cracks initiated at windows and were caused by insufficient local reinforcement,

combined with square corners which produced severe stress concentrations.

Another early application of Fracture Mechanics occurred in General Electric in 1957.

Winnie and Wundt used Irwin’s energy release rate approach to investigate the failure

large steam turbine rotors. They were able to predict the bursting behavior of large

disks extracted from rotor forgings, and applied this knowledge to the prevention of

FRACTURE in actual rotors.

In 1960, Paris and coworkers failed to find a receptive audience for their ideas on the

Fracture Mechanics approach to Fatigue Crack growth Analysis.

Linear Elastic Fracture Mechanics (LEFM) is not valid when significant plastic

deformation precedes FRACTURE. In 1960 – 61, several researchers developed

analysis to correct for yielding at the crack tip. Irwin’s plastic zone correction was

simple extension of LEFM. Dugdale and Barenblaat developed elaborate models based

on a narrow strip of yielded material at the crack tip.

Wells proposed in 1961, Crack Tip Opening Displacement (CTOD) as an alternative

fracture parameter when significant plastic deformation at the crack tip precedes

FRACTURE.

In 1968, Rice developed another parameter to account for nonlinear material behavior

around the crack tip. By idealizing plastic deformation as nonlinear elastic, Rice was

able to generalize the energy release rate to nonlinear material behavior. He showed

that this nonlinear energy release rate can be expressed as a line integral, which he

called the J-integral, evaluated along an arbitrary contour around the crack tip.

The same year, Hutchinson, Rice and Rosengren related the J-integral to crack tip

stress fields in nonlinear materials. This showed that J- integral can also be viewed as

non linear Stress Intensity Factor as well as a non linear energy release rate.

Fracture Mechanics analysis is widely applied in the design of Nuclear Reactor

components. One major difficulty in applying Fracture Mechanics in this case was

that most nuclear pressure vessel steels were too tough to be characterized with

LEFM without resorting to very large test specimens for Fracture Toughness Testing

to measure KIC

Begley and Landers at Westinghouse, decided to characterize fracture toughness of

Nuclear Pressure vessel steels with the J - integral. Their experiments were successful

and led to the publication of a Standard Test procedure to measure JIC of materials .

Ten years later ! JIC is also a measure of Fracture Toughness of materials.

Material Toughness characterization is only one aspect of Fracture Mechanics. In

order to apply Fracture Mechanics concepts to modern design one must have a

mathematical relation between Toughness, applied stress and flaw (crack) size. This

is provided by Phenomenological Fracture criteria.

Shih and Hutchinson provided the theoretical frame work for Elastic – Plastic

Fracture Mechanics Analysis based on the J – integral. An engineering approach for

EPFM analysis was then developed at EPRI (1981).

In the UK, Well’s CTOD parameter was applied extensively to Fracture Mechanics

Analysis of welded structures.

Shih in 1981 demonstrated a relationship between the J – integral and CTOD

implying that both parameters are equally valid for EPFM analysis.

Much of the theoretical foundations of dynamic fracture mechanics was also

developed during 1960 – 1980.

Recent trends in Fracture Mechanics research

More sophisticated material models are being included in Fracture Mechanics

Analysis.

To incorporate time – dependent non linear material behavior into Fracture

Mechanics Analysis, Viscoplasticity or Viscoelasticity is employed.

Vicoplasticity is motivated by the use of tough, creep – resistant high

temperature materials.

Viscoelasticity reflects the increasing proportion of Polymeric materials in

engineering applications.

Fracture Mechanics has also been used (and sometimes abused ) in the

characterization of laminated composite materials.

Development of micro structural models and models to relate local and global

fracture behavior of materials. A related topic is the effort to characterize and

predict geometry dependence of fracture toughness.

New approaches where traditional single – parameter fracture mechanics breaks

down.


The Energy Release Rate Criterion

Crack extension ( FRACTURE) occurs when the energy available for crack growth is

sufficient to overcome the resistance of material to crack growth. The resistance may

include the Surface energy, Plastic work, or other type of energy dissipation associated

with a propagating crack.

The energy release rate, G , is defined as the rate of change in potential energy with

crack area for a linear elastic material. At the moment of fracture G = G c the critical

energy release rate, is a measure of the material fracture toughness.

For a through crack of length 2a in an infinite plate subjected to a remote tensile stress

σ, the energy release rate is

G = σ2 a / E

where, E is the Youngs modulus of Elasticity of the material.

At fracture G = Gc and

Gc = f2 ac / E

where, σf is the fracture stress and ac is the measured crack length at

the onset of Fracture.

The energy release rate is a driving force, while Gc is the material resistance to crack

propagation.


The Stress Intensity Factor Criterion

The singular stress field around a crack tip

X

KI is the Mode I Stress Intensity Factor. It is the AMPLITUDE of stress singularity at

the crack tip. The singularity of the type γ-1/2 .

Fracture occurs when KI = KIC. KIC is a measure of the fracture toughness of the

material.

For an infinite plate with a central crack of length 2a, the SIF is

KI = σ

KI is the driving force and KIC is the resistance of the material to crack propagation.

KIC is assumed to be a size independent material property.

Relation between KI and G

G = KI2 / E

The energy release rate and stress intensity factor approaches to predict fracture ( as

failure due to crack propagation) are equivalent for linear elastic material behavior.


Time Dependent Crack Growth

Fracture Mechanics plays a key role in Life prediction of component that are subjected

to time – dependent crack growth mechanisms such as fatigue or stress – corrosion

cracking.

The fatigue crack growth rate in metals is described by the Paris law

is the crack growth per cycle, is the SIF range

C and m are material dependent constants.

Damage Tolerance Approach Design is illustrated in this figure. The initial crack size a0

is inferred from NDT, and the critical crack size ac is computed using applied stress and

fracture toughness. An allowable crack size is then defined by dividing the critical size

by a safety factor. The service life of the component can then be inferred by calculating

the time required for the flaw to grow from initial size to the maximum allowable size.

ac

The Fracture Mechanics Approach to Design Vs Traditional Approach

In the traditional approach to design and material selection a material is assumed to be

adequate , if its strength (yield or ultimate) is greater than the maximum allowed stress.

This approach may guard against brittle fracture by imposing a safety factor on stress,

combined with minimum tensile elongation requirements of material.

The Fracture Mechanics approach has three important variables as seen in the following fig.

FRACTURE TOUGHNESS

FLAW SIZE

APPLIED STRESS

Fracture Mechanics quantifies the critical combinations of these three variables

There are two alternative approaches to Fracture Analysis: The energy release rate

criterion and the Stress Intensity Factor criterion. These two are equivalent in certain

circumstances.

FRACTURE MECHANICS APPROACH TO FATIGUE DESIGN

Invokes a “defect – tolerant” philosophy based on the premise that all engineering

components are inherently flawed. The size, shape and location of a pre-existing

flaw(s) is determined by NDT.

If no flaw is found in the component, Proof tests are conducted at a stress level

slightly higher than the service stress. If no cracks are detected by the NDT and if

catastrophic failure does not occur during the proof test, the largest (undetected)

initial crack size is estimated from the resolution of the NDT.

The fatigue life is then defined as the number of cycles (or time) to propagate the

dominant cracks from the initial size to some critical size. The critical size based on

the Fracture Toughness of the material, the LIMIT load for the component, the

design allowable strain or the permissible change in compliance of the component.

The prediction of crack propagation life using the defect – tolerant approach involves

empirical Fatigue Crack Growth Laws based on Fracture Mechanics.

Various methods are available to include the effect of mean stress, stress

concentrations, environments, variable amplitude loading spectra and multiaxial

stress state in the estimation of Fatigue Crack Growth.

This intrinsically conservative approach to fatigue is widely used in fatigue – critical

applications. Examples, Aerospace and Nuclear Power Engineering.

Optimization of materials microstructure to improve resistance to both crack initiation

and crack growth would require a trade-off.

SAFE – LIFE AND FAIL – SAFE APPROACHES TO FATIGUE DESIGN

Developed by Aerospace Engineers

In the safe – life approach to fatigue design, the typical cyclic load spectra, which are

imposed on a structure / component in service are first determined. The components are

either analyzed or tested in the laboratory under load conditions which are typical of

service load spectra, and a useful fatigue life is estimated for the components.

The estimated fatigue life is suitably modified with a factor of safety (or a factor of

ignorance) then provides a prediction of safe - life for the component.

At the end of ‘safe - life’, the component is automatically retired from service, even if

no failure has occurred during service and the component has considerable residual life.

Although an estimate of fatigue life may be obtained from tests on the actual component,

the safe – life method is intrinsically theoretical in nature. This procedure has to account

for several unknowns; unexpected changes in loading conditions; errors in the

estimation of typical service load spectra; scatter in test results; variability in properties

among different batches of the same material; existence of initial defects in the

production process; corrosion of the parts; and human errors in the operation.

By selecting a large margin of safety a safe operating life can be guaranteed.

The approach is conservative and may not be desirable from the view point of economy

and performance. However, if fatigue cracks are nucleated in the component in service,

the component may fail catastrophically. In the safe – life approach the emphasis is

therefore on the prevention of crack initiation!

The fail – safe approach to fatigue design, by contrast, is based on the argument that,

even if an individual member of a structure fails, there should be sufficient structural

integrity in the remaining parts to enable the structure to operate safely until the crack

is detected. Components with multiple load paths are generally fail – safe because of

redundancy. In addition, the component may contain crack arresters to prevent

undesirable levels of crack growth.

The fail – safe approach therefore mandates PERIODIC INSPECTION along with a

requirement that the NDT techniques be capable of identifying flaws to enable prompt

REPAIRS or REPLACEMENTS.

Whatever philosophy is employed in fatigue design, it is preferable that the critical

components of a structure be inspected periodically. This step eliminates dangerous

consequences arising from false estimates and errors in the design stage, especially

with the safe – life approach

FRACTURE MECHANICS : FAMILY TREE

Linear Elastic Fracture Mechanics

LEFM

Linear elastic

time – independent material behavior

Elastic – Plastic Fracture Mechanics

EPFM

Non linear

time – independent material behavior

Dynamic Fracture Mechanics

Non linear time – dependent material behavior

Viscoelastic Fracture Mechanics

Viscoplastic Fracture Mechanics

The specific branch of Fracture Mechanics, one should use in a particular problem

that obviously depends on material behavior, component geometry, applied loads,

operating environment, etc.,

It is unlikely that all these topics can be covered in a single module. This module is

limited in scope to the study of Linear Elastic Fracture Mechanics. However, it

should form the foundation for the study of EPFM, DFM, VEFM, etc., in future

modules.

PRACTICAL USES OF FRACTURE MECHANICS

Provide a conceptually different approach to Engineering Design Practice; Namely The Damage Tolerance Design Methodology

Enables to quantify toughness of the materials as Resistance to Fracture (a failure mode due to crack propagation)

Enables stress analysis of components/structures with cracks

Helps to evaluate Fracture Mechanics parameters

Crack-tip Stress Intensity Factors (Ki) (i= 1,2,3)

Strain Energy Release Rates (Gi) (i=1,2,3)

Path-Independent Integral (J)

Crack Tip Opening Displacement (CTOD)

Identifies Fracture Criteria to predict residual strength of the cracked materials, components and structures as well as the direction of crack propagation.

Defines Fatigue Crack Growth Laws that enable life estimation of cracked components/structures

Helps to fix Non destructive Inspection Intervals

Supports Service failure Investigations involving fatigue and Fracture.

PRACTICAL USES OF FRACTURE MECHANICS

ASSIGNMENTS

1. FE modeling and SIF evaluation of different cracked bodies (with known target solutions)

2. Prediction of Residual strength of cracked bodies (comparison of different fracture criteria)

3. Prediction of fatigue crack growth using different FCG laws (for a given cracked body with known SIF solutions)

4. Fatigue Analysis – fatigue life of notched components using ANSYS (low cycle fatigue)

5. Fatigue Analysis – fatigue life of components using ANSYS (High cycle fatigue)

6. G evaluation (Penny shaped crack at interface)

7. J evaluation (DCB test)

8. CTOD evaluation (Compact Tension Test)

9. Critical study of standards for K, J, CTOD testing

10. Commentary on Analytical, experimental, Phenomenological and Computational aspects of Fracture Mechanics

11. Material Information System for Fracture Mechanics analysis and Fatigue Analysis: Surey & Assessment

12. Compendium of SIF solutions: 3D cracks.

13. Atlas of Fatigue Curves:Study and Update

14. Prediction of Crack Tip Plastic Zone(Shape or Size)

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