Introduction to Fracture

Fracture Mechanics

Introduction to FracturePresented by

Calvin M. Stewart, PhD

MECH 5390-6390

Fall 2020

Outline

• Definition

• Failure of Structures

• Fracture Mechanics Approach to Design

• Significance

• Review of Approaches• Energy Approach, Stress Intensity Factor Approach,

Crack Tip Plasticity, Fracture Toughness, The J Integral, Classification, Fatigue

• Methods• Analytical, Experimental, Computational

Definition of Fracture Mechanics

• Fracture – field of study focused on characterizing the behavior of crack in cracked structures.

• Notes:• Understanding how crack behaves equips engineers with the tools needed to

design against the initiation and propagation of cracks

• Fracture behavior is dependent on material, load/displacement, and geometric factors

Failure of Structures

Fracture Mechanics Approach to Design

Significance

In the nineteenth century, it was realized that pre-existing flaws could initiate cracking and fracture.

It was discovered that brittle fracture in steels was promoted by low temperatures.

Significance

There was considerable development of new high strength alloys.

In the 1950’s, it was recognized that although these materials are not intrinsically brittle, the energy required for fracture is comparatively low.

The possibility, and indeed occurrence, of this low energy fracture in high strength materials stimulated the modern development of fracture mechanics.

Significance

Consider a structure containing pre-existing flaws and/or in which crack initiate in service.

The crack may growth with time owing to various causes (fatigue, stress corrosion, creep, etc.) and will generally grow progressively faster.

The residual strength of the structure, which is the failure strength as a function of crack size, decreases with increasing crack size.

After a time, the residual strength becomes so low that the structure may failure in service.

Significance

• Fracture Mechanics should attempt to provide quantitative answers to the following questions:

1. What is the residual strength as a function of crack size?2. What crack size can be tolerated under service loading, i.e. what is the

maximum permissible crack size?3. How long does it take for a crack to grow from a certain initial size, for

example the minimum detectable crack size, to the maximum permissible crack size?

4. What is the service life of a structure when a crack-like flaw (e.g. a manufacturing defect) with a certain size is assumed to exist?

5. During the period available for crack detection how often should the structure be inspected for cracks?

Review of ApproachesEnergy Approach, Stress Intensity Factor Approach, Crack Tip Plasticity, Fracture Toughness, The J Integral, Classification, Fatigue

Linear Elastic Fracture Mechanics

LEFM• Assumes that the material is isotropic and linear elastic

• Stress field near the crack tip is calculated using the theory of elasticity

• Valid only when the plastic deformation is “small” compared to the length scale of the crack (i.e., small-scale yielding)

• Key Concepts: Elastic energy release rate, G, Stress Intensity Factor,K.

The Energy Approach

• For Linear Elastic Materials,

• The energy approach, developed by Griffith 1920 and improved by Irwin 1950, states that crack extension (i.e., fracture) occurs when the energy available for crack growth is sufficient to overcome the resistance of the material.

• Energy release rate, G – energy per unit of new crack area

• applied stress, σ

• Crack length, a

• Young’s modulus, E

• Critical value of energy release rate, Gc - material property

• Crack resistance, R

2

c

aG G R

E

= =

Through-thickness crack in an infinite plate subject to a remote tensile stress. In practical terms, “infinite ”

means that the width of the plate is >>2a.

The Stress-Intensity Approach

• In the 1950’s, owing to the practical difficulties of calculating the energy approach, Irwin developed the stress intensity approach, where Linear Elastic theory shows that the stresses in the vicinity of a crack tip take the form

• where r and θ are the radius and angle with respect to the crack tip.

( )2

ij ij

Kf

r

=

The Stress-Intensity Approach

• Example for an infinite width plate with central crack.

• The elastic stress field equations are dependent on loading and geometry.

• Notice the term K.

The Stress-Intensity Approach

• The Stress Intensity Factor, K completely characterizes the crack tip conditions in a Linear

Elastic material.

• If one assumes that the material fails locally at some critical combination of stress and

strain, then it follows that fracture must occur at a critical value of stress intensity, Kc.

• Stress Intensity Factor, K

• applied stress, σ

• Crack length, a

• Critical value of SIF, Kc

cK a K =

The Stress-Intensity Approach

• Comparing the Energy Release Rate and Stress Intensity Factor we find a

relationship exists between them. The same relationship exists for the critical

values.

• The critical values can be determined experimentally by measuring the fracture stress, σf of a component with a known crack length, ac.

2KG

E=

2

cc

KG

E=

2

f c

c

aG

E

=

c f cK a =

Crack Tip Plasticity

• The elastic stress distribution in the vicinity of a crack tip, shows that as r tends to zero the stresses become infinite, i.e. there is a stress singularity at the crack tip.

• Since structural materials deform plastically above the yield stress, there will in reality be a plastic zone surrounding the crack tip.

• Thus the elastic solution is not unconditionally applicable.

Crack Tip Plasticity

• Irwin considered a circular plastic zone exists at the crack tip under tensile loading. He showed for plane stress, the zone size is

• and for plane strain,

• Yield strength, σys

• Correction factor, C usually 1.7

2

1

2

cy

ys

Kr

=

2

1

2

cy

ys

Kr

C

=

Fracture Toughness

• The value of Kc at a particular temperature depends on the specimen thickness.

• It is customary to write the asymptotic value of Kc as the Plane Strain fracture toughness as KIc.

• Critical stress intensity factor, Kc

• report with the thickness.

• Plane Strain Fracture Toughness, KIc

• insensitive to thickness.

Asymptote is Kic

Plane Stress Transitional Plane Strain

Elastic-Plastic Fracture Mechanics

EPFM• Assumes that the material is isotropic and elastic-plastic

• Method is appropriate for structures with relatively large plastic zones

• Strain energy fields or crack tip opening displacements (CTOD) are used to predict crack behavior

• Key Concepts: Strain Energy Release Rate, J-integral, Crack Opening Displacement (COD), Crack Tip Opening Displacement (CTOD)

Elastic-Plastic Fracture Mechanics

• The LEFM approach only deals with limited crack tip plasticity.

• Due to its complexity, the concepts of EPFM are not so well developed as LEFM, a fact that is reflected in the approximate nature of the eventual solutions.

• In 1961, Wells introduced the crack opening displacement (COD) approach. This approach focuses on the strain in the crack tip instead of the stresses.

Elastic-Plastic Fracture Mechanics

• In the presence of plasticity a crack tip will blunt when it is loaded in tension.

• Wells proposed to use the crack flank displacement at the tip of the blunt crack, the so-called crack tip opening displacement (CTOD) as a fracture parameter.

• It was shown to be difficult to determine the required CTOD for a given load and geometry or alternatively to calculate the critical crack lengths or loads.

Elastic-Plastic Fracture Mechanics

• In 1968, Rice considered the potential energy changes involved in crack growth in non-linear elastic material.

• Rice derived a fracture parameter called J, a contour integral that can be evaluated along any arbitrary path enclosing the crack tip.

• He showed J to be equal to the energy release rate for a crack in non-linear elastic material, analogous to G for linear elastic material.

Elastic-Plastic Fracture Mechanics

Elastic-Plastic Fracture Mechanics

• For simple geometries and load cases the J integral can be evaluated analytically. However, in practice finite element calculations are often required.

• In spite of this, J has found widespread application as a parameter to predict the onset of crack growth in elastic-plastic problems.

• Later it was found that J could also be used to describe a limited amount of stable crack growth.

Time-Dependent Fracture Mechanics

TDFM• Assumes that the load-displacement behavior of the material is time-

dependent due to dynamic loading or due to creep, stress relaxation, and other dynamic effects

• Crack tip stress fields vary with time

• Key Concept: C*-integral, Ct-parameter

Time-Dependent Fracture Mechanics

• The J Integral can be written as

• In 1980’s, Ashok Saxena proposed the C(t) integral that encapsulates the time-dependent behavior at the crack tip as follows

( )lim

r

i

i

duC t Wdy T ds

dx→

= −

i

i

duJ Wdy T ds

dx= −

Classification

Classification

For low toughness materials, brittle fracture is the governing failure mechanism, and critical stress varies linearly with KIc.

At very high toughness values, LEFM is no longer valid, and failure is governed by the flow properties of the material.

At intermediate toughness levels, there is a transition between brittle fracture under linear elastic conditions and ductile overload.

Classification

Fatigue

( ), , , *,da

f G K J CdN

=

• Fatigue is the cyclic application of loads which can contribute to crack growth.

• Crack growth increases the stress intensity factor, K at the crack tip, which eventually leads to fracture.

• The fatigue crack growth rate, da/dN is defined as the crack extension over a small number of cycles.

Fatigue

• At high stress, the crack grows quicker and the critical crack length, 2acis shorter when compared to low stress.

• The da/dN versus ΔKdata, is virtual indentical.

• Threshold, ΔKth

• Critical, Kc

MethodsAnalytical, Experimental, Computational

Methods

• Analytical• Applying of theories of elasticity, plasticity, or viscoplasticity

• Develop math expression for specific loading and geometry

• Experimental• Subjecting small samples to mechanical test conditions simulating the service

environment. Measure the crack resistance, R, Gc, Kc, Jc, C*c, etc.

• Computational• Using finite element analysis (FEA) or the boundary element method (BEM) to

analyze cracked structures

Analytical Methods

• Books and Various Design Codes and Standards

• Sih, G.C. (Editor), Methods of Analysis and Solutions of Crack Problems, Noordhoff International Publishing (1973): Leiden.

• Paris, P.C., McMeeking, R.M. and Tada, H., The Weight Function Method for Determining Stress

• Intensity Factors, Cracks and Fracture, ASTM STP 601, American Society for Testing and Materials, pp. 471 489 (1976): Philadelphia.

• Rooke, D.P. and Cartwright, D.J., Compendium of Stress Intensity Factors, Her Majesty’s Stationery Office (1976): London.

• Tada, H., Paris, P.C. and Irwin, G.R., The Stress Analysis of Cracks Handbook, Paris Productions Incorporated (1985): St. Louis, Missouri.

• Murakami, Y., Stress Intensity Factors Handbook, Vols. 1, 2 and 3, Pergamon Press (1987 vols. 1 and 2, 1992 vol. 3): Oxford.

Experimental Methods

• Def: Standard – an accepted guide that governs concepts, procedure, definitions, etc.

• Ex: ASTM E399 - Standard Test Method for Plane- Strain Fracture Toughness of Metallic Materials

• Ex: ISO 12135:2002- Metallic materials -- Unified method of test for the determination of quasistatic fracture toughness

Computational Methods➢ Finite Element Method (FEM)➢ Boundary Element Method (BEM)➢ Extended Finite Element Method (XFEM)

Summary➢ Fracture Mechanics is the study of cracks in cracked structures.

➢ Most structures exhibit pre-existing flaws and/or develop flaws while in service.

➢ If fracture mechanics is not considered in design, accidents and engineering disasters can occur.

➢ Engineers must provide quantitative answers to the fracture resistance of structures (before and during service).

➢ Engineers must determine which Branch of Fracture Mechanics is most appropriate for a structure while including the service environment effects.

Homework 1

• Please tell me a few things about yourself. For example from where did you originate? Who is your academic Advisor?

• Do you have a prior background with fracture mechanics? In what ways is fracture relevant to your research activities or job functions?

• Download and Install ANSYS Student Edition.

• Using the equation and materials properties on page 16 of Janssen, plot Figure 1.9 using MATHCAD on My.Apps. Provide a written critical analysis of the two conditions poised in the textbook.

References

• Janssen, M., Zuidema, J., and Wanhill, R., 2005, Fracture Mechanics, 2nd Edition, Spon Press

• Anderson, T. L., 2005, Fracture Mechanics: Fundamentals and Applications, CRC Press.

• Sanford, R.J., Principles of Fracture Mechanics, Prentice Hall

• Hertzberg, R. W., Vinci, R. P., and Hertzberg, J. L., Deformation and Fracture Mechanics of Engineering Materials, 5th Edition, Wiley.

• https://www.fracturemechanics.org/

CONTACT INFORMATION

Calvin M. Stewart

Associate Professor

Department of Mechanical Engineering

The University of Texas at El Paso

500 W. University Ave, Suite A126, El Paso, TX 79968-0521

Ph: 915-747-6179

cmstewart@utep.edu

me.utep.edu/cmstewart/