Fracture Mechanics

FractureMechanicsIntroduction to Fracture

Introduction to FracturePresented byCalvin M. Stewart, PhDMECH 5390-6390Fall 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 ofcrack in cracked structures. Notes: Understanding how crack behaves equips engineers with the tools needed todesign against the initiation and propagation of cracks Fracture behavior is dependent on material, load/displacement, andgeometric factors

Failure of Structures

Fracture Mechanics Approach to Design

SignificanceIn the nineteenth century, itwas realized that pre-existingflaws could initiate crackingand fracture.It was discovered that brittlefracture in steels waspromoted by lowtemperatures.

SignificanceThere was considerable development of new high strength alloys.In the 1950’s, it was recognized that although these materials are not intrinsically brittle, the energyrequired for fracture is comparatively low.The possibility, and indeed occurrence, of this low energy fracture in high strength materials stimulatedthe modern development of fracture mechanics.

SignificanceConsider a structure containing pre-existingflaws and/or in which crack initiate in service.The crack may growth with time owing tovarious causes (fatigue, stress corrosion, creep,etc.) and will generally grow progressivelyfaster.The residual strength of the structure, which isthe failure strength as a function of crack size,decreases with increasing crack size.After a time, the residual strength becomes solow that the structure may failure in service.

Significance Fracture Mechanics should attempt to provide quantitative answersto 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 themaximum permissible crack size?3. How long does it take for a crack to grow from a certain initial size, forexample the minimum detectable crack size, to the maximum permissiblecrack size?4. What is the service life of a structure when a crack-like flaw (e.g. amanufacturing defect) with a certain size is assumed to exist?5. During the period available for crack detection how often should thestructure be inspected for cracks?

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

Linear Elastic Fracture MechanicsLEFM Assumes that the material is isotropic and linear elastic Stress field near the crack tip is calculated using the theory ofelasticity Valid only when the plastic deformation is “small” compared to thelength 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 byGriffith 1920 and improved by Irwin 1950,states that crack extension (i.e., fracture)occurs when the energy available for crackgrowth is sufficient to overcome theresistance of the material.G 2 aE Gc R 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, RThrough-thickness crack in an infinite plate subject to aremote 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 theenergy approach, Irwin developed the stress intensity approach,where Linear Elastic theory shows that the stresses in the vicinity of acrack tip take the form ij Kfij ( )2 r where r and θ are the radius and angle with respect to the crack tip.

The Stress-Intensity Approach Example for an infinite width plate withcentral crack. The elastic stress field equations aredependent 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 LinearElastic material. If one assumes that the material fails locally at some critical combination of stress andstrain, then it follows that fracture must occur at a critical value of stress intensity, Kc.K a Kc Stress Intensity Factor, Kapplied stress, σCrack length, aCritical value of SIF, Kc

The Stress-Intensity Approach Comparing the Energy Release Rate and Stress Intensity Factor we find arelationship exists between them. The same relationship exists for the criticalvalues.K c2K2G Gc EE The critical values can be determined experimentally by measuring the fracturestress, σf of a component with a known crack length, ac. f 2 acGc EK c f ac

Crack Tip Plasticity The elastic stress distribution in thevicinity of a crack tip, shows that asr tends to zero the stresses becomeinfinite, i.e. there is a stresssingularity at the crack tip. Since structural materials deformplastically above the yield stress,there will in reality be a plastic zonesurrounding the crack tip. Thus the elastic solution is notunconditionally applicable.

Crack Tip Plasticity Irwin considered a circular plasticzone exists at the crack tip undertensile loading. He showed for planestress, the zone size is1ry 2 Kc ys 2 and for plane strain,1ry 2 Kc C ys 2 Yield strength, σys Correction factor, C usually 1.7

Fracture ToughnessPlane StressTransitionalPlane Strain The value of Kc at a particular temperaturedepends on the specimen thickness. It is customary to write the asymptoticvalue of Kc as the Plane Strain fracturetoughness as KIc. Critical stress intensity factor, Kc report with the thickness. Plane Strain Fracture Toughness, KIc insensitive to thickness.Asymptote is Kic

Elastic-Plastic Fracture MechanicsEPFM 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) areused to predict crack behavior Key Concepts: Strain Energy Release Rate, J-integral, Crack OpeningDisplacement (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 welldeveloped as LEFM, a fact that is reflected in the approximate natureof the eventual solutions. In 1961, Wells introduced the crack opening displacement (COD)approach. This approach focuses on the strain in the crack tip insteadof the stresses.

Elastic-Plastic Fracture Mechanics In the presence of plasticity acrack tip will blunt when it isloaded in tension. Wells proposed to use thecrack flank displacement at thetip of the blunt crack, the socalled crack tip openingdisplacement (CTOD) as afracture parameter. It was shown to be difficult to determine the required CTOD for agiven load and geometry or alternatively to calculate the critical cracklengths or loads.

Elastic-Plastic Fracture Mechanics In 1968, Rice considered the potential energy changes involved incrack growth in non-linear elastic material. Rice derived a fracture parameter called J, a contour integral that canbe evaluated along any arbitrary path enclosing the crack tip. He showed J to be equal to the energy release rate for a crack in nonlinear 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 evaluatedanalytically. However, in practice finite element calculations are oftenrequired. In spite of this, J has found widespread application as a parameter topredict the onset of crack growth in elastic-plastic problems. Later it was found that J could also be used to describe a limitedamount of stable crack growth.

Time-Dependent Fracture MechanicsTDFM Assumes that the load-displacement behavior of the material is timedependent 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 asduiJ Wdy Tidsdx In 1980’s, Ashok Saxenaproposed the C(t) integral thatencapsulates the timedependent behavior at the cracktip as followsC (t ) limr duiWdy Tidsdx

Classification

ClassificationFor low toughness materials, brittle fractureis the governing failure mechanism, andcritical stress varies linearly with KIc.At very high toughness values, LEFM is nolonger valid, and failure is governed by theflow properties of the material.At intermediate toughness levels, there is atransition between brittle fracture underlinear elastic conditions and ductileoverload.

Classification

Fatigue Fatigue is the cyclic application ofloads which can contribute to crackgrowth. Crack growth increases the stressintensity factor, K at the crack tip,which eventually leads to fracture. The fatigue crack growth rate,da/dN is defined as the crackextension over a small number ofcycles.da f ( G, K , J , C*,dN)

Fatigue At high stress, the crackgrows quicker and thecritical crack length, 2acis shorter whencompared 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 serviceenvironment. Measure the crack resistance, R, Gc, Kc, Jc, C*c, etc. Computational Using finite element analysis (FEA) or the boundary element method (BEM) toanalyze 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 Methodfor Determining Stress Intensity Factors, Cracks and Fracture, ASTM STP 601, American Societyfor 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 CracksHandbook, 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 acceptedguide that governs concepts,procedure, definitions, etc. Ex: ASTM E399 - Standard TestMethod for Plane- Strain FractureToughness of Metallic Materials Ex: ISO 12135:2002- Metallicmaterials -- Unified method oftest for the determination ofquasistatic 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 engineeringdisasters can occur. Engineers must provide quantitative answers to the fracture resistance ofstructures (before and during service). Engineers must determine which Branch of Fracture Mechanics is mostappropriate for a structure while including the service environment effects.