mechanics of solid materials

5/27/2018 mechanics of solid materials

1/27


2/27

FRACTUREMECHANICSANINTRODUCTION


3/27

FRACTURE MECHANICSMECHANICSOFFRACTURE


4/27

WHAT IS MECHANICS

WHAT IS FRACTURE


5/27

UNEXPECTED FAILURE OF WEAPONS, BUILDINGS,BRIDGES, SHIPS, TRINS, AIRPLANES, AND VARIOUSMACHINES HAS OCCURRED THROGHOUT THEINDUSTRIAL WORLD.

NO SOUBT SOME OF THESE FAILURES HAVE BEENDUE TO POOR DESIGN. HOWEVER, IT HAS BEENDISCOVERED THAT MANY FAILURES HAVE BEENCAUSED BY PRE-EXISTING FLAWS IN MATERIALS. THESE FLAWS INITIATE CRACKS THAT GROW ANDLEAD TO FRACTURE. THIS DISCOVERY LED TO THEFIELD OF STUDY KNOWN AS FRACTURE MECHANICS.

THE FIELD OF FRACTURE MECHANICS IS EXTREMELYBROAD.

FAILURE OF STRUCTURES - FRACTURE

MECHANICS


6/27

FRACTURE MECHANICS INCLUDES APPLICATIONS INENGINEERING, STUDIES IN APPLIED MECHANICS,ELASTICITY AND PLASTICITY, AND MATERIALSCIENCE. IN FACT FRACTURE MECHANICS IS COMBINATIONSOF MATERIAL BEHAVIOUR, SERVICE ENVIRONMENT,LOADING CONDITIONS, CRACK ORIENTATION, ANDPART GEOMETRY. DEPENDING UPON THE BEHAVIOUR OF MATERIALSUNDER APPLICATION OF LOADS CAN BE DIVIDEDINTO MAJOR PARTS; LINEAR ELASTIC FRACTURE

MECHANICS LEFM), AND ELASTIC-PLASTICFRACTURE MECHANICS EPFM). FOUNDATION OF FRACTURE MECHANICS WAS FIRST

ESTABLISHED BY GRIFFITH IN 1921. HE WAS THEFIRST ONE TO PRESENT A THEORY FOR FRACTURE.


7/27

GRIFFITH FORMULATED THAT AN EXISTING CRACKWILL PROPAGATE IF THEREBY THE TOTAL ENERGY OFTHE SYSTEM IS LOWERED, CONSISTING OF A DECREASEIN ELASTIC STRAIN ENERGY WITHIN THE STRESSEDBODY AS THE CRACK EXTENDS.

THE GRIFFITH CONCEPT WAS FIRST RELATED TOBRITTLE FRACTURE OF METALLIC MATERIALS BYZENER AND HOLLOMON 1944. SOON AFTER, IRWIN POINTED OUT THAT THE GRIFFITH-TYPE ENERGY BALANCE MUST BE BETWEEN 1) THESTORED STRAIN ENERGY AND 2) THE SURFACEENERGY PLUS THE WORK DONE IN PLASTICDEFORMATION. IN THE MIDDLE 1950s IRWIN CONTRIBUTED ANOTHERMAJOR ADVANCE BY SHOWING THAT THE ENERGY

APPROACH IS EQUIVALENT TO A STRESS INTENSITYAPPROACH.


8/27

AS WE KNOW THAT FAILURE OF STRUCTURALSYSTEMS MAY OCCUR BY EXCESSIVE DEFLECTION,YIELDING, OR FRACTURE. HOWEVER, THESE MODES OF FAILURES DO NOTOCCUR IN A SINGULAR FASHION. PRIOR TO FAILUREBY FRACTURE YIELDING OF A MEMBER MAY OCCUR. IN A SIMILAR WAY A MEMBER MAY UNDERGOCONSIDERABLE DEFLECTION BEFORE IT FAILS BYEXCESSIVE YIELDING. THIS IS THE REASON THAT FAILURE CRITERIA ARE

USUALLY BASED ON THE DOMINANT FAILURE MODE. FRACTURE MAY ALSO OCCUR IN A SUDDEN MANNER,IT MAY OCCUR AS BRITTLE FRACTURE OF CRACKED

OR FLAWED MEMBERS, OR IT MAY OCCUR INPROGRESSIVE STAGES.


9/27

THERE ARE MANY THEORIES / CRITERIA OFSTATIC FAILURE WHICH CAN BE POSTULATED

AFTER TENSILE TESTING. WHEN THE TENSILESPECIMEN BEGINS TO YIELD, THE FOLLOWINGEVENTS OCCUR:

THE MAXIMUM-PRINCIPAL-STRESS THEORY:

THE MAXIMUM PRINCIPAL STRESS REACHESTHE TENSILE YIELD STRENGTH Sy

THE MAXIMUM-SHEAR-STRESS THEORY, ALSOKNOWN AS THE TRESCA THEORY: THE

MAXIMUM SHEAR STRESS REACHES THESHEAR YIELD STRENGTH, 0.5Sy

THE MAXIMUM-PRINCIPAL-STRAIN THEORY:THE MAXIMUM PRINCIPAL STRAIN REACHESTHE YIELD STRAIN Sy/E.


10/27

THE MAXIMUM-STRAIN-ENERGYTHEORY: THESTRAIN ENERGY PER UNIT VOLUME REACHESA MAXIMUM OF 0.5Sy/E.

THE MAXIMUM-DISTORTION-ENERGY THEORY,ALSO KNOWN AS VON MISES THEORY; THEENERGY CAUSING A CHANGE IN SHAPE(DISTORTION) REACHES [(1 + )/(3E)Sy.

THE MAXIMUM-OCTAHEDRAL-SHEAR-STRESSTHEORY: THE SHEAR STRESS ACTING ON EACHOF EIGHT FACES CONTAINING A HYDOSTATICNORMAL STRESS ave = (1 + 2 + 3)/3REACHES A VALUE OF 2Sy/3.

THE APPLICATION OF THE MAXIMUMPRINCIPAL STRAIN THEORY AND THEMAXIMUM STRAIN-ENERGY THEORY TO REALMATERIALS IS QUITE LIMITED.


11/27

THE MAXIMUM-SHEAR-STRESS THEORY ANDTHE MAXIMUM-DISTORTION-ENERGY THEORYARE GENERALLY APPLIED WHEN THESTRUCTURAL MATERIAL IS DUCTILE,

THE MAXIMUM-DISTORTION-ENERGY THEORYGENERALLY PREDICTS FAILURE MOREACCURATELY, BUT THE MAXIMUM-SHEAR-STRESS THEORY IS OFTEN USED IN DESIGN AS

IT IS SIMPLER TO APPLY AND IS MORECONSERVATIVE.

FOR BRITTLE MATERIALS, THE COULOMB-MOHR FAILURE THEORY IS QUITE OFTEN USED

IN DESIGN.

FOR PLANE STRESS, THIS THEORY RESEMBLESA COMBINATION OF THE MAXIMUM-PRINCIPALSTRESS THEORY AND THE MAXIMUM-SHEAR

STRESS THEORY, AND IS QUITECONSERVATIVE.


12/27

GRIFFITH PRESENTED HIS THEORY OFFRACTURE BY USING THE STRESS FIELDCALCULATION FOR AN ELIPTRICAL FLAW IN AN

INFINITE PLATE LOADED BY AN APPLIEDUNIAXIAL STRESS .

IN THIS PARTICUALR CASE THE MAXIMUMSTRESS OCCURS AT (a, O) AND IS GIVEN BY

(y)max = ( 1 + 2a/b)

NOTE THAT WHEN a = b THE ELLIPSE BECOMESA CIRCLE AND ABOVE EQUATION GIVES ASTRESS CONCENTRATION FACTOR OF 3.

THIS RESULT AGREES WITH THE WELL-KNOWNRESULT FOR AN INFINITE PLATE WITH ACIRCULAR HOLE.


13/27


14/27

IRWIN, BETTER KNOWN AS FATHER OF FRACTUREMECHANICS, POINTED OUT THAT THE LOADSTRESSES NEAR A CRACK DEPEND ON THEPRODUCT OF THE NOMINAL STRESS AND THESQUARE ROOT OF THE HALF FLAW LENGTH.

IRWIN CALLED THIS RELATIONSHIP THE STRESS

INTENSITY FACTOR, DONOTED BY K, AND ISGIVEN AS:

K = a

THE STRESS INTENSITY IS A CONVENIENT WAY OFDESCRIBING THE STRESS DISTRIBUTION AROUNDA FLAW. IF TWO FLAWS OF DIFFERENT GEOMETRYHAVE THE SAME VALUE OF K, THEN THE STRESSFIELDS AROUND EACH OF THE FLAWS AREIDENTICAL.


15/27

STRESS INTENSITY FACTOR CAN ALSO BE GIVENAS FOLLOWS BY THE ADDITION OF A CRACK ANDGEOMETRY FACTOR

K = Ya

THE STRESS INTENSITY FACTOR IS, THEREFORE, AFUNCTION OF GEOMETRY, SIZE AND SHAPE OF THECRACK, AND ALSO THE LOADING. VALUES FOR

DIFFERENT CONFIGURATIONS ARE AVAILABLE INTHE LITERATURE.

THERE ARE THREE DISTINCT MODES OF CRACKPROPAGATION MODE I, THE OPENING CRACKPROPAGATION MODE, IS THE MOST COMMON INPRACTICE. TENSILE STRESS FIELDS GIVES RISE TOTHIS MODE OF CRACKING.

MODE II IS THE SLIDING MODE AND IS DUE TO IN-PLANE SHEAR LOADING. MODE III IS THE TEARING

MODE AND IT ARISES DUE TO OUT-OF-PLANESHEAR STRESSES.


16/27

STRESS INTENSITY FACTORS ARE ALSOCATEGORIZED WITH RESPECT TO MODE OF CRACKPROPAGATION, AND ARE REFERRED AS MODE ICRACK INTENSITY FACTOR OR MODE II CRACKINTENSITY FACTOR.

WHEN THE MODE I STRESS INTENSITY FACTOR

REACHES A CRITICAL VALUE, CRACKPROPAGATION INITIATES.

THIS CRITICAL STRESS INTENSITY FACTOR IS ALSOCALLED THE FRACTURE TOUGHNESS OF THEMATERIAL.

THE FRACTURE TOUGHNESS FOR PLANE STRAIN ISNORMALLY LOWER THAN THAT FOR PLANESTRESS. FOR THIS REASON THE TERM KIC ISTYPICALLY DEFINED AS THE MODE I, PLANE

STRAIN FRACTURE TOUGHNESS .

FRACTURE TOUGHNESS IS A MATERIAL PROPERTY


17/27

FRACTURE TOUGHNESS IS A MATERIAL PROPERTYIN THE SAME SENSE THAT YIELD STRENGTH IS AMATERIAL PROPERTY. THIS PROPERTY ISINDEPENDENT OF CRACK LENGTH, GEOMETY ANDLOADING SYSTEM.

ON THE OTHER HAND IT CAN VARY WITH THECRACK MODE, PROCESSING OF THE MATERIAL,TEMPERATURE, LOADING RATE AND THE STATE OFSTRESS AT THE CRACK SITE.

AS STATED EARLIER, KIC IS THE FRACTURETOUGHNESS FOR PLANE STRAIN CONDITIONS.PLANE STRESS ALWAYS EXISTS ON THE FREESURFACE PERPENDICULAR TO THE CRACKSURFACE.

HOWEVER, IF THE PART IS THICK ENOUGH AT THECRACK SITE, PLANE STRAIN WILL DOMINATE. ASTMRECOMMENDS THAT FOR PLANE STRAINCONDTIONS THICKNESS MUST BE GIVEN AS:

t = 2.5(KIC/Sy)


18/27


19/27


20/27

STRESS FIELD EQUATIONS PREDICT ASINGULARITY AT THE CRACK TIP. AS WE DUCTILEMATERIALS EXHIBIT A YIELD STRESS ABOVE

WHICH THEY DEFORM PLASTICALLY.

THIS MEANS THAT THERE WILL EXIST A REGION ATTHE CRACK TIP WHERE PLASTIC DEFORMATIONOCCURS AND THE SINGULARITY CAN NOT EXIST.

THE EXTENT OF THE PLASTIC ZONE IS DENOTEDBY rp, AND THE PLASTIC ZONE CORRECTION TOTHE CRACK LENGTH WILL REQUIRE AN ESTIMATEOF THE EXTENT OF PLASTIC ZONE.

IRWIN PROPOSED THAT THE EXISTENCE OF APLASTIC ZONE MAKES THE CRACT ACT AS IFDISPLACEMENTS ARE LARGER AND THE STIFFNESSIS LOWER THAN FOR THE STRICTLY ELASTICSITUATION.


21/27

IT MEANS THE USUAL CORRECTION IS TO ASSUMETHAT THE EFFECTIVE CRACK LENGTH IS THEACTUAL LENGTH PLUS THE RADIUS OF THEPLASTIC ZONE. HENCE

aeff = a + rp

WHERE, FOR PLANE STRESS CONDITIONS

rp = 1/2(K/Sy)

LEFM APPROACH WORKS WELL FOR HIGH-STRENGTH MATERIALS, BUT IT IS LESSUNIVERSALLY APPLICABLE FOR LOW-STRENGTH

STRUCTURAL MATERIALS.

THERE IS A LIMIT TO THE EXTENT TO WHICH KCAN BE ADJUSTED FOR CRACK TIP PLASTICITY BYTHE OTHER METHODS. IF rp BECOMES ANAPPRECIABLE FRACTION OF CRACK LENGTH,OTHER APPROACHES BECOME NECESSARY.


22/27

THE CONCEPT OF CRACK-TIP DISPLACEMENTCONCEPT CONSIDERS THAT THE MATERIAL AHEAD OFTHE CRACK CONTAINS A SERIES OF MINIATURE

TENSILE SPECIMENS HAVING SOME GAUGE LENGTHAND WIDTH.

ACCORDING TO THIS CONCEPT CRACK GROWTHOCCURS WHNE THE SPECIMEN ADJACENT TO THECRACK IS FRACTURED. WHEN ALL SPECIMENS FAIL

IMMEDIATELY WE HAVE A SITUATION OF SLOW CRACKGROWTH.

IN THIS SITUATION THE APPLIED STRESSES MUST BEINCREASED FOR STABLE CRACK GROWTH TO

CONTINUE.

CRACK TIP OPENING DISPLACEMENT FOR A CRACK OFLENGTH 2a IN AN INFINITE THIN PLATE SUBJECTEDTO UNIFORM TENSION IN A MATERIAL WHERE PLASTICDEFORMATION OCCURS AT THE CRACK TIP MAY BE

CALCULATED BY AN EXPRESSION.


23/27

A MORE COMPREHENSIVE APPROACH TO THEFRACTURE MECHANICS OF LOWER-STRENGTHDUTILE MATERIALS IS PROVIDED BY THE J

INTEGRAL.

IT WAS SHOWN THAT THE LINE INTEGRAL RELATEDTO THE ENERGY IN THE VICINITY OF A CRACK CANBE USED TO SOLVE TWO-DIMENSIONAL CRACK

PROBLEMS IN THE PRESENCE OF PLASTICDEFORMATION.

FRACTURE OF SUCH MATERIALS WOULD OCCURWHEN THE J INTEGRAL REACHES A CRITICALVALUE. J HAS UNITS OF MN / m.

THE J INTEGRAL CAN IN FACT BE INTERPRETED ASTHE POTENTIAL ENERGY DIFFERENCE BETWEENTWO IDENTICALLY LOADED SPECIMENS HAVINGSLIGHTLY DIFFERENT CRACK LENGTHS.


24/27

THE STRESS INTENSITY FOR A PARTIAL-THROUGHTHICKNESS FLAW IS GIVEN BY K = a sec a/2tWHERE a IS THE DEPTH OF PENETRATION OF THE

FLAW THROUGH A WALL THICKNESS t. IF FLAW IS5 mm DEEP IN A WALL 0.5 INCH THICK, DETERMINEWHETHER THE WALL WILL SUPPORT A STRESS OF25,000 psi IF IT IS MADE FROM 7075-T6 ALLUMINUMALLOY. FROM TABLE KIC = 24 MPa m

(a) K = a sec a/2t

(b) 25,000 psi = 172 MPa


25/27

CONSIDER A 200-MM WIDE AND 20-MM THICKPLATE MADE OF A HIGH-STRENGTH STEEL ALLOYWITH THE PROPERTIES KIC= 80 Mpa M AND y =

1500 MPa. USING A FACTOR OF SAFETY OF n = 2,DETERMINE (a) THE MAXIMUM ALLOWABLETENSILE FORCE THAT CAN BE APPLIED TO THEPLATE BASED ON YIELDING AND (b) THE MAXIMUMALLOWABLE TENSILE FORCE THAT CAN BEAPPLIED TO THE PLATE IF THE PLATE WITH A

CRACK SIZE 2a = 15 MM.

(a) = P / A = y / n, P = y A / n

(b) KI = KIC/ n KI= a, P = A


26/27

A PLATE MADE OF TITANIUM IS GIVEN SUCH THATKIC= 110 MPa M AND y = 820 MPa AND b = 100.DETERMINE THE LARGEST STABLE CRACK SIZE IF

THE APPLIED STRESS IS LIMITED TO 0.5y.

(a) KI= a, rp = 1/6(KI/y) = y / n,

(b) a 2.5 (KI/ y)

(b) a = aeff - rp


27/27

QUESTIONS AND QUERIES

IF ANY!

IF NOT THEN

GOOD BYE

SEE ALL OF YOU IN NEXT LECTUREON-------------------------

mechanics of solid materials

Documents

fact fracture mechanics

field of fracture mechanics

applied mechanics

failureby fracture yielding

stress theory

strain theory

failure of structures

failure of structuralsystems