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MECHANICS OF MATERIALS - II

MECHANICS OF MATERIALS - II. FRACTURE MECHANICS - AN INTRODUCTION. FRACTURE MECHANICS MECHANICS OF FRACTURE. WHAT IS MECHANICS? WHAT IS FRACTURE?.

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MECHANICS OF MATERIALS - II

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  1. MECHANICS OF MATERIALS - II

  2. FRACTURE MECHANICS - AN INTRODUCTION

  3. FRACTURE MECHANICS MECHANICS OF FRACTURE

  4. WHAT IS MECHANICS? WHAT IS FRACTURE?

  5. UNEXPECTED FAILURE OF WEAPONS, BUILDINGS, BRIDGES, SHIPS, TRINS, AIRPLANES, AND VARIOUS MACHINES HAS OCCURRED THROGHOUT THE INDUSTRIAL WORLD. NO SOUBT SOME OF THESE FAILURES HAVE BEEN DUE TO POOR DESIGN. HOWEVER, IT HAS BEEN DISCOVERED THAT MANY FAILURES HAVE BEEN CAUSED BY PRE-EXISTING FLAWS IN MATERIALS. THESE FLAWS INITIATE CRACKS THAT GROW AND LEAD TO FRACTURE. THIS DISCOVERY LED TO THE FIELD OF STUDY KNOWN AS FRACTURE MECHANICS. THE FIELD OF FRACTURE MECHANICS IS EXTREMELY BROAD. FAILURE OF STRUCTURES - FRACTURE MECHANICS

  6. FRACTURE MECHANICS INCLUDES APPLICATIONS IN ENGINEERING, STUDIES IN APPLIED MECHANICS, ELASTICITY AND PLASTICITY, AND MATERIAL SCIENCE. IN FACT FRACTURE MECHANICS IS COMBINATIONS OF MATERIAL BEHAVIOUR, SERVICE ENVIRONMENT, LOADING CONDITIONS, CRACK ORIENTATION, AND PART GEOMETRY. DEPENDING UPON THE BEHAVIOUR OF MATERIALS UNDER APPLICATION OF LOADS CAN BE DIVIDED INTO MAJOR PARTS; LINEAR ELASTIC FRACTURE MECHANICS (LEFM), AND ELASTIC-PLASTIC FRACTURE MECHANICS (EPFM). FOUNDATION OF FRACTURE MECHANICS WAS FIRST ESTABLISHED BY GRIFFITH IN 1921. HE WAS THE FIRST ONE TO PRESENT A THEORY FOR FRACTURE.

  7. GRIFFITH FORMULATED THAT AN EXISTING CRACK WILL PROPAGATE IF THEREBY THE TOTAL ENERGY OF THE SYSTEM IS LOWERED, CONSISTING OF A DECREASE IN ELASTIC STRAIN ENERGY WITHIN THE STRESSED BODY AS THE CRACK EXTENDS. THE GRIFFITH CONCEPT WAS FIRST RELATED TO BRITTLE FRACTURE OF METALLIC MATERIALS BY ZENER AND HOLLOMON 1944. SOON AFTER, IRWIN POINTED OUT THAT THE GRIFFITH-TYPE ENERGY BALANCE MUST BE BETWEEN (1) THE STORED STRAIN ENERGY AND (2) THE SURFACE ENERGY PLUS THE WORK DONE IN PLASTIC DEFORMATION. IN THE MIDDLE 1950s IRWIN CONTRIBUTED ANOTHER MAJOR ADVANCE BY SHOWING THAT THE ENERGY APPROACH IS EQUIVALENT TO A STRESS INTENSITY APPROACH.

  8. AS WE KNOW THAT FAILURE OF STRUCTURAL SYSTEMS MAY OCCUR BY EXCESSIVE DEFLECTION, YIELDING, OR FRACTURE. HOWEVER, THESE MODES OF FAILURES DO NOT OCCUR IN A SINGULAR FASHION. PRIOR TO FAILURE BY FRACTURE YIELDING OF A MEMBER MAY OCCUR. IN A SIMILAR WAY A MEMBER MAY UNDERGO CONSIDERABLE DEFLECTION BEFORE IT FAILS BY EXCESSIVE 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 IN PROGRESSIVE STAGES.

  9. FAILURE THEORIES / CRITERIA THERE ARE MANY THEORIES / CRITERIA OF STATIC FAILURE WHICH CAN BE POSTULATED AFTER TENSILE TESTING. WHEN THE TENSILE SPECIMEN BEGINS TO YIELD, THE FOLLOWING EVENTS OCCUR: THE MAXIMUM-PRINCIPAL-STRESS THEORY: THE MAXIMUM PRINCIPAL STRESS REACHES THE TENSILE YIELD STRENGTH Sy THE MAXIMUM-SHEAR-STRESS THEORY, ALSO KNOWN AS THE TRESCA THEORY: THE MAXIMUM SHEAR STRESS REACHES THE SHEAR YIELD STRENGTH, 0.5Sy THE MAXIMUM-PRINCIPAL-STRAIN THEORY: THE MAXIMUM PRINCIPAL STRAIN REACHES THE YIELD STRAIN Sy/E.

  10. THE MAXIMUM-STRAIN-ENERGYTHEORY: THE STRAIN ENERGY PER UNIT VOLUME REACHES A MAXIMUM OF 0.5Sy²/E. THE MAXIMUM-DISTORTION-ENERGY THEORY, ALSO KNOWN AS VON MISES THEORY; THE ENERGY CAUSING A CHANGE IN SHAPE (DISTORTION) REACHES [(1 + ν)/(3E)Sy². THE MAXIMUM-OCTAHEDRAL-SHEAR-STRESS THEORY: THE SHEAR STRESS ACTING ON EACH OF EIGHT FACES CONTAINING A HYDOSTATIC NORMAL STRESS σave = (σ1 + σ2 + σ3)/3 REACHES A VALUE OF √2Sy/3. THE APPLICATION OF THE MAXIMUM PRINCIPAL STRAIN THEORY AND THE MAXIMUM STRAIN-ENERGY THEORY TO REAL MATERIALS IS QUITE LIMITED.

  11. THE MAXIMUM-SHEAR-STRESS THEORY AND THE MAXIMUM-DISTORTION-ENERGY THEORY ARE GENERALLY APPLIED WHEN THE STRUCTURAL MATERIAL IS DUCTILE, THE MAXIMUM-DISTORTION-ENERGY THEORY GENERALLY PREDICTS FAILURE MORE ACCURATELY, BUT THE MAXIMUM-SHEAR-STRESS THEORY IS OFTEN USED IN DESIGN AS IT IS SIMPLER TO APPLY AND IS MORE CONSERVATIVE. FOR BRITTLE MATERIALS, THE COULOMB-MOHR FAILURE THEORY IS QUITE OFTEN USED IN DESIGN. FOR PLANE STRESS, THIS THEORY RESEMBLES A COMBINATION OF THE MAXIMUM-PRINCIPAL STRESS THEORY AND THE MAXIMUM-SHEAR STRESS THEORY, AND IS QUITE CONSERVATIVE.

  12. GRIFFITH PRESENTED HIS THEORY OF FRACTURE BY USING THE STRESS FIELD CALCULATION FOR AN ELIPTRICAL FLAW IN AN INFINITE PLATE LOADED BY AN APPLIED UNIAXIAL STRESS “σ”. IN THIS PARTICUALR CASE THE MAXIMUM STRESS OCCURS AT (±a, O) AND IS GIVEN BY (σy)max = ( 1 + 2a/b)σ NOTE THAT WHEN a = b THE ELLIPSE BECOMES A CIRCLE AND ABOVE EQUATION GIVES A STRESS CONCENTRATION FACTOR OF 3. THIS RESULT AGREES WITH THE WELL-KNOWN RESULT FOR AN INFINITE PLATE WITH A CIRCULAR HOLE.

  13. IRWIN, BETTER KNOWN AS FATHER OF FRACTURE MECHANICS, POINTED OUT THAT THE LOAD STRESSES NEAR A CRACK DEPEND ON THE PRODUCT OF THE NOMINAL STRESS “σ” AND THE SQUARE ROOT OF THE HALF FLAW LENGTH. IRWIN CALLED THIS RELATIONSHIP THE STRESS INTENSITY FACTOR, DONOTED BY “K”, AND IS GIVEN AS: K = σ√πa THE STRESS INTENSITY IS A CONVENIENT WAY OF DESCRIBING THE STRESS DISTRIBUTION AROUND A FLAW. IF TWO FLAWS OF DIFFERENT GEOMETRY HAVE THE SAME VALUE OF K, THEN THE STRESS FIELDS AROUND EACH OF THE FLAWS ARE IDENTICAL. STRESS INTENSITY FACTOR

  14. STRESS INTENSITY FACTOR CAN ALSO BE GIVEN AS FOLLOWS BY THE ADDITION OF A CRACK AND GEOMETRY FACTOR K = Yσ√πa THE STRESS INTENSITY FACTOR IS, THEREFORE, A FUNCTION OF GEOMETRY, SIZE AND SHAPE OF THE CRACK, AND ALSO THE LOADING. VALUES FOR DIFFERENT CONFIGURATIONS ARE AVAILABLE IN THE LITERATURE. THERE ARE THREE DISTINCT MODES OF CRACK PROPAGATION MODE I, THE OPENING CRACK PROPAGATION MODE, IS THE MOST COMMON IN PRACTICE. TENSILE STRESS FIELDS GIVES RISE TO THIS 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-PLANE SHEAR STRESSES.

  15. STRESS INTENSITY FACTORS ARE ALSO CATEGORIZED WITH RESPECT TO MODE OF CRACK PROPAGATION, AND ARE REFERRED AS MODE I CRACK INTENSITY FACTOR OR MODE II CRACK INTENSITY FACTOR. WHEN THE MODE I STRESS INTENSITY FACTOR REACHES A CRITICAL VALUE, CRACK PROPAGATION INITIATES. THIS CRITICAL STRESS INTENSITY FACTOR IS ALSO CALLED THE FRACTURE TOUGHNESS OF THE MATERIAL. THE FRACTURE TOUGHNESS FOR PLANE STRAIN IS NORMALLY LOWER THAN THAT FOR PLANE STRESS. FOR THIS REASON THE TERM “KIC” IS TYPICALLY DEFINED AS THE MODE I, PLANE STRAIN FRACTURE TOUGHNESS . FRACTURE TOUGHNESS

  16. FRACTURE TOUGHNESS IS A MATERIAL PROPERTY IN THE SAME SENSE THAT YIELD STRENGTH IS A MATERIAL PROPERTY. THIS PROPERTY IS INDEPENDENT OF CRACK LENGTH, GEOMETY AND LOADING SYSTEM. ON THE OTHER HAND IT CAN VARY WITH THE CRACK MODE, PROCESSING OF THE MATERIAL, TEMPERATURE, LOADING RATE AND THE STATE OF STRESS AT THE CRACK SITE. AS STATED EARLIER, “KIC” IS THE FRACTURE TOUGHNESS FOR PLANE STRAIN CONDITIONS. PLANE STRESS ALWAYS EXISTS ON THE FREE SURFACE PERPENDICULAR TO THE CRACK SURFACE. HOWEVER, IF THE PART IS THICK ENOUGH AT THE CRACK SITE, PLANE STRAIN WILL DOMINATE. ASTM RECOMMENDS THAT FOR PLANE STRAIN CONDTIONS THICKNESS MUST BE GIVEN AS: t = 2.5(KIC/Sy)²

  17. STRESS FIELD EQUATIONS PREDICT A SINGULARITY AT THE CRACK TIP. AS WE DUCTILE MATERIALS EXHIBIT A YIELD STRESS ABOVE WHICH THEY DEFORM PLASTICALLY. THIS MEANS THAT THERE WILL EXIST A REGION AT THE CRACK TIP WHERE PLASTIC DEFORMATION OCCURS AND THE SINGULARITY CAN NOT EXIST. THE EXTENT OF THE PLASTIC ZONE IS DENOTED BY “rp”, AND THE PLASTIC ZONE CORRECTION TO THE CRACK LENGTH WILL REQUIRE AN ESTIMATE OF THE EXTENT OF PLASTIC ZONE. IRWIN PROPOSED THAT THE EXISTENCE OF A PLASTIC ZONE MAKES THE CRACT ACT AS IF DISPLACEMENTS ARE LARGER AND THE STIFFNESS IS LOWER THAN FOR THE STRICTLY ELASTIC SITUATION. PLASTIC ZONE CORRECTION

  18. IT MEANS THE USUAL CORRECTION IS TO ASSUME THAT THE EFFECTIVE CRACK LENGTH IS THE ACTUAL LENGTH PLUS THE RADIUS OF THE PLASTIC ZONE. HENCE aeff = a + rp WHERE, FOR PLANE STRESS CONDITIONS rp = 1/2π(K/Sy)² LEFM APPROACH WORKS WELL FOR HIGH-STRENGTH MATERIALS, BUT IT IS LESS UNIVERSALLY APPLICABLE FOR LOW-STRENGTH STRUCTURAL MATERIALS. THERE IS A LIMIT TO THE EXTENT TO WHICH “K” CAN BE ADJUSTED FOR CRACK TIP PLASTICITY BY THE OTHER METHODS. IF “rp” BECOMES AN APPRECIABLE FRACTION OF CRACK LENGTH, OTHER APPROACHES BECOME NECESSARY.

  19. THE CONCEPT OF CRACK-TIP DISPLACEMENT CONCEPT CONSIDERS THAT THE MATERIAL AHEAD OF THE CRACK CONTAINS A SERIES OF MINIATURE TENSILE SPECIMENS HAVING SOME GAUGE LENGTH AND WIDTH. ACCORDING TO THIS CONCEPT CRACK GROWTH OCCURS WHNE THE SPECIMEN ADJACENT TO THE CRACK IS FRACTURED. WHEN ALL SPECIMENS FAIL IMMEDIATELY WE HAVE A SITUATION OF SLOW CRACK GROWTH. IN THIS SITUATION THE APPLIED STRESSES MUST BE INCREASED FOR STABLE CRACK GROWTH TO CONTINUE. CRACK TIP OPENING DISPLACEMENT FOR A CRACK OF LENGTH “2a” IN AN INFINITE THIN PLATE SUBJECTED TO UNIFORM TENSION IN A MATERIAL WHERE PLASTIC DEFORMATION OCCURS AT THE CRACK TIP MAY BE CALCULATED BY AN EXPRESSION. CRACK OPENING DISPLACEMENT

  20. A MORE COMPREHENSIVE APPROACH TO THE FRACTURE MECHANICS OF LOWER-STRENGTH DUTILE MATERIALS IS PROVIDED BY THE “J” INTEGRAL. IT WAS SHOWN THAT THE LINE INTEGRAL RELATED TO THE ENERGY IN THE VICINITY OF A CRACK CAN BE USED TO SOLVE TWO-DIMENSIONAL CRACK PROBLEMS IN THE PRESENCE OF PLASTIC DEFORMATION. FRACTURE OF SUCH MATERIALS WOULD OCCUR WHEN THE J INTEGRAL REACHES A CRITICAL VALUE. “J” HAS UNITS OF MN / m. THE J INTEGRAL CAN IN FACT BE INTERPRETED AS THE POTENTIAL ENERGY DIFFERENCE BETWEEN TWO IDENTICALLY LOADED SPECIMENS HAVING SLIGHTLY DIFFERENT CRACK LENGTHS. J INTEGRAL

  21. THE STRESS INTENSITY FOR A PARTIAL-THROUGH THICKNESS FLAW IS GIVEN BY K = σ√π a √sec πa/2t WHERE “a” IS THE DEPTH OF PENETRATION OF THE FLAW THROUGH A WALL THICKNESS “t”. IF FLAW IS 5 mm DEEP IN A WALL 0.5 INCH THICK, DETERMINE WHETHER THE WALL WILL SUPPORT A STRESS OF 25,000 psi IF IT IS MADE FROM 7075-T6 ALLUMINUM ALLOY. FROM TABLE KIC = 24 MPa √m (a) K = σ√π a √sec πa/2t (b) 25,000 psi = 172 MPa DESIGN PROBLEM - I

  22. CONSIDER A 200-MM WIDE AND 20-MM THICK PLATE MADE OF A HIGH-STRENGTH STEEL ALLOY WITH THE PROPERTIES KIC = 80 Mpa √M AND σy = 1500 MPa. USING A FACTOR OF SAFETY OF n = 2, DETERMINE (a) THE MAXIMUM ALLOWABLE TENSILE FORCE THAT CAN BE APPLIED TO THE PLATE BASED ON YIELDING AND (b) THE MAXIMUM ALLOWABLE TENSILE FORCE THAT CAN BE APPLIED TO THE PLATE IF THE PLATE WITH A CRACK SIZE 2a = 15 MM. (a) σ = P / A = σy / n, P = σy A / n (b) KI = KIC / n KI = σ√πa, P = σA DESIGN PROBLEM - II

  23. A PLATE MADE OF TITANIUM IS GIVEN SUCH THAT KIC = 110 MPa √M AND σy = 820 MPa AND b = 100. DETERMINE THE LARGEST STABLE CRACK SIZE IF THE APPLIED STRESS IS LIMITED TO 0.5σy. (a) KI = σ√πa, rp = 1/6π(KI/σy)² σ = σy / n, (b) a ≥ 2.5 (KI / σy)² (b) a = aeff - rp DESIGN PROBLEM - III

  24. QUESTIONS AND QUERIES IF ANY! IF NOT THEN GOOD BYE SEE ALL OF YOU IN NEXT LECTURE ON-------------------------

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