A. BASIC CONCEPTS 6
INTRODUCTION The final fracture of structural components is associated with the presence of macro or microstructural defects that affect the stress state due to the loading conditions. Fracture occurs when this state reaches at local level a critical condition. component Material + Geometry and defects + loading Vs. Toughness Stress state (local) Vs. Critical local condition to fracture 7
INTRODUCTION Processing Crack (defect): produced by Cracking processes Operation damage component Material + Geometry and defects + loading Vs. Toughness Stress state (local) Vs. Critical local condition to fracture Fracture analysis FRACTURE MECHANICS 8
INTRODUCTION Fracture Mechanics σ a Crack growth a 0 a σ design. σ use σ C Safety Critical loading evolution t FRACTURE classic design use Crack growth a (t) 9
INTRODUCTION Fracture Modes F: Loading forces F a: crack advance n n: crack plane normal a n a F // n F // a F n F a F MODE I MODE II MODE III Tensile Shear Torsion 10
FRACTURE CRITERIA Stress state in a crack front Stress concentration factor, Kt Stress concentration Stress X X Stress concentration factor, Kt Stress concentration factor, Kt 11
FRACTURE CRITERIA Local stress and strain states in a crack front (Irwin) σ Semiinfinite plate, uniform stress (MODE I) σ x =σ a r STRESSES Plane solution cos θ 1 sen θ 3θ sen σ y =σ a r cos θ 1 + sen θ τ xy =σ a cos θ r sen θ cos 3θ Plane stress (PSS) sen 3θ I σ Plane strain (PSN) σ z = 0 σ z = ν (σ x + σ y ) ij = σ a f r I ij ( θ ) a Y τ xy r θ X Ζ σ y σ x σ z u = σ E v = σ E κ = 3 4ν DISPLACEMENTS ar 1 +ν ( )cos θ ( ) κ 1 ar 1+ν ( ) κ +1 (PSS) cos 3θ ( )sen θ sen 3θ κ = 3 ν 1+ν (PSN) σ w = ν E ( σ x +σ y ) dz 1
FRACTURE CRITERIA Stress state in a crack front. Stress Intensity Factor Y r σ τ xy σ y σ x σ z σ I ij = σ a r σ I ij f I ij = ( θ) K I 1 I = σ πa f ( θ) πr ij 1 I f ( θ) ij πr Position a θ Ζ X Stress Intensity Factor K I = σ πa σ K I defines the stress state in the crack front 13
FRACTURE CRITERIA Stress state in a crack front Stress Intensity Factor For any component (geometry + defects) σ a σ K I = fσ πa or fσ πg(a) f: geometric factor For other modes analogously... K II = f II τ πg(a) 14
FRACTURE CRITERIA Stress fracture criterion If σ K I σ ij (local) If σ ij (local) = σ ij (critical) Local fracture criterion K I = K I C Sress fracture criterion If fracture critical conditions (K IC ) only depend on material K Ic (Fracture Toughness) Stress Fracture Criterion K I = K Ic 15
FRACTURE CRITERIA Stress fracture criterion σ σ c (a o ) σ K I = K Ic a o 4a o σ c (ao) σ c (4ao) = a o σ Fracture occurs with σ c (a) Another observation: strain The compliance of the component increases with the length of the defects. Compliance: Indicates the length of the defects. 16
FRACTURE CRITERIA Energetic fracture criterion (Griffith) Comparison between the energy that is released in crack extension and the energy that is necesary to generate new surfaces because of that extension. d( U ) d γ E da da unitary thickness d( U) Bda Crack grows (Fracture) deγ = γ Bda γ: surface energy of the material (Energy per unit of generated surface) 17
FRACTURE CRITERIA Energetic fracture criterion (Griffith) As a geometry function Semiinfinite plate G f ( σ, a, E) γ πσ a = E G: Energy release rate G c : Fracture Toughness G c G = G c Fracture criterion Where: G G c = = α K I E α =1 (Plane stress) ( 1 α = υ ) (Plane strain) γ in very brittle materials >> γ in materials with plasticity before fracture 18
FRACTURE TOUGHNESS Fracture Toughness Characterisation A) Standardised specimens B) Fatigue precracked specimens : a (a as initial crack length) 19
FRACTURE TOUGHNESS Fracture Toughness Characterisation C) Mechanical Testing P Q (Load on Fracture initiation) Pmax Pmax Pmax K Q = f (P Q, a, geometry) K Q = PQ B W 1 f a W a measured on fracture surface for CTs K Q = K Ic (toughness), if some normalised conditions are fulfilled 0
FRACTURE TOUGHNESS Fracture Toughness Characterisation Fatigue crack profile a 6 Load (kn) 4 Fracture mechanisms 0 0 1 3 4 5 6 COD (mm) UCQ-13 50 μm 1
FRACTURE TOUGHNESS Thickness effect Fracture at 45º with respect to the Rotura formando 45 con el plano original original crack de plane fisura 45 c K I K Ic PLANE Tensión STRESS plana Deformación plana PLANE STRAIN σ X σ Y Fisura crack Z TENSION PLANE STRESS PLANA σ Z = 0 Z + Effect of Plastic Zone Rotura Fracture en in crack plano plane de fisura Labios a 45 en extremos B min Thickness Espesor (B) (B) B min (P.Strain) K.5 Ic σ Y = σ X σ Z σ Y DEFORMACION PLANE STRAINPLANA
FRACTURE TOUGHNESS Effect of temperature and loading rate a TOUGHNESS STATIC TEST INCREASING LOADING RATE IMPACT TEMPERATURE b c K Ic K Ic TOUGHNESS STATIC K Id DYNAMIC TEMPERATURE LOADING RATE 3
MATERIAL TOUGHNESS Value It depends on microstructure and external variables Environment (temperature) (degradation) Loading rate 4
FRACTURE TOUGHNESS Impact Toughness: Charpy Test E ductile Absorbed Energy (E) % Ductile fracture Scale brittle % Lateral expansion pointer Initial position Hummer T T Final position specimen 5
FRACTURE TOUGHNESS Impact Toughness: Examples of the effect of different variables Influence of Irradiation 300 50 Longitudinal Transversal 00 Influence of Carbon Content E (J) 150 100 50 0-00 -150-100 -50 0 50 T (ºC) Influence of microstructural orientation 6
PLASTICITY ON FRACTURE Plasticity in a crack front σ y = K I πr θ cos For θ = 0 (crack plane): θ 1 + sen 1 st plasticzonemodel σ y = θ cos 3 K I πr Linear elastic solution (LEFM) If σ y σ Y 1 K ry π σ I = Y plastic zone: σ y = σ Y (problem: there is no stress equilibrium) nd plasticzonemodel(irwin correction) r P 1 K I = π σy (stress redistribution) (approximate solution) 7
PLASTICITY ON FRACTURE Plastic Zones on Plane Stress and Plane Strain Plastic Perfilzone de la zona plástica profile y z Deform Plane strain ación plana x Fisura crack r p Crack Fisura Plane Tensión stress plana Plane Stress. Yield stress for σ y = σ Y r P 1 K = π σ I Y Plane Strain. Yield stress for σ y 3σ Y 1 K I r = P 9π σ Y 8
PLASTICITY ON FRACTURE Corrections on Linear Elastic Fracture Mechanics (LEFM) If the plastic zone is small and it is constrained: Plastic Zone r P << a, defect r P << B, thickness r P << (W-a), residual ligament K I = K I(a ef ) = K I(a + Δa P ) Crack Fisura r p Effective defect = Real defect + Δa P Δa P : plastic correction to crack length Δa P = f (r P ) = 1 nπ K σ I Y n = 6 Plane Strain n = Plane Stress An iterative calculation is required to obtain K I 9
PLASTICITY ON FRACTURE Elastic-Plastic Fracture Mechanics (EPFM) If plastic zone has important dimensions: a Frente Original original crack de la fisura front Parameters and fracture criteria change because of local condition changes - Physical parameters and criteria CTOD = CTOD c - Energetic parameters and criteria CTOD J-Integral J = J Ic (equivalent to G in linear elastic conditions) 30
PLASTICITY ON FRACTURE Elastic-Plastic Fracture Mechanics (EPFM) The non linear energy release rate, J, can be written as a path-independent line integral. Considering an arbitrary counter-clockwise path (Γ) arround the tip of the crack, the J integral is given by: Arbitrary contour around the tip of the crack 31
PLASTICITY ON FRACTURE Elastic-Plastic Fracture Mechanics (EPFM) J can also be seen as a Stress Intensity Parameter for Elastic-Plastic problems as long as the variation of stress and strain ahead of the crack tip can be expressed as: Where k 1 and k are proportionally constants and n is the strain hardening component. 3
PLASTICITY ON FRACTURE Elastic-Plastic Fracture Mechanics (EPFM) Many materials with high toughness do not fail catastrophically at a particular value of J or CTOD. In contrast, these materials exhibit a rising R curve, where J and CTOD increase with crack growth. The figure illustrates a typical J resistance curve for a ductile material. 33
PLASTICITY ON FRACTURE Elastic-Plastic Fracture Mechanics (EPFM) J R,J app CRACK DRIVING FORCE DIAGRAM P 1 < P = P crit J app (P,a o ) a o J app (P 1,a o ) J R a Local conditions in the component J app (P,a) = J e (P,a) + J p (P,a) Characterises the local state Critical conditions in the material J R (Δa) Characterises the strength of the material to cracking 34
FRACTURE MICROMECHANISMS Brittle Fracture: Cleavage It occurs on metallic material with brittle behaviour favoured by low temperatures and high loading rates favoured in materials with high σ Y 35
FRACTURE MICROMECHANISMS Ductile Fracture: Void nucleation and coalescence Fibrous appearance Shear Metallic materials with plastic behaviour favored by T, σ Y, σ 36
FRACTURE MICROMECHANISMS Ductile Fracture: Void nucleation and coalescence 37
FRACTURE MICROMECHANISMS Intergranular fractures Because of the environment or grain boundary segragations 38
BIBLIOGRAPHY / REFERENCES Anderson T.L, Fracture Mechanics. Fundamentals and Applications, nd Edition, CRC Press, Boca Raton (1995) Broeck D., Elementary Engineering Fracture Mechanics, Martinus Nijhoff Pub., La Haya, 198. Broeck, D., The Practical Use of Fracture Mechanics, Kluwer Academic Publisher, Dordrecht, Teh Netherlands, 1989 Kanninen, M.F. and Popelar, C.H., Advanced Fracture Mechanics, Oxford University Press, New York, 1985. Thomason, P.F., Ductile Fracture of Metals, Pergamon Press, Oxford, UK, 1990 39