Lecture #7: Basic Notions of Fracture Mechanics Ductile Fracture by Dirk Mohr ETH Zurich, Department of Mechanical and Process Engineering, Chair of Computational Modeling of Materials in Manufacturing 015 1 1 1
Basic Notions of Fracture Mechanics
Fracture Mechanics Fracture mechanics is a branch of mechanics that is concerned with the study of the propagation of cracks and growth of flaws. The starting point of a fracture mechanics analysis therefore is a structure with a pre-existing crack or flaw. Central questions in fracture mechanics are for example: Under which mechanical loads does a preexisting crack propagate? What is the maximum size of a crack that can be tolerated in a structure that is subject to a known mechanical load such that the crack does not propagate? crack 3 3 3
Griffith theory an energy approach Griffith s (191) made an attempt to come up with a fracture criterion for brittle solids by writing down the energy balance during the growth of a crack. Consider an isotropic linear elastic plate subject to uniaxial tension and let W 0 denote the elastic strain energy stored in that system. According to Inglis (1913) analysis, the introduction of a through thickness crack of length a reduces the elastic strain energy (energy release) by a t E da a da t 4 4 4
Griffith theory Introducing the free surface energy per unit area, s, the total energy of the system then reads U tot t E U0 a 4at s After differentiating with respect to a, we obtain the rate of change of the total energy as a function of the crack length and the applied stress du da tot t a E 4t s da a da t 5 5 5
du da tot t a E Griffith theory 4t Note that for small flaws and for low applied stresses, the surface energy term dominates, i.e. the total energy of the system would increase as the crack advances. However, when the critical condition du tot /da=0 is met, the change in total energy becomes negative (energy release). s 151-0735: Dynamic behavior of materials and structures 6 6 6 da a da According to Griffith, this condition defines the onset of unstable crack growth. The critical far field stress for fracture initiation therefore reads t a 4t s 0 E c s E a t
Linear Elastic Fracture Mechanics (LEFM) The stress fields in the vicinity of cracks in elastic media can be calculated using linear elastic stress analysis. In particular, the analytical solutions have been developed for three basic modes of fracture: Mode I opening mode Mode II in-plane shear mode Mode III out-of-plane shear mode 7 7 7
Linear elastic fracture mechanics The leading terms of analytical solutions for the crack tip stress fields are typically expressed through the product of a radial and circumferential term. For example, the stress field for Mode I plane stress loading reads x y a r a r cos cos 1 sin sin 1 sin sin 3 3 xy a r cos sin cos 3 http://www.fracturemechanics.org 8 8 8
Linear elastic fracture mechanics y a r cos 1 sin sin 3 Note that the governing terms all exhibit a singularity at the crack tip, lim r0 ij crack r / a 9 9 9
Stress Intensity Factor The limit r0 K I : lim r y 0 is called stress intensity factor. It characterizes the magnitude of the stresses at the crack tip. An analog factor can be defined for Mode II and Mode III fracture. In the above example, we have y[ 0] a r and thus K I a The expressions of the stress intensity factor for different crack shapes and loading conditions can be found in many textbooks. 10 10 10
Fracture toughness 151-0735: Dynamic behavior of materials and structures In linear elastic fracture mechanics, it is assumed that a crack propagates if the stress intensity factor reaches a critical value, KI K c The critical value K C for Mode I fracture under plane strain conditions is called fracture toughness. Its units are MPa m. It is considered as a material property which is measured using Single Edge Notch Bend (SENB) or Compact Tension (CT) specimens, see ASTM E-399 standard. 11 11 11
K-dominance 151-0735: Dynamic behavior of materials and structures The stress intensity factor characterizes only the leading term of the stress field near the crack tip. The exact solution for the neartip fields includes also non-singular higher order terms K ij[, r] fij[ ] r higher order terms The annular region within which the singular terms (socalled K-fields) dominate is described by the radius r K of the zone of K-dominance. crack r / a exact K-field 1 1 1
Small scale yielding condition The K-fields can still be meaningful even if the real mechanical system is different from that assumed in the theoretical analysis. Examples include situations where the crack is not sharp; the material deforms plastically micro-cracks are present near the crack tip The condition of applicability of linear elastic fracture mechanics is that the radius r p of the zone of inelastic deformation at the crack tip must be well confined inside the region of K-dominance. rp r K 13 13 13
An alternative to fracture mechanics The classical fracture mechanics approach is based on the assumption that failure is the outcome of the growth of a preexisting crack (and that all materials contain flaws). An alternative approach consists of assuming that a solid is initially crack-free. Phenomenological fracture criteria in terms of macroscopic stresses and strains are then often employed to predict the onset of fracture. A simple example of a phenomenological criterion for brittle solids is to assume that fracture initiates when the maximum principal stress exceeds a critical value, I crit 14 14 14
Ductile Fracture 15 15 15
Fracture in Automotive Applications Shear induced fracture (Courtesy of ThyssenKrupp) FLD Bending under tension (Courtesy of US Steel) Fractures on tight radii during stamping cannot be predicted by Forming Limit Diagram (FLD) Usually termed as shear fracture, presents little necking, shows slant fracture 16 16 16
Interrupted tension experiments Flat notched tensile specimens (1.4mm initial thickness) 0mm 500mm 17 17 17
RD RD Th Th 50mm 18 18 18 18
50mm RD Th 19 19 19 19 19
Surface versus Cross-section View Signature of voids on fracture surface! BUT: Almost no voids just below fracture surface! 0 0 0
Failure Mechanism Summary e=0. 1 e~0.9 3 1. Onset of necking 4 e>1.0. Void volume fraction increases (more nucleation) 3. Shear localization 4. Void sheet failure Define strain to fracture as strain at the onset of shear localization 1 1 1
A Think Model of the Ductile Fracture Process 1 3 4 5 6 7 initial porosity growth & nucleation primary localization growth & nucleation secondary localization nucleation & growth final fracture
Definition of strain to fracture 1 3 4 5 6 7 initial porosity growth & nucleation primary localization growth & nucleation secondary localization nucleation & growth final fracture Strain to fracture = macroscopic equiv. plastic strain at instant of first localization RVE 3 3 3
Tomographic observations xxx aluminum 4 4 4
Void evolution in a plastic solid. void evolution depends on stress state 5 5 5
Decomposition of the Stress Tensor HYDROSTATIC PART (average stress) DEVIATORIC PART (differences among stresses) σ III (σ III σ m ) σ I σ II σ m σ m = + σ m (σ I σ m ) (σ II σ m ) σ m = σ I + σ II + σ III 3 6 6 6
Effect of Stress State on Void Evolution HYDROSTATIC PART controls void growth is characterized by: σ m STRESS TRIAXIALITY η = σ m തσ DEVIATORIC PART controls shape change is characterized by: LODE PARAMETER 7 7 7
Definition of Lode Parameter τ σ N σ III σ II σ I σ II σ N Maximum shear stress (radius of biggest circle): Normal stress on plane of max. shear (center of biggest circle) Position of the intermediate principal stress: τ = σ I σ III σ N = σ I + σ III L = σ II σ N τ LODE PARAMETER 8 8 8
Which states have the same Lode parameter? Uniaxial tension σ σ σ m = σ/3 η = 1/3 തσ = σ σ II = σ III = 0 L = 1 Plane strain tension σ σ/ σ m = σ/ η = 1/ 3 തσ = 3/σ σ II = (σ I + σ III )/ L = 0 Pure shear σ/ σ/ σ m = 0 η = 0 തσ = 3/σ σ II = (σ I + σ III )/ L = 0 9 9 9
Lode angle parameter Stress triaxiality: m III s III plane I II III Normalized third stress invariant 7 J 3 3 Lode angle parameter 1 arccos( ) I II s I +1 0-1 s II 30 30 30
Lode angle parameter Lode parameter (Lode, 196) II I L I III III III s III plane I II III s II Lode angle parameter 1 arccos( ) L I II s I +1 0-1 1 axisymmetric tension 0 generalized shear 1 axisymmetric compression III II I III II I III II I 31 31 31
Plane stress states For isotropic materials, the stress tensor is fully characterized by three stress tensor invariants, { I1, J, J 3} or { I, II, III} while the stress state is characterized by the two dimensionless ratios of the invariants, e.g. { I1 / J, J 3 / J 3/ } or {, } or /, / } { II I III I with I 1 3 3 J and 3 3 1 arccos 3 3/ J J 3 3 3
Plane stress states Under plane stress conditions, one principal stress is zero. The stress state may thus be characterized by the ratio of the two nonzero principal stresses. As a result, the stress triaxiality and the Lode angle parameter are no longer independent for plane stress, i.e. we have a functional relationship [] axisymmetric tension (1 Biaxial comp. ( I 0 Tensioncompression ( II 0 generalized shear (0 Biaxial tension ( III 0 axisymmetric compression (1 33 33 33
Unit Cell with Central Void 151-0735: Dynamic behavior of materials and structures Results from Localization Analysis Stresses on Plane of Localization Linear Mohr-Coulomb approximation 34 34 34
Hosford-Coulomb Ductile Fracture Model Principal stress space,, } { I II III τ + c(σ I + σ III ) = b തσ Hf Hosford- Mohr-Coulomb Coordinate transformation Haigh-Westergaard Mixed strain-stress space {,, } space,, e } f [, ] f Isotropic hardening law k[ e p ] e f 1 k { p [, ] f f e f 35 35 35
Hosford-Coulomb Ductile Fracture Model General form von Mises equivalent plastic strain to fracture e e [,, a, b, c] f f e f Stress triaxiality Lode angle parameter 3 material parameters Detailed expressions e g f HC b g 1 c [, ] HC 1 n 1 1 a 1 a 1 a f f f f f f a c f f I II II III I III I III f I [ ] cos (1 ) 3 6 f II [ ] cos (3 ) 3 6 [ ] cos (1 ) 3 6 3 36 36 36 f
Hosford-Coulomb Ductile Fracture Model Influence of parameter b a=1.3 c=0.05 b=0.5 b=0.4 b=0.3 b=0. b = strain to fracture for uniaxial tension (or equi-biaxial tension) 37 37 37
Hosford-Coulomb Ductile Fracture Model Influence of parameter a c 0.1 a a a 1.5 1. Can easily adjust the depth of the plane strain valley Compare: Mohr-Coulomb a 1 c 0. c 0.35 a 0.8 c 0 c 0.1 a 1 38 38 38
Hosford-Coulomb Ductile Fracture Model Influence of parameter c c=0.1 c=0. a=1.3 n=0.1 c=0.05 c=0 39 39 39
Hosford-Coulomb Ductile Fracture Model Rapid calibration guideline Step I: Identify b Step II: Identify a Step III: Identify c a= c=0 b=0.4 c=0 b=0.4 a=1.15 b=0.4 a=1.15 c=0.14 40 40 40
Hosford-Coulomb Ductile Fracture Model Excel program for calibration 41 41 41
Hosford-Coulomb Ductile Fracture Model 3D View e f D View e f plane stress plane stress heart of the model: e e [, ] f f 4 4 4
Damage Accumulation Define damage indicator e f e [, ] f D de e [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial tension VIDEO 43 43 43
Damage Accumulation Define damage indicator e f e [, ] f D de e [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial compression followed by tension VIDEO 44 44 44
Damage Accumulation Define damage indicator e f e [, ] f D de e [, ] f p D D 0 1 (initial) (fracture) Example: uniaxial compression followed by tension Non-linear loading path effect! 45 45 45
Reading Materials for Lecture #7 S. Suresh, Fatigue of Materials, Cambridge University Press, 1998. and S.J. Marcadet, http://www.sciencedirect.com/science/article/pii/s000768315000700 46 46 46