Linear Elastic Fracture Mechanics
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1 Measure what is measurable, and make measurable what is not so. - Galileo GALILEI Linear Elastic Fracture Mechanics Krishnaswamy Ravi-Chandar Lecture presented at the University of Pierre and Marie Curie April 30, 2014 Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering and Engineering Mechanics
2 The plan Introduction to fracture mechanics April 30, 2014 Quasi-static instability problems (JB Leblond) May 6, 2014 Introduction to dynamic fracture mechanics May 15, 2014 Dynamic instabilities during fast fracture May 21,
3 How strong is a solid? 1. An atomic point of view Energy, E Force, F Repulsion Attraction Tension Compression F F max sin 2 ( a a ) 0 a 0 Interatomic distance, a 2 a0 ( a a0) max sin, a 0 a 0 Interatomic distance, a E d 2a 2a 0 0 max cos d a E 0 max max 2a0 Material Modulus GPa Strength GPa ~ E Strength/Modulus Steels Glass Carbon fibers Glass fibers Macroscopic strength is significantly smaller than the theoretical strength
4 How strong is a solid? 2. The role of defects (,0) 1 2 a 22 a a ( a,0) 2 where b / a b 1 f a max constant 2 ~ 1E 10 m a~ 1E 6 m 2b 2a x 2 f ~ x 1 max 1 Griffith s experiments 1. Used experiments on glass tubes and glass bulbs loaded under internal pressure to show that σ a was constant 2. Manufactured fresh glass fibers with diameters in the range of 1 mm to 3 microns to show that small fibers had strength of about 11 GPa 4
5 The continuum view of fracture L p Process of fracture can be cleavage, intergranular/ transgranular fracture (polycrystalline materials), cavitation (ductile metals), disentanglement (polymers), microcracking (glasses, ), fiber breakage, Details of processes within L p are not important; only the total energy needed for the fracture process is assumed to play a role in the development of the fracture L p is small small-scale process zone what does this mean? 5
6 The energy balance a continuum view Total energy of the system: where define U s W R R Ga ( ) du da U s R E U s potential energy of the body strain energy stored in the body d Energy Release Rate da a At equilibrium, E 0 fracture resistance R Fracture criterion G a c surface energy (or fracture energy) W U R Stable if E a c 0 R work done by the external forces on the body 2a x 2 x 1 6
7 Remarks 1: R Fracture criterion G a c a c is the equilibrium crack length (reversible) Fracture resistance R(a) includes the effect of all dissipative fracture processes and is typically calibrated from experiments. For Linearly Elastic Fracture Mechanics (LEFM), the region outside L p must exhibit linear elastic behavior, but this is not a general requirement. Other than L p being small, there is no length scale here! The theory works at length scales from the atomic to the tectonic. Need methods to calculate G(a) for specific crack problems 7
8 Remarks 2: R Fracture criterion G a c R varies over several orders of magnitude True surface energy is ~ O(1) J/m 2 Glasses and ceramics~ 10 J/m 2 Polymers ~1 kj/m 2 Metals ~ 100 kj/m 2 Differences arise due to different mechanisms of deformation and failure Must be determined through calibration experiments, such as the pioneering work of Obreimoff (1930) 8
9 9 C Calculation of G(a) M compliance of the loading system CM 0 fixed displacement C fixed load M C( a) / P compliance of the specimen T total displacement (fixed) C P C( a) C P T M M T 1 1 Strain Energy: U R CMP C( a) P 2 2 d 1 2 G( a) P C( a) da ( ) 2 P C a R equilibrium crack length 2 2 P, T C M P, a Stable if E a c 0 C a C a C a C ( ) ( ) ( ) M Stability depends on C M! Can cause stick-slip and other unstable crack growth effects
10 Example 1: G(a) for a double cantilever beam P, T C M P, P, d 2d a a 8a 24a C( a) ; C( a) Ed Ed G a R a P c unstable ER 12d 3 C a 2 C a C a C ( ) ( ) ( ) M 4a 12a C( a) ; C( a) Ed Ed Ed G ac R a 8R stable 10
11 Example 2: G(a) for an infinite strip specimen G a E 2 1 h a 2h stable 11
12 Fracture mechanics the global point of view The global point of view works quite well for a number of problems. It circumvents detailed calculation of stress/strain states in the vicinity of the crack. Has been applied successfully in a number of structural applications Difficulty in calculating the compliance, C(a) Difficulty in calibrating the fracture energy, R Difficulty in selecting/identifying fracture path Modern numerical simulations incorporate the energy approach through the phase-field methodology. 12
13 Fracture mechanics why a local point of view? Provides a systematic way of calculating G(a) Provides a method for analyzing different loading symmetries Local approach based on stress and strain fields permits decoupling of path selection from failure characterization 13
14 Loading symmetries Mode I or Opening mode Mode II or In-plane shear Mode III or Anti-plane shear 14
15 Linear elasticity y( x) x u( x) 1 ε( x) ( ) 2 u u σ( x) 1 2ε kk σ f 0 T - boundary conditions u( x) u*( x) on sx ( ) σ( x) n s*( x) on Anti-plane shear u 0; u u ( x, x ) 2 u R R Plane strain u u ( x, x ); u x ( x1, x2 ) u, u, 2 33 const; 3 0 ( x1, x2) 2 33( x1, x2) 3 ( x1, x2) 0 If ; ; 11,22 22,11 12,12 4 then 0 15
16 The J-integral u J U Rn1 n ds x1 x 2 n 1 1. The integral is zero if contour is closed inside the body without enclosing singularities 2. If the contour goes from below to above the crack surface as indicated, the integral is independent of the path R x 1 3. This integral can be interpreted in terms of the energy release rate: J G( a) d / da 4. Path independence implies that and therefore, 1/ 2 1/ 2 σ ~ r σ ; ε ~ r ε σ : ε ~ r 1 16
17 Anti-plane shear K III ( ) 3 h3 2 r K lim 2 r r,0 III r0 32 mode III stress intensity factor 1. Anti-plane shear can exist only in specimens without bounding planes in axisymmetric geometries or infinitely thick plates Mode III or Anti-plane shear 2. Free surfaces in finite thickness plates introduce coupling to mode II 3. Connection to J and G obtained by using the path independent integral: 4. Failure criterion for mode III is still being debated (more on this later!) J G( a) K III 17
18 In-plane loading symmetries r0 Mode I or Opening mode mode I stress intensity factor K lim 2 r r,0 I 22 Mode II or In-plane shear mode II stress intensity factor K lim 2 r r,0 II r0 12 KI I r, F T 2 r KII II F 2 r 1 1 T represents nonsingular stress and plays a role in crack path stability 18
19 Calculation of the stress intensity factors Elastic boundary value problem to be solved Numerous examples exist in handbooks Robust numerical methods based on FEM, BEM, available Considered a solved problem: Given a geometry, loading, etc, there is no difficulty in determining K I, K II, and K III. K I K I load, crack length, geometry P, Example: Single-edge-notched specimen K I P a a f tw W P a g displacement twe W a w 19
20 Fracture criterion for in-plane loading 1. Connection to J and G obtained by using the path independent integral: J G( a) KI KII E 2. For pure mode I loading, the crack grows along the line of symmetry and the energy based fracture criterion can be restated in terms of the stress intensity factor K I K IC ER Residual strength (load carrying capacity) can be determined for structural applications 4. Stability of structures can be evaluated 20
21 Snap-back instability K I P a a f tw W P a g displacement twe W P, a 1 a w a a 2 1 Crack is unstable in load control and displacement control 21
22 Fracture criterion for mixed mode I + II crack 1. For combined modes I and II, we need other criterion (criteria?) that dictates the crack path selection a. Maximize energy release rate b. Maximum hoop stress c. Principle of local symmetry: K K, K 0 Crack tilting I IC II 2. Maximum hoop stress criterion is simplest to use 3. Experimental scatter is large and unable to discriminate between the different criteria Principle of Local Symmetry: Goldstein and Salganik, Int J Fract, Crack kinking
23 Crack path evolution under Mode I + II Yang and Ravi-Chandar, J Mech Phys Solids, 2001 Photograph Courtesy of Dov Bahat Ben Gurion University Tectonofractography, Springer Principle of Local Symmetry works very well for this problem 23
24 Possible fracture criteria for mixed mode I + III Criterion I: Goldstein and Salganik, Int J Fract, 1974 K II f K I 0, K 0 III Criterion II: Lin, Mear and Ravi- Chandar, Int J Fract, 2010 K K, K 0, K I IC II III tan 2 K K III I Hull, Int J Fract, 1995 Cooke and Pollard, J Geophy Res,
25 Mixed mode I + III crack problem Below a threshold of K III /K I, the crack front twists Above the threshold, crack front fragments cr = 3.3 Sommer, Eng Frac Mech,
26 Mixed mode I + III crack problem Knauss, Int J Fract,
27 Summary and plan Energy based method can provide a simple way of analyzing fracture problem (with some residual difficulty regarding the path selection) Stress-intensity factor based method provides an effective way of designing fracture critical structures residual strength diagram 27
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