1 CH-3 Energetic of fracture HUMBERT Laurent Thursday, march 11th 2010

2 FRACTURE MECHANICS PROJECTS Group 1: Experimental project #1 Berglund Niklas Fogelfors Oscar Tunçer Gözde Group 2: Experimental project #2 Farmand Ashtiani Ebrahim Khoushabi Azadeh Group 3: Numerical project #3 Baader Jakob Bernet Adeline Uriarte Amaia RESPONSIBLES Laurent HUMBERTROOM ME C1 365 Samuel STUTZ ROOM ME C1 375 (group 2)

3 SESSIONS S1 Thursday, March 11 th 10h-12h group1 and 3 Friday, March 12 th 10h-12hgroup2 S2 Thursday, March 18 th 10h-12h group1 and 3 Friday, March 19 th 10h-12hgroup2 S3 Thursday, March 25 th 10h-12h group1 and 3 Friday, March 26 th 10h-12hgroup2 S4 Thursday, April 1 st 10h-12h group1 and 3 ? Friday, April 2 nd 10h-12hgroup2Good Friday S5 Thursday, April 15 th 10h-12h group1 and 3 Friday, April 16 th 10h-12hgroup2 S6 Thursday, April 22 nd 10h-12h group1 and 3 Friday, April 23 rd 10h-12hgroup2 S7 Thursday, April 29 th 10h-12h group1 and 3 Friday, April 30 th 10h-12hgroup2 S8 Thursday, May 6 th 10h-12h group1 and 3 Friday, May 7 th 10h-12hgroup2 S9 ? Thursday, May 13 th 10h-12h group1 and 3Ascension Day Friday, May 14 th 10h-12hgroup2 S10 Thursday, May 20 th 10h-12h group1 and 3 Friday, May 21 st 10h-12hgroup2 S11 Thursday, May 27 th 10h-12h groups 1, 2 and 3 Oral presentation Proposed work schedule...

4 3.1 Theoretical strength (recalls) Ideal crystal structure Equilibrium at inter-atomic distance a=a 0 Atomic scale Bond energy where l assumed to be approximately equal to twice the atomic spacing a 0 P = applied force Force-displacement relationship idealized as one half of the period of the sine wave Cohesive force P c Derivative of the potential energy interaction ° Lennard-Jones potential

5 - Lattice deformation: : critical stress (theoretical strength) Approximate Stress-strain relationship: - Dividing P by the number of bonds per unit area, - Assuming small strains (linear elastic), expanding the sin function in Taylor series: Young’s modulus:

6 And by taking, From the definition of the Young’s modulus : with more accurate calculations Introducing the surface energy : Surface energy of a solid = energy it costs to make it 2 new surfaces →

7 In reality : In practice, real materials do not achieve the ideal value Why ? Because all crystals contain defects : Such as vacancies, dislocations, imperfect grain boundaries depending on the environmental conditions

8 3.2 Griffith’s analysis of strength Stresses concentration appears in holes, slots, threads and huge change in section c : characteristic dimension associated with the defect → lower the global strength by magnifying the stress locally

9 Maximum local stress ? : minimum radius of curvature : stress concentration factor (dimensionless !) for tension for torsion and bending S 0 : nominal section Equation inexact but adequate working approximation for many design problems →

10 Examples : Circular hole of radius R in a large plate loaded in tension Local yielding occurs when ! Elliptical cavity (semi-axes a > b) traversing a plate loaded in tension R a b Result given by Inglis (1913)

11 Cracked material Cracks highly concentrate stress ! a

12 The engineering problem of a crack in a structure

13 Non uniform local stress : Cracks are sharp : - previous equation for holes, notches of finite radius of curvature not relevant here ! - their radius at the crack tip are essentially zero rises deeply at the crack tip With the minimum radius at crack tip Assuming failure when remote stress at failure → rough estimate : continuum approach of Inglis not valid at the atomic level

14 Griffith energy balance: Slender ellipse Total energy of the plate : : potential energy = strain energy U - work of external forces F : work required to create two new surfaces Griffith energy balance (equilibrium) : for an increment in the crack area dA B = Through crack in a plate of thickness B model based on a global energy approach

15 For the cracked plate configuration (plane stress conditions), Moreover with a constant thickness B and a through crack of length 2a : surface energy per unit area of the material Crack area = projected area of the crack = 2aB (here) Surface area = two matching surfaces = 4aB

16 Solving for a, equilibrium Energy crack length a

17 Solving for the stress : Type of equilibrium ? → sign of the second derivative of the energy → stable or unstable propagation Always unstable configuration ! fracture stress

18 Griffith approach applicable to other crack configurations : Ex: penny-shaped crack configuration Fracture stress : : Poisson’s ratio E : Young’s modulus a circular crack of radius a Remarks :

19 plastic work per unit area of surface created Large underestimation of the fracture strength for metals : → modified Griffith equation for materials that exhibit some plastic deformation (Irwin and Orowan) account for additional energy dissipation (e.g. dislocation motion) only valid for brittle solids (e.g. glass) → total energy of broken bonds in a unit area

20 General effect of temperature on the fracture energy of structural metals : Generalization for any type of energy dissipation : w f : fracture energy Limited to elastic global behavior (because of P ) and w f assumed to be constant ! → plastic, viscoelastic, viscoplastic effects

21 The force P does no work during a virtual crack increment : Glass wedge Strain energy : calculated by considering the thin layer as a cantilever beam ( of unit thickness, B=1) 3.3 Obreimoff’s experiment

22 Thus, Always stable configuration !

23 Cracked specimen under load : 3.4 Energy analysis – compliance method Potential energy : Total energy : Work F done by the applied forces ? Linear elastic behaviour : U ?

24 Compliance : Strain energy expressed by From the linear law, Clearly, Total energy (with unit thickness B=1) Crack propagation when Sign of Positive : stable propagation Negative : unstable propagation Note that when W s is a linear function of a

25 Energy release rate (ERR): By definition (Irwin 1956), Crack extension occurs when : Stability of the crack growth ? Stable crack propagation (controllable manner) Unstable crack propagation (uncontrollable manner) Ex : For the plate of with a center crack of length 2a Always unstable ! “crack driving force”

26 ERR under constant load : B Thus, Graphical interpretation : O D E

27 Expression with the compliance C : By reporting in the expression of G, one has differencing with respect to a, P constant :  constant :

28 ERR under constant displacement Thus, Becauseand expression of G at constant load recovered ! Graphical interpretation : O A C Difference with load control negligible

29 Generalization : O a a +da P : possible relationship between the load P and the displacement D for a moving crack Identification of the crack driving force G A B C D E Thus, Having measured and knowing the loading curves for neighbouring crack areas, the development of the ERR can be followed

30 Example : Double Cantilever Beam (DCB) Double cantilever beam specimen at fixed load (a) or fixed grips (b) (a) (b)

31 Cracked structure with a finite system compliance : d and G are assumed to depend only of Structure held at a fixed remote displacement Differentiating,(1) Dividing by dA and fixing D, (2) (1) + (2) implies cf Anderson’s book