Planes in Materials. Demirkan Coker Oklahoma State University. March 27, Department of Aerospace Engineering

Transcription

1 Dynamic Shear Failure of Weak Planes in Materials Demirkan Coker Oklahoma State University March 27, 2009 Middle East Technical University, Ankara, Turkey Department of Aerospace Engineering

2 Outline 1. Examples of dynamic failure along weak planes 2. Fracture: Shear failure of coherent interfaces 1. Dynamic fracture experiments 2. How fast can cracks propagate p in materials with weak planes? 3. Friction: Shear failure of incoherent interfaces 1. Rate State friction laws and the finite element model 2. What are the frictional sliding modes? 4. Summary

3 Dynamic fracture and friction: Shear failure along weak planes or interfaces Coherent interface: Fracture Incoherent interface: Friction σ o V tip (~km/s) (km/s) V tip (~km/s) V i V impact (~10 m/s) σ o V tip (~km/s) propagation velocity of discontinuity tip V impact Driving ii speed (e.g. projectile impact)

4 Aircraft Hardening (FAA/Boeing) Damage sustained by aircraft fuselage during explosive loading experiment. Dynamic crack initiation i i i and growth criteria in ductile metals. Formulation of local/global methodology to predict dynamic crack initiation and growth from pre existing fatigue cracks thereby quantify the susceptibility of fuselage structures to global dynamic loading Evaluation of existing and future structural design concepts for their resistance to internal explosive loading (aircraft hardening).

5 Composite Fan Blades (LANL/GE/Boeing) Five foot long composite fan blades of the GE 90 engine used in Boeing 777. Incorporation of dynamic fracture criteria and dynamic crack growth toughness values into elaborate 3 D numerical codes Codes utilized to model composite fan blade bird impact test for FAA certification.

6 Navy composite hull structures Blast or Shock Response Impact Response Fatigue High strain rate effects and properties of thermoset composites Fracture mechanics (joints, strain energy release rates, ) Material failure models/complex stress states DYNAMIC DEFORMATION & FAILURE OF COMPOSITE LAMINATES Cracks running along a weak plane in a multi layered material system under dynamic loading. Model lsystem Unidirectional Graphite/Epoxy composite laminates

7 Interface Failure in Engineering Site of shear dominated failure Lightweight Tomahawk MissileCapsule Steel/S Glass Composite Joint Co molded Hybrid FRP Steel Joint The integrity of structures are often limited by failure at interfaces.

8 San Andreas fault as an example of crack growth along a weak plane

9 Part I. Fracture Crack Growth V tip (~km/s) V impact (~10 m/s)

10 Modes of crack growth Mode-I I(Opening) Mode-II (Shear) Crack growth in homogeneous materials can only occur by Mode I In homogeneous materials, Mode II cracks will change direction such that crack tip locally becomes mode I. To grow Mode II cracks, we need a weak plane that will trap it and force it to grow an interface.

11 Stresses near a crack tip

12 LINEAR ELASTIC FRACTURE MECHANICS Stationary and Growing cracks Stationary Crack: Stress field: K σ ij = I, II 2πr f I, II ij ( θ ) K I,II : Stress Intensity Factor Failure Criterion: Elasticity K ( Q, a) K ( Material) I = Ic Experiments Energy release rate: G I = 2 K I E Growing Cracks: Equations for slow crack growth is the same except a velocity dependence is added.

13 Dynamic crack growth criterion d I d ( t ) = K I ( Q ( t ), a, v ) = K D ( v ) for t t f K > K D (v). Dynamic Crack Growth Toughness (Crack tip driving force) Depends D d on local l strain rate through hcrack tip velocity only. σ σ Steady-state tt singular stress field fildfor a subsonically growing crack kin orthotropic materials: (Liu, Rosakis, Stout and Ellis (1996)) d K ( t) A1 ( b, v) B1 ( b, v) x = 1, x2, v, t) I ij ij cos( θ1 / 2) cos( / 2) 2 2π 1/ 2 1/ 2 r1 r2 d K ( t) A2 ( b, v) B2( b, v) x = 1, x2, v, t) I ij ij cos( θ1 / 2) cos( / 2) 2 2π 1/ 2 1/ 2 r1 r2 11( θ2 22( θ2 Scaled Coordinates r α = 2 1 x θ = tan θ α μ = 2 2 α 2 + μ x μ x α x1 (, v) 1 2 α μ α b ij

14 Question Can a crack travel faster than any of the characteristic waves in the material? Initial Answer: NO!

15 Mach wave for a disturbance traveling faster than the characteristic speed FLUIDS Subsonic Supersonic 0 C L η 1 β β V η 2 V = C S * Sinβ SOLIDS Sub Shear Intersonic Supersonic 0 R S 2 c s C C CL

16 Fiber reinforced unidirectional graphite/epoxy composite laminate Fiber Direction x 2 x 3 x 1 Homogenized Elastic Properties Characteristic Wave Speeds E 1 80 GPa c l // 7500 m/s E GPa c l 2700 m/s ν c s 1560 m/s μ 12 36GPa 3.6 c R 1548 m/s 50 μm

17 Experimental set up for dynamic fracture testing using coherent gradient sensing (CGS) optical technique Gas Gun Grating 1 Grating 2 Lens Aperture Collimated Laser beam (50 mm diameter) Rotating ti Mirror type high h speed camera 2x10 6 frames/sec

18 Mode I Opening Crack CGS fringe pattern Surface Deformation h ( b σ b σ ) u 3 =

19 EXPERIMENTAL SET UP FOR DYNAMIC FRACTURE TEST USING OPTICAL TECHNIQUE OF CGS Camera Gratings Specimen Gas Gun IR Camera

20 Mode I (Opening) crack propagation 5 mm -0.6 μs 1.2 μs

21 Crack tip speeds for dynamic mode I crack propagation in Gr/Ep unidirectional composites Homogeneous material with a weak plane (Washabaugh & Knauss, 1994) HOMALITE HOMALITE MODE-I C R C s C l

22 Experimental CGS Interferogram of a fast moving shear crack

23 Shear dominated intersonic crack growth in a unidirectional graphite epoxy composite laminate

24 Intersonic shear crack propagation in unidirectional composites C rack tip speed (m/s s) c l v c c R Time (μs) Fiber Direction 50 mm Field of View

25 Crack tip speeds for mode I and mode II dynamic crack propagation in unidirectional composites 9000 (m/s) c l pl- σ v c tip Speed Crack Mode-II 2000 c R =099c 0.99 s 1000 Mode-I Crack Extension (mm)

26 Crack tip stress singularities for intersonically growing cracks in orthotropic materials Huang, Wang, Liu, Rosakis; Mode-I q q I I (v) E1 vc = cs = 6580m / s μ ( + ν ) AI σαβ = f (, /, ) q ( v) αβ θ v cs cij I r Energy needed for Mode-I fracture: G I = - for c s < v < c l q II q II (v) v c / c s = σ αβ = Mode-II A II f (, /, q ( v) αβ θ v cs cij r II Energy needed for Mode-II fracture: G II = 0 for c s <v<c < l G II = finite for v = v c only ) V/c v/c s s Stable and unstable intersonic crack growth is possible under shear (mode-ii) conditions only. Stable intersonic growth is possibly at v=v c.

27 Steady state crack tip speed for intersonic shear crack growth Orthotropic Materials: v c = 2 c11c22 c12 E1 = ρ ( c12 + c22 ) ρ ( 1+ ν ( 12 = 6600 m/s ) Isotropic Materials (Freund, 1979): E 2μ v = = = 2 ρ ( 1+ ν ) ρ c C S

28 Fracture along a weak plane: experiments Gas Gun 125 mm Collimated Laser beam (50 mm diameter) Homalite φ Projectile Lens 150 mm Circular Polarizer 150 mm Specimen Circular Polarizer 125 mm 50φ 150 mm Homalite 100 Rotating Mirror type high-speed camera 150 mm

29 Intersonic shear crack growth in a homogeneous materials with a weak plane Isochromatic Fringe Patterns Homalite Homalite 28 m/s Homalite Homalite Rosakis, Samudrala, Coker; SCIENCE, 1999

30 Intersonic Mode II Crack Propagation 125 mm Homalite 150 mm 50Φ 150 mm Experiment Rosakis, Samudrala & Coker 99 Theory Freund 79 Homalite GIMP Daphalapurkar, Lu, Coker, Komanduri 07 MD Simulations Abraham 04

31 Field evidence of intersonic rupture during the 1999 Izmit and Duzce earthquakes in Turkey M. Bouchon, M. Bouin, H. Karabulet, M. Toksöz, M. Dietrich and A. Rosakis, Geophysical Research Letters, 2001 V = C R V = 2 C S = 4.9 km/s S

32 Question Can a crack travel faster than any of the characteristic waves in the material? Initial Answer: Depends! For Mode I cracks: NO For Mode II cracks: YES.

33 Part II. Friction "God made solids but surfaces were the work of the devil" Wolfgang Pauli Tribological investigations of LIGA microstructures t T. Bieger, U. Wallrabe

34 Frictional sliding: Homogeneous and Heterogeneous Slip Davis and Reynolds, Structural Geology of Rocks and Regions

35 Earthquakes San Andreas Fault in California Earthquakes can be viewed as frictional sliding of tectonic plates at two time scales: Stick slip at year time scale and dynamic frictional sliding at 100 seconds time scale. (Heaton, 1990) Years Seconds

36 Frictional sliding in composite materials Frictionalsliding is animportanttoughening toughening mechanism during fiber pull out in composite materials (Tsai & Kim, 1996)

37 Effect of history and sliding speed on friction μ o μ s Amontons Coulomb Law τ = μ σ μ d V 37

38 Modeling Continuum mechanics Elastic material properties Existence of an interface or weak plane Mathematically straight interface Cohesive zone models dlused for the interface/weak plane Cohesive law for fracture simulations Rate and state dependent friction law for friction simulations No contact model is used

39 Isochromatic Fringe Patterns during Frictional Sliding showing shear Mach Waves Σ o =6 MPa, V imp =2 m/s

40 Isochromatic Fringe Patterns during Frictional Sliding showing shear Mach Waves Periodic Slip Pulses Σ o =10 MPa, V imp =20 m/s

41 Molecular Dynamic Simulations of Sliding J. Ma, H. Lu, B. Wang, R. Hornung, A. Wissink, and R. Komanduri, 2006 (a) t = 56 (b) t = 64 (c) t = 72

42 Summary Sliding and Fracture of Interfaces show similar characteristics Discontinuity tip travels at speeds faster than the shear wave speed Shear Mach Waves are observed through optical techniques Frictional sliding in the form of multiple l self healing l pulses traveling at intersonic speeds are observed These characteristics are observed at different length scales from the atomic to tectonic.