Transition from Quasi-Static to Dynamic Fracture Carlos G. Dvila - PowerPoint PPT Presentation

https://ntrs.nasa.gov/search.jsp?R=20160006280 2018-07-31T20:38:53+00:00Z Damage Instability and Transition from Quasi-Static to Dynamic Fracture Carlos G. Dávila Structural Mechanics & Concepts Branch NASA Langley Research Center Hampton, VA ICCST/10 USA Instituto Superior Técnico Lisbon, Portugal, 2-4 September 2015 1

Quasi-Static Loading and Rupture Loading Phases: • 0) to A) – Quasi-static (QS) loading • A) to B) – Dynamic response Force A) No QS solution exists A) B) 0) Displacement F, d Snapback behavior: B) • More strain energy available than necessary for fracture 2

Failure Criteria and Material Degradation Progressive Failure Analysis Stress E Elastic property Residual=E/100 e/e 0 1 1 Failure criterion Strain Benefits • Simplicity (no programming needed) • Convergence of equilibrium iterations Drawbacks • Mesh dependence • Dependence on load increment • Ad-hoc property degradation • Large strains can cause reloading • Errors due to improper load redistributions 3

Failure Criteria and Material Degradation Progressive Failure Analysis Stress E Elastic property Residual=E/100 e/e 0 1 1 Failure criterion Strain Progressive Damage Analysis – Regularized Softening Laws Increasing l elem Stress E Increasing l elem Elastic property e/e 0 1 1 Failure criterion Strain 4

Strength-Dominated Failure      K , c   i F,         ( 1 ) , d K  i f K G c (1-d)K L+ + F Before damage After damage 2 G   c Unstable Stable    F       c F A EA F A EA  0 2 EG E E      c L L  2 K K c   F 2 EG For stable fracture under  control:   c L 0    2 c For “long” beams, the response is unstable, dynamic, and independent of Gc 5

Fracture-Dominated Failure G    2   G(  , a 0 ) 2 Slope : a  E G E 2 a R = G Ic  max  unstable  a a 0 Crack propagates unstably once driving force G(  , a 0 ) reaches G Ic 6

Fracture-Dominated Failure  G c G G(  , a 0 ) G R (  a) 2( a+  a)   2 a  G E 2 a  max G Init  init unstable  a  a stable a 0 Crack propagates stably when driving force G(  , a 0 ) > G Init   G G G Unstable propagation initiates at Init c 7

Mechanics of Crack Arrest  G R = G Ic  max unstable arrest  a a 0 a arrest Crack arrest due to decreasing G 8

Mechanics of Crack Arrest  G R rate sensitive arrest?  max unstable  a a 0 Large strain rates often result in lower fracture toughness and delayed arrest 9

Griffith Criterion and Stability Griffith growth criterion Stability of equilibrium propagation Wimmer & Pettermann J of Comp. Mater, 2009 10

Stability of Propagation with Multiple Crack Tips P, v P, v Curved laminate with through-the-width delamination 0/90/…/90/0 15 plies total a 2 2.25 mm a 1 Force, N a 1 , mm Wimmer & Pettermann J of Comp. Mater, 2009 a 2 , mm Displacement, mm 11

Scaling: The Effect of Structure Size on Strength Scaling from test coupon to structure Structural size, in. log  n Yield or Strength Criteria Scaling Laws (Z. Bažant) Normal testing log D 12

Cohesive Laws Two material properties: • G c Fracture toughness •  c Strength Final debond length t  0 Yield or Strength Criteria log  n Bilinear Traction-Displacement Law       c ( ) d G log D  c c 0 Characteristic Length: K p E G G c   c l p unloading reloading  2 c  c  0 13

Crack Length and Process Zone F,  Force, F E G G c = constant   c l  p a 0 2 c Brittle: LEFM error  a 100 l 0 p Quasi- brittle Quasi-brittle:   100 l a 5 l p 0 p Ductile:  Applied displacement,  5 l a p 0 14

Crack Length and Process Zone F,  Force, F E G LEFM error   c l  p a 0 2 c Short crack  Ductile Brittle: LEFM error  a 100 l 0 p Quasi- brittle Quasi-brittle: LEFM error   100 l a 5 l p 0 p Long crack  Brittle Ductile:  Applied displacement,  5 l a p 0 15

Strength and Process Zone F,  Force, F E G G c = constant   c l  p a 0 2 c Decreasing  c LEFM error As the strength  c decreases, 1. the length l p of the process zone increases a 0 2. the error of the Linear a 0 +l p LEFM solution increases Applied displacement,  16

Size Effect and Material Softening Laws Damage Evolution Laws: Two material properties: •  c Strength Each damage mode has its • G c Fracture toughness own softening response Strength Tests  c Fracture Tests E Material length scale E G G c / l   c l  c 2 e c 17

Progressive Damage Analysis (Maimí/Camanho 2007) Damage Modes:   F F Tension Compression LaRC04 Criteria • In-situ matrix strength prediction • Advanced fiber kinking criterion • Prediction of angle of fracture (compression) • Criteria used as activation functions within   framework of continuum damage mechanics y y M M (CDM)     1    d 1 exp A 1 f i i i f Damage Evolution: i f i : LaRC04 failure criteria as activation functions Thermodynamically-consistent material      y y s degradation takes into account energy i F ; F ; M ; M ; M release rate and element size for each mode  * 2 Bazant Crack Band Theory: 2 l X  i A  i * 2 2 E G l X i i i 2 E G Critical (maximum) finite l  * i i element size: 2 X i e 18

Predicting Scale Effects with Continuum Damage Models Prediction of size effects in notched composites • Stress-based criteria predict no size effect • CDM damage model predicts scale effects w/out calibration (P. Camanho, 2007) Hexcel IM7/8552 [90/0/45/-45] 3s CFRP laminate log (Strength, MPa) Experimental (mean) Analysis log (diameter, mm) 19

Process Zone and Scale Effect in Open Hole Tension Cohesive law Stress distribution log (Strength, MPa) (P. Camanho, 2007) Scale effect is due to relative size of process zone log (diameter, mm) 20

Length of the Process Zone (Elastic Bulk Material) Short Tensile Test E G   Lexan Polycarbonate c l 0 . 6 3 . 4 mm  c 2h 2 CT Sun, 2a o c Cohesive Purdue U elements Symmetry A D h/a o = 1 h  l 4 . 7 mm pz Symmetry a o B E Maximum Load C F 21

Cohesive Laws - Prediction of Scale Effects Observations: • The use of cohesive laws to predict the • LEFM overpredicts tests for h/a<1 fracture in complex stress fields is explored 6000 • The bulk material is modeled as either Fmax, N elastic or elastic-plastic LEFM 5000 4000 Lexan Plexiglass tensile specimens (CT Sun) 3000 Test (CT Sun) 2000 2h 1000 2a Cohesive 0 h/a=4 h/a=2 h/a=1 h/a=0. 5 h/a=0.25 0 1 2 3 4 h/a h/a=1 (short process zone) h/a = 0.25 (long process zone)  l Width cz  l 4 . 7 mm. cz 22

Study of size effect: measuring the R-curve Double-notched compression specimens By FEM analysis From test 2w  0    a 2 a a 0 . 5     u G a 0 a 0 w   eff w E   2 a  (Similar to ) G E   2 , MPa -2 *10 -5 Catalanotti, et al., Comp A , 2014 G u 3.2 G R (  a) 2.8 2.4 2.0 a 1.6 Increasing w 1.2 0.8 0.4 0.0  a  a stable 2 3 4 5 6 7 8 9 a 0 a 0 , mm 23

Characterization of Through-Crack Cohesive Law Compact Tension (CT) Specimen Characterization Procedure: 𝐻 𝑆 = 𝑄 2 𝜖𝐷 𝑄 1. Measure R-curve from CT 0 ° Thin 2𝑢 𝜖𝑏 test multidirectional 1.18𝑋 laminate 2. Assuming a trilinear 𝑜 𝑡 cohesive law, fit analytical 𝑗 − 𝐻 𝑆 𝑗 𝜃 = 𝐾 fit R-curve to the measured 𝑏 0 ∆𝑏 𝑗 R-curve 𝑧 𝑋 𝑦 3. Obtain the cohesive law 𝜏 𝜀 = 𝜖𝐾 fit by differentiating the Experimental setup 𝜖𝜀 analytical R-curve 𝜏 Specimen Bergan, 2014 Clevis 𝜏 𝑑 𝑧 𝐿 Trilinear cohesive law 𝐻 𝑑 𝑦 𝑨 Anti- buckling 𝜀 guide 24

Size-Dependence of R-Curve Large (b) ‘L’ Small (a) ‘S’ G R , N/mm G R , N/mm 25 mm Displacements  a eff , mm  a eff , mm measured through digital image correlation (DIC) G R , N/mm Bergan, 2014 Plotting the R-curve as a function of the notch displacement removes the size-dependency 𝜀 , mm 25

R-Curve Effect in Fiber Fracture 𝜀 𝑑 𝐾 𝑆 = 𝜏 𝜀 𝑒𝜀 0 G R , N/mm Curve fit assuming bilinear  P,  Cohesive elements w/ characterized cohesive law 𝜀 , mm ∆𝑏 𝑏 0 6. 800 600 4.  , MPa P , kN Cohesive response 𝑄 [kN] for fiber failure 400 Bergan, 2014 [MPa] 2. 200 Test Analysis 0. 0 0.0 1.0 2.0 3.0 4.0 0.0 0.5 1.0 1.5 2.0 𝜀 , mm 𝜀 , mm 26 [mm]

Mode II-Dominated Adhesive Fracture Teflon Adhesive thickness: 0.13 mm Tip of adhesive 27

ENF J-Integral from DIC

MMB Test - Analysis Results Mixed mode bending (MMB) test fixture Applied load, N Displacement, mm Nominally identical bonded MMB specimens sometimes fail in quasi-static mode and others dynamically. Why? 29