Numerical Simulation of Fatigue Crack Growth: Cohesive Zone Models vs. XFEM

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1 Numerical Simulation of Fatigue Crack Growth: Cohesive Zone Models vs. XFEM Thomas Siegmund Purdue University 1

2 Funding JOINT CENTER OF EXCELLENCE for ADVANCED MATERIALS, FAA Cooperative Agreement 04-C-AM-PU. Damage Tolerance and Durability of Adhesively Bonded Composite Structures Technical Monitor: Curt Davies Cost share: Purdue University Additional Funding: Catterpillar Inc.

3 Research Accomplishements Compare cohesive zone model approach to XFEM. Implement fatigue crack growth in XFEM Study contact fatigue failure

4 XFEM-ABAQUS extended Finite Element Method Extension of conventional FEM based on the concept of partition of unity; 4

5 u= N ( x) u + H( x) a N u a I I I N I = 1 XFEM-ABAQUS shape function [ ] I I I conventional nodal dispalcement enriched nodal displacements 5

6 Traction-Sparation Law Tn,0 K0 0 Δn = Tt,0 0 K0 Δ t T n,0 Tt,0 max, = 1 σ σ max,0 max,0 T = (1 D) T, T = (1 D) T n n,0 t t,0 D m = Δ m 0 m T dδ φ eff φ T. Siegmund, Δ Purdue 0 m el

7 Example: CZM vs. XFEM Plate with a central hole Remote displacements CZ elements along several radial lines XFEM enrichment throughout

8 Damage Accumulation Rule Damage accumulation starts if a deformation measure, accumulated or current, is greater than a critical magnitude. There exists an endurance limit which is a stress level below which cyclic loading can proceed infinitely without failure. The increment of damage is related to the increment of absolute value of deformation as weighted by the ratio of current load level relative to strength.

9 Damage Accumulation Rules Δ m T eff CZM : dd = C, D = dd δ σmax c f c c σ f..cohesive endurance limit δ Σ.cyclic cohesive length 1 1 c f c c ε p σ p XFEM : dd = C, D = dd ε σmax δ ε Σ Σ K = 0 E0 σ f..cohesive endurance limit ε Σ cyclic reference strain 9

10 Fatigue Damage Model Cyclic damage variable: Effective tractions: Tˆ F Tn = = A (1 D ) (1 D ) 0 0 ( σmax,0 δ0 ) Since: Tˆ = Tˆ, n n D n C = A damaged A Tn = Tn σmax,0(1 DC), δ0 C C Current Cohesive Properties: σ = σ (1 D ), max max,0 φ = χσ δ = χσ (1 D ) δ n C max 0 max,0 C 0 Need to define : D = D D, Δ, Δ,T,T... ( ) C C C n t n t 10

11 Monotonic Loading 1.4E E+07 ELASTIC XFEM Σ 22 [Pa] 1.0E E E E E E E 22 CZM

12 Cyclic Loading: Damage Evolution in Mode I Cyclic Damage [ ] the first three elements of the row of elements emerging from the hole at st element CZM (both integration points shown), st element XFEM, nd element CZM (lagging integration point shown), nd element XFEM, rd element CZM (lagging integration point shown), rd element XFEM. Normalized Time [ ]

13 Cyclic Loading: Mixed Mode

14 Contact Fatigue Multiaxial Fatigue Criteria: Critical Plane Approach E.g.: Findley Criterion: max τ a σ n,max { },max τ + α σ = β a f n f shear stress amplitude on a plane max. normal stress on that plane 14

15 Contact Fatigue Multiaxial Fatigue Criteria: Critical Plane Approach E.g.: Findley Criterion: max τ a σ n,max { },max τ + α σ = β a f n f shear stress amplitude on a plane max. normal stress on that plane 15

16 Example Result Provides location of crack initiation but not: -- number of cycles to failure -- not applicable to variable amplitude loading 16

17 CZM Approach Cohesive Zone Model CE LE CE LE LE CE LE CE LE CE LE CE CE LE LE CE LE CE LE CE Mesh generator available LE CE LE Tie CZ mesh to main model 17

18 Redefine: Effective Traction Common Fatigue Crack Growth ( ) ( ) 2 2 T = T + T n t Contact Fatigue ( η ) ( β ) n 2 2 T = T + T t 0.64 T = Tn + T 3 if T > T = Tn + T 3 if T < 0 n n 2 ( ) t t 2 18

19 P max = N at Failure Contact Radius 19

20 Damage Evolution Subsurface Crack Initiation Site Contact Radius 20

21 Damage Evolution P max = 2800 N Subsurface Crack Initiation Site Contact Radius 21

22 Comparison Damage Accumulation Rate Subsurface Surface Load

23 Summary CZM vs. XFEM: Provide close results if correctly calibrated CZM is mesh dependent, XFEM less Contact Fatigue: Multifacet CZM Effective traction Variable amplitude loading or tilted geometry 23