Introduction & Motivation experiments and numerical model Superposed cohesive laws approach for bridging Numerical i

2. CONTENTS • Introduction & Motivation • experiments and numerical model • Superposed cohesive laws approach for bridging • Numerical identification • Conclusions • Cohesive zone models and fibre bridging • DCB tests on fiberglass specimens and numerical model • Superposition of cohesive elements and analytical identification of material parameters • Response surface and optimization approaches to material parameter identification

3. INTRODUCTION AND MOTIVATION Bi-linear cohesive laws can be successfully in FE models of delaminations They are adequate when toughness is constant with crack length. Material model Application Characterisation Verification Analysis of crack growth in curved fabric laminates

4. INTRODUCTION AND MOTIVATION The crack growth resistance can significantly increase in the presence of fibre bridging In large scale fibre bridging a very long process zone develops before toughness reaches a steady level GC Cohesive laws with linear softening are inadequateto model the G-a curve effect.

5. INTRODUCTION AND MOTIVATION The measurement of bridging tractions in the wake of crack confirms that they do not have a linear softening (Sorensen et al. 2008). Other shapes must be employed for the softening law The superposition of two linear softening laws has been proposed for intralaminar fracture (Davila et. Al 2009). It can be considered an appealing practical approach (conventional cohesive elements can be used)

6. INTRODUCTION AND MOTIVATION Objectives: • Apply the superposed element approach to model the R-a curve effects in interlaminar fracture in glass fiber reinforced laminates • Develop an analytical approach for the calibration of material parameters from the experimental R-a curve • Apply numerical techniques for the automatic identification of such parameters based on the force vs. displacement response of DCB tests

7. EXPERIMENTS AND NUMERICAL MODEL DCB tests have been performed on [0]48 laminates of S2 Glass fibre reinforced tape with an Epoxy Cycom SP250 matrix (5 Tests) • Pre-crack has been obtained by means of a PTFE insert • Pre-opening test were performed • Subsequent opening tests • Crack advance monitored by dye penetrant inspection.

8. EXPERIMENTS AND NUMERICAL MODEL Four data reduction techniques: Beam Theory (BT), Compliance Calibration (CC), Modified Beam Theory (MBT), Modified Compliance Calibration (MCC) Large scale fibre bridging and a marked G-a curve effect. The length of the process zone (LPZ) is approximately 80 mm

9. EXPERIMENTS AND NUMERICAL MODEL A 2 mm wide strip of the specimen has been analysed in Abaqus Standard Incompatible modes C3D8I elements Imposed displacement 0.5 mm equispaced grid COH3D8 cohesive elements Material stiffness from previous characterisation and transverse isotropy assumptions

10. EXPERIMENTS AND NUMERICAL MODEL Preliminary numerical evaluation: • cohesive law with linear softening • GIC = 1.0 KJ/m2 • 0 =20 MPa and 0=50 MPa Bi-linear cohesive law largely overestimates the force in DCB tests Peel strength has a little influence on DCB response as expected

11.  SUPERPOSED COHESIVE LAWS APPROACH In the presence of bridging, the softening law is non-linear  the complete cohesive law is approximated by means of two superimposed cohesive laws .

12. Gc G1 SUPERPOSED COHESIVE LAWS APPROACH reference length of the process zone Linearised expression of the G-a curve by Davila et al. 2009 Parameter m is G1/Gc nis obtained by imposing GR = GC in correspondance of the experimental

13. SUPERPOSED COHESIVE LAWS APPROACH The previous formulation has been applied and verified for a compact tension specimen (Davila et al. 2009) In DCB test adherends are thin and LPZ becomes much shorter than Turon et al. (2008) suggested a correction of reference process zone based on an undetermined factor H A refined model using a single cohesive (linear softening law) has been used to asses an appropriate expression of reference LPZ

14. SUPERPOSED COHESIVE LAWS APPROACH Two corrections are considered: FEM 2D LPZ 1 LPZ 1 LPZ 2 LPZ 2 • The errors in the uncorrected lc are very large when LPZ is long • For large LPZ a correction factor with the additional parameter  provides the best results •  is set to 0.48 for best correlation

15. SUPERPOSED COHESIVE LAWS APPROACH Using and m=2 superposed cohesive elements model: Numerical G(a) LPZ and Force vs. Displacement curves captured for Sigma = 15 and 25 MPa

16. NUMERICAL IDENTIFICATION The presented model proved effective to accurately capture the forces and the process zone lenght for moderate values of peel strength Analytical calibration of material parameters requires the knowledge of the G-a curve An alternative strategy is explored, based on a numerical identification technique The objective is the identification of material parameters considering the Force vs. Displacement curve • A cost function is defined • response surfaces techniques is applied to explore the feasibility of the approach • Optimization procedures is applied to minimize the error

17. NUMERICAL IDENTIFICATION Cost Functions Mean Square Error between numerical and average test Average MSE values in 4 selected zones d4 d3 d1 d2 Global error index

18. Abaqus runs Ei Matlab post-processing variables Total error Error zones NUMERICAL IDENTIFICATION Implementation Ichrome/NEXUS Optimisation Suite

19. NUMERICAL IDENTIFICATION Response surface techniques Response surfaces have been built by means of a Kriging approximation (second order polynomial + local gauss functions) The surface has been created by allocating 300 points within the domain Steady state toughness has been set at 1.0 kJ/m2 The database allows the creation of different surfaces of the cost function in the space m-n at a given value of peel strength (Sigma)

20. NUMERICAL IDENTIFICATION Response surface for Sigma = 15 MPa Minimum of cost function is found along a valley for high values of n An interval 0.05 < m < 0. 2 can be identified along the valley

21. NUMERICAL IDENTIFICATION Response surface for Sigma = 25 MPa optimal m seems to be lower than m=0.2, but derivatives are small in such direction As Sigma is increased optimal n slightly moves towards 1.0

22. NUMERICAL IDENTIFICATION Response surface for Sigma = 35 MPa For Sigma = 35 MPa qualitative tendencies are confirmed. Overall minimum values of cost function are about 20 N.

23. NUMERICAL IDENTIFICATION Following the meta-model indications three solutions have been selected Meta-model allows identifying acceptable approximations

24. NUMERICAL IDENTIFICATION Optimization: Gradient-based method Sigma = 15 Mpa, Gc = 1 kJ/m2 Initial guess m=0.3, n=0.7 (meta-model indications ignored) Optimized Solution Evolution of m,n, Objective

25. NUMERICAL IDENTIFICATION For Sigma =25 and 35 MPa meta-model indication have been used as initial guess for a gradient based method The application of different weights to error indices in the different zones of the curve has been investigated Initial Guess Interesting results have been found by increasing the weights in the first 2 zones of the domain

26. NUMERICAL IDENTIFICATION minimization of cost function lead to increase m Sigma = 35 Mpa Sigma = 25 Mpa Improvement of Force-displacement and G-a correlation in the initial part of the response Final GC is almost unchanged (imposed value of 1 kJ/m2)

27. CONCLUSIONS • Bi-linear softening laws can model delamination processes in the presence of fibre bridging • Ananalytical calibration procedure of the model has been assessed for moderate values of peel strength (more refined models could be required for higher values) • Numerical identification (response surface/optimization) can obtain approximate solutions without requiring the knowledge of the G-a curve • Numerical procedures can be extended to multi-linear softening laws which could be more flexible for capturing both force response, G-a curve and process zone lengths