th U.S. National Congress on Computational Mechanics Extraction of Cohesive Properties of Elasto-Plastic material using Inverse Analysis Arun Lal Gain, Jay Carroll, Glaucio H. Paulino, John Lambros University of Illinois at Urbana Champaign 7/8/9 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 9 Board of Trustees of the University of Illinois
Contents Introduction Cohesive Zone Modeling Elasto-Plastic Forward v/s Inverse Problem Modeling Approaches Shape Regularization PPR model Numerical Simulations Summary & Conclusions Future Work 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Introduction: Cohesive Zone Model Cohesive Zone Modeling - fracture seen as phenomenon of gradual separation taking place across cohesive zone (path of crack) and resisted by cohesive tractions Four staged failure Stage : Material homogeneous Stage : Crack Initiation Criterion: Stress reaching tensile strength (simplified) Stage 3: Crack propagation based on traction v/s separation curve Stage : Complete failure Criterion: e.g. Crack width reaches certain predefined value 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 3
Introduction: Cohesive Zone Model Various approaches to obtain cohesive zone model are available in literature Obtain through experiments Direct tension test van Mier, van Vliet, Uniaxial tension test for determination of fracture parameters of concrete, Fracture of Concrete & Rock, Assume the shape CZM shape significantly affects fracture analysis results should be chosen carefully Song S.H, Paulino G.H, Buttlar G.H, Influence of cohesive zone model shape parameter on asphalt concrete fracture behavior, American Institute of Physics, 8 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Introduction: Cohesive Zone Model Indirect method: Inverse analysis Development in experimental stress analysis techniques like photo-elasticity, DIC have made Inverse Analysis attractive van Mier, Fracture processes of concrete : assessment of material parameters for fracture models, CRC Press, 997 Hanson J. H., An experimental - computational evaluation of the accuracy of the fracture toughness tests on concrete, PhD Thesis, Cornell university, Hanson J.H., Ingraffea A.R. Using numerical simulations to compare the fracture toughness values for concrete from the size-effect, twoparameter and fictitious crack models, Engineering Fracture Mechanics, 3 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 5
T n (Mpa) T n (Mpa) Load (N) Introduction: Forward v/s Inverse Problem P P P P Forward Problem 5 5 CMOD(m)...3 Global Response u x, y Inverse Problem u P x, y DIC / Synthetic Data from forward problem P Optimization Nelder-Mead Scheme 5 5 CMOD(m) Constitutive Response 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Elasto-Plastic Forward Problem + coh, coh K u K u u F j j j ext b i i i i F F K b Plane Stress J Plasticity K element coh k c n d d, d x x x NN T T t ld 3 Cohesive Element f coh d x d x =K u element coh l s element n (Mpa) n k c 5 5 n (m) T d N N t ld x Shen, B., 9, Functionally Graded Fiber Reinforced Cementitious Composites : Manufacturing and Extraction 7/8/9of Cohesiveth Properties U.S. National usingcongress Finite Elements on Computational and Digital Mechanics, ImageColumbus, Correlation, Ohio PhD Thesis, UIUC 7
Modeling Approaches: Shape Regularization Elasto-Plastic Inverse Problem +, K u K u u F b coh coh ext Y n n c Y i i X, Y i i Nelder-Mead Optimization λ λ w F F w f Y w f X min, : R R ext int M f f X i - X i X i+ X n n nc f Y i ψ γ Y i, f X i ψ γ i λ coh Cohesive Parameters X,X,..,X,Y,Y,..,Y n n where, i i X X X i i i X X i i i γ <<, ψ >> Constraints X X.. X n 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 8
T n (Mpa) Modeling Approaches: PPR model Unified potential based model: PPR (Park-Paulino-Roesler) m n m n ψ n, t min n, t n n t n n n t n t t t n t t T ψ, Tt n, t n n t n ψ t =.3 =. =. Park K., Paulino G.H., Roesler J.R., 9, A unified potential-based cohesive model of mixed-mode fracture, Journal of the Mechanics and Physics of Solids, 57, 89 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 3 (m) n 9
Modeling Approaches: PPR model Elasto-Plastic Forward Problem F F + coh, coh K u K u λ u F j j j ext b i i i i λ coh Cohesive Parameters,, n max 3 d x K element coh k c n NN T t ld Cohesive Element d x l s n (Mpa) k c 5 5 n (m) d d, d x x x T f coh element element coh T x =K u d N N t ld 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Modeling Approaches: PPR model Elasto-Plastic Inverse Problem +, K u K u u F b coh coh ext Nelder-Mead Optimization λ λ F F w f min w, : R R ext int 3 f λ Cohesive Parameters Constraints:,, n max Barrier Function: f ψ, ψ >> 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Numerical Simulations Problem Details P P 38. 5. E 7 GPa,. 5, app. mm Isotroplic Hardening MPa, K y D MPa 578 Nodes 53 Q Elements 3 Cohesive Elements Displacement Ctrl: Steps 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Load (N) Numerical Simulations: Shape Regularization Forward Problem 5 Elastic bulk material Elasto-plastic bulk material 5 3 (MPa).5.. Linear softening CZM.5..5..5 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 3
Load (N) Numerical Simulations: Shape Regularization Inverse Problem: Different Loading Points 5 (MPa) 5 3 Point A Point B Point C Point D Point E Point F (MPa).5.. Linear softening CZM Initial Guess.5..5. Control Points Piecewise Cubic Hermite interpolation Synthetic data without any noise 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio A X:.88 Y: 35.5 X:.37 Y: X: 7.5.3 X:.39 Y: 3 X:.3 Y: Y:. X:.8 Y: 5.7 9.5 B C DE F....8 Forward Problem Plot
Load (N) Numerical Simulations: Shape Regularization Inverse Problem: Various Control Points 5 5 3 - Control Points - Control Points 5 - Control Points - Control Points (MPa).5 (MPa) 3 Initial Guess.5..5. Displacement field taken from loading point C =. N Piecewise Cubic Hermite interpolation Synthetic data without any noise 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio.. Linear softening CZM X:.39 Y:.....8 C Forward Problem Plot 5
Numerical Simulations: Shape Regularization Load (N) Inverse Problem: Different Initial Guess 5 8 (MPa).5 (MPa) Initial Guess.. Linear softening CZM Initial Guess...3..5 Displacement field taken from loading point C =. N Control Points Piecewise Cubic Hermite interpolation Synthetic data without any noise 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio X:.39 Y:. C....8 Forward Problem Plot
Numerical Simulations: Shape Regularization Load (N) Inverse Problem: Noise in Synthetic Data 5 (MPa) 5 3.5..5. Displacement field taken from loading point C =. N Control Points Initial Guess Max Noise. % Max Noise.% Max Noise. % Max Noise. % Piecewise Cubic Hermite interpolation Synthetic data with varying amount of noise 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio (MPa).5.. Linear softening CZM X:.39 Y:. C....8 Forward Problem Plot 7
Load (N) Numerical Simulations: PPR model Forward Problem 5 Elastic bulk material Elasto-plastic bulk material = 3. (Mpa) 3.5..5..5.3.35...3 n 5 N / m max =3 5 MPa 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 8
Load (N) Numerical Simulations: PPR model Inverse Problem: Different Loading Points (MPa) 5 3 Initial Guess Point A Point B Point C Point D Point E Point F (Mpa) = 3....3 Initial Guess...3. X:.5 Y:. X:.59 Y: 37. X:.8 Y: 33.9 X:.3 Y: 9. X:.39 Y:. X:.79 Y: 7. A B C D E F Synthetic data without any noise...3. Forward Problem Plot 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 9
Numerical Simulations: PPR model Inverse Problem: Various Initial Guesses Load (N) 8 (Mpa) = 3. (MPa) Initial Guess Initial Guess...3 Initial Guess 3...3..5 D X:.3 Y: 9. Displacement field taken from loading point D = 9. N Synthetic data without any noise...3. Forward Problem Plot 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Numerical Simulations: PPR model Load (N) Inverse Problem: Noise in Synthetic Data (MPa) 5 3 Initial Guess Max Noise =. % Max Noise =.5 % Max Noise =.5 % Max Noise = 5. % (Mpa) = 3....3 Initial Guess...3. D X:.3 Y: 9. Displacement field taken from loading point D = 9. N Synthetic data with varying amount of noise...3. Forward Problem Plot 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Load (kn) Inverse Analysis using DIC Load v/s CMOD from experiment.8. Image 7 DIC Data used for simulation Load vs. COD P P...8.. Crack Propagation Observed in Images, Image Preliminary results using PMMA. 8 8 Load Crack Line Opening Displacement (mm)(um) 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Inverse Analysis using DIC Extracted Cohesive Relation using Inverse Analysis 8 Shape of Cohesive Relation Results from different simulation runs Shape similar to the one used in reference below n, MPa 8 Preliminary results using PMMA.5..5..5.3.35. n, mm 7/8/9 Elices M., Guinea G.V., Gomez th J. U.S. & Planas National J.,, Congress The Cohesive on Computational Zone Model: Advantages, Mechanics, Limitations Columbus, and Ohio Challenges, EFM, 9, 37 3
Inverse Analysis using DIC Traction Separation relation for PMMA used by Elices et al. 7/8/9 Elices M., Guinea G.V., Gomez th J. U.S. & Planas National J.,, Congress The Cohesive on Computational Zone Model: Advantages, Mechanics, Limitations Columbus, and Ohio Challenges, EFM, 9, 37
Summary & Conclusions Developed inverse analysis techniques to extract cohesive fracture properties of elasto-plastic materials Shape regularization PPR model Verified implementation for various conditions Ongoing collaborative work: Hybrid technique (Experimental DIC + Inverse analysis) for polymers and metal/metal composites such as Ti/Ti composites 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 5
Future Work Inverse analysis for fatigue loading Validation of elasto-plastic inverse analysis using DIC experiments Extension of elasto-plastic inverse analysis to plates and shells 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio
Thank You! Acknowledgements:: Bin Shen, Jason Patric 7/8/9 th U.S. National Congress on Computational Mechanics, Columbus, Ohio 7