and Fragmentation in Two- and Three-Dimensions Daniel Spring Sofie - PowerPoint PPT Presentation

USNCCM 13 Unstructured Methods for Simulating Pervasive Fracture and Fragmentation in Two- and Three-Dimensions Daniel Spring Sofie E. Léon, Glaucio H. Paulino Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign

Fracture and Failure is Prevalent in Materials Science 2 and Across the Engineering Disciplines Materials (Soft) Materials (Quasi-Brittle) Biomechanical Renner, 2010 und.nodak.com pixshark.com Structural Earthquake Geomechanical imrtest.com usgs.gov total.com In this presentation, I discuss the use of unstructured methods with interfacial cohesive elements for investigating dynamic fracture problems in quasi-brittle materials.

Outline of Presentation Normal Traction 40.0 Dynamic Fracture with Interfacial CZMs 20.0 0 10 5 2 0 0 Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) Unstructured Methods for 2D Dynamic Fracture Using Polygonal Finite Elements Unstructured Methods for 3D Pervasive Fragmentation

Cohesive Zone Models Are Used To Capture the Nonlinear Behavior in the Zone in Front of the Macro Crack-Tip 𝑈 In ductile or quasi-brittle 𝑜 Macro Crack-tip materials, the nonlinear zone ahead of a crack tip is ∆ not negligible, and LEFM principles may not be ∆ appropriate Nonlinear zone, voids and micro-cracks Cohesive region  A macro-crack forms when the traction in the traction-separation relation goes to zero  In a numerical setting, we use zero thickness cohesive elements to capture the cohesive zone ahead of the crack-tip Barenblatt, GI, The formation of equilibrium cracks during brittle fracture: General ideas and hypotheses. Axially symmetric cracks. Journal of Applied Mathematics and Mechanics , Vol. 23, pp. 255-265, 1959. Dugdale, DS, Yielding of steel sheets containing slits, Journal of the Mechanics and Physics of Solids, Vol. 8, pp. 100-104, 1960. 4

In a Finite Element Setting, Zero-Thickness Cohesive 5 Elements are Adaptively Inserted Between Bulk Elements Cohesive elements consist of two facets that can separate from each other by means of a traction-separation relation Element Opening ∆ 𝑜 , ∆ 𝑢 Traction (MPa) Bulk 2D: Bulk Δ 𝑜 Δ 𝑢 Cohesive element of Separation of Traction-Separation initially zero thickness Cohesive Elements Relation 𝑈 Traction (MPa) 3D: ∆ Δ 𝑜 Δ 𝑢 Bulk Bulk Element Opening ∆ 𝑜 , ∆ 𝑢

Park-Paulino-Roesler (PPR) Potential-Based Cohesive Model The cohesive model is defined by a potential function 200 Potential (N/m) 𝛽 Δ 𝑜 Ψ Δ 𝑜 , Δ 𝑢 = min 𝜚 𝑜 , 𝜚 𝑢 + Γ 𝑜 1 − + 𝜚 𝑜 − 𝜚 𝑢 𝜀 𝑜 100 𝛾 Δ 𝑢 × Γ 𝑢 1 − + 𝜚 𝑢 − 𝜚 𝑜 𝜀 𝑢 0 10 10 1 User inputs: 𝜚 𝑜 , 𝜚 𝑢 (Fracture energy) 5 5 0 0 𝜏 𝑜 , 𝜐 𝑢 Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) (Cohesive strength) 𝛽 , 𝛾 (Softening shape parameter) From the cohesive potential, one can determine the traction-separation relations by taking the respective derivatives. Normal Traction Tangential Traction 𝑜 Δ 𝑜 , Δ 𝑢 = 𝜖Ψ 40.0 30.0 Traction (MPa) Traction (MPa) 𝑈 , 𝜖Δ 𝑜 20.0 20.0 𝑈 𝑢 Δ 𝑜 , Δ 𝑢 = 𝜖Ψ 10.0 , 𝜖Δ 𝑢 0 0 10 10 10 1 5 2 5 5 0 0 0 0 Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) Park K, Paulino GH, Roesler JR, 2009. A unified potential-based cohesive model for mixed-mode fracture. Journal of the Mechanics and Physics of Solids . Vol. 57, No. 6, pp. 891-908. Park K, Paulino GH, 2011. Cohesive zone models: A critical review of traction-separation relationships across fracture surfaces. Applied Mechanics Reviews . Vol. 6 64, pp 1-20

The Cohesive Element Contribution in a Dynamic Setting 7 In a typical dynamic fracture code, the active cohesive elements contribute an additional term to the external forces acting on the model (during explicit time integration). Bulk Elements Dynamically Inserted Traction-Separation Relation Create model (Continuum Behavior) Cohesive Elements 40.0 Traction (MPa) Read input file 20.0 0 10 Initialize 5 2 model data 0 0 Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) Explicit time integration for extrinsic fracture Conduct explicit Initialization: displacements ( 𝐯 0 ), velocity ( 𝐯 0 ), acceleration ( 𝐯 0 ) time integration for 𝑜 = 0 to 𝑜 𝑛𝑏𝑦 do 𝐯 𝑜 + ∆𝑢 2 /2 Update displacement: 𝐯 𝑜+1 = 𝐯 𝑜 + ∆𝑢 𝐯 𝑜 Write output Check the insertion of cohesive elements ext + 𝐒 𝑜+1 coh − 𝐒 𝑜+1 𝐯 𝑜+1 = 𝐍 −1 𝐒 𝑜+1 int Update acceleration: 𝐯 𝑜+1 = 𝐯 𝑜 + ∆𝑢/2 𝐯 𝑜 + 𝐯 𝑜+1 Update velocity: Delete model Update boundary conditions end for Celes W, Paulino GH, Espinha R, A compact adjacency-based topological data structure for finite element mesh representation. International Journal for Numerical Methods in Engineering , vol. 64, pp. 1529-1556, 2005.

Outline of Presentation Normal Traction 40.0 Dynamic Fracture with Interfacial CZMs 20.0 0 10 5 2 0 0 Δ 𝑜 ( 𝜈 m) Δ 𝑢 ( 𝜈 m) Unstructured Methods for 2D Dynamic Fracture Using Polygonal Finite Elements Unstructured Methods for 3D Pervasive Fragmentation

Structured Meshes for Dynamic Cohesive Fracture 9 One of the primary critiques of the cohesive element method is its mesh dependency. 4  The structured 4k mesh is commonly used Crack in dynamic cohesive fracture simulation 8 However, structured meshes may introduce artifacts into the fracture behavior, presenting preferred paths for cracks to propagate along. 45° 45° 45° 4k meshes are anisotropic , but have a many choices of crack path at each node. Zhang Z, Paulino GH, Celes W, 2007. Extrinsic cohesive zone modeling of dynamic fracture and microbranching instability in brittle materials. International Journal for Numerical Methods in Engineering . vol. 72, pp. 893-923.

Unstructured Meshes for Dynamic Cohesive Fracture 10 Alternatively, we propose using a randomly generated Centroidal Voronoi Tessellation (CVT). New Seeds Random Seeds Calculate Seeds = ≈ Centroids of Cells Centroids of Cells Unstructured meshes produce random fracture paths with no discernible patterns ~ 3 Polygonal meshes are isotropic , but have a limited number of crack paths at each node. Talischi, C., Paulino, G. H., Periera, A., and Menezes, I. F. M. PolyMesher: A general-purpose mesh generator for polygonal elements written in Matlab. Journal of Structural and Multidisciplinary Optimization . Vol. 45, pp. 309-328, 2012.

How to Quantify Mesh Isotropy/Anisotropy (Mesh Bias) 11 Dijkstra’s algorithm is used to compute the shortest path between two points in the mesh. Euclidean distance end Element edges 𝑀 𝐹 Euclidean distance Element edges 𝑀 𝑕 start L    g The path deviation is computed as: 1 L E Dijkstra EW, A note on two problems in connexion with graphs. Numerische Mathematik . Vol. 1, pp. 269 – 227, 1959. Rimoli JJ, Rojas JJ, Meshing strategies for the alleviation of mesh-induced effects in cohesive element models. International Journal of Fracture , Vol. 193, pp. 29-42, 2015. Spring DW, Léon SE, Paulino GH, Unstructured polygonal meshes with adaptive refinement for the numerical simulation of dynamic cohesive fracture. International Journal of Fracture , Vol. 189, pp. 33-57, 2014.

Quantifying Mesh Isotropy/Anisotropy 12 L A study was conducted on the path deviation over a range of 180°    g 1 L E 4k Mesh Polygonal Mesh 90  90  120  60  120  60  150  30  150  30  180  180  0  0  0.05 0.02 0.1 0.15 0.04 0.06 0.2 0.08 0.25 0.1 210  330  330  210  240  300  240  300  270  270  The structured 4k mesh is anisotropic, while the unstructured polygonal discretization is isotropic. However, the path deviation in the polygonal mesh is significantly higher that that in the structured mesh. Rimoli JJ, Rojas JJ, Meshing strategies for the alleviation of mesh-induced effects in cohesive element models. International Journal of Fracture , Vol. 193, pp. 29-42, 2015. Léon SE*, Spring DW*, Paulino GH, Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements. International Journal for Numerical Methods in Engineering , Vol. 100, pp. 555-576, 2014.

Crack Propagation Within the Element 13 In order to reduce the path deviation in the unstructured polygonal mesh, we propose using an element-splitting topological operator to increase the number of fracture paths at each node in the mesh. We restrict elements to be split A 1 A along the path which minimizes A 1 1 the difference between the areas A 2 of the resulting split elements. A A 2 2 The propagating crack now has twice as many paths on which it could travel at each node. Triple junction All Potential Paths Allowable Paths Split Element Léon SE*, Spring DW*, Paulino GH, Reduction in mesh bias for dynamic fracture using adaptive splitting of polygonal finite elements. International Journal for Numerical Methods in Engineering , Vol. 100, pp. 555-576, 2014.