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

Daniel Spring Sofie E. Léon, Glaucio H. Paulino

Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign

USNCCM 13

Unstructured Methods for Simulating Pervasive Fracture and Fragmentation in Two- and Three-Dimensions

Fracture and Failure is Prevalent in Materials Science and Across the Engineering Disciplines

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In this presentation, I discuss the use of unstructured methods with interfacial cohesive elements for investigating dynamic fracture problems in quasi-brittle materials.

und.nodak.com

Materials (Quasi-Brittle) Biomechanical

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Geomechanical

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Earthquake Structural

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Materials (Soft)

Renner, 2010 usgs.gov

Outline of Presentation

Dynamic Fracture with Interfacial CZMs Unstructured Methods for 2D Dynamic Fracture Using Polygonal Finite Elements Unstructured Methods for 3D Pervasive Fragmentation

Δ𝑢 (𝜈m) Normal Traction Δ𝑜 (𝜈m) 0 0 20.0 40.0 10 5 2

Cohesive Zone Models Are Used To Capture the Nonlinear Behavior in the Zone in Front of the Macro Crack-Tip

Nonlinear zone, voids and micro-cracks Macro Crack-tip

In ductile or quasi-brittle materials, the nonlinear zone ahead of a crack tip is not negligible, and LEFM principles may not be appropriate

Cohesive region

∆ 𝑈

𝑜

∆

 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.

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Δ𝑢 Traction (MPa) Δ𝑜

∆ 𝑈

Δ𝑢 Traction (MPa) Δ𝑜 5

In a Finite Element Setting, Zero-Thickness Cohesive 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

Cohesive element of initially zero thickness Bulk Bulk Bulk Bulk Separation of Cohesive Elements

2D: 3D:

Element Opening ∆𝑜, ∆𝑢 Element Opening ∆𝑜, ∆𝑢 Traction-Separation Relation

The cohesive model is defined by a potential function

Ψ Δ𝑜, Δ𝑢 = min 𝜚𝑜, 𝜚𝑢 + Γ𝑜 1 −

Δ𝑜 𝜀𝑜 𝛽

+ 𝜚𝑜 − 𝜚𝑢 × Γ𝑢 1 −

Δ𝑢 𝜀𝑢 𝛾

+ 𝜚𝑢 − 𝜚𝑜

From the cohesive potential, one can determine the traction-separation relations by taking the respective derivatives.

𝑈

𝑜 Δ𝑜, Δ𝑢 = 𝜖Ψ

𝜖Δ𝑜 , 𝑈𝑢 Δ𝑜, Δ𝑢 = 𝜖Ψ 𝜖Δ𝑢 ,

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. 64, pp 1-20

User inputs: 𝜚𝑜, 𝜚𝑢 (Fracture energy) 𝜏𝑜, 𝜐𝑢 (Cohesive strength) 𝛽, 𝛾 (Softening shape parameter)

Park-Paulino-Roesler (PPR) Potential-Based Cohesive Model

Δ𝑢 (𝜈m) Traction (MPa) Normal Traction Δ𝑜 (𝜈m) 0 0 20.0 40.0 10 5 2 Traction (MPa) Tangential Traction 0 0 20.0 30.0 1 10.0 10 5 5 10 Δ𝑢 (𝜈m) Δ𝑜 (𝜈m) Potential (N/m) 0 0 100 200 1 10 5 5 10 Δ𝑢 (𝜈m) Δ𝑜 (𝜈m) 6

Δ𝑢 (𝜈m) Traction (MPa) Δ𝑜 (𝜈m) 0 0 20.0 40.0 10 5 2 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). Create model Read input file Conduct explicit time integration Initialize model data Write output Delete model

The Cohesive Element Contribution in a Dynamic Setting

Bulk Elements (Continuum Behavior) Dynamically Inserted Cohesive Elements Traction-Separation Relation Explicit time integration for extrinsic fracture Initialization: displacements (𝐯0), velocity ( 𝐯0), acceleration ( 𝐯0) for 𝑜 = 0 to 𝑜𝑛𝑏𝑦 do Update displacement: 𝐯𝑜+1 = 𝐯𝑜 + ∆𝑢 𝐯𝑜 + ∆𝑢2/2 𝐯𝑜 Check the insertion of cohesive elements Update acceleration: 𝐯𝑜+1 = 𝐍−1 𝐒𝑜+1

ext + 𝐒𝑜+1 coh − 𝐒𝑜+1 int

Update velocity: 𝐯𝑜+1 = 𝐯𝑜 + ∆𝑢/2 𝐯𝑜 + 𝐯𝑜+1 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

Dynamic Fracture with Interfacial CZMs Unstructured Methods for 2D Dynamic Fracture Using Polygonal Finite Elements Unstructured Methods for 3D Pervasive Fragmentation

Δ𝑢 (𝜈m) Normal Traction Δ𝑜 (𝜈m) 0 0 20.0 40.0 10 5 2

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Structured Meshes for Dynamic Cohesive Fracture

One of the primary critiques of the cohesive element method is its mesh dependency.

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.

45° 45° 45°

4k meshes are anisotropic, but have a many choices of crack path at each node. However, structured meshes may introduce artifacts into the fracture behavior, presenting preferred paths for cracks to propagate along.

 The structured 4k mesh is commonly used

in dynamic cohesive fracture simulation 4 8 Crack

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Unstructured Meshes for Dynamic Cohesive Fracture

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.

~3 Alternatively, we propose using a randomly generated Centroidal Voronoi Tessellation (CVT).

Random Seeds Calculate Centroids of Cells New Seeds Seeds ≈ Centroids of Cells =

Unstructured meshes produce random fracture paths with no discernible patterns

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Dijkstra’s algorithm is used to compute the shortest path between two points in the mesh.

How to Quantify Mesh Isotropy/Anisotropy (Mesh Bias)

The path deviation is computed as:

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.

Element edges Euclidean distance start end Element edges Euclidean distance

𝑀𝐹 𝑀𝑕

1

g E

L L   

0.05 0.1 0.15 0.2 0.25 30 210 60 240 90 270 120 300 150 330 180 0 0.02 0.04 0.06 0.08 0.1 30 210 60 240 90 270 120 300 150 330 180 0

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Quantifying Mesh Isotropy/Anisotropy

A study was conducted on the path deviation over a range of 180° 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. 4k Mesh Polygonal 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.

g E

L L







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Crack Propagation Within the Element

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.

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.

We restrict elements to be split along the path which minimizes the difference between the areas

• f the resulting split elements.

All Potential Paths Allowable Paths Split Element

Triple junction

The propagating crack now has twice as many paths on which it could travel at each node.

A

A

A

A

A

A

Adaptive Refinement Ahead of the Crack-Tip

Additionally, we propose the use of an adaptive refinement operator, wherein each polygon around the crack tip is removed and replaced with a set of unstructured quads; which meet at the centroid of the original polygon.

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.

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Unrefined Mesh Refined Mesh

The mesh is adaptively refined in front of the propagating crack-tip.

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Quantification of Improvement in Path Deviation

0.05 0.1 0.15 0.2 0.25 30 210 60 240 90 270 120 300 150 330 180 0 Voronoi, No Splitting Refined, With Splitting 0.05 0.1 0.15 0.2 0.25 30 210 60 240 90 270 120 300 150 330 180 0 Voronoi, No Splitting Voronoi, With Splitting 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. 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.

Meshing Strategy Average Standard Deviation Improvement Polygonal 0.1931 0.0013

• Polygonal with Splitting

0.0445 0.0009 77% Polygonal with Refinement 0.0698 0.0021 64% Polygonal with Refinement and Splitting 0.0171 0.0004 91%

g E

L L







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Example: Pervasive Fracture and Fragmentation

𝐹 210𝐻𝑄𝑏 𝜍 7850 𝑙𝑕 𝑛3 𝜚 2000 𝑂 𝑛 𝜏 850𝑁𝑄𝑏 𝛽 2 160 mm 300 mm Internal pressure (P)

Pervasive fracture of an internally impacted thick cylinder

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. 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.

20 40 60 200 250 300 350 400 Time (s) Impact pressure, P (MPa)

𝑄 𝑢 = 400𝑓

− 𝑢−1 100

Internal Pressure, P (MPa) Time (𝜈𝑡)

Expected result contains complete fragments

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When we only use a geometrically unstructured mesh, we get unbiased fracture behavior, but unrealistic fracture patterns. When we use both a geometrically and topologically unstructured mesh, we get unbiased fracture behavior and realistic fracture patterns. Without element-splitting With element-splitting

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.

Example: Influence of Element-Splitting Operator

~21 Fragments

5 10 15 200 400 600 800 Time (𝜈𝑡) Crack Velocity

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Example: Dominant Crack with Microbranching

Sharon E, Fineberg J, Microbranching instability and the dynamic fracture of brittle materials. Physical Review B, vol. 54, pp. 7128-7139, 1996. 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.

Conductive Layer

𝜏∞ 𝜏∞

L

Experimental setup and results: Numerical model:

Fracture Pattern 𝐹 3.24𝐻𝑄𝑏 𝜍 1190 𝑙𝑕 𝑛3 𝜚 352.4 𝑂 𝑛 𝜏 129.6𝑁𝑄𝑏 𝛽 2 2 𝑛𝑛 16 𝑛𝑛 4 𝑛𝑛 Initial notch

PMMA Properties

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Fracture Patterns: Influence of Meshing Strategy

Adaptively refined unstructured meshes produce a smooth crack path with large macrobranches and a uniform distribution of microbranching. The adaptively refined meshes produce results in good agreement with experiments. Coarse Polygonal Element- Splitting Adaptively Refined

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.

20

Influence of Meshing Strategy on the Crack-Tip Velocity

Numerical Results

Adaptively Refined Mesh Coarse Polygonal Mesh

5 10 15 200 400 600 800 Time (𝜈𝑡)

Experimental Result

Sharon E, Fineberg J, Microbranching instability and the dynamic fracture of brittle materials. Physical Review B, vol. 54, pp. 7128-7139, 1996. 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.

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Comparison of Computational Cost

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.

Case Elements Nodes Cost (min) Iterations to Fracture Cost/Iteration (10-3s) 1 4,000 7,188 21.5 28,200 45.7 2 4,000 7,188 20.8 23,000 54.3 3 4,000 7,188 19.1 21,000 54.6

Polygonal Element-Splitting Adaptive Refinement Case 1 Case 2 Case 3

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Some Remarks on 2D Cohesive Fracture

 Unstructured polygonal meshes produce an isotropic discretization of the problem domain.  Without careful design considerations, polygonal meshes are inherently poorly suited to

dynamic fracture simulation with the cohesive element method.

 The newly proposed topological operators are designed to increase the number of paths a

crack can propagate along, and result in a meshing strategy on par with the best, fixed meshing strategy available in the literature.

 The adaptive refinement with element splitting scheme increases the problem size, but can

decrease the computational cost.

 By combining geometrically and topologically unstructured methods, the model is truly

random and reduces numerically induced restrictions. Thus, reducing uncertainty in numerical simulations.

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. 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.

Outline of Presentation

Dynamic Fracture with Interfacial CZMs Unstructured Methods for 2D Dynamic Fracture Using Polygonal Finite Elements Unstructured Methods for 3D Pervasive Fragmentation

Δ𝑢 (𝜈m) Normal Traction Δ𝑜 (𝜈m) 0 0 20.0 40.0 10 5 2

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Pervasive Fracture and Fragmentation in 3D

Pervasive cracking and fragmentation comprises the entire spectrum of fracture behavior.

 Crack branching  Crack coalescence  Complete fragmentation  Sensitivity to material heterogeneity  Any structure introduced to the mesh

will bias fragmentation behavior Characteristics Issues

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

The computational framework for 3D cohesive fracture is an extension of that for 2D cohesive fracture.

Single/Dominant Cracks Pervasive Cracking and Fragmentation

Range of fracture behavior Crack Branching Crack Coalescence

swanston.com wellcoll.nl

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

Most materials contain heterogeneity (or defects) at the microscale.

Weibull W, 1939. A statistical theory of the strength of materials, Proceedings of the Royal Academy of Engineering Sciences, Vol. 151, pp. 1–45.

Issue 1: Sensitivity to Material Heterogeneity



Defects naturally arise in materials due to grain boundaries, voids, or inclusions.



Defects may also be introduced through the act of processing or machining the material.



Microscale defects constitute potential regions where stresses can concentrate and lead to damage or failure.

𝜏𝑛𝑗𝑜 = 264𝑁𝑄𝑏, 𝜇 = 50, 𝑛 = 2

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Constitutively Unstructured Through a Statistical Distribution of Material Properties

The material strength is assumed to follow a modified Weibull distribution: 𝜏 = 𝜏𝑛𝑗𝑜 + 𝜇 −ln 1 − 𝜍

1 𝑛 200 400 600 800 0.005 0.01 0.015 0.02 Strength (MPa) PDF 𝜇 = 50 𝜇 = 100 𝜇 = 200 250 300 350 400 0.02 0.04 0.06 0.08 Strength (MPa) PDF 𝑛 = 10 𝑛 = 5 𝑛 = 2 𝑛 = 2 𝜇 = 50

𝜏𝑛𝑗𝑜

Weibull W, 1939. A statistical theory of the strength of materials, Proceedings of the Royal Academy of Engineering Sciences, Vol. 151, pp. 1–45.

27

Issue 2: Structured Artifacts in Automatically Generated Meshes

 Automatic mesh generators often conduct additional post-processing of the mesh; to

remove elements with degenerate edges and sliver elements.

 In some cases, this additional post-processing leads these (initially random) meshes to

contain an underlying structure.

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

 To remove this structure, we propose using the technique of nodal perturbation.

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Geometrically Unstructured Through Nodal Perturbation

 We conduct a set of geometric studies to quantify the effect of the magnitude of the

nodal perturbation factor on the quality of the mesh.

 Nodes are randomly perturbed by a

multiple of dmin:

Paulino GH, Park K, Celes W, Espinha R, 2010. Adaptive dynamic cohesive fracture simulation using nodal perturbation and edge-swap operators. International Journal for Numerical Methods in Engineering , Vol. 84, pp 1303-1343. Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

dmin

NP = 0.0 (Unperturbed) NP = 0.2 NP = 0.4 NP = 0.6

Nodal Perturbation: 𝐘𝑜 = 𝐘𝑝 + dmin × NP × 𝐨random

0.2 0.4 0.6 0.8 5000 10000 15000 Coefficient of Variation of Facet Areas (per Element) Number of Occurrences NP = 0.0 (Unperturbed) NP = 0.2 NP = 0.4 NP = 0.6

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Lo SH, 1991, Volume discretization into tetrahedra - II. 3D triangulation by advancing front approach, Computers & Structures. Vol. 39, pp. 501–511. Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

First, we investigate the influence of nodal perturbation on the coefficient of variation of facet areas

 The dynamic time step is a

function of the element size.

 Small element facets reduce

the size of the critical time step.

NP = 0.4 NP = 0.0 (Unperturbed)

Geometrically Unstructured Through Nodal Perturbation

NP = 0.0 NP = 0.4

COV

60 70 80 90 100 110 120 130 140 150 160 170 5000 10000 15000 Maximum Angle (  ) Number of Occurrences NP = 0.0 (Unperturbed) NP = 0.2 NP = 0.4 NP = 0.6 5 10 15 20 25 30 35 40 45 50 55 60 5000 10000 15000 Minimum Angle (  ) Number of Occurrences NP = 0.0 (Unperturbed) NP = 0.2 NP = 0.4 NP = 0.6

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Summary Similar trends are observed for the study on the maximum and minimum interior angles in the mesh

Geometrically Unstructured Through Nodal Perturbation

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

Ex: The commercial software Abaqus qualifies elements with interior angles less than 10° or greater than 160° as distorted

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Example: Radial Fragmentation of a Hollow Sphere

Here, we consider the pervasive fragmentation of a hollow sphere with symmetric boundary conditions.

𝐹 = 370 𝐻𝑄𝑏 𝜍 = 3900 𝑙𝑕 𝑛3 𝜚 = 50 𝐾 𝑛2 𝜏𝑛𝑗𝑜 = 264 𝑁𝑄𝑏 ~100,000 Elements ~25,000 Nodes

The sphere is impacted with an impulse load, which is converted to an initial nodal velocity

9.25mm 0.75mm

𝐰𝟏 𝑦, 𝑧, 𝑨 = ε𝐲

Sarah Levy, “Exploring the physics behind dynamic fragmentation through parallel simulations.“ PhD. Dissertation, Ecole Polytechnique Federale de Lausanne, 2010. Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

32

Regularization Through Geometric Features

We investigate the influence of idealized surface features, namely bumps and dimples, on the fragmentation behavior of the hollow sphere. Bumps or Protrusions Dimples or Depressions

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

 Alternatively, surface features may be viewed as geometric defects, and investigating

their impact on the global fragmentation response is equally significant.

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Numerical Definition of a Fragment

A fragment is defined as a mass of bulk elements which are surrounded by a free boundary and/or fully separated cohesive elements. Free Boundary Fully separated cohesive elements

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

34

Geometric Features Can Be Used to Regularize Fragmentation Patterns

Dimples Bumps Smooth

 The initial impact velocity is set at: 𝐰0 𝑦, 𝑧, 𝑨 = 2500𝐲  Similar trends are observed at higher impact velocities and with different statistical

distributions of material strength.

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

2mm CP = Core Part OL = Outer Layer

35

Example: Kidney Stone Fragmentation by Direct Impact

This example considers the direct impact of a kidney stone. We use this example to investigate the use functionally graded materials to regularize fragmentation behavior.

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation. Caballero A, Molinari JF, Finite element simulations of kidney stones fragmentation by direct impact: Tool geometry and multiple impacts. International Journal of Engineering Science, vol. 48, pp. 253-264, 2010. Zhong P, Chuong CJ, Goolsby RD, Preminger GM, Microhardness measurements of renal calculi: Regional differences and effects of microstructure. Journal of Biomedical Materials Research, vol. 26, pp. 1117-1130, 1992.

COM: 𝐹 = 25.16 𝐻𝑄𝑏 𝜍 = 2038 𝑙𝑕 𝑛3 𝜚 = 0.735 𝐾 𝑛2 𝜏 = 1.0 𝑁𝑄𝑏 CA: 𝐹 = 8.504 𝐻𝑄𝑏 𝜍 = 1732 𝑙𝑕 𝑛3 𝜚 = 0.382 𝐾 𝑛2 𝜏 = 0.5 𝑁𝑄𝑏 Materials considered: COM: calcium oxalate monohydrate CA: carbonate apatite 𝑠 = 10𝑛𝑛 𝑠 = 0.325𝑛𝑛

36

Fragmentation of a Homogeneous Stone

To develop a baseline, we first consider the fragmentation of a homogeneous stone with different levels of variation in material properties.

CA CA(50)/COM(50) COM

Small variation Large variation CA: 𝐹 = 8.504 𝐻𝑄𝑏 𝜍 = 1732 𝑙𝑕 𝑛3 COM: 𝐹 = 25.16 𝐻𝑄𝑏 𝜍 = 2038 𝑙𝑕 𝑛3 𝜚 = 0.382 𝐾 𝑛2 𝜏 = 0.5 𝑁𝑄𝑏 𝜚 = 0.735 𝐾 𝑛2 𝜏 = 1.0 𝑁𝑄𝑏

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

37

Next, we show that fragmentation behavior can be regularized by modelling the graded distribution of material in the stone.

Soft Inner Core Soft Inner Core Hard Inner Core

Fragmentation of a Functionally Graded Stone

CA: 𝐹 = 8.504 𝐻𝑄𝑏 𝜍 = 1732 𝑙𝑕 𝑛3 COM: 𝐹 = 25.16 𝐻𝑄𝑏 𝜍 = 2038 𝑙𝑕 𝑛3 𝜚 = 0.382 𝐾 𝑛2 𝜏 = 0.5 𝑁𝑄𝑏 𝜚 = 0.735 𝐾 𝑛2 𝜏 = 1.0 𝑁𝑄𝑏

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

Some Remarks on 3D Pervasive Fracture & Fragmentation

 The cohesive element method constitutes a framework which allows us to capture the full

spectrum of fracture mechanisms.

 A statistical distribution of material properties can be used to account for microscale defects

and inhomogeneities.

 A random perturbation of the nodes reduces structure created by automatic mesh generators.  By incorporating constitutive and geometric heterogeneity in the model we can reduce

numerically induced artifacts into the simulated results and increase the certainty in our simulations.

 We can use simple geometric and constitutive design features to regularize pervasive fracture

and fragmentation behavior in three-dimensions.

Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

Acknowledgements

Collaborator: Dr. Sofie E. Léon Advisor: Prof. Glaucio H. Paulino Funding Agencies: NSERC, UIUC and NSF Additional details may be found in the related publications:



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.

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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.

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Spring DW, Paulino GH, Achieving pervasive cohesive fracture and fragmentation in three-dimensions: Theory, computation and applications. In Preparation.

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Daniel Spring spring2@illinois.edu

Thank You, Questions?