unstructured methods for simulating pervasive fracture and

Daniel Spring

Sofie E. Léon, Glaucio H. PaulinoDepartment 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

2

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

pixshark.com

Geomechanical

total.com

EarthquakeStructural

imrtest.com

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

Δ𝑡 (𝜇m)

Normal Traction

Δ𝑛 (𝜇m) 0 0

0

20.0

40.0

105 2

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

Nonlinear zone, voids and micro-cracks

Macro Crack-tipIn 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

∆

𝑇𝑛

∆

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 cohesivezone 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

Δ𝑡

Trac

tio

n (

MP

a)

Δ𝑛

∆

𝑇

Δ𝑡

Trac

tio

n (

MP

a)

Δ𝑛

5In 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∆𝑛, ∆𝑡

Element Opening∆𝑛, ∆𝑡

Traction-Separation Relation

The cohesive model is defined by a potential function

Ψ Δ𝑛, Δ𝑡 = min 𝜙𝑛, 𝜙𝑡 + Γ𝑛 1 −Δ𝑛

𝛿𝑛

𝛼+ 𝜙𝑛 − 𝜙𝑡

× Γ𝑡 1 −Δ𝑡

𝛿𝑡

𝛽+ 𝜙𝑡 − 𝜙𝑛

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

𝑇𝑛 Δ𝑛, Δ𝑡 =𝜕Ψ

𝜕Δ𝑛,

𝑇𝑡 Δ𝑛, Δ𝑡 =𝜕Ψ

𝜕Δ𝑡,

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: 𝜙𝑛, 𝜙𝑡 (Fracture energy) 𝜎𝑛, 𝜏𝑡 (Cohesive strength)𝛼, 𝛽 (Softening shape parameter)

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

Δ𝑡 (𝜇m)

Trac

tio

n (

MP

a)

Normal Traction

Δ𝑛 (𝜇m)0 0

0

20.0

40.0

105 2

Trac

tio

n (

MP

a)

Tangential Traction

0 0

0

20.0

30.0

1

10.0

105 5

10

Δ𝑡 (𝜇m)Δ𝑛 (𝜇m)Po

ten

tial

(N

/m)

0 0

0

100

200

110

5 510

Δ𝑡 (𝜇m)Δ𝑛 (𝜇m)

6

Δ𝑡 (𝜇m)

Trac

tio

n (

MP

a)

Δ𝑛 (𝜇m) 0 0

0

20.0

40.0

105 2

7

In a typical dynamic fracture code, the active cohesive elements contribute an additionalterm 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 (𝐮0), velocity ( 𝐮0), acceleration ( 𝐮0)

for 𝑛 = 0 to 𝑛𝑚𝑎𝑥 do

Update displacement: 𝐮𝑛+1 = 𝐮𝑛 + ∆𝑡 𝐮𝑛 + ∆𝑡2/2 𝐮𝑛Check the insertion of cohesive elements

Update acceleration: 𝐮𝑛+1 = 𝐌−1 𝐑𝑛+1ext + 𝐑𝑛+1

coh − 𝐑𝑛+1int

Update velocity: 𝐮𝑛+1 = 𝐮𝑛 + ∆𝑡/2 𝐮𝑛 + 𝐮𝑛+1Update 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.





Δ𝑡 (𝜇m)

Normal Traction

Δ𝑛 (𝜇m) 0 0

0

20.0

40.0

105 2

9Structured 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, presentingpreferred paths for cracks to propagate along.

The structured 4k mesh is commonly used in dynamic cohesive fracture simulation

4

8Crack

10Unstructured 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

11

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 edgesEuclidean distance

start

end

Element edges

Euclidean distance

𝐿𝐸

𝐿𝑔

1g

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

12Quantifying 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.

1g

E

L

L

13Crack Propagation Within the Element

In order to reduce the path deviation in the unstructured polygonal mesh, we proposeusing an element-splitting topological operator to increase the number of fracture pathsat each node in the mesh.


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

1A

1A

1A

2A 2

A 2

A

Adaptive Refinement Ahead of the Crack-Tip

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


14

Unrefined Mesh Refined Mesh

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

15Quantification 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



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%

1g

E

L

L

16Example: Pervasive Fracture and Fragmentation

𝐸 210𝐺𝑃𝑎

𝜌 7850 𝑘𝑔 𝑚3

𝜙 2000 𝑁 𝑚

𝜎 850𝑀𝑃𝑎

𝛼 2160 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.


0 20 40 60200

250

300

350

400

Time (s)

Imp

act

pre

ssu

re,

P (

MP

a)

𝑃 𝑡 = 400𝑒 − 𝑡−1 100

Inte

rnal

Pre

ssure

, P

(MP

a)

Time (𝜇𝑠)

Expected result contains complete fragments

17

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


Example: Influence of Element-Splitting Operator

~21 Fragments

0 5 10 15

200

400

600

800

Time (𝜇𝑠)

Crack Velocity

18Example: 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.


Conductive Layer

𝜎∞

𝜎∞

L

Experimental setup and results:

Numerical model:

Fracture Pattern

𝐸 3.24𝐺𝑃𝑎

𝜌 1190 𝑘𝑔 𝑚3

𝜙 352.4 𝑁 𝑚

𝜎 129.6𝑀𝑃𝑎

𝛼 2

2 𝑚𝑚

16 𝑚𝑚

4𝑚𝑚

Initial notch

PMMA Properties

19Fracture Patterns: Influence of Meshing Strategy

Adaptively refined unstructured meshes produce a smooth crack path with largemacrobranches and a uniform distribution of microbranching.

The adaptively refined meshes produce results in good agreement with experiments.

Coarse Polygonal

Element-Splitting

Adaptively Refined


20Influence of Meshing Strategy on the Crack-Tip Velocity

Numerical Results

Adaptively Refined Mesh

Coarse Polygonal Mesh

0 5 10 15

200

400

600

800

Time (𝜇𝑠)

Experimental Result

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


21Comparison of Computational Cost


Case Elements Nodes Cost (min) Iterations to Fracture Cost/Iteration (10-3 s)

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

22Some 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 todynamic fracture simulation with the cohesive element method.

The newly proposed topological operators are designed to increase the number of paths acrack can propagate along, and result in a meshing strategy on par with the best, fixedmeshing strategy available in the literature.

The adaptive refinement with element splitting scheme increases the problem size, but candecrease the computational cost.

By combining geometrically and topologically unstructured methods, the model is trulyrandom and reduces numerically induced restrictions. Thus, reducing uncertainty in numericalsimulations.







Δ𝑡 (𝜇m)

Normal Traction

Δ𝑛 (𝜇m) 0 0

0

20.0

40.0

105 2

24Pervasive 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/DominantCracks

Pervasive Cracking and Fragmentation

Range of fracture behavior

Crack Branching Crack Coalescence

swanston.com wellcoll.nl

25

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.

𝜎𝑚𝑖𝑛 = 264𝑀𝑃𝑎, 𝜆 = 50, 𝑚 = 2

26Constitutively Unstructured Through a Statistical Distribution of Material Properties

The material strength is assumed to follow a modified Weibull distribution:

𝜎 = 𝜎𝑚𝑖𝑛 + 𝜆 −ln 1 − 𝜌 1 𝑚

200 400 600 8000

0.005

0.01

0.015

0.02

Strength (MPa)

PD

F

𝜆 = 50

𝜆 = 100

𝜆 = 200

250 300 350 4000

0.02

0.04

0.06

0.08

Strength (MPa)

PD

F

𝑚 = 10

𝑚 = 5

𝑚 = 2

𝑚 = 2

𝜆 = 50

𝜎𝑚𝑖𝑛

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

27Issue 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.


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

28Geometrically 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.


dmin

NP = 0.0 (Unperturbed) NP = 0.2

NP = 0.4 NP = 0.6

Nodal Perturbation:

𝐗𝑛 = 𝐗𝑜 + dmin × NP × 𝐧random

0 0.2 0.4 0.6 0.80

5000

10000

15000

Coefficient of Variation of Facet Areas (per Element)

Nu

mb

er o

f O

ccu

rren

ces

NP = 0.0 (Unperturbed)

NP = 0.2

NP = 0.4

NP = 0.6

29

Lo SH, 1991, Volume discretization into tetrahedra - II. 3D triangulation by advancing front approach, Computers & Structures. Vol. 39, pp. 501–511.


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

The dynamic time step is afunction of the element size.

Small element facets reducethe size of the critical timestep.

NP = 0.4NP = 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 1700

5000

10000

15000

Maximum Angle ( )

Nu

mb

er o

f O

ccu

rren

ces


NP = 0.2

NP = 0.4

NP = 0.6

0 5 10 15 20 25 30 35 40 45 50 55 600

5000

10000

15000

Minimum Angle ( )

Nu

mb

er o

f O

ccu

rren

ces


NP = 0.2

NP = 0.4

NP = 0.6

30

Summary

Similar trends are observed for the study on the maximum and minimum interior anglesin the mesh

Geometrically Unstructured Through Nodal Perturbation


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

31Example: Radial Fragmentation of a Hollow Sphere

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

𝐸 = 370 𝐺𝑃𝑎 𝜌 = 3900 𝑘𝑔 𝑚3

𝜙 = 50 𝐽 𝑚2 𝜎𝑚𝑖𝑛 = 264 𝑀𝑃𝑎~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

𝐯𝟎 𝑥, 𝑦, 𝑧 = ε𝐱

Sarah Levy, “Exploring the physics behind dynamic fragmentation through parallel simulations.“ PhD. Dissertation, Ecole Polytechnique Federale de Lausanne, 2010.


32Regularization Through Geometric Features

We investigate the influence of idealized surface features, namely bumps and dimples, onthe fragmentation behavior of the hollow sphere.

Bumps or ProtrusionsDimples or Depressions


Alternatively, surface features may be viewed as geometric defects, and investigating their impact on the global fragmentation response is equally significant.

33Numerical Definition of a Fragment

A fragment is defined as a mass of bulkelements which are surrounded by afree boundary and/or fully separatedcohesive elements.

FreeBoundary

Fullyseparatedcohesiveelements


34Geometric Features Can Be Used to Regularize Fragmentation Patterns

Dimples BumpsSmooth

The initial impact velocity is set at: 𝐯0 𝑥, 𝑦, 𝑧 = 2500𝐱

Similar trends are observed at higher impact velocities and with different statistical distributions of material strength.


2mm

CP = Core PartOL = Outer Layer

35Example: Kidney Stone Fragmentation by Direct Impact

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


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: 𝐸 = 25.16 𝐺𝑃𝑎 𝜌 = 2038 𝑘𝑔 𝑚3

𝜙 = 0.735 𝐽 𝑚2 𝜎 = 1.0 𝑀𝑃𝑎

CA: 𝐸 = 8.504 𝐺𝑃𝑎 𝜌 = 1732 𝑘𝑔 𝑚3

𝜙 = 0.382 𝐽 𝑚2 𝜎 = 0.5 𝑀𝑃𝑎

Materials considered:

COM: calcium oxalate monohydrate CA: carbonate apatite

𝑟 = 10𝑚𝑚

𝑟 = 0.325𝑚𝑚

36Fragmentation of a Homogeneous Stone

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

CA CA(50)/COM(50) COM

Small variation

Largevariation

CA: 𝐸 = 8.504 𝐺𝑃𝑎 𝜌 = 1732 𝑘𝑔 𝑚3 COM: 𝐸 = 25.16 𝐺𝑃𝑎 𝜌 = 2038 𝑘𝑔 𝑚3

𝜙 = 0.382 𝐽 𝑚2 𝜎 = 0.5 𝑀𝑃𝑎 𝜙 = 0.735 𝐽 𝑚2 𝜎 = 1.0 𝑀𝑃𝑎


37

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

Soft Inner Core Soft Inner Core Hard Inner Core

Fragmentation of a Functionally Graded Stone

CA: 𝐸 = 8.504 𝐺𝑃𝑎 𝜌 = 1732 𝑘𝑔 𝑚3 COM: 𝐸 = 25.16 𝐺𝑃𝑎 𝜌 = 2038 𝑘𝑔 𝑚3

𝜙 = 0.382 𝐽 𝑚2 𝜎 = 0.5 𝑀𝑃𝑎 𝜙 = 0.735 𝐽 𝑚2 𝜎 = 1.0 𝑀𝑃𝑎


Some Remarks on 3D Pervasive Fracture & Fragmentation

The cohesive element method constitutes a framework which allows us to capture the fullspectrum of fracture mechanisms.

A statistical distribution of material properties can be used to account for microscale defectsand 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 reducenumerically induced artifacts into the simulated results and increase the certainty in oursimulations.

We can use simple geometric and constitutive design features to regularize pervasive fractureand fragmentation behavior in three-dimensions.


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:




39

Daniel Spring

[email protected]

Thank You, Questions?

unstructured methods for simulating pervasive fracture and

Documents