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Mesh-independent Fracture Modeling (XFEM) Lecture 9

Frac l09 Xfem

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Page 1: Frac l09 Xfem

Mesh-independent Fracture Modeling (XFEM)

Lecture 9

Page 2: Frac l09 Xfem

L9.2

Modeling Fracture and Failure with Abaqus

Overview

• Introduction

• Basic XFEM Concepts

• Damage Modeling

• Creating an XFEM Fracture Model

• Example 1 – Crack Initiation and Propagation

• Example 2 – Propagation of an Existing Crack

• Example 3 – Delamination and Through-thickness Crack Propagation

• Modeling Tips

• Current Limitations

• Workshop 6

• References

Page 3: Frac l09 Xfem

Introduction

Page 4: Frac l09 Xfem

L9.4

Modeling Fracture and Failure with Abaqus

Introduction

• The fracture modeling methods discussed so far only permit crack

propagation along predefined element boundaries

• This lecture presents a technique for modeling

bulk fracture which permits a crack to be located

in the element interior

• The crack location is independent of the mesh

Page 5: Frac l09 Xfem

L9.5

Modeling Fracture and Failure with Abaqus

Introduction

• This modeling technique…

• Can be used in conjunction with the cohesive zone model or the virtual

crack closure technique

• Delamination can be modeled in conjunction with bulk crack

propagation

• Can determine the load carrying capacity of a cracked structure

• What is the maximum allowable flaw size for safe operation?

• Applications of this technique include the modeling of bulk fracture and

the modeling of failure in composites

• Cracks in pressure vessels or engineering structures

• Delamination and through-thickness crack modeling in composite plies

Page 6: Frac l09 Xfem

L9.6

Modeling Fracture and Failure with Abaqus

Introduction

• Some advantages of the method:

• Ease of initial crack definition

• Mesh is generated independent of crack

• Partitioning of geometry not needed as when a crack is represented

explicitly

• Nonlinear material and nonlinear geometric analysis

• Arbitrary solution-dependent crack initiation and propagation path

• Crack path does not have to be specified a priori

• Mesh refinement studies are much simpler

• Reduced remeshing effort

• Improved convergence rate for the finite element solution (stationary

crack)

• Due to the use of singular crack tip enrichment

Page 7: Frac l09 Xfem

L9.7

Modeling Fracture and Failure with Abaqus

Introduction

• Mesh-independent Crack Modeling – Basic Ingredients

1. Need a way to incorporate discontinuous geometry – the crack – and

the discontinuous solution field into the finite element basis functions

• eXtended Finite Element Method (XFEM)

2. Need to quantify the magnitude of the discontinuity – the displacement

jump across the crack faces

• Cohesive zone model (CZM)

3. Need a method to locate the discontinuity

• Level set method (LSM)

4. Crack initiation and propagation criteria

• At what level of stress or strain does the crack initiate?

• What is the direction of propagation?

• These topics will be discussed in this lecture

Page 8: Frac l09 Xfem

Basic XFEM Concepts

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L9.9

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• eXtended Finite Element Method (XFEM) Background

• XFEM extends the piecewise polynomial function space of conventional finite

element methods with extra functions

• The solution space is enriched by the extra “enrichment functions”

• Introduced by Belytschko and Black (1999) based on the partition of unity

method of Babuska and Melenk (1997)

• Can be used where conventional FEM fails or is prohibitively expensive

• Appropriate enrichment functions are chosen for a class of problems

• Inclusion of a priori knowledge of partial differential equation behavior into

finite element space (singularities, discontinuities, ...)

• Applications include modeling fracture, void growth, phase change ...

• Enrichment functions for fracture modeling

• Heaviside function to represent displacement jump across crack face

• Crack tip asymptotic function to model singularity

Page 10: Frac l09 Xfem

L9.10

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• XFEM Displacement Interpolation

Heaviside enrichment term

H(x) Heaviside distribution

aI Nodal enriched DOF (jump discontinuity)

NG Nodes belonging to elements cut by crack

Crack tip enrichment term

Fa(x) Crack tip asymptotic functions

Nodal DOF (crack tip enrichment)

NG Nodes belonging to elements containing crack tip

b I

a

uI Nodal DOF for conventional shape functions NI

4

1

u (x) (x) u (x (x)b)ah

I I

I

II

I

N

N

I

N

FHN a

aa

G

Page 11: Frac l09 Xfem

L9.11

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• The crack tip and Heaviside enrichment functions are multiplied by the

conventional shape functions

• Hence enrichment is local around the crack

• Sparsity of the resulting matrix equations is preserved

• The crack is located using the level set method (discussed shortly)

• Heaviside function

• Accounts for displacement jump across crack

*1 if ( ) 0( )

1 otherwiseH x

x x ns

n

x

x*

Here x is an integration point, x* is the closest point to x on the crack face and n is the unit normal at x*

H(x) = 1 below crack

H(x) = 1 above crack

Page 12: Frac l09 Xfem

L9.12

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• Crack Tip Enrichment Functions (Stationary Crack Only)

• Account for crack tip singularity

• Use displacement field basis functions for sharp crack in an isotropic

linear elastic material

]2

cossinr ,2

sinsinr ,2

cosr ,2

sinr[4]-1 ),([

aa xF

Here (r, ) denote coordinate values from a polar coordinate system located at the crack tip

Page 13: Frac l09 Xfem

L9.13

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• Phantom Node Approach (Crack Propagation Implementation)

• Implementation of XFEM fitting into the framework of conventional FEM

• Discontinuous element with Heaviside enrichment is treated as a

superposition of two continuous elements with phantom nodes

• Does not include the asymptotic crack tip enrichment functions

• Introduced by Belytschko and coworkers (2006) based on the

superposed element formulation of Hansbo and Hansbo (2004)

Page 14: Frac l09 Xfem

L9.14

Modeling Fracture and Failure with Abaqus

Basic XFEM Concepts

• Level Set Method for Locating a Crack

• A level set (also called level surface or isosurface) of a real-valued function

is the set of all points at which the function attains a specified value

• Example: the zero-valued level set of f (x, y) : x2 y2 r2 is a circle of

radius r centered at the origin

• Popular technique for representing surfaces in interface tracking problems

• Two functions F and Y are used to completely describe the crack

• The level set F = 0 represents the crack face

• The intersection of level sets Y = 0 and F = 0 denotes the crack

front

• Functions are defined by nodal values whose spatial variation is

determined by the usual finite element shape functions (example

follows)

• Function values need to be specified only at nodes belonging to

elements cut by the crack

Page 15: Frac l09 Xfem

L9.15

Modeling Fracture and Failure with Abaqus

• Calculating F and Y

• The nodal value of the function F is the signed distance of the node from

the crack face

• Positive value on one side of the crack face, negative on the other

• The nodal value of the function Y is the signed distance of the node from

an almost-orthogonal surface passing through the crack front

• The function Y has zero value on this surface and is negative on the

side towards the crack

Basic XFEM Concepts

Y = 0F = 0

1 2

3 4

0.5

1.5

Node F Y

1 0.25 1.5

2 0.25 1.0

3 0.25 1.5

4 0.25 1.0

Page 16: Frac l09 Xfem

Damage Modeling

Page 17: Frac l09 Xfem

L9.17

Modeling Fracture and Failure with Abaqus

Damage Modeling

• Damage modeling is achieved through the use of a traction-separation

law across the fracture surface

• It follows the general framework introduced in earlier lectures

• Damage initiation

• Damage evolution

• Traction-free crack faces at failure

• Damage properties are specified as part of the bulk material definition

Damage initiation

Failure

Page 18: Frac l09 Xfem

L9.18

Modeling Fracture and Failure with Abaqus

Damage Modeling

• Damage Initiation

• Two criteria available at present

• Maximum principal stress criterion (MAXPS)

• Initiation occurs when the maximum principal stress reaches

critical value

• Maximum principal strain criterion (MAXPE)

• Initiation occurs when the maximum principal strain reaches

critical value

• Crack plane is perpendicular to the direction of the maximum principal

stress (or strain)

• Crack initiation occurs at the center of the element

• However, crack propagation is arbitrary through the mesh

• The damage initiation criterion is satisfied when 1.0 ≤ f ≤ 1.0 + ftol

where f is the selected damage criterion and ftol is a user-specified

tolerance value

max

0

max

f

max

0

max

f

Page 19: Frac l09 Xfem

L9.19

Modeling Fracture and Failure with Abaqus

Damage Modeling

• Damage Evolution

• Any of the damage evolution models for traction-separation laws

discussed in the earlier lectures can be used

• However, it is not necessary to specify the undamaged traction-

separation response

Page 20: Frac l09 Xfem

L9.20

Modeling Fracture and Failure with Abaqus

Damage Modeling

• Damage Stabilization

• Fracture makes the structural response nonlinear and non-smooth

• Numerical methods have difficulty converging to a solution

• As discussed in the earlier lectures, using viscous regularization helps

with the convergence of the Newton method

• The stabilization value must be chosen so that the problem definition

does not change

• A small value regularizes the analysis, helping with convergence

while having a minimal effect on the response

• Perform a parametric study to choose appropriate value for a class

of problems

Page 21: Frac l09 Xfem

L9.21

Modeling Fracture and Failure with Abaqus

Damage Modeling

• Damage stabilization can currently be defined in Abaqus/CAE only

through the keyword editor

Page 22: Frac l09 Xfem

Creating an XFEM Fracture Model

Page 23: Frac l09 Xfem

L9.23

Modeling Fracture and Failure with Abaqus

Creating an XFEM Fracture Model

• Steps

1. Define damage criteria in the material model

2. Define an enrichment region (the associated material model should

include damage)

• Crack type – stationary or propagation

3. Define an initial crack, if present

4. If needed, set analysis controls to aid convergence

• Steps will be illustrated later through examples

• Crack initiation and propagation in a plate with a hole

• Propagation of an existing crack

• Delamination and through-thickness crack propagation in a double

cantilever beam

• The next few slides describe step-dependent enrichment activation

and postprocessing

Page 24: Frac l09 Xfem

L9.24

Modeling Fracture and Failure with Abaqus

Creating an XFEM Fracture Model

• Step-dependent Enrichment Activation

• Crack growth can be activated or deactivated in analysis steps

*STEP...

*ENRICHMENT, NAME=Crack-1, ACTIVATE=[ON|OFF]

1

2

Page 25: Frac l09 Xfem

L9.25

Modeling Fracture and Failure with Abaqus

Creating an XFEM Fracture Model

• Output Quantities

• Two output variables are especially useful

• PHILSM

• The signed distance function F used to represent the crack

surface

• Needed for visualizing the crack

• STATUSXFEM

• Indicates the status of the element with a value between 0.0

and 1.0

• A value of 1.0 indicates that the element is completely cracked,

with no traction across the crack faces

• Any other output variable available in the static stress analysis

procedure

Page 26: Frac l09 Xfem

L9.26

Modeling Fracture and Failure with Abaqus

Creating an XFEM Fracture Model

• Postprocessing

• The crack location is specified by the zero-valued level set of the signed

distance function F

• Abaqus/CAE automatically creates an isosurface view cut named

Crack_PHILSM if an enrichment is used in the analysis

• The crack isosurface is displayed by default

• Contour plots of field quantities should be done with the crack isosurface

displayed

• Ensures that the solution is plotted from the active parts of the

overlaid elements according to the phantom nodes approach

• If the crack isosurface is turned off, only values from the “lower”

element are plotted (corresponding to negative values of F)

• Probing field quantities on an element currently returns values only from

the “lower” element (on the side with negative values of F)

Page 27: Frac l09 Xfem

Example 1 – Crack Initiation and

Propagation

Page 28: Frac l09 Xfem

L9.28

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

• Model crack initiation and propagation in a plate with a hole

• Crack initiates at the location of maximum stress concentration

• Half model is used taking advantage of symmetry

Page 29: Frac l09 Xfem

L9.29

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Define the damage criteria

• Damage initiation

1

Damage initiation tolerance (default 0.05)

*MATERIAL...

*DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05

Page 30: Frac l09 Xfem

L9.30

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Define the damage criteria (cont’d)

• Damage evolution

*DAMAGE INITIATION, CRITERION=MAXPS, TOL=0.05

*DAMAGE EVOLUTION, TYPE=ENERGY, MIXED MODE BEHAVIOR=POWER LAW, POWER=1.0

2870.0, 2870.0, 2870.0

1

Page 31: Frac l09 Xfem

L9.31

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Define the damage criteria (cont’d)

• Damage stabilization

Keyword interface

*DAMAGE STABILIZATION

1.e-5

Coefficient of viscosity m

• Abaqus/CAE interface currently not available

• The keyword editor may be used to add stabilization through

Abaqus/CAE.

1

Page 32: Frac l09 Xfem

L9.32

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Define the enriched region

Pick enriched region

Specify contact interaction

(frictionless small-sliding contact only)

Propagating crack

2

Page 33: Frac l09 Xfem

L9.33

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Define the enriched region (cont’d)

Keyword interface

No initial crack definition is needed

• Crack will initiate based on specified damage criteria

3

*ENRICHMENT, TYPE=PROPAGATION CRACK, NAME=CRACK-1,

ELSET=SELECTED_ELEMENTS, INTERACTION=CONTACT-1

Frictionless small-sliding contact interaction

2

Page 34: Frac l09 Xfem

L9.34

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Set analysis controls to improve convergence behavior

• Set reasonable minimum and maximum increment sizes for step

• Increase the number of increments for step from the default value of

100

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

4

Page 35: Frac l09 Xfem

L9.35

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Set analysis controls to improve convergence behavior (cont’d)

• Use numerical scheme applicable to discontinuous analysis

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

*CONTROLS, ANALYSIS=DISCONTINUOUS

4

Page 36: Frac l09 Xfem

L9.36

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

Set analysis controls to improve convergence behavior (cont’d)

• Increase value of maximum number of attempts before abandoning

increment (increased to 20 from the default value of 5)

*STEP

*STATIC, inc=10000

0.01, 1.0, 1.0e-09, 0.01...

*CONTROLS, ANALYSIS=DISCONTINUOUS

*CONTROLS, PARAMETER=TIME INCREMENTATION

, , , , , , , 20

8th field

4

Page 37: Frac l09 Xfem

L9.37

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

• Output Requests

• Request PHILSM and STATUSXFEM in addition to the usual output for

static analysis

Page 38: Frac l09 Xfem

L9.38

Modeling Fracture and Failure with Abaqus

Example 1 – Crack Initiation and Propagation

• Postprocessing

• Crack isosurface (Crack_PHILSM) created and displayed automatically

• Field and history quantities of interest can be plotted and animated as

usual

Page 39: Frac l09 Xfem

Example 2 – Propagation of an Existing

Crack

Page 40: Frac l09 Xfem

L9.40

Modeling Fracture and Failure with Abaqus

Example 2 – Propagation of an Existing Crack

• Model with crack subjected to mixed mode loading

• Initial crack needs to be defined

• Crack propagates at an angle dictated by mode mix ratio at crack tip

Page 41: Frac l09 Xfem

L9.41

Modeling Fracture and Failure with Abaqus

Example 2 – Propagation of an Existing Crack

Define damage criteria in the material model as described in Example 1

Specify the enriched region as in Example 1

Define the initial crack

• Two methods are available to define initial crack in Abaqus/CAE

1. Create a separate part representing the crack surface or line and

assemble it along with the part representing the structure to be

analyzed

2. Create an internal face or edge representing the crack in the part

• Method 1 is preferred as it takes full advantage of the mesh-

independent crack representation possible using XFEM

• Meshing is easier using this method

• Method 2 will create nodes on the internal crack face

• Element faces/edges are forced to align with the crack

1

2

3

Page 42: Frac l09 Xfem

L9.42

Modeling Fracture and Failure with Abaqus

Example 2 – Propagation of an Existing Crack

Define the initial crack (cont’d)3

The crack location can be an edge or a

surface belonging to the same

instance as the enriched region or to a

different instance (preferred)

** Model data

*INITIAL CONDITIONS, TYPE=ENRICHMENT

901, 1, Crack-1, -1.0, -1.5

901, 2, Crack-1, -1.0, -1.4

901, 3, Crack-1, 1.0, -1.4

901, 4, Crack-1, 1.0, -1.5

Element Number

Relative Node Order in Connectivity

Enrichment Name

F Y

Page 43: Frac l09 Xfem

L9.43

Modeling Fracture and Failure with Abaqus

Example 2 – Propagation of an Existing Crack

• The other steps are as described in Example 1 and are in line with

those necessary for the usual static analysis procedure

Page 44: Frac l09 Xfem

Example 3 – Delamination and

Through-thickness Crack Propagation

Page 45: Frac l09 Xfem

L9.45

Modeling Fracture and Failure with Abaqus

Example 3 – Delamination and Through-thickness Crack

• Model through-thickness crack propagation using XFEM and

delamination using surface-based cohesive behavior in a double

cantilever beam specimen

• Interlaminar crack grows initially

• Through-thickness crack forms once interlaminar crack becomes long

enough and the longitudinal stress value builds up due to bending

• The point at which the through-thickness crack forms depends upon the

relative failure stress values of the bulk material and the interface

Page 46: Frac l09 Xfem

L9.46

Modeling Fracture and Failure with Abaqus

Example 3 – Delamination and Through-thickness Crack

• This model is the same as the double cantilever beam model presented

in the surface-based cohesive behavior lecture except:

• Enrichment has been added to the top and bottom beams to allow

XFEM crack initiation and propagation

Page 47: Frac l09 Xfem

Modeling Tips

Page 48: Frac l09 Xfem

L9.48

Modeling Fracture and Failure with Abaqus

Modeling Tips

• General Information

• Averaged quantities are used in an element for determining crack

initiation and the propagation direction

• The integration point principal stress or strain values are averaged

• A new crack always initiates at the center of the element

• Within an enrichment region, a new crack initiation check is performed

only after all existing cracks have completely separated

• This may result in the abrupt appearance of multiple cracks

• Complete separation is indicated by STATUSXFEM=1

• Cracks cannot initiate in neighboring elements

• Crack propagates completely through an element in one increment

• Only the initial crack tip can lie within an element

Page 49: Frac l09 Xfem

L9.49

Modeling Fracture and Failure with Abaqus

Modeling Tips

• The enrichment region must not include “hotspots” due to boundary

conditions or other modeling artifacts

• Otherwise, unintended cracks may initiate at such locations

• Damage initiation tolerance

• A larger value may result in multiple cracks initiating in a region

• Small value results in small increment size and convergence difficulty

• Damage stabilization

• As mentioned earlier, judicious use of viscous regularization can aid in

convergence

• Initial crack should bisect elements if possible

• Convergence is more difficult if crack is tangential to element boundaries

• Use displacement control rather than load control

• Crack propagation may be unstable under load control

Page 50: Frac l09 Xfem

L9.50

Modeling Fracture and Failure with Abaqus

Modeling Tips

• Limit maximum increment size and start with a good guess for initial

increment size

• In general, this is a good approach for any non-smooth nonlinearity

• Analysis controls

• Can help obtain a converged solution and speed up convergence

• Contour plots of field quantities should be done with the crack

isosurface displayed

• Ensures that the solution is plotted from the active parts of the overlaid

elements according to the phantom nodes approach

• If the crack isosurface is turned off, only values from the “lower” element

are plotted (on the side with negative values of F)

Page 51: Frac l09 Xfem

L9.51

Modeling Fracture and Failure with Abaqus

Modeling Tips

• When defining the crack using Abaqus/CAE, extend the external crack

edges beyond base geometry

• This helps avoid incorrect identification of external edges as internal due

to geometric tolerance issues

Defining a through-thickness crack in a cylindrical vessel

Top View

Page 52: Frac l09 Xfem

Current Limitations

Page 53: Frac l09 Xfem

L9.53

Modeling Fracture and Failure with Abaqus

Current Limitations

• Implemented only for the static stress analysis procedure

• Can use only linear continuum elements

• CPE4, CPS4, C3D4, C3D8 and their reduced integration/incompatible

counterparts

• Element processing is not done in parallel

• On SMP machines, only the solver runs in parallel

• Cannot run in parallel on DMP machines

• Contour integrals for stationary cracks not currently supported

• Cannot model fatigue crack growth

• Intended for single or a few non-interacting cracks in the structure

• Shattering cannot be modeled

• An element cannot be cut by more than one crack

• Crack cannot turn more than 90 degrees in one increment

• Crack cannot branch

Page 54: Frac l09 Xfem

L9.54

Modeling Fracture and Failure with Abaqus

Current Limitations

• The first signed distance function F must be non-zero

• If the crack lies along an element boundary, a small positive or negative

value should be used

• This slightly offsets the crack from the element boundary

• Only frictionless small-sliding contact is considered

• The small-sliding assumption will result in nonphysical contact behavior

if the relative sliding between the contacting surfaces is indeed large

• Only enriched regions can have a material model with damage

• If only a portion of the model needs to be enriched define an extra

material model with no damage for the regions not enriched

• Probing field quantities on an element currently returns values only

from the “lower” element (corresponding to negative values of F)

Page 55: Frac l09 Xfem

Workshop 6

Page 56: Frac l09 Xfem

L9.56

Modeling Fracture and Failure with Abaqus

Workshop 6

• In this workshop, you will

continue with the analysis of a

cracked beam subjected to pure

bending using XFEM

• This workshop demonstrates:

• The ease of meshing and initial

crack definition compared to the

techniques presented in earlier

lectures

• The use of analysis controls

Page 57: Frac l09 Xfem

References

Page 58: Frac l09 Xfem

L9.58

Modeling Fracture and Failure with Abaqus

References

1. I. Babuska and J. Melenk, Int. J. Numer. Meth. Engng (1997), 40:727-758

2. T. Belytschko and T. Black, Int. J. Numer. Meth. Engng (1999), 45:601-620

3. A. Hansbo and P. Hansbo, Comp. Meth. Appl. Mech. Engng (2004),

193:3523-3540

4. J. H. Song, P. M. A. Areias and T. Belytschko, Int. J. Numer. Meth. Engng

(2006), 67:868-893