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Mesh-independent Fracture Modeling (XFEM)
Lecture 9
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
Introduction
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
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
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
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
Basic XFEM Concepts
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
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
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
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
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)
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
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
Damage Modeling
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
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
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
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
L9.21
Modeling Fracture and Failure with Abaqus
Damage Modeling
• Damage stabilization can currently be defined in Abaqus/CAE only
through the keyword editor
Creating an XFEM Fracture Model
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
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
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
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)
Example 1 – Crack Initiation and
Propagation
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
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
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
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
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
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
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
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
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
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
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
Example 2 – Propagation of an Existing
Crack
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
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
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
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
Example 3 – Delamination and
Through-thickness Crack Propagation
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
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
Modeling Tips
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
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
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)
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
Current Limitations
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
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)
Workshop 6
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
References
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