The eXtended Finite Element Method (XFEM) for … eXtended Finite Element Method (XFEM) for brittle and cohesive fracture Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos

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The eXtended Finite Element Method (XFEM) forbrittle and cohesive fracture

Prof. Dr. Eleni ChatziDr. Giuseppe Abbiati, Dr. Konstantinos Agathos

Lecture 7 - 16 November, 2017

Institute of Structural Engineering, ETH Zurich

November 16, 2017

Institute of Structural Engineering Method of Finite Elements II 1

Outline1 Introduction2 Discontinuities-High gradients3 Discretization methods4 Partition of unity enrichment5 Extended Finite Element Method for brittle fracture

Representation of discontinuitiesEnrichment functionsShape functions and derivativesStiffness matrixNumerical integrationEnrichment shiftingStress intensity factorsSolution overview

6 XFEM for cohesive fracture7 Extension to 3D


Learning goals

Introducing partition of unity enrichment

Introducing the eXtended/Generalized Finite Element Method(XFEM/GFEM)

Reviewing some implementation isuues

Application of XFEM in brittle fracture

Application of XFEM in cohesive fracture


Applications

Fracture in brittle materials

Fatigue fracture

Fracture in concrete


Discontinuities-High gradients

The methods presented are of interest for problems involvingdiscontinuities and high gradients

Those usually occur in the vicinity of, or along points, lines andinterfaces

The nature of these phenomena is therefore mostly local

Often those interfaces evolve over time

Representation of moving interfaces may become an importantissue


Strong discontinuitiesStrong discontinuities:

Discontinuities in the main solution variable

Example: Displacement jump along crack faces

crack

Displacement

Strain


Weak discontinuitiesWeak discontinuities:

Discontinuities in the derivatives of the main solution variable

Example: Strain discontinuity in bi-material interfaces

bimaterial interface

Displacement

Strain

material 1

material 2


High gradients

High gradients:

Large variation of a variable in a small area

Example: Strain singularity around the tip of a crack


Standard FEMFEM solution of a problem involving discontinuities and singularitiesalong moving interfaces (e.g. crack propagation):

Domain



Initial mesh



Crack appears → Re-meshing and mesh refinement is required



Crack propagates → Re-meshing is required











Standard FEM

Difficulties in the above approach:

In every step of crack propagation re-meshing is required

The procedure cannot be fully automated since the generatedmeshes need to be inspected

For nonlinear problems projection of solution parameters isrequired between meshes


Numerical solution of PDESBasic ingredients of numerical methods for PDEs:

Problem formulation:

Weak form, e.g.: Principle of Virtual Work, Galerkin Method,Hu-Washizu principle

Strong form

Discretization scheme, e.g.:

Piecewise polynomials

Splines

Radial basis functions


Numerical solution of PDES

By employing combinations of the above, different methods can beobtained, e.g.:

Weak form + Piecewise polynomials → FEM

The methods presented in the following mainly employ improveddiscretization schemes based on the standard FEM


Partition of unity

DefinitionA set of functions N∗I (x) defined in a domain Ω such that:∑

∀IN∗I (x) = 1, ∀I ∈ Ω

Is called a partition of unity (PU).

Example: FE shape functions


Partition of unity

Any function Ψ (x) can be exactly represented in Ω by theproduct: ∑

∀IN∗I (x) Ψ (x)

If the additional parameters bI are introduced, then the functioncan be spatially adjusted:∑

∀IN∗I (x) Ψ (x) bI


PU FEM

If a FE mesh is considered then the PU property can be exploited toenrich the approximation:

u (x) =∑∀I

NI (x) uI︸︷︷︸FE approximation

+∑∀I

N∗I (x) Ψ (x) bI︸︷︷︸enriched part

where:

u (x) The main solution variable (e.g. displacements)NI (x) The FE shape functions

uI Nodal dofsN∗I (x) A set of functions forming a PUΨ (x) The enrichment function

bI Additional unknowns


PU FEM

The above, for N∗I ≡ NI , becomes the Partition of Unity FiniteElement Method (PU FEM):

u (x) =∑∀I

NI (x) uI︸︷︷︸FE approximation

+∑∀I

NI (x) Ψ (x) bI︸︷︷︸enriched part

PU FEM allows for:

Incorporating known features of the problem in the solutionthrough Ψ (x)Spatially adjusting Ψ (x) to the specific problem solved throughbI

Preserving some desired properties of the FE method such assparsity of the system matrices


XFEM

Enrichment in PU FEM is global, i.e. it is applied to all nodesof the mesh

Phenomena such as cracks, holes and discontinuities are of localnature

The eXtended Finite Element Method (XFEM) was introducedto model discontinuities independently of the FE mesh used

The XFEM employs local enrichment, i.e. enrichment appliedonly to nodes in the vicinity of the discontinuity


Representation of discontinuities

In XFEM:

Discontinuities are not part of the mesh

The geometrical representation of evolving surfaces assumes animportant role

The Level Set Method (LSM) has been established as animportant tool for tracking discontinuities


Level Set Method

We consider a domain Ω divided in subdomains Ω1 and Ω2 by aninterface Γ, then:

→ Γ can be implicitly definedby a level set function φ (x)such that:

φ (x) > 0 in Ω1

φ (x) < 0 in Ω2

φ (x) = 0 on Γ


Level Set Method

Typically the signed distance function is used as a level set function:

φ (x) = minx∈Γ‖x− x‖ sign (n · (x− x))

where n is the normal vector to the interface and sign is the signfunction:

sign (x) =

1 for x > 0− 1 for x < 0

Signed distance functions have the property:

‖∇φ‖ = 1


Level Set Method

Examples of level set functions:

Circular level set function: φ (x) = ‖x− xc‖ − r


Level Set Method

Examples of level set functions:

Elliptical level set function: φ (x , y) =(x

a

)2+(y

b

)2− 1


Level Set Method

Typically, level set values are computed for a set of points, e.g.all nodal points in an FE mesh

In many cases the initial LS function is not a signed distancefunction, e.g. for the elliptical LS:

φ (x , y) =(x

a

)2+(y

b

)2− 1

Then a Jacobi-Hamilton type equation is employed to imposethe condition ‖∇φ‖ = 1:

∂φ

∂τ+ sign (φ) (‖∇φ‖ − 1) = 0


Level Set Method

For the case of evolving interfaces:

The level set function is not constant over time

Initial values need to be updated

The velocity field normal to the interface vn needs to be known

Jacobi-Hamilton type equations are employed to update LSvalues:

∂φ

∂t + Vn (‖∇φ‖) = 0


Crack representation

To represent crack surfaces two LS functions are needed:

φ→ the signed distancefrom the crack surface

ψ → the signeddistance from thenormal to the cracksurface crack surface crack extension

) = 0x(ψ

)x(ψ

)x(φ

x

r

θ

) = 0x(φ

+n

crack front


Crack representation

The following relations should hold for the LS description of thecrack:

φ = 0 should give the crack surface

ψ = 0 should give the normal crack surface

φ = 0, ψ = 0 should give the crack front/ crack tip

‖∇φ‖ = 1, ‖∇ψ‖ = 1

∇φ · ∇ψ = 0


Propagating cracks

The propagation criterion is only evaluated at the cracktip/front

The velocity field is only known at those locations

To update the LS representation of the crack the velocity fieldneeds to be extended to the whole domain

Along with the previous conditions, an additional one needs tobe imposed such that the existing crack surface remainsunaltered


Propagating cracks

All the conditions needed for the representation of propagatingcracks need to be enforced through Jacobi-Hamilton typeequations

The above significantly complicates the LS update procedure

Simple expressions for updating the LS functions can beobtained by analytically solving some of the equations involvedand employing some simple geometrical relations


Level set interpolation

Typically level sets are computed and updated at nodal points andFE interpolation is used for the rest of the domain:

φ =∑

INI (x)φI

ψ =∑

INI (x)ψI

Derivatives and gradients of the level sets can be computed usingderivatives of the FE shape functions:

φ,x =∑

INI,x (x)φI , φ,y =

∑I

NI,y (x)φI , φ,z =∑

INI,z (x)φI

ψ,x =∑

INI,x (x)ψI , ψ,y =

∑I

NI,y (x)ψI , ψ,z =∑

INI,z (x)ψI


Jump enrichment functions

The discontinuity along the crack surface is represented using amodified step function:

H(φ) =

1 for φ > 0− 1 for φ < 0


Jump enrichment functions

The set of jump enriched nodes can be defined using the level setfunctions as:

All elements are looped

The signs of the nodal level set values are examined

If for an element the first level set changes sign and the secondis negative then the element is intersected by the crack

All nodes belonging to elements being intersected by the crackare jump enriched


Tip enrichment functions

To derive the enrichment functions used to represent the singularityat the tip of a crack, the displacement expression of the Westergaardsolution is employed:

u1 = KI2G√

r/2π cos(θ/2)[κ− 1 + 2 sin2(θ/2)

]+ KII

2G√

r/2π sin(θ/2)[κ+ 1 + 2 cos2(θ/2)

]u2 = KI

2G√

r/2π sin(θ/2)[κ+ 1− 2 cos2(θ/2)

]− KII

2G√

r/2π cos(θ/2)[κ− 1− 2 sin2(θ/2)

]where κ and G are the Kolosov constant and shear modulus



The functions:

Fj (r , θ) =[√

r sin θ2 ,√

r cos θ2 ,√

r sin θ2 sin θ,√

r cos θ2 sin θ]

are used as enrichment functions

→ Fj form a basis that can represent the solution exactly.



3D plots of the enrichment functions:

√r sin θ

2√

r cos θ2

√r sin θ

2 sin θ√

r cos θ2 sin θ



The coordinates r , θ are defined using the level sets as:

r =√φ2 + ψ2

θ = arctan(φψ

)crack surface crack extension

) = 0x(ψ

)x(ψ

)x(φ

x

r

θ

) = 0x(φ

+n

crack front



The set of tip enriched nodes can be defined by using the level setfunctions as:

All elements are looped

The signs of the nodal level set values are examined

If for an element both level sets change sign then it contains thecrack tip/front

All nodes belonging to elements containing the crack tip/frontare tip enriched


XFEM approximation

The displacement approximation for XFEM for fracture mechanics is:

u (x) =∑I∈N

NI (x) uI︸︷︷︸continuous part

+∑

J∈N j

NJ (x) H (φ) aJ +∑

T∈N t

∑j

NT (x) Fj (r , θ) bTj︸︷︷︸discontinuous part

where:

N is the set of all nodes in the mesh

N j is the set of jump enriched nodes as defined previously

N t is the set of tip enriched nodes as defined previously


XFEM approximation

Enrichment in 2D and 3D meshes:

Tip enriched node

Jump enriched node


XFEM approximation

Shape function derivatives can be obtained from the displacementapproximation for the standard, jump and tip enriched nodes:

BuI , Ba

I , BbI

where

BuI =

NI,x 0 00 NI,y 00 0 NI,z

NI,y NI,x 0NI,z 0 NI,x

0 NI,z NI,y

corresponds to the standard part


XFEM approximation

BaI =

(NIH),x 0 00 (NIH),y 00 0 (NIH)z

(NIH),y (NIH),x 0(NIH),z 0 (NIH),x

0 (NIH),z (NIH),y

corresponds to the jump enriched part


XFEM approximation

BtipI =

[Bb

I1 BbI2 Bb

I3 BbI4

]corresponds to the tip enriched part and the part corresponding toeach enrichment function is:

BbIj =

(NIFj),x 0 00 (NIFj),y 00 0 (NIFj)z

(NIFj),y (NIFj),x 0(NIFj),z 0 (NIFj),x

0 (NIFj),z (NIFj),y


XFEM approximation

In the above expressions, derivatives are evaluated using the chainrule. For instance:

Jump enrichment function:

(NIH),x = NI,xH + NIH,x

where

H(φ),i = δ (φ) =

1 for φ = 00 otherwise


XFEM approximation

The jump enriched part becomes:

BaI =

NI,xH 0 00 NI,y H 00 0 NI,zH

NI,y H NI,xH 0NI,zH 0 NI,xH

0 NI,zH NI,y H


XFEM approximation

Tip enrichment functions:

(NIFJ),x = NI,xFj + NIFj,x

where

Fj,x = Fj,r (r,φφ,x + r,ψψ,x ) + Fj,θ (θ,φφ,x + θ,ψψ,x )

and for instance

F1,r = 12√

r sin θ2 , F1,θ =√

r2 cos θ2

The rest of the derivatives are computed in a similar fashion


Discrete equilibrium equations

The discrete equilibrium equations are obtained by plugging theshape function derivatives into the weak form:∫

Ωδε : D : ε dΩ =

∫Ωδu · b dΩ +

∫Γtδu · t dΓ

where:

ε are the strains obtained through the shape functionderivatives

D is the Hooke tensoru are the displacements obtained through the XFEM

approximationb, t are body forces and tractions



From the above the usual system of equations results:

Kun = f

with the following structure: Kuu Kua Kub

Kau Kaa Kab

Kbu Kba Kbb

u

ab

=

fu

fa

fb

where elements of the stiffness matrix are obtained as:

KklIJ =

∫Ω

(Bk

I

)TDBl

JdΩ, k, l = u, a, b



The different parts of the load vector are obtained as:

fuI =

∫Ω

NIb dΩ +∫

ΓtNI t dΓ

faI =

∫Ω

NIHb dΩ +∫

ΓtNIH t dΓ +

∫Γt

c

NIH tc dΓc

fbIj =

∫Ω

NIFjb dΩ +∫

ΓtNIFj t dΓ +

∫Γt

c

NIFj tc dΓc


Numerical integration

In order to obtain the stiffness matrix and load vectors for theenriched approximation:

Discontinuous and singular functions need to be integrated

Usual Gauss integration cannot be used since it is only accuratefor polynomials

Element partitioning and special transformations are typicallyemployed


Numerical integration - Jump enrichment

Jump enriched elements are integrated using element partitioning:

Gauss point Integration element

FE approximation

of the crack

Crack


Numerical integration - Jump enrichment

For tip enriched elements polar integration is typically employed:


Enrichment shifting

By reviewing the displacement approximation:

u (x) =∑

I∈N NI (x) uI +∑

J∈N j NJ (x) H (φ) aJ +∑

T∈N t∑

j NT (x) Fj (r , θ) bTj

we observe that:

Enrichment functions do not vanish at nodal points

The standard FE dofs do not correspond to nodal displacements


Enrichment shifting

This can be fixed by employing “shifted” enrichment functions:

u (x) =∑I∈N

NI (x) uI +∑

J∈N j

NJ (x) (H (φ)− H (φJ)) aJ+

+∑

T∈N t

∑j

NT (x) (Fj (r , θ)− Fj (rT , θT )) bTj

The nodal values of the enrichment functions are subtractedfrom the enrichment functions themselves

Enrichment functions vanish at nodal points

Standard FE dofs correspond to nodal displacements


Stress Intensity Factors

Stress intensity factors are typically obtained using the interactionintegral:

I =∫A

[(σaux : ε) e1 −

(σ · ∂uaux

∂x + σaux · ∂u∂x

)]∇qdA

In the above:

Stress, strain and displacement fields are obtained from theXFEM solutionAuxiliary fields are obtained from the Westergaard solutionA system of coordinates defined from the LS gradients is usedFunction q needs to be defined


Stress Intensity Factors

To define q:

A radius rd is selectedNodes within that radius fromthe crack tip are given a valueqI = 1Nodes outside that radius aregiven a value qI = 0Values in the element interiorsare interpolated using the FEshape functions:

q =∑

INIqI

Interaction integral domain

q

crack

rd


Solution overview

The level sets are initialized for an initial crack geometry

The displacement, stress and strain fields are obtained withXFEM

SIFs are computed, the fracture criterion (e.g. the maximumcircumferential stress criterion) is evaluated and the propagationdirection is computed

The crack is propagated by an increment in the computeddirection

The LS crack description is updated

The procedure is repeated for several steps


Comparison to FEM

FEM and XFEM solution of a problem involving discontinuities andsingularities along moving interfaces (e.g. crack propagation):

Domain


Comparison to FEM


Initial mesh


Comparison to FEM


Crack appears → Re-meshing and mesh refinement is required


Comparison to FEM




Comparison to FEM




Comparison to FEM




Comparison to FEM




Cohesive zone models

In those models some forces are introduced which resist separation ofthe surfaces:

Those forces gradually reduce to a value of zero which correspondsto full separation


Weak form

The weak form in the presence of cohesive forces becomes:

∫Ωδε : D : ε dΩ︸︷︷︸

fint

+∫

Γcδ(u+ − u−

)· tc dΓc︸︷︷︸

−fc

= λ

∫Γtδu · t dΓ︸︷︷︸fext

where:

tc are the cohesive forces: tc = τcnn is the normal vector to the crack surface

u+,u− are the displacements at the two crack facesλ is a load factor


Enrichment functions

The modified step function is used for jump enrichment

Several alternatives have been proposed for tip enrichment, forinstance:

Fj (r , θ) = r sin θ2 or r3/2 sin θ2 or r2 sin θ2

we observe that a singularity is not present in this type offracture!


Cohesive laws

Different traction separation laws are possible:

Linear Law

Normal displacement jump

Norm

al t

ract

ion

Nonlinear Law

Normal displacement jumpN

orm

al t

ract

ion


Integration of cohesive forces

Cohesive forces:

Have to be integrated along the crack faces

Are non linear functions of the displacements

Introduce a non linearity even for linear elastic materials


Nonlinear solution

During the nonlinear solution the load factor needs to bedetermined

An additional condition is needed for the determination of theload factor

Usually one of the following conditions is used:

The mode I SIF at the tip should be zero: g = KI = 0

The stress projection normal to the surface of the crack shouldbe equal to the tensile strength of the material:g = n · σ · n− σc = 0


SIFs and propagation direction

Stress Intensity Factors are obtained using a modified version ofthe interaction integral

The propagation direction is determined using a criterion suchas the maximum circumferential stress criterion


Nonlinear solution

The residual and tangent stiffness for the Newton-Raphson methodare:

K− ∂fc∂u −fext

∂g∂u 0

· [ uλ

]= −

[Ku− λfext − fc

g

]

We observe the similarities to arc length methods!


Solution overview

Given an initial crack the displacements and load factor arecomputed using the nonlinear solution method presented

SIFs are computed and the direction of propagation is obtainedusing the maximum circumferential stress criterion

The crack is propagated by an increment in the computeddirection

The procedure is repeated for as many crack propagation stepsas needed


Extension to 3D

The methods presented mostly apply to 2D crack propagation

Extension of the discretization schemes (XFEM) isstraightforward

Extension of the fracture mechanics methods (failure criteriaetc.) is somehow more involved

Several alternatives exist in the literature each with itsrespective advantages and drawbacks


Documents

The eXtended Finite Element Method (XFEM) for … eXtended Finite Element Method (XFEM) for brittle and cohesive fracture Prof. Dr. Eleni Chatzi Dr. Giuseppe Abbiati, Dr. Konstantinos