An introduction to Computationnal Fracture

MechanicsJ.Cugnoni, LMAF-EPFL, 2014

Stress based criteria (like Von Mises) usually define the onset of “damage” initiation in the material

Once critical stress is reached, what happens?

In this case, a defect is now present (ie crack)

The key question is now: will it propagate? If yes, will it stop by itself or grow in an unstable manner.

Stress based design vs Fracture Mechanics approach

A crack is formed…Will it extend further?If yes, will it propagate abruptly until catastrophic failure?

Stress concentrator:Critical stress is reached…

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Crack propagation : stress intensity factor◦ Stress intensity factors KI, KII, KIII measure the

intensity of stress singularity at crack tip. “Stress intensity factor”:

◦ Constants of the 1/sqrt(r) term in stress field at crack tip:

“Critical Stress intensity factor”: ◦ Maximum K that a material can sustain,

considered as a material property and indentified in standard fracture tests. Units: Pa/sqrt(m), symbol: KIc KIIc KIIIc

Crack propagation occurs if K > Kc with K=Kic +KIIc+KIIIc

In many materialy, propagation is mode I dominated:

KI>KIc

Propagation criteria of a crack

AssumedCrack extension dA

Stress singularity

Fracture test: measure force at failure and calculate KIcFrom analytical solutions or FE

Crack propagation : an “energetic” process◦ Extend crack length: energy is used to create a

new surface (break chemical bonds). ◦ Driving “force”: internal strain energy stored in

the system “Energy release rate”:

◦ Change in potential energy P (strain energy and work of forces) for an infinitesimal crack extension dA. Units: J/m2, Symbol: G

◦ measure the crack “driving force”

“Critical Energy release rate”: ◦ Energy required to create an additionnal crack

surface. Is a material characteristic (but depends on the type of loading). Units: J/m2, symbol: Gc

Crack propagation occurs if G > Gc

Propagation criteria of a crack

New crack surface dA:Dissipates Ed=Gc*dA

AssumedCrack extension dA

Potential energy: P0=U0-V0

Potential energy: P1= P0 –ErAnd Er = G*dA

Using numerical simulation method we can:◦ Apply Finite Element, Finite Volume/Difference,

Boundary Element or Meshfree methods

◦ To obtain approximations of displacement, force, stress and strain fields in arbitrary configuration.

And then? To apply fracture mechanics, we need: ◦ to compute fracture mechanics parameters

(SIF K, G) in 2D and 3D configurations;

◦ to compute J integral in elastic-plastic analyses ;

◦ to simulate crack growth (under general mixed-mode conditions);

Computational Fracture Mechanics: Why?

FE simulation of a compact tension fracture test

Calculation of K (linear elasticity):◦ Stress or displacement field matching (single / mixed mode)

◦ Indirectly from G (Interaction integrals in mixed mode)

Calculation of G:◦ Finite difference of potential energy (linear, evtl. non linear)

◦ Compliance method (linear)

◦ Virtual Crack Closure Technique VCCT (linear, mixed mode)

◦ J-integral (non-linear)

Simulate crack growth◦ to VCCT criterion, node release, remeshing or XFEM

◦ Cohesive elements / interfaces (Damage mechanics)

Overview of the different techniques

Idea:◦ compare FE stress field at crack tip with theory,

◦ fit KI from the numerical stress value

◦ In LEFM (for r 0):

Step 1: From FE: Extract stress field Syy at =0 (along crack direction)

Calculation of KI: stress matching

( 0) / 2 ( )yy IK r O r (eq 4.36a)

Fitting method 1: Plot (as a function of r, for r 0)

Calculation of KI: stress matching

0lim 2I yyr

K r

Fit a line over the quasi constant region of the plot, identify KI either as average value

or as the intercept at r=0

Fit region

Fitting method 2: Plot as a function of In theory the plot should be linear as:

Calculation of KI: stress matching' 1 / 2x r

Fit a line over the quasi linear region of the plot, identify KI either as the slope of the

line

yy

(1/ 2 ) ( ) 'yy I IK r O r K x b

Note : the same method can be applied to get Kii from shear component at theta=0

Idea:◦ compare FE crack displacement opening field with theory,

◦ fit KI from the numerical displacement value

◦ In LEFM (for r 0):

Step 1: From FE: Extract stress field Syy at =0 (along crack direction)

Calculation of KI: displacement matching

( 1)( ) / 24I

yKu u r

(eq 4.40b)

x

31

For plane stress: 3 4 For plane strain: = shear modulus= 2(1 )E

Fitting method 1: Plot (as a function of r=-x, for r 0)

Calculation of KI: displacement matching

Fit a line over the quasi constant region of the plot, identify either as the average value or as the intercept at r=0

Fit region

( 1)2 /4I

yKu r

( 1)4IK

Fitting method 1: Plot uy as a function

In theory, the plot should be linear as :

Calculation of KI: displacement matching

Fit a line over the quasi linear region of the plot, the slope is

' 1 / 2 /x r

( 1)4IK

( 1) ( 1)1 2 / '4 4I I

yK Ku r x

Evaluation of K from stress field requires very fine mesh. To better capture the 1/sqrt(r) stress singularity, one can use singular quadratic elements with shifted mid side nodes at ¼ of edge length (Barsoum,1976, IJNME, 10, 25; Henshell and Shaw, 1975, IJNME, 9, 495-507)

Improving mesh convergence: singular elements

By collapsing all nodes of one edge (degenerate a quadrangle to a triangle), the 1/sqrt(r) singularity covers then the whole element.

Can be extended into 3D for 20 node hexahedrons

Using 8 node second order quadrangular elements, if we move the midside nodes at ¼ of edge, we make the Jacobian transformation of the element singular as 1/sqrt(r) along the element edges.

For linear elastic isotropic materials, the following relations link G to K:

Computing K from G (and vice-versa)

2 2 21 1 18 8 2I I II II III IIG K G K G K

31

For plane stress: 3 4 For plane strain: = shear modulus= 2(1 )E

For Mode I in plane stress and plane strain, this simplifies to:

2 2 2(1 ): :I II IK Kplane stress G plane strain GE E

So knowing either K or G, the other can be determined directly. This can be extended to orthotropic material.

◦ compare the strain energy of several FE models with different crack length, calculate the ERR as the derivative of potential energy P = U - F

◦ For a linear elastic material: the potential energy F = PD 2U thus P = U – F = - U with the strain energy

◦ Compute ERR or as the slope of U(a)

Calculation of G: Strain energy method

Different FE models for a0, a1, a2…

12 ij ij

V

U dV

a

( ) ( )U a a U aGa a

P D

D

Equations are for a unit depth, if not the case, compute use U/b instead of U

a

P, D

Idea (similar to experimental test data reduction !)◦ Calculate the compliance C(a)=D(a)/P(a) for different crack lengths

a0, a1 (several FE models required) and fit it with an appropriate function

◦ For a linear elastic material: the ERR can be obtained from the C(a)

◦ Compute ERR where dC/da is obtained from the fitted curve

Calculation of G: Compliance method

Different FE models for a0, a1, a2…

2 2

2 2P cst cst

P dC P dCG or Gb da b da D

(eq 3.41 & 3.43b)

Using the “J-Integral” approach (see course), it is possible to calculate the ERR G as G = J in linear elasticity.

If we know the displacement and stress field around the crack tip, we can compute J as a contour integral:

J-integral is path independent for all continuum materials. But G must be perpendicular to the interface if dissimilar materials are used.

Calculation of G: J-integral method

t σ n

W=strain energy densityu = displacement fields = stress fieldG = contour: ending and starting at crack surface

Calculation of G: J-integral method J-integrals are usually computed from a volume/surface integral in

FE, for example in Abaqus with q = virtual crack extension vector:

With on G and on C. Using divergence and equilibrium equations we can obtain:

J-integral in 3D: 3D effectsJ-integral can be extend in 3D by computing J on several “slices” of the model.

However, there are notable 3D effects affecting crack propagation:in 3D, the external surface is free of normal stress, so in plane stress stateFor thick specimens, the center of the specimen is closer to plane strain state

S22 on external surfacePlane stress

S22 on symmetry planePlane strain => higher stresses

J-integral in 3D: 3D effectsAs a consequence, J-integral and G are not constant across the width.This will lead to a slightly curved crack front during propagation

G is max in the center => propagate earlier

-Need to be careful about specimen size effects when characterizing G or K- In 2D FE simulation: use plane stress for very thin specimens, and plane strain for thick ones.

G is min on the side => propagate last

Results obtained on a relatively coarse 3D mesh to highlight which methods are less mesh sensitive

Comparison of methods

External surface, plane stress

  Stress fit1 Stress fit2 Displacement fit1 Displacement fit2From J-integral

K from Abaqus

K1 3870 4345 4380 5079 4337 4540(units: Mpa*sqrt(mm))

Inside, plane strain (* please note that the FE data for u2 and s22 were extracted on free surface, a source of error)

  Stress fit1 Stress fit2 Displacement fit1 Displacement fit2 J-integralK from Abaqus

K1 4345 3870 4813 5582 5144 5140

 

Strain energy, forward finite

difference

Strain energy, backward finite

difference

Strain en. centered finite

differenceStrain energy fit Compliance

method J integ. Avg

G 322 282 302 302.5 301.5 296(units: mJ/mm2)

Conclusion: K determination is more sensitive to mesh ! G calculation using compliance, J-integral or strain energy fit are the most reliable

Can be calculated in elasticity / plasticity in 2D plane stress, plane strain, shell and 3D continuum elements.

Requires a purely quadrangular mesh in 2D and hexahedral mesh in 3D.

J-integral is evaluated on several “rings” of elements: need to check convergence with the # of ring)

Requires the definition of a “crack”: location of crack tip and crack extension direction

J-integral in Abaqus

Rings 1 & 2

Crack plane

Crack tip andextension direction

Quadrangle mesh

Create a linear elastic part, define an “independent” instance in Assembly module

Create a sharp crack: use partition tool to create a single edge cut, then in “interaction” module, use “Special->Crack->Assign seam” to define the crack plane (crack will be allowed to open)

In “Interaction”, use “Special->Crack->Create” to define crack tip and extension direction (can define singular elements here, see later for more info)

In “Step”: Define a “static” load step and a new history output for J-Integral. Choose domain = Contour integral, choose number of contours (~5 or more) and type of integral (J-integral).

Define loads and displacements as usual Mesh the part using Quadrangle or Hexahedral elements, if possible quadratic.

If possible use a refined mesh at crack tip (see demo). If singular elements are used, a radial mesh with sweep mesh generation is required.

Extract J-integral for each contour in Visualization, Create XY data -> History output.

!! UNITS: J = G = Energy / area. If using mm, N, MPa units => mJ / mm2 !!!

By default a 2D plane stress / plane strain model as a thickness of 1.

J-Integral in Abaqus: application notes and demonstration

See demo1.cae example file

To create a 1/sqrt(r) singular mesh:◦ In Interaction, edit crack definition and

set “midside node” position to 0.25 (=1/4 of edge) & “collapsed element side, single node”

◦ In Mesh: partition the domain to create a radial mesh pattern as show beside. Use any kind of mesh for the outer regions but use the “quad dominated, sweep” method for the inner most circle. Use quadratic elements to benefit from the singularity.

◦ Refine the mesh around crack tip significantly.

Singular elements & meshing tips

J-integral Convergence vs # contours

Contour 1 Contour 2 Contour 3 Contour 4 Contour 537.6

37.8

38

38.2

38.4

38.6

38.8

39Regular meshSingular Mesh

Fracture mechanics oriented design

Stress analysis•Perform a stress analysis•Locate stress critical regions

Crackanalysis

•Assume the presence of a defect in those regions (one at a time)•Consider different crack lengths and orientation•For each condition, check if the crack would propagate and if yes if it is

stable or not

Design evaluation •Define operation safety conditions: maximum stress / crack length,…

before failure occurs•Define damage inspection intervals / maintainance plan

Abaqus tutorials:◦ http://lmafsrv1.epfl.ch/CoursEF2012

Abaqus Help:◦ http://lmafsrv1.epfl.ch:2080/v6.8◦ See Analysis users manual, section 11.4 for

fracture mechanics Presentation and demo files:

◦ http://lmafsrv1.epfl.ch/jcugnoni/Fracture Computers with Abaqus 6.8:

◦ 40 PC in CM1.103 and ~15 in CM1.110

Resources & help