Finite element analysis and fracture mechanics: an brief

introduction

J.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…

Str

ess

analy

sis

Fract

ure

mech

.

Fracture mechanics oriented design

Stress analys

is

•Perform a stress analysis

•Locate stress critical regions

Crackanalys

is

•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 evalua

tion

•Define operation safety conditions: maximum stress / crack length,… before failure occurs

•Define damage inspection intervals / maintainance plan

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

(see the rest of the course for more explanation on these concepts)

Stability of a crack: an introduction

New crack surface dA:Dissipates Ed=Gc*dA

AssumedCrack extension dA

Potential energy: P0=U0-V0

Potential energy: P1= P0 –Er

And Er = G*dA

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:

How to get these fields: a Finite Element simulation can be used to evaluate the displacement and stress field in any condition!

J-integral can be calculated in Abaqus.

Finite Element evaluation of ERR

t σ n

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

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

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

Mesh Convergence

Contour 1 Contour 2 Contour 3 Contour 4 Contour 537.6

37.8

38

38.2

38.4

38.6

38.8

39

Regular mesh

Singular Mesh

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