computational fracture mechanics (2)

دانشکده مکانیک -دانشگاه صنعتي اصفهان مکانیک شکست

دانشگاه صنعتي اصفهان دانشکده مکانیک

1

Computational Fracture Mechanics (2)

دانشکده مکانیک -دانشگاه صنعتي اصفهان مکانیک شکست 2

Introduction to Finite Element method

Singular Stress Finite Elements

Extraction of K (SIF), G

J integral

Finite Element mesh design for fracture mechanics

Computational crack growth

Traction Separation Relations

Computational fracture mechanics


Extraction of K

K from local fields

1. Displacement

or alternatively from the first quarter point element:

Recall for 1D:


Extraction of K

K from local fields

1. Displacement

Mixed mode generalization:


Extraction of K

K from local fields

2. Stress

or can be done for arbitrary angle (q) taking

s angular dependence f (q) into account

Stress based method is less accurate because:

• Stress is a derivative field and generally is one order less accurate than displacement

• Stress is singular as opposed to displacement

• Stress method is much more sensitive to where loads are applied(crack surface or far field)


Extraction of K

K from energy approaches

1. Elementary crack advance (two FEM solutions for a and a + Da)

2. Virtual Crack Extension: Stiffness derivative approach

3. J-integral based approaches (next section)

After obtaining G (or J =G for LEFM) K can be obtained from:

2 IK E G

2

plane stress

plane strain1

E

E E


Extraction of K


1. Elementary crack advance

Drawback:

Requires two solutions

Prone to Finite Difference (FD) errors

0edU

da

For fixed grip boundary condition perform two simulations (1, a) and (2, a+Da):

All FEM packages can compute strain (internal) energy Ui

1 2

Fixed grips:

( ) ( )1 1

i i idU U a a U aG

B da B a


Extraction of K


2. Virtual crack extension

1

2 u K u u p

Furthermore when the loads are constant:

2 1

2

TIKK

G u uE a

1

2

u K u pG K u u u p u

a a a a a

1

2

u K pK u p u u u

a a a

1

2

K pG u u u

a a

0

Potential energy is given by:


Extraction of K


2. Virtual crack extension

Only the few elements that are distorted contribute to .

K

a

We may not even need to form elements and assemble K for a and a+Da to

obtain . We can explicitly obtain for elements affected by crack growth

by computing derivatives of actual geometry of the element to parent geometry.

K

a

eK

a

This method is equivalent to J integral method (Park 1974)


Extraction of K

K from energy approaches 2. Virtual crack extension: Mixed mode

For LEFM energy release rates G1 and G2 are given by:

Obtain KI and KII from:

2 2 2

1 12

I II IIIK K K

J GE

2 2

2

I IIK K

J GE

Using Virtual crack extension (or elementary crack advance) compute G1 and G2 for

crack lengths a, a + Da (along θ = 0, and determine G1 and along θ = π determine G2).

2 1

2 2

8

4

8

4

I

II

Gs s

K

Gs s

K

1 22 ,

G G

s

(1 )(1 )

E

Note that there are two

sets of solutions!

that:


J integral

Methods to evaluate J integral:

Uses of J integral:

1. LEFM: Can obtain KI and KII from J integrals (G = J for LEFM)

2. Still valid for nonlinear (NLFM) and plastic (PFM) fracture mechanics.

2 2

2

I IIK K

J GE

2 2 2

1 12

I II IIIK K K

J GE

Contour integral:

utJ wdy d

x

Equivalent (Energy) domain integral (EDI):

• Gauss theorem: line/surface (2D/3D) integral surface/volume integral

• Much simpler to evaluate computationally

• Easy to incorporate plasticity, crack surface tractions, thermal strains, etc.

• Prevalent method for computing J-integral


J integral

Contour integral

Stresses are available and also more accurate

at Gauss points

Integral path goes through Gauss points

Cumbersome to formulate the integrand, evaluate normal vector, and integrate over lines

(2D) and surfaces (3D)

utJ wdy d

x2 2

1

1 2 1 21

1

1

1( ) ( )

2

x xy y x xy xy y

u u v v y u v x yJ n n n n d

x y x y x x

Id

s s s s

w dy

ut

xds

Not commonly used


J integral

Equivalent Domain Integral

General form of J integral

00

10 1

( )lim

j

i ij i

uJ w T n d

x s

0

mkl m

ij ijw d

s

Inelastic stress Kinetic energy density Can include (visco-) plasticity,

and thermal stresses

Elastic Plastic

Thermal (Q temperature)

total el pl m t

ij ij ij ij ij kk

: J contour approaches Crack tip

Accuracy of the solution deteriorates at Crack tip

Inaccurate/Impractical evaluation of J using contour integral

0 0


J integral

Divergence theorem: Line/Surface (2D/3D) integral Surface/Volume Integral

2D mesh covers

crack tip

Original J integral

contour Surface integral after

using divergence theorem

• Contour integral added to create closed surface

• By using q = 0 this integral in effect is zero

Application in FEM meshes

0 0

Equivalent Domain Integral

computational fracture mechanics (2)

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