Computational FM 1

COMPUTATIONAL FRACTURE MECHANICS

WHY?

- to compute fracture mechanics parameters (SIF, G) in 2Dand 3D configurations;

- to compute J integral and CTOD in elastic-plasticanalyses;

- to simulate crack growth (under general mixed-modeconditions);

- to solve special problems: dynamic fracture, ductilefracture, cohesive fracture, fracture at interfaces, ….

NUMERICAL METHODS:

To determine the distribution of stresses and strains in acracked body subject to external loads or displacements(when close-form analytical solutions are not available)

- The Finite Difference Method;- The Finite Element Method;- The Boundary Element Methods.

Computational FM 2

TECHNIQUES FOR COMPUTING FRACTUREMECHANICS PARAMETERS

POINT MATCHING APPROACHES:

1) STRESS MATCHING

2) DISPLACEMENT MATCHING(The Displacement Correlation Technique)

3) STRESS FUNCTION MATCHING

ENERGY APPROACHES:

4) THE GLOBAL ENERGY RELEASE METHOD

5) THE STIFFNESS DERIVATIVE TECHNIQUE (for FEM)

6) CONTOUR INTEGRATION

7) THE ENERGY DOMAIN INTEGRAL METHOD

8) THE MODIFIED CRACK-CLOSURE INTEGRAL TECHNIQUE

Computational FM 3

POINT MATCHING APPROACHES:

Idea: to correlate numerical solutions for stresses ordisplacements at specific locations with analytic solutionsdepending on the stress intensity factors.

Hp: homogeneity, isotropylinearly elastic material

1) STRESS MATCHING

hp: mode I loading.

( ) 0)( r 2 0r

limKI =

→= θπσ 22

2) DISPLACEMENT MATCHING

hp: mode I loading.

)( r

2u 0r

lim

1κ2GK 2I πθπ =

→+

=

κ = 3−4ν (plane strain)κ = (3−ν)/(1+ν) (plane stress)

better than 1): higher precision in the numerical calculationof displacements (continuity between elements)

Computational FM 4

3) MATCHING STRESS FUNCTIONS(The Boundary Collocation Method)

- The Airy stress function is expressed in terms of complexpolynomials. The coefficients of the polynomials are inferredfrom nodal quantities. SIF's are inferred from the stressfunctions. (highly cumbersome)

Drawbacks:

- High degree of mesh refinement is required forengineering accuracy even for simple geometry, loadingand a single crack.

- Use of special elements at the crack tip that exhibit the1/√r singularity is necessary (enriched elements, quarterpoint elements).

Computational FM 5

2) THE DISPLACEMENT CORRELATION TECHNIQUE

(Ref. Chan, Tuba and Wilson, Eng. Fract. Mech., 2(1),1970)

Hp: homogeneity, isotropylinearly elastic material

General form of displacement function near crack tip:

.... 2

sin 2- 1)- 2

cos 2r

2GK -

.... 2

2cos - 1) 2

sin 2r

GK =u

2II

2 I 2

+

+

+

θκθπ

θκθπ

(

(2

.... 2

cos 2+ 1) 2

sin 2r

2GK

.... 2

2sin + 1)- 2

cos 2r

GK = u

2II

2 I1

+

++

+

θκθπ

θκθπ

(

(2

κ = 3−4ν (plane strain)κ = (3−ν)/(1+ν) (plane stress)

Consider a FE mesh at crack tip:

Computational FM 6

Pure Mode I and plane stress:

.... 2

cos - 1

2 2

sin 2r

GK =u 2 I

2 +

+θ

νθ

π

.... 2

sin +- 2

cos 2r

GK = u 2 I

1 +

+θ

ννθ

π 11

taking θ = ± π and r = rAB:

)u -(u 4

)G(1 r2 K 2C 2BAB

*I

νπ +=

Pure Mode II and plane stress:

2

sin - 2

cos 2r

GK =u 2II

2 ....1

1 +

+

+θ

ννθ

π

... 2

cos + 1

2 2

sin 2r

GK = u 2II

1 +

+θ

νθ

π

taking θ = ± π and r = rAB:

)u -(u 4

)G(1 r2 K 1C 1BAB

*II

νπ +=

Mixed Mode and plane stress:

Mode I and mode II problems uncouple at θ = ± π.Consequently previous equations are still applicable.

Computational FM 7

For Quarter Points Singular Elements:(Shih, de Lorenzi and German, 1976, IJF, 12, 647-651)

- If FE mesh is composed of quarter point elements at cracktip:

]1

)u-(u -)u -[4(u G r2 K 2E2D2C2BAD

*I +

=κ

π

)u-(u -)u -[4(u G r2 K 1E1D1C1BAD

*II ]

1+=

κπ

(from shape functions of element and analytical expressionsfor COD and CSD).

Computational FM 8

FINITE ELEMENT MESH DESIGN

- In homogeneous, isotropic, elastic material, the stress fieldat crack tip exhibits a 1/√r singularity.

- The FE mesh must be such that the stress fieldcharacteristics are reproduced in the numerical solutions.

Solutions:

- Mesh refinement at crack tip (slow convergence of local parameters)

- Use of “enriched” elements at crack tip (Wilson element;Tracey element, …):

stress / displacement variations around crack tip areembedded in the shape functions of element

(Tracey, 1971, Eng. Fract. Mech., 3, 255-266)

(edge displacement incompatibility with surroundingelements)

- Use of Quarter Point Singular Elements:

in 8-nodes isoparametric quadrilateral elements, placing themid-side nodes on or outside the ¼ of the side causes theJacobian of transformation to become non positive definite.The 1/√r singularity is obtained.

(Barsoum,1976, IJNME, 10, 25; Henshell and Shaw, 1975, IJNME, 9, 495-507)(only nodal coordinate input data are altered. Elementsatisfies essential converge criteria)

Computational FM 9

- If the 8-nodes element is first degenerated to anisoparametric triangle with 3 nodes at crack tip and thendistorted by moving the side nodes, improved results areobtained: 1/√r singularity also within element not only alongedges.

- In 3D, analogous results obtained by distorting threedimensional 20-nodes isoparametric elements(Ingraffea and Manu,1980, IJNME, 15, 1427-1445)

Computational FM 10

QUARTER POINT ELEMENTS: PROPERTIES

Computational FM 11

Computational FM 13

ENERGY APPROACHES:

1) THE GLOBAL ENERGY RELEASE METHOD:

Hp: 2D linear elastic body.

From energy balance:

dadW-=G

W = total potential energy per unit width.

- Perform two analyses, one for crack length a, the secondfor a +∆a, for constant load. Compute G from:

aU

aU-

aL

aW-

∆∆=

∆∆

∆∆=

∆∆=G

L = potential of the applied loads per unit width in a 2Dbody;U = elastic strain energy stored in the body per unit width

- For pure mode I, compute SIF from:

GE = KI (plane stress)

2I - 1E = K

νG

(plane strain)

(hp: isotropy, homogeneity)

Computational FM 14

Drawbacks:

- two analyses required for KI computation. At least threeanalyses for KI and KII computations. Four analyses for KI,KII and KIII.

- SIF's not directly computed. Postprocessing of results isrequired.

- For mixed-mode fracture it is very difficult to separate Ginto its mode I and mode II components.

Advantages:

- no special crack-tip elements are necessary. Relativelycoarse meshes can be used.

- When used in conjunction with the Stiffness DerivativeTechnique, only partial reanalysis is necessary.

Computational FM 15

2) THE STIFFNESS DERIVATIVE TECHNIQUE

(Parks, IJF, 10, 1974, 487-502)

- Formulated in terms of FE stiffness matrix. Not compatiblewith boundary element analysis.

- Total potential energy of FE solution:

W = 1/2 uT[K]u - uTP

[K] = structure stiffness matrix;u = vector of nodal displacements;P = vector of applied nodal forces.

- Differentiate to obtain G:

( )a

a

21 -

a -= TT

T

∂∂+

∂∂−

∂∂ PuuKuPuKu ][][G

where first term on right hand side must be zero.

- If applied load is independent of crack length and puremode I:

uKu a

21 -

E'K= T

2I

∂∂= ][

G

Computational FM 16

-If only elements surrounding the crack tip are modified bycrack advance ∆a (see figure), then [K] remains unchangedin the region outside:

( )uKKuuKu a

21 -

a

21 -

E'K=

nc

1ia

ciaa

ci

Tnc

1i

ciT

2I ∑∑

=∆+

=−

∆≅

∂∂= ][][1][

G

[Kci] = stiffness matrix of the ith element surrounding the

crack tip (within Γ1 in the figure);nc= number of such elements;[Kc

i]a element stiffness matrix for original crack configuration;[Kc

i]a+∆a element stiffness matrix for new crack configuration;

Computational FM 17

6) CONTOUR INTEGRATION

The J integral can be evaluated numerically along a contour(in 2D) or a surface (in 3D) surrounding the crack tip. Thenin an elastic body:

J= G

Advantages:

- Path independence enables the user to calculate J at aremote contour (surface) where numerical accuracy isgreater.- No mesh refinement is required.- J can be separated into its mode I and mode IIcomponents to deduce KI and KII.

Drawbacks:

- Contour/surface integrals are difficult to implementnumerically and the accuracy is not high.

Solution:

- The Energy Domain Integral: the J integral is formulated interms of area (for 2D) or volume (for 3D) integrals.

Better accuracy. Easier to implement.

Computational FM 18

7) THE ENERGY DOMAIN INTEGRAL METHOD

(Shih, Moran and Nakamura, 1986, IJF, 30, 79-102)

The J integral for a 2D body is:

∫∫ ∂∂=

∂∂=

ΓΓJ dsn

xu-nUds

x-dxU j

1

iijdd σ1

12

uT

- Consider the close path Γn = Γ0 + ΓS+

+ ΓS- - Γ .

- Introduce a weight function q(x1,x2) that is equal to unity onΓ and zero on Γ0 and Γs.

The J integral along the new path is:

dsn q xu-U- j

iijjd

n∫

∂∂=

ΓJ

11 σδ

Computational FM 19

- Apply the divergence theorem to transform the integralalong the close contour into a domain integral:

dA q xu

x-

xU-dA

xq

xu-

xqU-

1

iij

j

d

A j

iijd ∫∫

∂∂

∂∂

∂∂

∂∂

∂∂

∂∂=

AJ σσ

111

where the second term vanishes for elastic problems.

- Implement the domain integral into a finite element code(e.g. Franc).

Mode separation:(Bui, J.M.P.S., 31(6), 439-448, 1983)

- The displacement field in the crack-tip region isdecomposed into symmetric and antisymmetriccomponents: u = usymm. + uantisymm.

- The J integral is evaluated twice: once with the symmetricdisplacements to find J1 and a second time with theantisymmetric displacements to find J2.

- For elastic problems:GI = J1

GII = J2

SIF’s are found from GI and GII.

Computational FM 20

8) THE MODIFIED CRACK-CLOSUREINTEGRAL TECHNIQUE

(Rybicki and Kanninen, 1977)

- The strain energy release rate G is estimated in terms ofthe work done by the stresses ahead of the crack tip overthe displacements produced by the introduction of a virtualcrack extension (Irwin concept of crack-closure integral):

1121

a

22 ,0)dxx-a(u ,0)(x21

a2

0alim

I∆

∆→∆= ∫

∆

0σG

1111

a,0)dxx-a(u ,0)(x

21

a2

0alim

II∆

∆→∆= ∫

∆

012σG

- Two numerical analyses are required to obtain the stressfield ahead of the crack before propagation and to computethe displacement field after a virtual crack extension isintroduced.

Computational FM 21

- Simplification proposed by Rybicki and Kanninen:

the displacement field ahead of the crack is approximatedby the the displacement field behind the crack tip.

The problem is then solved with one single analysis step.

- In the finite element approach the crack closure integralsare rewritten in terms of equivalent nodal forces and relativenodal displacements (different expressions for each cracktip element).

Advantages:

- no assumption of isotropy or homogeneity around thecrack tip is necessary.

Computational FM 22

MIXED-MODE CRACK GROWTH

- The direction of crack growth is defined using theMaximum Circumferential Stress Criterion.(Erdogan and Sih, 1963, J. Basic Engrg., 85, 519-527)

- The crack is propagated of ∆a along the calculateddirection.

- Remeshing: a new mesh is generated around the newcrack tip. Different techniques.

Franc3D code:

Solid Modeling Techniques to represent the desiredgeometrical aspects of the structure explicitly. Thediscretization is mapped onto this original solid model.

Remeshing: the solid model remains valid and the newmesh is remapped into the solid model and inherits thenecessary attributes.

(see Franc3D manual for more details and references)

- New fracture parameters at a + ∆a are calculated.