Computational Fracture Mechanics

COMPUTATIONAL FRACTURE MECHANICS:

Overview:

Computers have had an enormous influence in virtually all branches

of engineering, and fracture mechanics is no exception.

Finite element modeling has become an indispensable step in

computational fracture mechanic problems since few cracked body

Problems! have closed form analytical solutions. (K, G, J, CTOD)

Stress Intensity Factor solutions for literally hundreds of cracked

body problems have been compiled, the majority of which were

inferred from Finite Element Analysis. Elastic-Plastic FEA to

compute the the J-integral and Crack Tip Opening

Displacement(CTOD) are also quite common.

Researches are developing Advanced Numerical Techniques to

special problems; such as fracture at interfaces, dynamic fracture

ductile crack growth,etc.

Rapid advances in computer hardware technology are primarily

responsible for exponential growth in Application of Computational

Fracture Mechanics.

Commercial FEM systems are highly user friendly and have

incorporated Computational Fracture Mechanics capabilities.

example:ANSYS with K-CAL command.

Experimental stress analysis methods such as photo elasticity, Moire

Interferrometry and Caustics are available, but even these techniques

require a numerical analysis to interpret experimental measurements/

observations.

A variety of numerical methods have been applied to problems in Solid

Mechanics including the FDM, FEM, and BEM. In recent years, the

FEM has been applied almost exclusively for the analysis for cracked

body problems. However, a number of special techniques are necessary

to compute fracture mechanics parameters (K, J, G, CTOD) from the

results of FEA (nodal displacements/element stresses).

Accurate SIF solutions for through cracks in plates and shells can be

obtained from Finite Element Analysis only.

Early researchers in CFM attempted to introduce “special finite

elements” at the crack tip that exhibit the 1/ singularity.

• Tracey,D.M. Finite elements for determination of crack tip

stress intensity factors, Engineering Fracture Mechanics,

vol.3,1971, pp.255-266

r

Barsoum later achieved the same effect by using doubly distorted

Quadratic Isoparametric element.This approach is now universal for FE

modeling of cracked solid /structures, since commercial FEM system

can be directly used.

Barsoum,R.S. On the use of isoparametric finite elements in LEFM, IJNME,

vol.10,1976.pp25-37.

However, there is real need to develop and add post processing programs to compute

the SIF’s / Energy released rates / J / CTOD / using standard output

from a FEA program.

More recent formulations of J computation apply an area integration for

2D problems and volume integration for 3D problems.This approach

provide much better accuracy than contour and surface integrals and

also much easier to implement in the post processor of FEA programs.

J-evaluation by the virtue crack extension(VCE) technique is an

example.

For crack problems, the design of a finite element mesh is as much

an art form as it is a science.Many commercial FEA codes have

automatic mesh generation modification commands. However,

realization of an appropriate FE mesh invariably requires some

human intervention. In particular, require a certain amount of

judgment on the part of the analyst.

8 noded quadrilateral 2D elements and 20 noded hexahedral 3D

elements are widely used in FE Modeling of cracked bodies. At the

cracktip, the quadrilateral elements are degenerated to triangular

elements.Note that the three nodes at the crack tip occupy the same

point in space.

In LEFM, the 3 nodes at the crack tip normally tied and mid nodes

moved to quarter point locations.such modifications results in a 1/

strain singularity within the element.Use of such singular elements

enhance accuracy of computed SIF’s / ERR’s.

When a plastic zone forms, the 1/ strain singularity is no longer valid at

the crack tip.Consequently use of elastic singular elements is not

appropriate for EPFM.Figure 11.15(b) shows a special element that

exhibits the desired strain singularity under fully plastic conditions.

The element is degenerated to a triangle shape as before, but the crack

tip nodes are untied and the location of mid-side nodes unchanged.This

element produces a 1/ r strain singularity, which corresponds to the crack

tip strain field for fully plastic, non-hardening materials.

r

r

(a) Elastic singularity element (b) Plastic singularity element

Fig.11.15. Crack tip elements for elastic and elastic-plastic analysis

Element (a) produces a 1/√r strain singularity, while (b) exhibits

a 1/r strain singularity

Fig.11.13. Degeneration of quadrilateral element in to a triangle at the crack tip

Fig11.14 Degeneration of a brick element in to wedge

One side benefit of the use of plastic singularity elements is that it

allows the Crack Tip Opening Displacement (CTOD) to be

computed from the deformed mesh as fig 11.16 illustrates.The

CTOD can be inferred from the deformed crack profile by means

of 90º intercept mesh.

Fig. 11.16. Deformed shape of plastic singularity elements ( fig. 11.15(b))

The crack tip elements model blunting and it is possible to measure CTOD

The most efficient mesh design for the crack tip region has proven

to be “spider web” configuration, which consist of concentric rings

of quadrilateral elements that are focused towards the crack tip.The

innermost ring are degenerated to triangular elements.Since the

crack tip region features steep stress and strain gradients, the mesh

refinement should be greatest at the crack tip.The “spider web”

design facilitates a smooth transition from a fine mesh at the crack

tip region to coarser mesh remote to the crack tip. Fig 11.17 shows

a half-symmetry FE Model of a single edge cracked panel.

Fig. 11.17. Half-symmetry model of a cracked panel

COMPUTATIONAL FRACTURE MECHANICS:

Benchmark for Mixed Mode Membrane Stress Intensity Factor Evaluation

Central circular arc crack in a rectangular panel:

Target SIF solution:

KI/σ0 , KII/σ0 α (0<α<90)

FE Modeling: Using singular and regular isoparametric elements

(STRIA6,QUAD8)

SIF evaluation : using K-VALUES a post processing subprogram

Convergence study: Model #1 NS=36, =a/100

Model #2 NS=72, =a/100

Results:

1)crack tip stress field

2)crack tip SIF’s

3)crack tip plastic zone(shape and size)

a a

aa

Vs

COMPUTATIONAL FRACTURE MECHANICS:

Limitations

FE Modeling of a cracked body can compute the crack tip FM

parameters. But this alone cannot predict when fracture will occur.

To predict fracture,we need a validated fracture criterion and

associated material property data (Fracture Toughness, etc.,.)

FEA relies on continuum mechanics which cannot model voids, micro

cracks, second phase particles, grain boundaries, dislocations, or any

other microscopic or submicroscopic features that in reality control the

fracture behavior of engineering materials.

Fracture process by itself can be modeled, but a separate

FRACTURE CRITERION is required. For example, one might

model cleavage fracture by using a stress based fracture criterion

in which FEA would predict fracture when user specified stress is

reached at a specified point ahead of crack tip.

FEA will undoubtedly play a major role in developing

micro mechanical models for fracture. Numerical simulation of

processes such as micro-crack nucleation,void growth, and

interface fracture should lead to new insights into fracture /damage

mechanisms. Such research may then lead to rational fracture

criteria that can be incorporated into global continuum models of

cracked bodies.