frac l03 contour

Fracture Analysis

Lecture 3

L3.2

Modeling Fracture and Failure with Abaqus

Overview

• Calculation of Contour Integrals

• Examples

• Nodal Normals in Contour Integral Calculations

• J-Integrals at Multiple Crack Tips

• Through Cracks in Shells

• Mixed-Mode Fracture

• Material Discontinuities

• Numerical Calculations with Elastic-Plastic Materials

• Workshop 1

• Workshop 2

Calculation of Contour Integrals

L3.4



• Abaqus offers the evaluation of J-integral values, as well as several

other parameters for fracture mechanics studies. These include:

• The KI, KII, and KIII stress intensity factors, which are used mainly

in linear elastic fracture mechanics to measure the strength of local

crack tip fields;

• The T-stress in linear elastic calculations;

• The crack propagation direction: an angle at which a preexisting

crack will propagate; and

• The Ct-integral, which is used with time-dependent creep behavior.

• Output can be written to the output database (.odb), data (.dat), and

results (.fil) files.

L3.5



• Domain representation of J

• For reasons of accuracy, J is evaluated

using a domain integral.

• The domain integral is evaluated over

an area/volume contained within a

contour surrounding the crack tip/line.

• In two dimensions, Abaqus defines the

domain in terms of rings of elements

surrounding the crack tip.

• In three dimensions, Abaqus defines a

tubular surface around the crack line.

L3.6



• Different contours (domains) are

created automatically by Abaqus.

• The first contour consists of the

crack front and one layer of

elements surrounding it.

• Ring of elements from one

crack surface to the other (or

the symmetry plane).

• The next contour consists of the

ring of elements in contact with the

first contour as well as the

elements in the first contour.

• Each subsequent contour is

defined by adding the next ring of

elements in contact with the

previous contour.

Contour 1 Contour 2

Contour 3 Contour 4

L3.7



• The J-integral and the Ct-integral at steady-state creep should be

path (domain) independent.

• The value for the first contour is generally ignored.

• Examples of contour domains:

2nd

contour

1st

contour

Crack-tip node crack-front nodes

1st contour2nd contour

Crack-tip node

L3.8



• Usage:

*CONTOUR INTEGRAL, CONTOURS= n,

TYPE={J, C, T STRESS, K FACTORS},

DIRECTION = {MTS, MERR, KII0}

Note: In this lecture, we focus on the output-specific parameters of the *CONTOUR INTEGRAL

option. The crack-specific parameters SYMM and NORMAL were discussed in the previous lecture.

Specifies the number of contours (domains)

on which the contour integral will be

calculated

This is the output

frequency in

increments

L3.9



• Usage (cont’d):




• J for J-integral output,

• C for Ct-integral output.

• T STRESS to output T-stress

calculations

• K FACTORS for stress intensity

factor output

L3.10



• Usage (cont’d):




• Use with TYPE=K FACTORS to specify the criterion to be

used for estimating the crack propagation direction in

homogenous, isotropic, linear elastic materials:

• Maximum tangential stress criterion (MTS)

• Maximum energy release rate criterion (MERR)

• KII = 0 criterion (KII0)

Three criteria to calculate the crack

propagation direction at initiation

L3.11



• Output files

*CONTOUR INTEGRAL, OUTPUT

• Set OUTPUT=FILE to store the

contour integral values in the results (.fil) file.

• Set OUTPUT=BOTH to print

the values in the data and

results files.

• If the parameter is omitted, the

contour integral values will be printed in the data (.dat) file

but not stored in the results (.fil) file.

L3.12



• Loads

• Loads included in contour integral calculations:

• Thermal loads.

• Crack-face pressure and traction loads on continuum elements as well as those applied using user subroutines DLOAD and UTRACLOAD.

• Surface traction and crack-face edge loads on shell elements as well as those applied using user subroutine UTRACLOAD.

• Uniform and nonuniform body forces.

• Centrifugal loads on continuum and shell elements.

• Not all types of distributed loads (e.g., hydrostatic pressure and gravity

loads) are included in the contour integral calculations.

• The presence of these loads will result in a warning message.

L3.13



• Other loads not included in contour integral calculations:

• Contributions due to concentrated loads are not included.

• If needed, modify the mesh to include a small element and

apply a distributed load to the element.

• Contributions due to contact forces are not included.

• Initial stresses are not considered in the definition of contour

integrals.

Examples

L3.15


Examples

• Penny-shaped crack in an infinite space

• Model characteristics

• The mesh is extended far enough

from the crack tip so that the finite

boundaries will not influence the

crack-tip solution.

• The radius of the penny-shaped

crack is 1.

• Two types of loading are

considered:

• Uniform far-field loading

• Nonuniform loading on the

crack face: p = Arn.

L3.16


Examples

• Different mesh characteristics:

• Axisymmetric or three-dimensional

• Fine or coarse focused meshes

• With or without ¼ point elements

• Various element types used:

• First- and second-order

• With and without reduced integration

Axisymmetric model

20

20

Focused mesh around

crack tip

Crack tip

L3.17


Examples

• Fine mesh vs. coarse mesh (axisymmetric and 3D models)

~0.08

0.080.0004

The fine mesh is shown to the left;

the coarse mesh above. The length

perpendicular to crack line of the

crack-tip elements are indicated.

L3.18


Examples

• Axisymmetric model: geometry

Model geometry

Close up of crack tip region for

coarse mesh model (identical for

fine mesh model—only the inner

semicircular region is smaller)

Symmetry planes

L3.19


Examples

• Axisymmetric model: crack definition

Crack tip with extension direction

Set to 0.5 to use mid-

point rather than ¼ point

elements

L3.20


Examples

• 3D model: geometry and mesh

• A 90 sector is modeled because

of symmetry.

Additional partition

required for swept

mesh

On planes perpendicular to the crack

front, the mesh is very similar to the

axisymmetric mesh

In the circumferential direction around

the crack line, 12 elements are used.

Partitions used for coarse mesh model

(identical for fine mesh model—only

the inner semicircular region is smaller)

Fine 3D mesh

Symmetry planes

L3.21


Examples

• Why is the additional partition required?

• Without the additional partition, the region shown below would require

irregular elements at the vertex located on the axis of symmetry.

• This is not supported by Abaqus.

A 7-node element

is an example of an

irregular element.

Irregular elements

required here

because revolving

about a point

L3.22


Examples

• 3D model: crack definition

• Orphan mesh created to edit q

vectors.

L3.23


Examples

• Contour integral output requests (axisymmetric and 3D)

Separate output

requests are required

for J, K-factors, and the

T-stress.

L3.24


Examples

• Loads (axisymmetric and 3D)

The far-field load is suppressed.

L3.25


Examples

• Results

• MISES stress shown below for

the axisymmetric fine mesh.

Analytical Contour 1 Contour 2 Contour 3 Contour 4 Contour 5

5.796E-02 5.8169E-02 5.8095E-02 5.8121E-02 5.8104E-02 5.8084E-02

Contour 6 Contour 7 Contour 8 Contour 9 Contour 10

5.8064E-02 5.8044E-02 5.8024E-02 5.8005E-02 5.7985E-02

Deformation scale factor = 250

100%analytical numerical

analytical

J J

J

L3.26


Examples

J values from meshes with ¼ point elements (reduced integration)

• Abaqus values are based on the average of contours 3−5 in each mesh.

LoadingAnalytical

result

3-D Axisymmetric

C3D20R CAX8R

Coarse Fine Coarse Fine

Uniform

far field.0580 .0578 .0580 .0579 .0581

Uniform

crack face.0580 .0578 .0580 .0579 .0581

Nonuniform

crack face (n = 1).0358 .0356 .0357 .0356 .0358

Nonuniform

crack face (n = 2).0258 .0256 .0260 .0256 .0258

Nonuniform

crack face (n = 3).0201 .0199 .0206 .0200 .0202

L3.27


Examples

J values from meshes with ¼ point elements (full integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20 CAX8

Coarse Fine Coarse Fine

Uniform

far field.0580 .0577 .0572 .0578 .0580

Uniform

crack face.0580 .0577 .0572 .0578 .0580

Nonuniform

crack face (n = 1).0358 .0355 .0352 .0356 .0358

Nonuniform

crack face (n = 2).0258 .0255 .0253 .0255 .0258

Nonuniform

crack face (n = 3).0201 .0198 .0197 .0199 .0201

L3.28


Examples

J values from meshes without ¼ point elements (reduced integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20R C3D8R CAX8R CAX4R

Coarse Fine Coarse Coarse Fine Coarse

Uniform

far field.0580 .0574 .0580 .0563 .0574 .0581 .0562

Uniform

crack face.0580 .0574 .0580 .0563 .0574 .0581 .0562

Nonuniform

crack face (n = 1).0358 .0350 .0357 .0336 .0350 .0358 .0337

Nonuniform

crack face (n = 2).0258 .0250 .0260 .0234 .0250 .0258 .0236

Nonuniform

crack face (n = 3).0201 .0193 .0206 .0177 .0193 .0202 .0179

L3.29


Examples

J values from meshes without ¼ point elements (full integration)


LoadingAnalytical

result

3-D Axisymmetric

C3D20 C3D8 CAX8 CAX4

Coarse Fine Coarse Coarse Fine Coarse

Uniform

far field.0580 .0573 .0572 .0552 .0574 .0580 .0557

Uniform

crack face.0580 .0573 .0572 .0552 .0574 .0580 .0557

Nonuniform

crack face (n = 1).0358 .0350 .0352 .0329 .0350 .0358 .0333

Nonuniform

crack face (n = 2).0258 .0249 .0253 .0229 .0250 .0258 .0232

Nonuniform

crack face (n = 3).0201 .0193 .0197 .0172 .0193 .0201 .0175

L3.30


Examples

• Conclusions

• 3D fine meshes with second-order elements are more sensitive to the

choice of integration rule when determining J.

• The results are still very accurate (within 2% of analytical value).

• The inclusion of the singularity helps most in the coarser meshes.

• For mesh convergence in small strain, the singularity must be

included.

L3.31


Examples

• Conical crack in a half-space

• At each node set along the crack front, the crack propagation direction is

different.

L3.32


Examples

• Three-dimensional model

• Displaced shape and Mises stress distribution of full three-

dimensional model.

Deformation scale factor = 1.e6

L3.33


Examples

• J values of three-dimensional mesh

• There is some oscillation between J values evaluated at corner

nodes compared to J values evaluated at midside nodes.

Variation of J with angular position

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

1.338E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l 3D contour 5

3D contour 4

3D contour 3

3D contour 2

L3.34


Contours 3-5 have

converged

Examples

• Axisymmetric model and results

Axisymmetric results are

used as reference results.

L3.35


Examples

• Comparison of axisymmetric and 3D results


Contour 1

1.300E-07

1.320E-07

1.340E-07

1.360E-07

1.380E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 2

1.329E-07

1.330E-07

1.331E-07

1.332E-07

1.333E-07

1.334E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 3

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI


Contour 5

1.328E-07

1.330E-07

1.332E-07

1.334E-07

1.336E-07

1.338E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI

L3.36


Examples

• Since the three-dimensional mesh is quite coarse around the axis of

symmetry, these results are considered to be good—the error is less

than 0.5% for all but the first contour.

% difference in J between AXI and 3D results

0.00.51.01.5

2.02.53.03.5

0 45 90

Angle (degrees)

% d

iffe

ren

ce Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

L3.37


Examples

• Submodeling

• We can use submodeling to create

two meshes that are significantly

smaller than the full three-

dimensional model.

• The top-right figure is the

coarse mesh global model in

the vicinity of the crack.

• The bottom-right figure shows

the refined submodel mesh

overlaid on the global model

mesh.

L3.38


Examples

• J values of submodel:

• Inaccuracies are introduced

by the coarser mesh used in

the global model.

• Errors in J are less than 1%.

• CPU time was reduced by a

factor of 3.


1.318E-07

1.320E-07

1.322E-07

1.324E-07

1.326E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l 3D contour 5

3D contour 4

3D contour 3

3D contour 2


Contour 5

1.315E-07

1.320E-07

1.325E-07

1.330E-07

1.335E-07

0 45 90

Angle (degrees)

J-i

nte

gra

l

3D

AXI

% difference in J between AXI and 3D results

0.00.51.01.52.02.53.03.54.04.5

0 45 90

Angle (degrees)

% d

iffe

ren

ce Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

L3.39


Examples

• Compact Tension Specimen

• This is one of five standardized specimens defined by the ASTM for the

characterization of fracture initiation and crack growth.

• The ASTM standardized testing apparatus uses a clevis and a pin to

hold the specimen and apply a controlled displacement.

L3.40


Examples

• Model details

• Plane strain conditions assumed.

• The initial crack length is 5 mm.

• Elastic-plastic material

• Low alloy ferritic steel

Crack seam

q-vector

1/√r singularity modeled in

the crack-tip elements

Prescribed load line displacement

L3.41


Examples

• Results

Small strain analysis Finite strain analysis

L3.42


Examples

At small to moderate strain levels,

the small and finite strain models

yield similar results.

Finite strain effects must be

considered to represent this level of

deformation and strain accurately.

Nodal Normals in Contour Integral

Calculations

L3.44


Nodal Normals in Contour Integral Calculations

• Sharp curved cracks

• For sharp cracks, if the crack faces

are curved, Abaqus automatically

determines the normal directions of

the nodes on the portions of the crack

faces that lie within the contour

integral domains.

• This improves the accuracy of the

contour integral estimation.

• The normal is not used at the

crack-tip node, however.

Normals to top crack

surface nodes

n (normal to

crack plane)

Normals to bottom

crack surface nodes

q

L3.45



• Example: sharp curved crack

Contour # 1 2 3 4 5

J without normals 3.363 2.980 2.475 1.888 1.283

J with normals 3.600 3.602 3.605 3.605 3.605

L3.46



• Blunt cracks and notches

• All nodes on the notch should be included in the crack-tip node set.

• The J-integral results are more accurate since the q vector is

parallel to the crack surface in this case, as illustrated below.

Crack surfaceCrack surface

Single node in crack-tip node set;

normals calculated on nodes of

blunted surface; q not parallel to

crack surface.

All nodes on blunted surface in

crack-tip node set; q parallel to

crack surface.

Paths for contour

integrals

n

q q

J-Integrals at Multiple Crack Tips

L3.48


J-Integrals at Multiple Crack Tips

• Abaqus can calculate J (or Ct) at multiple crack tips

• Abaqus/CAE: multiple crack tips and history

output requests

• Input file: repeated use of the *CONTOUR

INTEGRAL option.

• If the domain for one crack tip envelopes the other

crack tip, the J value will go to zero (as it should).

Through Cracks in Shells

L3.50



• Second-order quadrilateral shell elements must be used if contour

integral output is requested.

• Sides of S8R elements should not be collapsed. If a focused mesh is

used, the crack tip must be modeled as a keyhole whose radius is small

compared to the other dimensions measured in the plane of the shell.

Crack-tip mesh for S8R elementsShell mesh

L3.51


• S8R5 elements can be collapsed and midside nodes moved to the 1/4 points.

• The q vector must lie in the shell surface.

• It should be tangent to the surface.


Crack-tip mesh for S8R5 elementsShell mesh

L3.52



• Example: Circumferential through crack under axial load

• Mean radius R = 10.5 in

• Wall thickness t = 0.525 in

• Crack half-angle q = p / 4

• Longitudinal membrane stress = 100 psi

L3.53



• Model details

• Axial load is applied using

a shell edge load

• Symmetry used to reduce

mode size

Edge loads

symmetry

L3.54



• Modeling a crack with a keyhole

Crack tip

Crack front

q vector

L3.55



• Results

Deformed shape—axial loading

J values—axial loading

L3.56



• In shell element meshes, mechanical loads which act normal to the shell

surface and are applied within the contour integral domain are not taken

into account in the calculation of the contour integral.

• For example, pressure loads are not considered because they act

normal to the shell surface

• Conversely, axial edge loads are considered because they act in

the shell surface.

• Two workarounds exist:

• Run successive shell models with differing crack lengths and

numerically differentiate the potential energy

• Use solid elements (if the response is membrane dominated)

L3.57



• Using numerical differentiation to obtain J:

• The PE values should be obtained from two separate analyses, with

crack lengths differing by Da.

• The values of PE in the Abaqus data (.dat) file are generally not

printed to a sufficient number of figures to be useful for this calculation and must be read from the results (.fil) file.

• A similar technique can be used to get Ct at long times.

Constant Load

Constant Load

( )

a a a

PEJ

a

PE PE

a

D

=

=

D.

Potential energy:

PE = ALLSE ALLWK

L3.58



• Using solid elements:

• If membrane deformation is dominant, the shell can be modeled

with a single layer of 20-node bricks since these solid elements

include loading contributions to contour integrals.

L3.59



• To obtain accurate values of J through the shell thickness with solid

elements, more than one element should be used in the thickness

direction.

J values will show significant path dependence unless

averaged.

• If only one element is used through the thickness, the values can be

averaged by thinking of J as a force per unit length:

• The average is calculated as if the J values were equivalent

nodal forces:

4

6

A B C

shell

J J JJ

= .

CB

A

L3.60



• Aside: Generating a solid element mesh from a shell mesh.

• A shell mesh can easily be converted to a solid one using the ―Offset

Mesh‖ tool.

• Creates solid layers from a shell mesh.

L3.61



• Example: Circumferential through crack in

an internally pressurized, closed-end pipe

• The same pipe discussed earlier, now

subjected to 10 psi internal pressure +

axial load (which simulates the closed

end).

• Comparison of J values using one layer

of C3D20R elements through the

thickness :

CONTOURJ values 100

1 2 3 4 5

At Node A 2.0965 2.1317 2.1505 2.1557 2.1697

At Node B 3.7396 3.6992 3.7004 3.6968 3.6904

At Node C 5.0226 5.0501 5.0813 5.1471 5.2373

Averaged 3.6796 3.6631 3.6722 3.6817 3.6948

CB

A

L3.62



• Example: Circumferential through crack under axial load revisited

• Now we revisit the problem in which the pipe is subjected to an axial

load.

• Comparison of J values using one layer of C3D20R elements through

the thickness:

CONTOURJ values 100

1 2 3 4 5

At Node A 2.2122 2.2524 2.2700 2.2740 2.2850

At Node B 3.7629 3.7202 3.7212 3.7184 3.7136

At Node C 4.9560 4.9893 5.0175 5.0737 5.1492

Averaged 3.7033 3.6871 3.6954 3.7036 3.7148

Analytical 3.7181

L3.63



• Comparing these results with the

shell element results presented

earlier:

• Errors with respect to the

analytical solution for the 3D

model are less than 1%.

• Much closer agreement because

transverse shear effects are

considered in the 3D model.

• Only in-plane stress and strain

terms are included in the Abaqus

J calculations for shells.

• Transverse shear terms are

neglected.

Mixed-Mode Fracture

L3.65


Mixed-Mode Fracture

• Abaqus uses interaction integrals to

compute the stress intensity factors.

• This approach accounts for

mixed-mode loading effects.

• Note that the J- or Ct-integrals

do not distinguish between

modes of loading.

• Usage:

*CONTOUR INTEGRAL,

TYPE=K FACTORS

• Stress intensity factors can

only be calculated for linear

elastic materials.

L3.66


Mixed-Mode Fracture

0K a p=

Element

type

22.5º CPE8 0.185 (2.9%) 0.403 (0.2%)

22.5º CPE8R 0.185 (2.9%) 0.403 (0.2%)

67.5º CPE8 1.052 (3.6%) 0.373 (1.0%)

67.5º CPE8R 1.053 (3.8%) 0.374 (1.3%)

22.5 = 67.5 =

• Example: Center slant cracked plate under tension

*Values enclosed in parentheses are

percentage differences with respect to

the reference solution. See Abaqus

Benchmark Problem 4.7.4 for more

information.

*

Material Discontinuities

L3.68



• The J-integral will be path independent if the material is homogeneous in

the direction of crack propagation in the domain used for the contour

integral calculation.

• If there is material discontinuity ahead of the crack in this region, the

*NORMAL option can be used to correct the calculation of J so that

it will still be path independent.

• The normal to the material discontinuity line must

be specified for all nodes on the material

discontinuity that will lie in a contour integral domain.

n

L3.69



• Example: J-integral analysis of a two material plate

• As an example, the figure shows a single-edge

notch specimen made from two materials in

which the material interface runs at an angle to

the sides of the specimen.

• The material containing the crack (left) has a

Young’s modulus of 2 105 MPa and a

Poisson’s ratio of 0.3.

• The uncracked material (right) has Young’s

modulus of 2 104 MPa and a Poisson’s ratio

of 0.1.

• The specimen is stretched by uniform

displacement at its ends.

L3.70



• J-integral analysis of a two material plate (cont’d)

• Along the material discontinuity, the normal to

the discontinuity is given using the *NORMAL

option.

• The normal needs to be defined on both

sides of the discontinuity.

*NORMAL

LEFT, NORM, 1.0, 0.125, 0.0

RIGHT, NORM, -1.0, -0.125, 0.0

L3.71



• The calculated J-integral values for 10 contours are as follows:

• The need for the normals on the interface (contours 5–10) is clear.

ContourJ (N/mm)

Without normals With normals

1 55681 55681

2 57085 57085

3 57052 57052

4 57058 57058

5 35188 57116

6 31380 57114

7 27536 57114

8 23512 57113

9 19172 57116

10 14181 57094

Numerical Calculations with

Elastic-Plastic Materials

L3.73


Numerical Calculations with Elastic-Plastic Materials

• For Mises plasticity the plastic deformation is incompressible.

• The rate of total deformation becomes incompressible (constant

volume) as the plastic deformation starts to dominate the response.

• All Abaqus quadrilateral and brick elements suitable for use in J-integral

calculations can handle this rate incompressibility condition except for

the ―fully‖ integrated quadrilaterals and brick elements without the

―hybrid‖ formulation.

• Do not use CPE8, CAX8, C3D20 elements with these materials.

They will ―lock‖ (become overconstrained) as the material becomes

more incompressible.

L3.74



• Second-order elements with reduced integration (CPE8R,

C3D20R, etc.) work best for stress concentration problems in

general and for crack tips in particular.

• If the displaced shape plot shows a regular pattern of deformation,

this state is an indication of mesh locking.

• Locking can be seen in quilt contour plots of hydrostatic

pressure for first-order elements—the pressure shows a

checkerboard pattern.

• Change to reduced integration elements if you are using fully

integrated elements.

• Increase the mesh density if you already using reduced

integration elements.

• If these steps do not help, use hybrid elements.

• Hybrid elements must be used for fully incompressible materials (such

as hyperelasticity, linear elasticity with n = 0.5).

L3.75



• Results with elastic-plastic materials (and nonlinear materials in general)

are more sensitive to meshing than for small-strain linear elasticity.

• Meshes adequate for linear elasticity may have to be refined.

• The more complex the solution, the more J values tend to be path

dependent.

• A lack of path dependence can be an indication of a lack of mesh

convergence; however, path independence of J does not prove

mesh convergence.

Workshop 1

L3.77


Workshop 1

• Crack in a three-point bend specimen

• Two-dimensional geometry

• Mesh sensitivity study

• Focus vs. unfocused mesh

• Quarter-point vs. mid-side nodes

Workshop 2

L3.79


Workshop 2

• Crack in a helicopter airframe component

• Three-dimensional geometry

• Create mesh and evaluate response for cracks at different locations

frac l03 contour

Documents

0258nonuniform

0358nonuniform

fine mesh

multiple crack

additional

tip node set

crack propagation

contour integral