FRAC L03 Contour

Fracture Analysis Lecture 3

L3.2

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

Modeling Fracture and Failure with Abaqus

Calculation of Contour Integrals

L3.4

Calculation of Contour Integrals • 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

Calculation of Contour Integrals • 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

Calculation of Contour Integrals • 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).

Contour 1

Contour 2

Contour 3

Contour 4

• 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.


L3.7

Calculation of Contour Integrals • 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

2nd contour

1st contour

Crack-tip node

1st contour crack-front nodes Crack-tip node


L3.8

Calculation of Contour Integrals • Usage: *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}

Specifies the number of contours (domains) on which the contour integral will be calculated

This is the output frequency in increments

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.


L3.9

Calculation of Contour Integrals • Usage (cont’d): *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0}

• 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

Calculation of Contour Integrals • Usage (cont’d): *CONTOUR INTEGRAL, CONTOURS= n, TYPE={J, C, T STRESS, K FACTORS}, DIRECTION = {MTS, MERR, KII0} Three criteria to calculate the crack propagation direction at initiation

• 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) Modeling Fracture and Failure with Abaqus

L3.11

Calculation of Contour Integrals • 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

Calculation of Contour Integrals • 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

Calculation of Contour Integrals • 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 = Ar n.


L3.16

Examples 20

• Different mesh characteristics:

• Axisymmetric or three-dimensional • Fine or coarse focused meshes

• With or without ¼ point elements • Various element types used:

20

• First- and second-order • With and without reduced integration

Axisymmetric model

Crack tip Focused mesh around crack tip Modeling Fracture and Failure with Abaqus

L3.17

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

0.08

0.0004

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.

~0.08 Modeling Fracture and Failure with Abaqus

L3.18

Examples • Axisymmetric model: geometry

Symmetry planes

Close up of crack tip region for coarse mesh model (identical for fine mesh model—only the inner semicircular region is smaller)

Model geometry


L3.19

Examples • Axisymmetric model: crack definition Crack tip with extension direction

Set to 0.5 to use midpoint rather than ¼ point elements


L3.20

Examples • 3D model: geometry and mesh

• A 90 sector is modeled because of symmetry.

Fine 3D mesh

Symmetry planes

Additional partition required for swept mesh

On planes perpendicular to the crack front, the mesh is very similar to the axisymmetric mesh Partitions used for coarse mesh model (identical for fine mesh model—only the inner semicircular region is smaller)

In the circumferential direction around the crack line, 12 elements are used.


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. Irregular elements required here because revolving about a point

A 7-node element is an example of an irregular element.


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.

J analytical  J numerical J analytical

100%

Deformation scale factor = 250

Analytical

5.796E-02

Contour 1

Contour 2

Contour 3

Contour 4

Contour 5

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


L3.26

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

Analytical 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


.0258

.0256

.0260

.0256

.0258


.0201

.0199

.0206

.0200

.0202

• Abaqus values are based on the average of contours 3−5 in each mesh. Modeling Fracture and Failure with Abaqus

L3.27

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

Analytical 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


.0358

.0355

.0352

.0356

.0358


.0258

.0255

.0253

.0255

.0258


.0201

.0198

.0197

.0199

.0201


L3.28

Examples J values from meshes without ¼ point elements (reduced integration) 3-D

Loading

Analytical result

C3D20R

Axisymmetric

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


.0358

.0350

.0357

.0336

.0350

.0358

.0337


.0258

.0250

.0260

.0234

.0250

.0258

.0236


.0201

.0193

.0206

.0177

.0193

.0202

.0179


L3.29

Examples J values from meshes without ¼ point elements (full integration) 3-D

Loading

Analytical result

C3D20

Axisymmetric

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


.0358

.0350

.0352

.0329

.0350

.0358

.0333


.0258

.0249

.0253

.0229

.0250

.0258

.0232


.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 threedimensional 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

J-integral

1.338E-07 1.336E-07

3D contour 5

1.334E-07

3D contour 4

1.332E-07

3D contour 3

1.330E-07

3D contour 2

1.328E-07 0

45

90

Angle (degrees)


L3.34

Examples • Axisymmetric model and results

Contours 3-5 have converged

Axisymmetric results are used as reference results.


L3.35

Examples • Comparison of axisymmetric and 3D results Variation of J with angular position Contour 1

Variation of J with angular position Contour 2

1.360E-07

3D

1.340E-07

AXI

1.320E-07

J-integral

J -integral

1.380E-07

1.300E-07 0

45

1.334E-07 1.333E-07 1.332E-07 1.331E-07 1.330E-07 1.329E-07

3D AXI

0

90



1.334E-07

3D

1.332E-07

AXI

1.330E-07

1.328E-07 90

J-integral

J -integral

1.336E-07

45

90

Angle (degrees)

Angle (degrees)

0

45

1.338E-07 1.336E-07 1.334E-07 1.332E-07 1.330E-07 1.328E-07

3D AXI

0

Angle (degrees)


45 Angle (degrees)

90

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

% difference in J between AXI and 3D results 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Contour 1

Contour 2 Contour 3

Contour 4 Contour 5 0

45

90

Angle (degrees)


L3.37

Examples • Submodeling

• We can use submodeling to create two meshes that are significantly smaller than the full threedimensional 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 % difference in J between AXI and 3D results

• Inaccuracies are introduced by the coarser mesh used in the global model.

% difference

• J values of submodel:

• Errors in J are less than 1%.

4.5 4.0 3.5 3.0 2.5 2.0 1.5 1.0 0.5 0.0

Contour 1 Contour 2 Contour 3 Contour 4 Contour 5 0

45

• CPU time was reduced by a factor of 3.

90

Angle (degrees)

Variation of J with angular position

Variation of J with angular position Contour 5 1.335E-07

3D contour 5

1.324E-07

3D contour 4

1.322E-07

3D contour 3

1.320E-07

3D contour 2

1.318E-07

J-integral

J -integral

1.326E-07

1.330E-07

3D

1.325E-07

AXI

1.320E-07 1.315E-07

0

45

90

0

Angle (degrees)


45 Angle (degrees)

90

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

Prescribed load line displacement

• Model details

Crack seam

• Plane strain conditions assumed. • The initial crack length is 5 mm. • Elastic-plastic material

• Low alloy ferritic steel

q-vector

1/√r singularity modeled in the crack-tip elements


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.

Normals to top crack surface nodes

n (normal to crack plane) Normals to bottom crack surface nodes

• The normal is not used at the crack-tip node, however.


q

L3.45

Nodal Normals in Contour Integral Calculations • Example: sharp curved crack

Contour # J without normals J with normals

1

2

3.363 3.600

2.980 3.602

3 2.475 3.605

4 1.888 3.605


5 1.283 3.605

L3.46

Nodal Normals in Contour Integral Calculations • 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 surface

Crack surface

Paths for contour integrals

n q Single node in crack-tip node set; normals calculated on nodes of blunted surface; q not parallel to crack surface.

q All nodes on blunted surface in crack-tip node set; q parallel to crack surface.


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

Through Cracks in Shells • 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.

Shell mesh

Crack-tip mesh for S8R elements


L3.51

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

Shell mesh

Crack-tip mesh for S8R5 elements

• The q vector must lie in the shell surface.

• It should be tangent to the surface.


L3.52

Through Cracks in Shells • 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

Through Cracks in Shells • Model details • Axial load is applied using a shell edge load • Symmetry used to reduce mode size

Edge loads

symmetry


L3.54

Through Cracks in Shells • Modeling a crack with a keyhole

Crack front

q vector

Crack tip


L3.55

Through Cracks in Shells • Results

Deformed shape—axial loading

J values—axial loading


L3.56

Through Cracks in Shells • 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

Through Cracks in Shells • Using numerical differentiation to obtain J:

 ( PE ) J = a Constant Load =

Potential energy:

PE = ALLSE  ALLWK

PE a Da  PE a Da

. Constant Load

• 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.


L3.58

Through Cracks in Shells • 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

Through Cracks in Shells • 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:

J

shell

=

J A  4J B  JC . 6


A B C

L3.60

Through Cracks in Shells • 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

Through Cracks in Shells • 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 : A

J values  100

CONTOUR 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


B C

L3.62

Through Cracks in Shells • 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: J values  100

CONTOUR

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

Through Cracks in Shells • 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 • Example: Center slant cracked plate under tension 

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%)

K0 =  p a *Values enclosed in parentheses are percentage differences with respect to the reference solution. See Abaqus Benchmark Problem 4.7.4 for more information.

 = 22.5

 = 67.5 Modeling Fracture and Failure with Abaqus

Material Discontinuities

L3.68

Material Discontinuities • 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

Material Discontinuities • 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

Material Discontinuities • 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

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

J (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

• The need for the normals on the interface (contours 5–10) is clear. Modeling Fracture and Failure with Abaqus

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

Numerical Calculations with Elastic-Plastic Materials • 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

Numerical Calculations with Elastic-Plastic Materials • 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

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