Computational Fracture Mechanics - Technische Fakultät · stress-strain curves reflect the measured load-elongation data. Except YA-1-28, which will be Except YA-1-28, which will

report_Brocks-Arafah, wbrocks, 30 March 2014, - 1 -

Computational Fracture Mechanics

Students Project

Final report

Wolfgang Brocks

Diya Arafah

January 2014

Dipartimento di Meccanica

Politecnico di Milano


Exercise #1: Determination of a true stress-strain curve for ABAQUS input from tensile test data.

Geometry: round tensile bar

diameter d0 = 6 mm (nominal)

measuring length L0 = 30 mm

Material 26NiCrMoV115

YOUNG’s modulus E = 204400 MPa

POISSON’s ratio ν = 0.3

0.2% proof stress Rp0.2 = 709 MPa (average)

Test results by Dip. Meccanica, PoliMi, data in “tensile_tests.xls”

Note that

• ABAQUS requires true stresses, � � ��, in dependence on logarithmic plastic strains, ��

� � �� ⁄ , as input for an elasto-plastic analysis,

• a σ(εp) curve beyond uniform elongation of the tensile bar is needed for the fracture mechanics analysis, which necessitates an extrapolation of the stress-strain curve by a power law

��

� � � ��

��

where ��, �� are normalisation values, commonly taken as yield limit �� 0� and �� ⁄ .

0

5

10

15

20

25

0 1 2 3 4 5

forc

e [k

N]

elongation [mm]

26NiCrMoV11

YA-1-26

YA-1-34

YA-1-28

YA-1-30

YA-1-33

YB-128


� Check the stress-strain curve by simulation of the tensile test beyond uniform elongation (maximum force) and comparing simulation with test data in terms of force, F, vs. elongation, ∆L .

Analysis of tensile test

• axisymmetric model,

• account for symmetries,

• displacement controlled,

• geometrically nonlinear (“large deformations”).

Strain and stress measures for large deformations

Linear strain �� ∆��

Logarithmic (HENCKY) strain ��

� � !1 # ��$

Nominal stress ��% � &'�

� 4&)*�+

“True” (CAUCHY) stress �,-./ � &'

Provided the existence of a uniaxial stress state for

��% 0 �% � &%12'�

and a uniform elongation, the requirement of isochoric plastic deformation yields '�� 4 '� ,

assuming that the elastic elongation is negligible,

�5 � �,�,1� � �/ 4 �,�,1� ,

true stresses can be calculated as

�,-./ � &' 4 &

'��

� �,-./!1 # ��$


1.1. Evaluation of test data

specimen YA-1-26 YA-1-34 YA-1-28 YA-1-30 YA-1-33 YB-1-28

d0 [mm] 6.00 6.01 5.96 5.99 6.04 6.02

The force elongation curves, F(∆L), measured in the tests are converted into engineering stress-strain curves, σ��%!ε��$, as presented in Fig. 1.1, and true stress-strain curves, σ,-./�ε��, next, Fig. 1.2, according to the formulas given above. The engineering nominal stress-strain curves reflect the measured load-elongation data. Except YA-1-28, which will be excluded in the following, the curves practically coincide up to maximum load or ��% � �%. Scatter of the test results occurs after uniform elongation when necking of the specimens starts. The conversion to “true” stresses holds only up to Rm, as for larger elongations, deformations are not uniform and the stress state is not uniaxial any more.

Fig. 1.1: Nominal stress vs linear strain (engineering stress-strain curves)

Fig. 1.2: Conversion to true stress vs logarithmic strain (specimen YA-1-26)

The curves of true stress vs. logarithmic plastic strain, �5 � �� ,-./ ⁄ , practically coincide for the five test specimens excluding YA-1-28, on the scale of Fig. 1.3. Whereas the technical definition of yielding by the 0.2% proof stress has been determined as �5�.+ �� 0.002� � 709 MPa in average, the “physical” yield strength, i.e. first occurrence of

plastic strain, �� ? 0�, varies between 640 MPa and 670 MPa, Fig. 1.4.

Fig. 1.3: True stress vs logarithmic plastic strain up to uniform elongation

Fig. 1.4: Determination of the yield strength �� 0�

0

200

400

600

800

1000

0 0,05 0,1 0,15

σno

m[M

Pa]

εlin [-]

YA-1-26

YA-1-34

YA-1-28

YA-1-30

YA-1-33

YB-1-28

0

200

400

600

800

1000

0 0,05 0,1 0,15

σ[M

Pa]

ε [-]

YA-1-26

sig_true

sig_nom

0

200

400

600

800

1000

0 0,02 0,04 0,06 0,08

σtr

ue[M

Pa]

εp [-]

YA-1-26

YA-1-34

YA-1-30

YA-1-33

YB-1-28

620

640

660

680

700

720

0 0,0005 0,001 0,0015 0,002

σtr

ue[M

Pa]

εp [-]

YA-1-26

YA-1-34

YA-1-30

YA-1-33

YB-1-28


A value of �� 650 MPa is chosen for the following.

Extrapolation beyond uniform elongation can be based on any of the stress strain curves of Fig 1.3, as they do not differ much. They cannot be fitted over the whole range of the logarithmic plastic strain by a unique power law

ptrue

0 0

nεσ ασ ε

=

, (1.1)

as their curvatures obviously change. Therefore, only the final part is used to define the power law. Normalisation is recommended in order to obtain a dimensionless equation so that α does not depend on the specific dimension used for the stresses. The normalising values can be arbitrarily selected, commonly taken as the yield-limit values, here �� 650 MPa and �� ⁄ � 0.00381. Depending on the range of εp used in the power-law fitting, different values of the parameters α and n are obtained (Figs. 1.5, 1.6). The effect on the extrapolation is negligible, however (Fig. 1.7).

Fig. 1.5: Power law fit #1 of true stress vs logarithmic plastic strain, see “extrapol(1)” in Fig. 1.7

Fig. 1.6: Power law fit #2 of true stress vs logarithmic plastic strain, see “extrapol(2)” in Fig. 1.7

Fig. 1.7: Extrapolation of true stress vs logarithmic plastic strain curve.

Fig. 1.8: True stress vs. logarithmic plastic strain data as input for an elasto-plastic analysis with ABAQUS

The data points used to describe the elasto-plastic material behaviour in ABQUS, Fig. 1.8, are established as

y = 1,0721x0,0786

1,32

1,33

1,34

1,35

1,36

1,37

1,38

14 16 18 20 22 24

σ/ σ

0

εp / ε0

YA-1-26

Pot.(YA-1-26)

y = 1,1012x0,0698

1,34

1,35

1,36

1,37

1,38

18 20 22 24

σ/ σ

0

εp / ε0

YA-1-26

Pot.(YA-1-26)

0

200

400

600

800

1000

1200

0 0,1 0,2 0,3 0,4

σtr

ue[M

Pa]

εp [-]

YA-1-26

YA-1-34

YB-1-28

extrapol(1)

extrapol(2)

0

200

400

600

800

1000

1200

0 0,2 0,4 0,6 0,8 1

σ[M

Pa]

εp [-]

ABAQUS


• first point: �� 0�, • second point �5�.+ � �� 0.002�, • 5 �� points from specimen YA-1-28 up to �� 0.039, • 10 equidistant �� points for 0,6 0 �� 0 1.0 from an extrapolation by power law fit

#2 with α = 1.10, n = 0.07.

1.2. Numerical simulation of the tensile test

An axisymmetric model of the tensile bar is applied, Fig. 1.9. The symmetry to the x1-axis is considered to reduce the computational effort; it guarantees that necking occurs at x2 = 0, in addition. Therefore only a quarter of the specimen is modelled, which is realised by applying the boundary conditions, E+!FG, 0$ � 0, along this line. The radial displacements, EG!FG, 0$, have to be free (roller support condition).

The specimen is loaded in displacement control, as the axial force decreases under still increasing elongation due to local necking. A prescribed displacement is applied at the upper cross section of the model by connecting all respective nodes to some reference point, Fig. 1.10, using the coupling constraint option in ABAQUS. This point moves in the x2-direction with displacement u2, whereas the boundary condition u1 = 0 is applied. The corresponding external force is obtained as reaction force at the reference point.

Fig. 1.9: Axisymmetric model of tensile bar Fig. 1.10: Application of prescribed dis-placement 1

Fig. 1.11: Mesh of tensile bar 1 Fig. 1.12: Iso-contours of radial displace-ment, u1, showing localisation of deformation after uniform elongation

In order to realise a homogeneous stress and strain state over the measuring length, L0, up to uniform elongation, the fixation of the specimen with a larger radius at the upper end has

1 courtesy of GIORGIA GOBBI, PhD student, Dipartimento di Meccanica


to be incorporated in the model. Fig. 1.11. For a cylinder of constant radius, necking might occur in the section where the prescribed displacement is applied instead of the centre section. The correct location of necking has to be checked, Figs. 1.12 to 1.14.

Fig. 1.13: Deformed mesh in final load-step displaying necking with iso-contours of axial displacement, u2

Fig. 1.14: 3D visualisation of necking, grey: plastic zone, �5 H 0.002

Fig 1.14 shows a comparison of the simulation results with the test data. The external force decreases due to local necking of the bar in the mid-section (Figs. 1.12, 1.13), which of course occurs only if a geometrically nonlinear analysis is performed. The corresponding value of uniform elongation is affected by any small imperfection of the bar, e.g. a reduction of diameter by 1% or a mesh refinement in the mid-section. As the bar is a geometrically ideal cylinder in the present model, necking is delayed compared to the tests.

Fig. 1.14: Comparison of test data with numerical simulation

The maximum force reached in the simulation is within the scatter of the test results, Tab. 1.1. The coincidence of the FE simulation with the tensile test data finally confirms the stress-strain input data.

Specimen YA-1-26 YA-1-34 YA-1-28 YA-1-30 YA-1-33 YB-1-28 FE

Fmax [kN] 23.3 23.6 23.8 23.4 23.6 23.7 23.5

Table 1.1: Maximum tensile force

0

5

10

15

20

25

0 1 2 3 4 5 6

forc

e [k

N]

elongation [mm]

26NiCrMoV11

YA-1-26

YA-1-34

YA-1-33

YB-1-28

ABAQUS


Exercise #2: Numerical analysis of a side-grooved C(T) specimen with stationary crack.

Geometry: width W = 50 mm

height H = 1.2 W = 60 mm

initial crack length a0 = 33.2 mm

thickness B = 25 mm

net thickness Bn = 20 mm

(20% side grooved)

Material: 26NiCrMoV115 as above for exercise #1



0.2% proof stress Rp0.2 = 709 MPa

yield limit �� 0�

Test results by Dip. Meccanica, PoliMi, data in “CT-tests.xls” 2

� Compare tests with FE simulation in terms of force vs. load-line displacement, F(VLL), up to approx. maximum force, i.e. IJJ 4 1.5KK (LM 4 0.5KK)

� Evaluate J-integral and plot J(VLL),

� Investigate path dependence of J.

Analysis of C(T)

• plane strain,

• account for symmetry,


• geometrically nonlinear (“large deformations”).

2 Note that the test results include crack extension, which is not considered in the present FE analysis! J is evaluated according to ASTM E 1820.

0

10

20

30

40

50

0 0,5 1 1,5 2 2,5 3

F[k

N]

VLL [mm]

YYB-19-1

YYB-19-3

YYB-19-4

YYB-19-6

YYB-19-9

YYB-19-10

0

100

200

300

400

500

600

0 0,5 1 1,5 2 2,5 3

J[k

J/m

2 ]

VLL [mm]

YYB-19-1

YYB-19-3

YYB-19-4

YYB-19-6

YYB-19-9

YYB-19-10


Mesh:

Mesh data in “CT_mesh”

2.1. FE model

The FE model disregards all technical details of the real C(T) specimen, as there are

• the pinhole for application of the force,

• the basically stress-free material volume left of the applied force and

• the machined notch used as starter for fatigue cracking and fitting for the clip gauge measuring VLL in testing.

It is thus reduced to a simple rectangle of size W×H/2. The mesh is refined towards the ligament, M� 0 FG 0 N, reaching a minimum element size of 0.125×0.125 mm2. The regular arrangement of square elements in the ligament is specific for simulations of crack extension (see exercise #3) but will yield strongly distorted elements at the crack tip and is hence inappropriate for calculating reliable stress and strain values at a stationary crack. Since the present analysis is focused on global values, namely load-line displacement and J-integral, the meshing at the crack tip is uncritical.

The FE-model accounts for symmetry with respect to the x-axis, which is realised by fixing all vertical displacements in the ligament, E+!M� 0 FG 0 N, F+ � 0$ � 0.

The specimen is loaded in displacement control mode by applying a vertical displacement u2 to a reference point as in exercise #1, which is connected either just to the node representing the midpoint of the pinhole or to some additional nodes around. The respective force is picked up as reaction force at the reference point. In order to avoid localisation of plastic deformation in the points of load application, the area around should be defined as “elastic” only (Fig. 2.1).

The load-line displacement, VLL, is picked up at the lower left edge of the model, IJJ �2E+!0,0$, which is slightly different from the clip position in the test. Due to the high stiffness and stresses close to zero in this part of the specimen, the respective effect on the VLL values is negligible, however.

The analysis is performed as plane strain with the specimen’s net thickness Bn.


Fig. 2.1: FE mesh of C(T) (half model)

Fig. 2.2: Automatic option of contour definition for J-integral calcul-ation: 1st contour = crack tip, last contour #24

2.2 J-Integral evaluation

The J-integral is defined as an integral over a closed contour, Γ, around the crack tip,

( )2 ,1ij j iJ wdx n u dsΓ

σ= −∫� , (2.1)

where w is the strain energy density, σij the stress tensor and nj the outer normal to the contour. It is equal to the energy release rate in a plane body of thickness, B, due to a crack extension, ∆a, in mode I,

LV

UJ

B a

∂= −∂

, (2.2)

which is exploited in the virtual crack extension (vce) method to calculate J numerically as well as in the experimental determination3 of J.

Applying the divergence theorem, any contour integral can be converted into an area (domain) integral in two dimensions or a volume integral in three dimensions, over a finite domain surrounding the crack. ABAQUS/Standard uses this domain integral method to evaluate the J-integral and automatically finds the elements that form each ring from the regions defined as the crack tip or crack line; commonly focused mesh design is used to provide such rings of elements. Each contour provides an evaluation of the contour integral. The number, n, of contours must be specified in the history output request.

*CONTOUR INTEGRAL, CONTOURS=n, TYPE=J

By definition, J is a “path-independent” integral, claiming that each contour should yield the same J-value. The derivation of its path-independence requires

(1) time independent processes,

(2) absence of volume forces and surface forces on the crack faces,

(3) small strains (geometrically linear),

(4) homogeneous hyper-elastic material (deformation theory of plasticity),

however. A geometrically nonlinear elasto-plastic analysis applying incremental theory of plasticity violates the two last-mentioned conditions, so that the values obtained for the various contours will differ and the J-integral becomes path-dependant, more precisely, J

3 ASTM E 1820-06: Standard test method for measurement of fracture toughness, Annual book

of ASTM Standards, Vol 03.01, American Society for Testing and Materials, Philadelphia.


increases with increasing size of the contour. The ABAQUS User’s Manual provides the following statement4 to this problem:

Domain dependence

“The J-integral should be independent of the domain used provided that the crack faces are parallel to each other, but J-integral estimates from different rings may vary because of the approximate nature of the finite element solution. Strong variation in these estimates, commonly called domain dependence or contour dependence, typically indicates an error in the contour integral definition. Gradual variation in these estimates may indicate that a finer mesh is needed or, if plasticity is included, that the contour integral domain does not completely include the plastic zone. If the “equivalent elastic material” is not a good representation of the elastic-plastic material, the contour integrals will be domain independent only if they completely include the plastic zone. Since it is not always possible to include the plastic zone in three dimensions, a finer mesh may be the only solution. If the first contour integral is defined by specifying the nodes at the crack tip, the first few contours may be inaccurate. To check the accuracy of these contours, you can request more contours and determine the value of the contour integral that appears approximately constant from one contour to the next. The contour integral values that are not approximately equal to this constant should be discarded. …”

Most of this is simply rubbish! The domain dependence has physical reasons and is not due to “the approximate nature of the finite element solution”. The derivation of the path-independence of the contour integral does not only require “that the crack faces are parallel” but also the assumptions mentioned above.5 Contour (or domain) dependence thus is physically significant due to the dissipative character of plastic deformation and not only some “error” or “inaccuracy” of the numerical calculation, and not only “the first few contours” display contour dependence of J. Suggesting a “finer mesh” at the crack tip is counter-productive as it just increases the number of contours automatically generated by ABAQUS and thus increases the number of contour dependent J-values. The manual is right suggesting “you can request more contours and determine the value of the contour integral that appears approximately constant from one contour to the next”. What is needed in a fracture mechanics analysis is a saturation value or “far-field” value of J corresponding to that evaluated from the mechanical work done by the external force at the load-line displacement, eq. (2.2). For further details see BROCKS & SCHEIDER6.

The user must define the crack front and the virtual crack extension in ABAQUS to request contour integral output; this can be done as follows:

• Defining the crack front

The user must specify the crack front, i.e., the region that defines the first contour. ABAQUS/Standard uses this region and one layer of elements surrounding it to compute the first contour integral. An additional layer of elements is used to compute each subsequent contour. The crack front can be equal to the crack tip in two dimensions or it can be a larger region surrounding the crack tip, in which case it must include the crack tip. By default ABAQUS defines the crack tip as the node specified for the crack front in the so called CRACK TIP NODES option of the contour integral command:

*CONTOUR INTEGRAL, CONTOURS=n, CRACK TIP NODES

4 “Contour Integral Evaluation”, ABAQUS User’s Manual, Section 11.4.2.

5 The violation of condition (4) is apparently addressed in the obscure clause “if the ‘equivalent elastic material’ is not a good representation of the elastic-plastic material”

6 W. BROCKS and I. SCHEIDER: Reliable J-values - Numerical aspects of the path-dependence of

the J-integral in incremental plasticity, Materialprüfung 45 (2003), 264-275.


Specify the crack front node set name and the crack tip node number or node set name.

Alternatively, a user-defined node set which include the crack tip can be provided to ABAQUS manually by omitting the CRACK TIP NODES option:

*CONTOUR INTEGRAL, CONTOURS=n

Specify the crack front node set name and the crack tip node number or node set name.

• Specifying the virtual crack extension direction

The direction of virtual crack extension must be specified at each crack tip in two dimensions or at each node along the crack line in three dimensions by specifying either the normal to the crack plane, n, or the virtual crack extension direction, q, see Fig. 2.3.

Fig. 2.3: Left: Crack front tangent, t, normal to the crack plane, n, and virtual crack extension direction, q, at a curved crack; Right: Typical focused mesh for fracture mechanics evaluation [ABAQUS user’s manual].

The normal to the crack plane, n, can be defined in ABAQUS using the so called (normal) option7 in the contour integral command:

*CONTOUR INTEGRAL, CONTOURS=n, NORMAL

nx-direction cosine, ny-direction cosine, nz-direction cosine(or blank), crack front node set name (2-D) or names (3-D)

ABAQUS will calculate a virtual crack extension direction, q, that is orthogonal to the crack front tangent, t, and the normal, n, see Fig. 2.3 (left).

Alternatively, the (normal) option can be omitted and the virtual crack extension direction, q, can be defined directly:

*CONTOUR INTEGRAL, CONTOURS=n

crack front node set name, qx-direction cosine, qy-direction cosine, qz-direction cosine (or blank)

Note, that the symmetry option (SYMM) must be used to indicate that the crack front is defined on a symmetry plane:

*CONTOUR INTEGRAL, CONTOURS=n, SYMM

7 The normal option can only be used if the crack plane is flat


The change in potential energy calculated from the virtual crack front advance is doubled to compute the correct contour integral values. Different combination of the above mentioned options can be used to request contour integral within ABAQUS, more details about this could be found in the ABAQUS documentation.

Fig. 2.4: Manual option of contour definition for J-integral calculation: 1st contour = „node set“

Fig. 2.5: Manual option of contour definition for J-integral calculation: last contour #12

Two options of defining the contours are used in the present study,

• The CRACK TIP NODES option and a sufficient number of contours8 to approach a saturated (or far-field) value of J are chosen, Fig. 2.2, hereafter called “automatic option”;

• A first contour around the crack tip of finite size (omitting the CRACK TIP NODES option) is defined via a crack tip node set, Fig. 2.4, hereafter called “manual option”.

The number of contours is much lower in the latter case. In the present mesh, n = 24 if the automatic option is used and the first contour is the first element ring around the crack tip, Fig. 2.2. The domain left of the crack tip is obviously unnecessarily large. In the manual option, n = 12 to reach the largest possible contour, Fig. 2.5.

2.3. Simulation Results

Fig. 2.6 compares the &!IJJ$ curves of the test specimen YYB-19-3 and the simulation. The simulation results meet the test data up to approximately maximum force, F = 41.6 kN, reached at a load line displacement VLL = 1.36 mm; the respective difference, !&O�% � &,/O,$/&,/O,, is less than 3%. The force keeps increasing in the simulation as no crack extension is modelled. Whereas the load decline in the tensile test was due to plastic necking, it is crack extension here which reduces the ligament area.

The coincidence between test and simulation results holds much longer for Q!IJJ$, Fig. 2.7, as the opposing effects of decreasing force and increasing ∆a on Jtest cancel each other. At VLL = 2.0 mm, the difference !QO�% � Q,/O,$/Q,/O, is just 1.8%. The curves J-24 and J-12 in Fig. 2.7 denote the “far-field” J-values obtained from contours #24 (Fig. 2.2) and #12 (Fig.2.5). There is obviously no difference between the two.

8 Note that the last contour must not touch the outer surface of the specimen, as the elements

outside this contour are taken for calculating the energy release rate.


Fig. 2.6: Comparison of test data with simulation: force vs. load-line displacement

Fig. 2.7: Comparison of test data with simulation: J-integral vs. load-line displacement

As described above, the first contours in the automatic option are element rings in the near field around the crack tip, where large plastic strains with corresponding rearrangement of stresses occur. These stress-strain fields violate the requirements of “deformation theory of plasticity”. In order to obtain meaningful far-field values of J, the corresponding contours should imbed the plastic zone emanating from the crack tip, which is not completely possible, however, as crack initiation occurs under gross yielding where the plastic zone extends from the crack tip to the free surface of the specimen. For VLL = 1 mm and 3 mm, the 11th and the 16th J-contour, respectively, are inclosing the plastic zone as far as possible, see Figs. 2.8 and 2.10.

Fig. 2.8: Plastic zone at VLL = 1 mm and 11th J-contour of automatic option

Fig. 2.9: Plastic zone at VLL = 1 mm and 3rd J-contour of manual option

In the manual option, the 1st contour is a user-defined node set including a comparably large area of elements (Fig. 2.4). Therefore, already the 3rd contour is large enough to embed the plastic zone, see Figs. 2.9 and 2.11.

Fig. 2.10: Plastic zone at VLL = 3 mm and 16th J-contour of automatic option

Fig. 2.11: Plastic zone at VLL = 3 mm and 3rd J-contour of manual option

0

10

20

30

40

50

0 1 2 3

F[k

N]

VLL [mm]

simulation

test YYB-19-3

0

200

400

600

800

0 1 2 3

J [k

J/m

2]

VLL [mm]

J 24 (automatic)

J 12 (manual)

test YYB-19-3


Due to plastic deformations, a significant domain dependence of J occurs close to the crack tip, which increases with increasing VLL, see curves J-1 to J-10 in Fig. 2.12. It needs about 10 to 12 contours in the automatic option to obtain a saturation (far-field) value of J. In the manual option starting from a 1st contour according to Fig. 2.3, domain independence of J is immediately achieved, Fig 2.13. This is also demonstrated in Fig. 2.14, where the J-values at VLL = 3.0 mm are plotted in dependence on the contour number. Note that J must increase with increasing domain size9., i.e. QRSTGU H QRSU, where k is the contour number.

Fig. 2.12: Contour dependence of J: auto-matic option, 1st contour = crack tip

Fig. 2.13: Contour dependence of J: manual option, 1st contour = node set

Fig. 2.14: Contour dependence of J Fig. 2.15: J from “far-field” domain integral and ASTM 1820

The J-values calculated by a contour (or domain) integral, eq. (2.1), in the far-field should equal the energy release rate calculated from the work of external forces, eq. (2.2) as it is done in the tests based on ASTM E 1820 3. The basic equations for the latter are as follows.

J is split into an elastic and a plastic part, where the elastic part is calculated from the stress intensity factor, K, and the plastic part from the area under the &!IJJ$ curve:

( )2 2

pe p

n 0

1K AJ J J

E B b

νη

−= + = + , (2.3)

0

n

aFK f

WBB W

=

, (2.4)

9 W. BROCKS and H. YUAN: On the J-integral concept for elastic-plastic crack extension, Nuclear Engineering and Design 131 (1991), 157-173.

0

200

400

600

800

0 0,5 1 1,5 2 2,5 3

J [k

J/m

2]

VLL [mm]

"automatic"J 24

J 12

J 10

J 8

J 7

J 6

J 5

J 4

J 3

J 2

J 1 (crack tip)0

200

400

600

800

0 0,5 1 1,5 2 2,5 3

J [k

J/m

2]

VLL [mm]

"manual"J 12

J 9

J 7

J 5

J 3

J 1 (node set)

0

200

400

600

800

0 5 10 15 20 25

J[k

J/m

2]

contour

VLL = 3.0 mm

automatic

manual

0

200

400

600

800

0 0,5 1 1,5 2 2,5 3

J [k

J/m

2]

VLL [mm]

J_ASTM

J 24 (automatic)

J 12 (manual)


e 21 1p LL LL LL LL2 2A F dV FV F dV C F= − = −∫ ∫ , (2.5)

02 0.522b

Wη = + , (2.6)

where a0 is the initial crack length, Bn the specimen net thickness10 and b0 the uncracked

ligament, V� � N � M�, W the width of the specimen, W XYZ[\ is a geometry function and CLL

the elastic compliance, both given in ASTM E 1820 as polynomials of !M N⁄ $ for C(T) specimens.

Fig. 2.16: Definition of area Ap for J

calculation.

Fig 2.15 shows that the far-field values of the contour integral and the J values calculated from the F(VLL) data indeed coincide. If analytical formulas are available as in the case of fracture mechanics specimens, this comparison is a convenient method to check either the calculated contour integral values or the analytical formula11.

The above equations (2.3) to (2.6) hold for constant crack length, a0, i.e. a stationary crack. Since crack length, M � M� # ∆M, and ligament width, V � N � M, change with crack extension, ∆a, they must be continuously updated during the J evaluation. An incremental procedure for calculating J(∆a) curves is given in ASTM 1820:

p p

( ) ( ) ( 1) ( ) ( 1)p p( ) ( 1) ( 1)

( ) n ( 1)

1i i i i ii i i

i i

A A a aJ J

b B b

ηγ− −

− −−

− −= + −

. (2.7)

10 Note that in the plane-strain analysis B = Bn = 20 mm

11 W. BROCKS, P. ANUSCHEWSKI and I. SCHEIDER: Ductile tearing resistance of metal sheets. Engineering Failure Analysis 17 (2010), 607-616


Exercise #3: Numerical analysis of a side-grooved C(T) specimen accounting for crack growth by applying cohesive elements

Geometry: width W = 50 mm

initial crack length a0 = 33.2 mm

thickness B = 25 mm

net thickness Bn = 20 mm

(20% side grooved)

Material: 26NiCrMoV115 as above in exercise #1



0.2% proof stress Rp0.2 = 709 MPa

yield limit �� 0�

Test results by Dip. Meccanica, PoliMi,data in “CT-tests.xls”

� Compare tests with FE simulations in terms of force vs. load-line displacement, F(VLL), and load-line displacement vs. crack extension, VLL (∆a)

� Investigate the effect of cohesive parameters σc, Γc

Analysis of C(T)

• plane strain,

• account for symmetry,


• geometrically nonlinear (“large deformations”),

• cohesive elements (UEL by SCHEIDER [2001])

0

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30

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50

0 0,5 1 1,5 2 2,5 3

F[k

N]

VLL [mm]

YYB-19-1

YYB-19-3

YYB-19-4

YYB-19-6

YYB-19-9

YYB-19-10

0

0,5

1

1,5

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2,5

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0 0,5 1 1,5 2 2,5 3

VLL

[mm

]

∆a [mm]

YYB-19-1

YYB-19-3

YYB-19-4

YYB-19-6

YYB-19-9

YYB-19-10


Cohesive law (traction-separation law)

For details of the implementation as UEL in ABAQUS and description of input see

The Cohesive Model – Foundation and Implementation

Report by INGO SCHEIDER, Release 2.0.3, Helmholtz Centre Geesthacht, 2011

”czm-doku-v2.0.3“

Suggested cohesive parameters:

(1) (2) (3)

cohesive strength σc MPa 2130 2840 3050

critical separation δc mm 0.10 0.01 0.01

separation energy Γc 12 kJ/m2 137 24.0 22.7

shape parameter δ1/δc - 0.02 0.02 0.02

shape parameter δ2/δc - 0.3 0.7 0.5

Mesh:

ABAQUS input file “CT_input”

12

regard the relation ]̂ � _`abcbXGd`

ef_fbTf`

fb\


3.1. FE model

The FE model is basically the same as in exercise #2, see Figs. 2.1 and 2.2.

Instead of fixing the vertical displacements in the ligament, M� 0 FG 0 N, F+ � 0, cohesive elements13 are applied in the ligament to allow for crack extension. Symmetry14 is realised by applying linear constraint equations, E+

!d$ � �E+!T$, Fig. 3.1, which guarantee a symmetric

opening of the cohesive element, g� � E+!T$ � E+

!d$ � 2E+!T$, implying delta_N = δc. These

constraint equations induce extra nodal forces15, however, which reduce the stresses in the respective element to one half, so that the cohesive strength traction_N = σc/2.

constraint equations

E+!d$ � �E+

!T$

EG!d$ � EG

!T$

cohesive parameters

delta_N = δc,

traction_N = σc/2

Fig. 3.1: Cohesive elements at symmetry line: Constraint equations and input variables

The variables of user elements cannot be displayed by ABAQUS/Viewer. They are hence transferred to the adjacent continuum elements as user output variables by providing an element mapping between cohesive and continuum elements which is given in the .cohes file,

*ELEMENT MAP cont_elem_1, coh_elem_1 cont_elem_2, coh_elem_2 …

cont_elem_n, coh_elem_n

The cohesive law of SCHEIDER13 is applied

( ) ( )

( ) ( )

n n

1 1

n 2 n 2

c 2 c 2

2

n 1

n n c 1 n 2

3 2

2 n c

2 for

( ) 1 for

1 2 3 for

δ δδ δ

δ δ δ δδ δ δ δ

δ δ

σ δ σ δ δ δ

δ δ δ− −− −

− ≤= ⋅ < ≤

+ − ≤ ≤

, (3.1)

13 I. SCHEIDER: The cohesive model – Foundation and implementation, Release 2.0.3, Report, Helmholtz Centre Geesthacht, 2011.

14 W. BROCKS, D. ARAFAH, M. MADIA: Exploiting symmetries of FE models and application to

cohesive elements, Report Milano / Kiel, November 2013.

http://www.tf.uni-kiel.de/matwis/instmat/departments/brocks/brocks_homepage_en.html 15 “Linear constraint equations introduce constraint forces at all degrees of freedom appearing in

the equations. These forces are considered external, but they are not included in reaction force output. Therefore, the totals provided at the end of the reaction force output tables may reflect an incomplete measure of global equilibrium.” - ABAQUS Analysis User's Manual (6.11): 33.2.1 Linear constraint equations.


Fig. 3.2: Traction-separation laws for

the three suggested parameter sets (R0 = 650 MPa)

with the cohesive energy

( )c

1 2c n n n c c

c c0

1 21- +

2 3d

δ δ δΓ σ δ δ σ δδ δ

= =

∫ . (3.2)

The J-integral is calculated as described above in Exercise #2 for the contour #12 of the manual option, Fig. 2.5.

3.2. Simulation Results - Comparison with Test Data.

Fig. 3.3: Force vs. load-line displacement Fig. 3.4: Load-line displacement vs. crack extension

Parameter set (1) results in a reasonably good coincidence of test and FE results in terms of force vs. load-line displacement, F(VLL), Fig. 3.3; the simulation results for VLL(∆a) do not meet the test data at all, however, Fig. 3.4. Initiation of crack extension occurs much too late, which is due to the large values of the critical separation, δc, see Fig. 3.2, and hence the separation energy, Γc. The graphs strongly indicate that a comparison of test and simulation data solely in terms of global quantities like F(VLL) can result in wrong conclusions with respect to the quality of the parameters.

Parameter sets (2) and (3) with a ten times smaller δc and a nearly six times smaller Γc 16

yield a reasonable estimate for the initiation of crack extension but underestimate the maximum force. The initial curvature of VLL(∆a) is better matched by parameter set (1). Increasing the cohesive strength, σc, obviously results in a higher slope of the VLL(∆a) curve compared to that of parameter set (1). There is little difference between the parameter sets

16 regard the relation between δc, Γc and σc according to eq. (3.2)

0

1

2

3

4

5

0 0,02 0,04 0,06 0,08 0,1

σ n/R

0[-

]

δn [mm]

(1)

(2)

(3)

0

10

20

30

40

50

0 0,5 1 1,5 2 2,5 3

F[k

N]

VLL [mm]

test YYB-19-3

FE param set (1)

FE param set (2)

FE param set (3)

0

0,5

1

1,5

2

2,5

3

0 0,5 1 1,5 2 2,5 3

VLL

[mm

]

∆a [mm]

test YYB-19-3

FE param set (1)

FE param set (2)

FE param set (3)


(2) and (3) as one might expect looking on Fig 3.2 and no conclusion is possible, which one is the better. As both the cohesive strength, σc, and the shape parameter, δ2, are varied, it is difficult to uniquely separate the respective effects on the simulation results.

Firm conclusions on the sensitivity of the simulation results to parameter variations require more systematic studies varying no more than one parameter at each time, see chapter 3.3. Risks with missing convergence may show up, however.

The JR-curve, Fig. 3.5, provides basically the same information as the plot VLL(∆a), Fig. 3.4, since the relation between J and the load-line displacement is linear over a wide range, see Fig. 3.6. It allows for relating the separation energy to a macroscopic fracture parameter, namely the J-value at initiation of crack extension, Ji. Parameter set (1) with Γc = 137 kJ/m2 yields Ji = 215 kJ/m2, whereas parameter sets (2) and (3) with a six times lower Γc = 23…24 kJ/m2 result in a value of Ji = 36 kJ/m2, which is also six times lower. The JR-curves for the parameter sets (2) and (3) are nearly straight lines after initiation. The plot J(VLL) in Fig. 3.6 shows nearly no sensitivity to the cohesive parameters and hence does not provide any significant information. From Fig. 2.7 which has been obtained for constant crack length no substantial effect of crack extension could be expected in any case.

Fig. 3.5: JR-curves Fig. 3.6: J-integral vs. load-line displace-ment

A systematic trial and error procedure based on an understanding of the effect of individual parameter variations on the mechanical response of the structure helps finding optimised parameters. Based on the sensitivity study described in the following section, a new parameter set (4) has been identified which yields a reasonable approximation of the test data for both global and local quantities, see Figs. 3.7 - 3.9.

Fig. 3.7: Force vs. load-line displacement Fig. 3.8: Load-line displacement vs. crack extension

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100

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400

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600

0 0,5 1 1,5 2 2,5 3

J[k

J/m

2]

∆a [mm]

test YYB-19-3

FE param set (1)

FE param set (2)

FE param set (3)

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600

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J[k

J/m

2 ]

VLL [mm]

test YYB-19-3

FE param set (1)

FE param set (2)

FE param set (3)

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N]

VLL [mm]

YYB-19-3

FE par-set (4)

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VL

L [m

m]

∆a[mm]

YYB-19-3

FE par-set (4)


parameter set (4): σc = 2130 MPa,

δc = 0.05 mm , Γc = 94.4 kJ/m2, δ1/δc = 0.04, δ1/δc = 0.8

Fig. 3.9: JR-curves

3.3 Sensitivity Study for Cohesive Parameters

The following graphs17 show the effects of a systematic variation of the four parameters cohesive strength, σc, critical separation, δc, and shape parameters, δ1/δc, δ2/δc, on the global response, F(VLL), and the R-curve in terms of VLL(∆a). One parameter is varied at a time, whereas the three others are kept constant. The respective separation energy, Γc, results from eq. (3.2). The sensitivity with respect to the cohesive parameters σc and δc is considerably more pronounced than that to the shape parameters δ1 and δ2.

3.3.1 Cohesive Strength

The cohesive strength is the maximum stress which the cohesive element can endure. It thus determines the maximum load which a structure can bear and the R-curve, see Fig. 3.10. If σc is low, e.g. σc = 1500 MPa ≈ 2.3 R0, fracture occurs quasi-brittle, the “R-curve” is nearly a constant and the specimen will fail unstable in small-scale yielding. If σc is high, e.g. σc = 3000 MPa ≈ 4.6 R0, the stresses in the ligament are limited by the yield condition and hardly reach σc so that nearly no crack extension occurs. The respective load-displacement curve of the specimen approaches that of Fig. 2.6. The parameter study suggests a range of 3 R0 < σc < 4 R0 for the present (plane strain) problem.

Fig. 3.10: Effect of σc [MPa], δc = 0.1 mm, δ1/δc = 0.02, δ2/δc = 0.3

17

The study presented in this paragraph has been performed by CAN IÇÖS, PhD student, Dipartimento di Meccanica, Note that the stress-strain curve input for ABAQUS is derived from the same tensile test data but is not identical to that shown in Fig. 1.8.

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2]

∆a [mm]

YYB-19-3

FE par-set (4)

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F[k

N]

VLL [mm]

YYB-19-1

sigma_c=1500

sigma_c=2000

sigma_c=2100

sigma_c=2130

sigma_c=3000

0

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1

1,5

2

2,5

3

0 0,5 1 1,5 2 2,5

VLL

[mm

]

∆a [mm]

YYB-19-1sigma_c=1500sigma_c=2000sigma_c=2100sigma_c=2130sigma_c=3000


3.3.2 Critical separation

The critical displacement at total local failure governs the “ductility” of the separation process. The effects of the two parameters σc and δc interact, however, via the separation energy, Γc ~ σc δc, according to eq. (3.2). Γc controls the initiation value of J as well as the slope of the R-curve. Ji can hence be taken as first approximation for Γc. If the slope of the R-curve is too high, F(VLL) approaches Fig. 2.6 again.

In the present parameter study, σc = 2130 MPa is kept unchanged, so that together with the chosen shape parameters, Γc ≈ 0.68 σc δc = 1450 MPa δc. The range of 0.01 mm ≤ δc ≤ 0.1 mm in Fig. .3.11 thus corresponds to 14.5 kJ/m2 ≤ Γc ≤ 145 kJ/m2.

Fig. 3.11: Effect of δc [mm], σc = 2130 MPa, δ1/δc = 0.02, δ2/δc = 0.5

3.3.3 Shape parameter δ1

The sensitivity of the simulation with respect to δ1 is minor but perceivable, see Fig. 3.12. However varying the shape parameter δ1 should not be part of the parameter identification. It determines the initial “elastic” stiffness, Kcoh, of the cohesive element, which is inevitable for the numerical stability of the solution but should be significantly larger than the elastic stiffness of the continuum elements in the ligament in order to avoid numerical artefacts,18

n

cncoh

n 1 elem0

2d EK

d hδ

σσδ δ

→

= = > , (3.3)

resulting in

c1 elem

20.025mmh

E

σδ < = (3.4)

for σc = 2130 MPa and helem = 0.125 mm. All values of δ1/δc in Fig. 3.11 meet this condition.

18 M. ELICES, G.V. GUINEA, J. GÓMEZ AND J. PLANAS: The cohesive zone model: advantages, limitations and challenges. Eng. Fract. Mech. 69 (2002), 137-163.

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F[k

N]

VLL [mm]

YYB-19-1

delta_c=0.01

delta_c=0.02

delta_c=0.05

delta_c=0.1

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1

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2

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3

0 0,5 1 1,5 2 2,5

VL

L[m

m]

∆a [mm]

YYB-19-1

delta_c=0.01

delta_c=0.02

delta_c=0.05

delta_c=0.1


Fig. 3.12: Effect of d1 = δ1/δc, σc = 2130 MPa, δc = 0.1 mm, δ2/δc = 0.5

3.3.4 Shape parameter δ2

The shape parameter δ2 affects the value of Γc for fixed σc and δc according to eq. (3.2). It can hence be used for a “fine tuning” of the slope of the R-curve, see Fig. 3.13.

Fig. 3.12: Effect of d2 = δ2/δc, σc = 2130 MPa, δc = 0.1 mm, δ1/δc = 0.02

Epilogue and Acknowledgement

The report presents the results of an exercise for the students of a PhD course on Computational Fracture Mechanics held at the Dipartimento di Meccanica of the Politecnico di Milano in October and November 2013. Beyond the direct results of the assignment which have been used as benchmark for the evaluation of the students’ reports, additional information and results are provided which have not been part of the definition of the project but may serve as background explanations, particularly the comments on the ABAQUS theory manual with respect to J calculation and the detailed sensitivity study for the cohesive parameters.

The authors gratefully acknowledge the contributions of GIORGIA GOBBI (Figs. 1.10, 1.11, 1.13), CAN IÇÖS, (Figs. 3.10 - 3.13), PhD students, and STEFANO FOLETTI (test data), professor at the Dipartimento di Meccanica.

The report can hopefully also give guidance to engineers and scientists working in the field of numerical simulations of fracture problems how to achieve reliable results.

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F[k

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VLL [mm]

YYB-19-1

d1=0.015

d1=0.02

d1=0.025

d1=0.05

d1=0.1

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2,5

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0 0,5 1 1,5 2 2,5

VLL

[mm

]

∆a [mm]

YYB-19-1

d1=0.015

d1=0.02

d1=0.025

d1=0.05

d1=0.1

0

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F[k

N]

VLL [mm]

YYB-19-1

d2=0.2

d2=0.3

d2=0.5

d2=0.7

0

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1,5

2

2,5

3

0 0,5 1 1,5 2 2,5

VLL

[mm

]

∆a [mm]

YYB-19-1

d2=0.2

d2=0.3

d2=0.5

d2=0.7

Documents

Computational Fracture Mechanics - Technische Fakultät · stress-strain curves reflect the measured load-elongation data. Except YA-1-28, which will be Except YA-1-28, which will