Overview of Problem Description

An
important element in the process of predicting residual strength of a structure
experiencing ductile tearing is having a criterion that predicts the onset and rate of
this phenomenon. Tests and numerical simulations have been performed to assess the
critical crack tip opening angle (CTOA_{c}) criterion for predicting residual
strength of structures containing MSD. The objectives of this problem are to describe the
bases for this criterion and to present example simulations that employ it. The next
section details the theoretical background behind the CTOA_{c} criterion, and
describes experimental and computational investigations into it. This section is followed by a review of findings
of these investigations. Those readers
wanting to go directly to a computational example application of this criterion can
proceed directly to the __Computational Models__
section.

__The
__*CTOA*_{c} Criterion for Ductile Tearing

The
local slope of the crack tip opening profile, or CTOA, has been suggested to characterize
ductile crack growth behavior [de Koning 1977]. Newman [1984], Rice and Sorensen [1978],
and Kanninen and Popelar [1985] further defined the CTOA as the crack tip opening angle
measured at a fixed distance behind the moving crack tip. The CTOA fracture criterion
asserts that this angle maintains a constant value during stable crack growth for a given
thickness of material.

The
definition of CTOA as suggested by Newman [1984] is adopted for this problem. For ModeI
only deformations, CTOA is defined as, Figure FAC-4.1:

_{}

where
*d* is the CTOD measured at a specific distance,
d, behind the crack tip.

Stable
crack growth is an inherent feature of elasticplastic materials because of the
occurrence of permanent plastic deformations during unloading [Rice 1975]. This effect can
be demonstrated by global energy dissipation or by the local residual plastic
deformations. Suppose two materials have the same uniaxial stressstrain curves; one is
an idealized nonlinear elastic material and the other is an elasticplastic material. For
cases without crack growth, the same CTOA and strain concentration will occur in the two
materials as illustrated in Figure FAC-4.2, STAGE 0. As the
crack propagates in the nonlinear elastic material, deformation fields need to be
readjusted and the same crack tip opening profile would occur for the new crack tip
location. This is not the case for the elasticplastic material because a large part of
the energy is consumed by plastic dissipation with far less strain recovered during
unloading. Thus, a smaller CTOA is obtained after crack growth (STAGE 1). Further increase
of the applied loading is needed to open the crack (STAGE 2) and causes stable crack
growth in the elasticplastic material. Fracture instability will occur as the crack
reaches a steadystate condition in which the crack continually advances without further
increase in load. If the analysis is performed under displacement control, then a
reduction in applied load is required to maintain a constant CTOA for continuous crack
growth. Hereafter, CTOA_{a} is the crack tip opening angle measured immediately
after propagation, STAGE 1. CTOA_{b} is denoted as the increase in crack tip
opening angle required to reach the critical value, CTOA_{c}. Thus, the condition

Figure FAC-4.1. Illustration of parameters used for CTOA
definition.

Figure FAC-4.2. Illustration of crack growth in nonlinear elastic
and elasticplastic materials.

CTOA_{a}
+ CTOA_{b} = CTOA_{c}

satisfies
the fracture criterion for crack propagation, and the condition

CTOA_{a}
= CTOA_{c}

indicates
the occurrence of fracture instability for the analysis under load control.

Another
related factor for stable crack growth is the plastic wake effect caused by the residual
plastic deformations [Newman 1984]. As the crack grows, the plastic zone behind the crack
tip unloads to an elastic state leaving the appropriate plastic wake behind the advancing
crack tip. This effect results in resistance to crack tip opening as illustrated in Figure FAC-4.3. The dashed curves in the plastic wake region show
what the crack opening profile would have been if residual plastic deformations had not
been retained in the material behind the advancing crack tip. This phenomenon is also
essential for simulating the initiation of stable crack growth associated with high
fatigue stress prior to tearing [Dawicke 1994b].

Laboratory
tests have been conducted on flat panels made of aluminum alloys to measure CTOA_{c}
values [Dawicke 1994a; Newman 1993]. Numerical simulations using these values have been
conducted using twodimensional [Newman, 1992, 1993; Dawicke 1994b, 1995, 1997a],
thinshell [Chen 1996, 1997, 1998], and threedimensional [Dawicke 1996, 1998, 1997b]
finite element elasticplastic crack growth analyses. These activities are first reviewed
to highlight important findings. The latest
results are then used as a starting point for the example simulations.

Figure FAC-4.3. Illustration of plastic wake effect caused by
crack growth.

__Review
of Findings on CTOA___{c} Criterion

A
series of fracture tests has been conducted using a 2024T3 aluminum alloy for MT, CT,
blunt notch, THCT and MSD specimens. Newman
*et al.* [1992]
conducted tests on 0.05, 0.07, and 0.09 inch thick, 3.0 and 11.8 inch wide MT and blunt
notch specimens as well as 0.09 inch thick, 10 inch wide THCT specimens. The bluntnotch
specimen is similar to the MT specimen except that a small hole is drilled at both ends of
the saw cut. It is intended to assess the suitability of elastic plastic finite element
analyses with the smallstrain assumption to model largescale plastic deformations.
Good agreement between predicted and measured load versus notchtip displacements
substantiates the assumption. The critical values of CTOA were measured for the MT and
THCT specimens to assert the specimen configuration independence of the fracture
criterion. The THCT specimen had a stress intensity factor solution like that of a
cracked, stiffened panel [Newman, 1995]. The measured CTOA_{c} values showed
higher angles at crack initiation, but reached the same constant value after a small
transition period of crack growth. The agreement of CTOA_{c} between MT and THCT
specimens indicated that the CTOA_{c} fracture criterion is independent of
specimen configuration; this was further confirmed by a followup study with measurements
from CT specimens [Dawicke 1995].

A
2D elasticplastic finite element code, ZIP2D [Newman1974], and a 6.1 degree CTOA_{c},
computed at 0.01875 inch behind the crack tip, were used to simulate fracture behavior of
the MT specimens [Newman 1992]. To model fatigue precracking, cyclic loading simulation
was conducted prior to stable tearing analyses. Experimental and predicted results showed
that a higher applied stress during the fatigue tests increased the resistance to stable
crack growth initiation. Predicted residual strengths under plane stress conditions were
within 4% of experimental results for 3.0 and 11.8 inch wide MT specimens. However, the
plane stress analyses overpredicted crack extension prior to limit load while the plane
strain analyses underpredicted crack extensions.

The
above studies raised two important questions:

1.
What is the governing mechanism that causes higher CTOA_{c} values during crack
initiation?

2.
What is the governing mechanism that causes the discrepancy between 2D predictions and
test results?

Dawicke
and Sutton [Dawicke 1994b] examined the higher values of measured CTOA_{c}
observed during crack initiation, i.e., question 1. Two independent techniques, optical
microscopy (OM) and digital image correlation (DIC) were used to measure surface CTOA_{c}
during crack growth. The results of the two methods agreed very well. Fatigue marker loads
and a scanning electron microscope were used to examine the fracture morphology and
sequences of crack front profiles. For specimens under low magnitude of fatigue stress
prior to tearing, crack surfaces underwent a transition from flattoslant crack growth.
A schematic of the transition is shown in Figure FAC-4.4.
During the transition period, the CTOA_{c} values were high and significant
tunneling occurred. After an amount of crack growth equal to about the specimen thickness,
CTOA_{c} reached a constant value. After crack growth equal to about twice the
thickness, crack tunneling stabilized. For specimens that were precracked under a high
magnitude of fatigue stress, a 45degree, slant, throughthickness initial crack was
formed prior to tearing. During the crack initiation period, the CTOA_{c} values
of specimens with high fatigue stress were lower than the ones with low fatigue stress.
But the same constant CTOA_{c} value was observed after crack growth equal to
about the specimen thickness.

The
discrepancy between 2D predictions and test results, i.e., question 2, was thought to be
related to the 3D constraint effect. Although thinsheet structures behave essentially in
plane stress, the constraint due to the finite thickness of the specimens can cause the
regions local to the crack tip to approach plane strain conditions [Hom 1990]. To
investigate the constraint effect, 2D and 3D analyses were conducted. In the 2D analyses,
a core of elements above and below the crack path were assigned as plane strain while all
other elements were assigned as plane stress. The
plane strain core concept is illustrated in Figure FAC-4.5.

In
their first attempt, Dawicke *et al.* [Dawicke
1995; Newman 1993] used 2D finite element analyses with a 6.0 degree CTOA_{c}
computed at 0.02 inch behind the crack tip and a plane strain core height equal to 0.2
inch to simulate fracture behavior with the constraint effect. They showed that the use of
a plane strain core was essential to accurately model crack growth. The predicted residual
strengths were within 2% for 3 and 12 inch wide, 0.09 inch thick MT specimens and within
4% for 6 inch wide, 0.09 inch CT specimens. For 20 inch wide, 0.04 inch thick MSD
specimens, 2D analyses with a 5.1 degree CTOA_{c} showed excellent agreement of
linkup and residual strength between predictions [Newman 1993] and test results [Broek
1994].

Figure FAC-4.4. Schematic of fracture surface indicating
transition from a flat to a slant crack plane (after [Newman 1992]).

Figure FAC-4.5. Schematic
of the plane strain core.

Dawicke
et al. [Dawicke1996,
1998] further studied the constraint effect using 3D finite element analyses with a 5.25
degree CTOA_{c} computed at 0.04 inch behind the crack tip.

The
3D analyses successfully simulated fracture behavior of 2.0, 4.0, 6.0, and 8.0 inch wide
CT specimens, 1.2, 3.0, 6.0, 12.0, 24.0, and 60.0 inch wide MT specimens, and 12.0 inch
wide MSD specimens made of 0.09 inch thick, 2024T3 aluminum alloy. A plane strain core
height of 0.12 inch was required for 2D analyses to match the measured results and the 3D
fracture predictions.

Computational Models

In
the following examples, tests on MT and MSD specimens are simulated. The FRANC3D/STAGS
system [www.cfg.cornell.edu] is used to simulate fracture behavior and to predict residual
strength using the guidelines derived from the 2D and 3D studies just described.

Fracture
tests of MT specimens were conducted by the Mechanics of Materials Branch at NASA Langley
Research Center [Dawicke 1994a, 1996,1998]. The test specimens were made of 0.09-inch
thick 2024T3 aluminum alloy. All specimens were fatigue precracked in the LT
orientation with a low stress level that results in a stress intensity factor range of DK
= 7 ksi Öinch.
For specimens with a single crack, different widths of panels equal to 3 inch, 12 inch,
and 24 inch with a cracklength to width ratio equal to 1/3 were tested, Figure FAC-4.6. For cases with multiple cracks, only the 12-inch
wide specimens with two to five near collinear cracks as illustrated in Figure FAC-4.7 were tested. All tests were conducted under
displacement control with guide plates to prevent outofplane buckling. Both OM and DIC
techniques were used to measure the CTOA_{c} during stable crack growth [Dawicke
1994]. Experimental results for MT and CT specimens are shown in Figure
FAC-4.8. The CTOA_{c} rapidly reaches a constant value with a scatter band
about ± 1
degree.

__Numerical
Simulations of MT Specimens__

Fracture processes in the MT specimens are simulated first. To
investigate panel size effects, numerical simulations of 60-inch wide panels with the same
cracklength-to-width ratio are also performed. Elasticplastic finite element analyses
based on incremental flow theory with the von Mises yield criterion and the small strain
assumptions are used to capture the active plastic zone and the plastic wake during stable
crack propagation. A piecewise linear representation is used for the uniaxial
stressstrain curve for 2024T3 aluminum, Figure FAC-4.9.
The CTOA_{c} used in this study was 5.25 degrees measured 0.04 inch behind the
crack tip. This particular value was provided by Dawicke and Newman [Dawicke 1996, 1998]
based on 3D simulations of CT specimens. Upon satisfaction of the fracture criterion,
nodal release and load (or displacement) relaxation techniques are employed to propagate
the crack. Because of the double symmetry of the geometry and loading, only one-quarter of
the specimen with imposed symmetry boundary conditions is modeled. Outofplane
displacements are suppressed. Displacementbased, fournoded and fivenoded
quadrilateral shell elements having C_{1} continuity are used [Rankin 1991]. These
elements are intended to model thin shell structures for which transverse shear
deformation is not important. Each node of the element has six degrees of freedom
including three translations and three rotations.

Figure FAC-4.6. Test
configuration for MT specimens.

Figure FAC-4.7. Schematic
of crack configurations for12- inch MT specimens.

* *

Figure FAC-4.8. Surface
measurements of CTOA_{c} [from Dawicke1998]

A
special fivenoded shell element, formulated by combining two fournoded elements and
using linear constraint along the edge to eliminate the dependent node, is used to
transition from locally refined zones around the crack path to a coarse mesh away from the
crack.

A
convergence study was conducted to determine the sensitivity of the predicted residual
strength to the element size along the crack extension path. Three meshes for the 24-inch
wide panel were created with crack tip element sizes of 0.04-inch, 0.02-inch, and 0.01-
inch. For all crack growth and residual strength analyses, the CTOA is evaluated at 0.04-
inch behind the crack tip to be consistent with experimental measurements. A finite
element mesh with 0.04-inch square crack tip elements for the 24-inch wide panel is shown
in Figure FAC-4.10. Predicted crack growth results for cases
with 0.04-inch and 0.02-inch crack tip elements as well as predicted residual strengths
for all three cases are shown in Figure FAC-4.11. Although
some discrepancy is observed at the early stage of stable tearing, the predicted results
exhibit little influence of mesh size after a relatively small amount of stable crack
growth. More importantly, the predicted residual strength is very insensitive to crack tip
element size. Thus, all the remaining meshes used in this example have 0.04 inch crack tip
elements.

Figure FAC-4.12 shows two predicted crack opening profiles for
the 24-inch wide panel. The angles are computed immediately after propagation (i.e., CTOA_{a},
see Figure FAC-4.2) with relaxation procedures completed and
before increasing the applied displacement. The two CTOA_{a} values correspond to
(1) the angle after the first increment of crack growth, and (2) the angle after the
specimen reaches its residual strength. As shown in the figure, CTOA_{a} is much
smaller than the critical angle after the first crack growth increment. This clearly
demonstrates the permanent plastic deformation effects on stable crack growth in the
elasticplastic material. As the crack propagates, CTOA_{a} increases. Since the
analyses are conducted under displacement control, the CTOA_{a} at residual
strength is less than, but approaching its critical value.

Comparisons
between numerical results and experimental measurements for the applied stress versus half
crack extension are shown in Figure FAC-4.13. Results of
predicted residual strength are comparable to experimental measurements, but as the width
of the panel increases, the relative difference between experimental measurements and
numerical predictions increases. Figure FAC-4.14 depicts the
predicted plastic zone as the specimens reach their ultimate strength.

Figure FAC-4.9. Piecewise
linear representation of uniaxial stress-strain relationship for 2023-T3 aluminum alloy
used in the present example.

Figure FAC-4.10. Finite
element mesh for 24- inch wide MT specimen and detail along crack path.

Figure FAC-4.11. Results
from convergence study: predicted crack growth and predicted residual strength for 24-
inch wide panel with different crack tip element sizes.

Figure FAC-4.12. Crack
opening profiles and CTOA a after the first crack growth increment and after reaching the
residual strength for 24 inch wide panels.

Two
distinct phenomena are observed. For small specimens, plastic zones reach the free edge
and the limit load is attained due to net section yielding. In contrast, for large
specimens, plastic zones are wellconfined by the elastic region and residual strength is
reached near the fracture instability of the specimens.

As
shown in Figure FAC-4.13, the relative difference in residual
strength between experimental and numerical results increases as the width of the panel
increases. This discrepancy is believed to be due to the threedimensional nature of the
stresses around the crack tip, a result of constraint effects due to the finite thickness
of the panels [Hom 1990; Dawicke 1995]. Numerical results using plane strain, plane stress
with a plane strain core height (see Figure FAC-4.5) equal to
0.12 inch, and threedimensional finite element analyses obtained from [Dawicke 1996,
1998] were studied to further demonstrate constraint effects on residual strength
predictions. Predicted results shown in Table FAC-4.1 and Figure FAC-4.15 suggest that:

· Thin
shell finite element analysis, behaving essentially in plane stress, tends to

overpredict
the residual strength as the width of the panel increases;

· Plane
strain analysis overpredicts the residual strength of small specimens, but
underestimates it for large specimens;

· 2D
plane stress analysis with a plane strain core and 3D analysis properly account for
constraint effects. The predicted results follow the trend of experimental measurements
even for wide panels.

Figure FAC-4.13.
Comparisons between experimental measurements and numerical predictions of applied stress
versus half crack extension for various sizes of specimens.

Figure FAC-4.14. Numerical
predictions of plastic zone for various sizes of specimens reaching their residual
strength.

Table
FAC-4.1 Comparisons of Residual Strength Predictions (ksi) for MT Specimens

Plate Width |
Thin Shell |
Plane Strain |
Plane Strain Core |
3D |
Experiment |

3 in. |
34.0 |
38 |
33.6 |
34.3 |
34.5 |

12 in. |
30.7 |
32.7 |
30.7 |
30.8 |
31.3 |

24 in. |
29.6 |
26.3 |
29.1 |
29.1 |
28.4 |

60 in. |
28.1 |
16.6 |
26.7 |
26.3 |
N/A |

The
cross-over between plane stress and plane strain in predicting residual strength as the
specimen size increases is an interesting topic. Based on the predicted plasticity
distribution in Figure FAC-4.14, the net section yielding
mechanism seems to dominate the residual strength prediction of small specimens. This may
explain why the plane strain analysis predicts a higher residual strength for small
specimens because the effective yield stress in plane strain is larger than that in plane
stress. Thus, a further increase of remote stresses under plane strain conditions is
needed for specimens to reach the point of net section yielding. For larger specimens,
residual strength is governed by stable crack growth and fracture. As one would expect
from the thickness effects on K_{c} in LEFM, materials in plane stress have higher
fracture toughness than materials in plane strain. Recent micromechanicsbased, 3D
analysis of ductile crack growth in a thin plate with a Gursontype model also showed
that, although the crack growth resistance at first increases with increasing plate
thickness, the resistance to crack growth decreases after a small amount of crack
extension [Mathur 1996]. For CTOAdriven ductile crack growth, stresses and strains under
plane stress and plane strain conditions have not been studied in sufficient detail to
clarify the issue. A possible cause of higher crack growth resistance in plane stress may
be related to the residual plastic deformation effects. Based on asymptotic solutions for
cracks growing in an incompressible elasticperfectly plastic material under Mode I
loading, larger residual plastic deformations would occur under plane stress than plane
strain conditions leading to higher crack growth resistance.

Figure FAC-4.15. Predicted
results of thin shell, plane strain, plane stress with a plane strain core, and 3D
analyses compared with experimental measurements.

** **

__Numerical
Simulations of Specimens with Multiple Cracks__

Numerical
simulations of tests with multiple cracks using the CTOA fracture criterion are
straightforward extensions of single crack specimen simulations. The same criterion (CTOA_{c}
= 5.25 degrees measured 0.04-inch behind the crack tip) is used to simulate stable crack
growth and the linkup of multiple cracks, and to predict residual strength. No
supplementary criterion is needed. Multiple crack test configurations as shown in Figure FAC-4.7 are modeled and the fracture processes are
simulated. Note that the symmetry conditions along the vertical central line of the
specimens are no longer valid due to the various lengths of fatigue precracks; thus, at
least one half of the specimen needs to be modeled. An example finite element mesh for
test configuration b is shown in Figure FAC-4.16. Mesh
patterns around the multiple cracks are similar to those of the single crack models.

Figure FAC-4.16. Finite
element mesh for the test configuration b (12-inch wide specimen with two cracks).

Figure FAC-4.17. Predicted
applied stress versus crack extension for test configurations b and d.

Numerical
results and experimental measurements for the applied stress versus half crack extension
for test configuration b and d are shown in Figure FAC-4.17.
Two distinct applied load versus crack growth history curves are predicted. For test
configurations a, b, and c, linkup of cracks happens before the specimens reach their
residual strength. For test configurations d and e, the limit load is attained before
linkup. These numerical predictions agree with observations from the fracture tests.
Again, plastic deformation plays an important role in the fracture process. Figure FAC-4.18 shows the plastic zone evolution of test
configuration b during stable crack growth. The inherent residual plastic deformations
during crack growth are clearly demonstrated through the deformed shapes.

Figure FAC-4.19 summarizes the relative difference between
predicted results and experimental measurements. The predicted residual strength of all
five MSD simulations agrees very well (within 3%) with experimental data. The predicted
linkup load is comparable to experimental measurements, but the difference is larger
than that for the residual strength. Reasons for the discrepancy may be related to the
difficulty in measuring linkup load during the fracture tests. It is of practical
importance to characterize the reduction in residual strength caused by MSD. Figure FAC-4.20 plots numerical predictions of residual strength
versus lead crack length for cases with and without small cracks. A loss of residual
strength due to the presence of multiple small cracks is observed.

Figure FAC-4.18. Crack
opening profile(s) and plastic zone evolution of test configuration b during crack growth:
(1) at the first increment, (2) before linkup, (3) after linkup, and (4) reaching
residual strength.

Figure FAC-4.19. Relative
difference of residual strength and linkup load between predicted results and
experimental measurements for specimens with multiple cracks.

Figure FAC-4.20. Loss of
residual strength due to the presence of small cracks.

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