The objects of the first, exploratory, stage of the project were listed as: (1) to make a detailed and critical review of the Boundary Element method as already published and with regard to elastic-plastic fracture mechanics, to assess its potential for handling present concepts in two-dimensional and three-dimensional cases. To this was subsequently added the Finite Volume method and certain aspects of the Finite Element method for comparative purposes; (2) to assess the further steps needed to apply the methods so far developed to the general field, covering a practical range of geometries, work hardening materials, and composites: to consider their application under higher temperature conditions; (3) to re-assess the present stage of development of the energy dissipation rate, crack tip opening angle and J-integral models in relation to the possibilities of producing a unified technology with the previous two items; and (4) to report on the feasibility and promise of this combined approach and, if appropriate, make recommendations for the second stage aimed at developing a generalized crack growth technology for its application to real-life problems

Curr, R. M.

Fenner, R. T.

Ford, Hugh

Ivankovic, A.

Turner, C. E.

English

NASA Technical Reports Server

NASA-CR-199256
IMPERIAL COLLEGE OF SCIENCE,
TECHNOLOGY AND MEDICINE
DEPARTMENT OF MECHANICAL
ENGINEERING
Development of Methods for Predicting Large Crack
Growth in Elastic-Plastic Work-Hardening Materials
in Fully Plastic Conditions
Sir Hugh Ford, C.E. Turner and R.T. Fenner
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https://ntrs.nasa.gov/search.jsp?R=19960009150 2020-06-16T05:54:32+00:00Z
Development of Methods of Predicting Large Crack
Growth in Elastic-Plastic Work-hardening Materials
in Fully Plastic Conditions
Introduction and Objectives
This report is in fulfilment of Research Grant Reference NAGAW - 3909 for NASA Ames
Research Center, Dryden Flight Research Facility, Edwards, CA 93523.
The objects of the first, exploratory, stage of the project were listed as:-
To make a detailed and critical review of the Boundary Element method as already
published and with regard to elastic-plastic fracture mechanics, to assess its potential
for handling present concepts in two-dimensional and three-dimensional cases. To
this was subsequently added the Finite Volume method and certain aspects of the
Finite Element method for comparative purposes.
To assess the further steps needed to apply the methods so far developed to the
general field, covering a practical range of geometries, work hardening materials and
composites: to consider their application under higher temperature conditions.
3.) To re-assess the present stage of development of the energy dissipation rate, crack tip
opening angle and J-integral models in relation to the possibilities of producing a
unified technology with items 1 and 2.
4. To report on the feasibility and promise of this combined approach and, if
appropriate, make recommendations for the second stage aimed at developing a
generalised crack growth technology for its application to real-life problems.
Outline of the methods used
To keep this exploratory study within reasonable bounds, only monotonic loading of metals
at room temperature has been used in the case studies. Seven such studies have been made,
four of the selected problems computed by all four methods used, followed by three problems
specific to one or two of the selected methods. The four methods were (1) The Boundary
Element [B.E]; (2) Finite Volume [F.V.]; (3) the Finite Element [F.E.] with a suite known
as Modified WHAMSE [F.E(Wh)] and (4) the widely used ABAQUS program [F.E(Ab)]. A
perhaps over-simplified statement on the differences in these approaches is that in the F.E.
formulation the classical differential relationships for equilibrium and compatibility are
discretised for each nodal point; in the F.V. formulation these relationships are discretised in
their integrated form for each sub-region in the domain (or control volume); in the B.E.
formulation they are discretised in an integrated form for the whole domain. The B.E. and
F.V. Methods are reviewed in Appendices A and B of the full report.
All the programs are usable for linear elasticity and for incremental plasticity following the
von-Mises criterion of yielding and the Prandtl-Reuss relationships, with work-hardening
according to plastic potential theory, referred to hereafter as elastic-plastic.
2.
The selected case studies
1. The cases-in-common to all methods used.
The purpose of the four cases-in-common is to give some measure of the effort
(preparation and computing time) and accuracy or suitability of results around which
to make a discussion of the methods. A simply supported beam is used, with a central
load, spread as a uniform pressure over a small distance on each side of the central
point of contact. Two cases are elastic - (a) un-notched and (b) notched (Figs. 1 and
2); and two elastic-plastic - (c) un-notched and (d) notched (Figs. 3 and 4). In all four
cases plane strain is used. A coarse mesh "sighting shot" followed by medium and
fine mesh computations were made to indicate convergence to about the 0.5% level.
Sample results for the four cases are given in Tables 1 to 4, using fine mesh
computations only together with a "datum" derived from other sources. There is no
reason to suppose that the "datum" values are any more reliable than the calculations
made by the team, but at least they give an idea of the correspondence with the best
calculated and experimental data available to us.
2. Further examples computed by particular methods are:
a) With the F.V. and F.E(Wh) methods, the same beam as in Fig. 4 was
computed for an impact loading at a constant striker speed of lOm/s. The results are
given in the full report.
b) Three-dimensional solutions of cases 1 to 4 were computed by
F.E(WHAMSE), using a more refined mesh. The differences in values between the
F.E(Wh) for the 3-D and 2-D cases are not large. The results are given in the full
report
c) Stable crack growth in an elastic-plastic beam. With the same beam as in
Case 4, the stable growth of a crack was computed using F.E(Wh). Fig. 5 is a
comparison of the computed and experimental load decay curves.
The work is discussed in detail in the full report.
Conclusions and recommendations for possible future work
For elastic problems of monotonic loading in fracture mechanics, there is little doubt that
linear elastic fracture mechanics is universally accepted as an appropriate model. Although
fatigue is predominantly elastic in terms of the applied stress, it is now generally accepted
that, for short cracks in relation to the micro-structure, it is inadequate, and inhomogeneities
of micro-structure dominate the formation and early growth of a fatigue crack.
For problems with plasticity there is less agreement but for many instances linear elastic
fracture mechanics has been extended by the inclusion of a plastic zone correction factor.
This is not the place to discuss the present stage of research into fracture, but merely to point
out that there is a strong case for further computational methods by which the considerations
of the physical models can be more easily carried on.
3.
With the further development of these computational methods, it is expected that attention
can be given to objectives 3 and 4 during the extension of the work recommended below. It
will therefore be important to keep a "watching brier to incorporate such fracture studies at
the right time. In particular, the J-integral procedure has been found wanting for both
initiation and crack growth with extensive plasticity; initiation is being tackled by the Two
Parameter methods, notably where crack tip fields are characterised by J and Q (hydrostatic
stress); studies to describe crack growth by energy dissipation rate D, and associated crack
tip opening angle (CTOA) exist, but are less well developed. We see a unification of J-Q and
CTOA-D methods as the most promising approach and, as work proceeds, an additional
project will be submitted to NASA when it is considered timely and promising to do so.
In this regard, there is a considerable amount of useful work that can be done using the Finite
Element method and there are many research groups pursuing this route. It is contended that,
on the evidence of this preliminary study, there are clearly benefits to be realised from
developing other methods and the Boundary Element and Finite Volume techniques are
particularly promising.
Our results show that the B.E method is an excellent choice for linear elastic stress analyses,
including fracture problems. While some of its advantages over other methods are lost when
non-linear material behaviour is introduced, it shows considerable promise for further
development for elastic-plastic continuum (but not for composites).
The Finite Volume method has potential for fracture problems, particularly non-linear such as
in large plastic deformations or when they are dynamic. The method is also promising in
regard to composite materials.
In view of our preliminary findings, it is recommended that:
1. Boundary Elements
First, to explore the application of existing programs to two-dimensional elastic-
plastic fracture mechanics problems. Attention would be given to the possible use of
special crack-tip boundary elements to optimise the representation of the singular
stress and strain fields there. There is also considerable scope for introducing
iterative equation solution techniques of the congregate gradient type. The method
should then be extended to three dimensional elastic-plastic problems.
This would be a two-year project for two post doctorals working under Professor
Fenner.
2. Finite Volume method
Future work on the F.V. method should concentrate on modelling non-linear
constitutive relationships and the development of large strain formulation in fracture
and should look at the extension to composite materials.
A one to two year project is envisaged.
3. Fracture Mechanics
A continuing assessment of the progress of the computational work in order to start
objectives 3 and 4 at the appropriate time.
Results
q/Pmm/MN
a(max) MPa
DoF
(DoF ratios
CPU time, s
(CPU ratios
CPU(s)/DoF
Table 1
Computational Study No. 1 : Comparisons of fine mesh data only
Un-cracked beam; elastic; plane strain
Notation used (see Fig. 1)
q/P compliance of C relative to D
a(max) maximum bending stress at A (for a unit load)
B.E.
31.09
42.70
272
1
0.3
1
0.0011
L\L
30.72
42.52
832
3.0
59
180
0.071
Datum
31.03 [1]
42.68*
[1] Underwood et al, 1985.
*using conventional engineering elastic beam theory with the central load spread over
± 1mm as in the computations.
Results
q/Pmm/MN
V/Pmm/MN
2/vPmm/MN
at a/32
at a/16
KI MPa/m
DoF
(DoF ratios
CPU time, s
(CPU ratios
CPU(s)/DoF
Table 2
Computational Study No. 2 : Comparisons of fine mesh data only
Cracked beam; elastic; plane strain
Notation used (see Fig. 2)
q/P the compliance of C relative to D
V/P the crack mouth opening compliance
2WP the compliances of the crack profile at two selected positions
K j the elastic crack tip intensity factor at unit load
B.E
5.26
3.24
0.385
0.563
0.936
224
1
1
1
0.0045
F.V.
5.28
3.31
0.433
0.601
0.988
1322
5.5
182
182
0.15
F.E.(WrT>
5.18
3.17
0.37
0.53
0.964
6240
27.8
900
900
0.29
F.E.(Ab1
5.24
3.21
0.381
0.538
0.942
6018
26.9)
64
64)
0.011
Datum
5.10 [1]
3.26 [2]
327 [3]
0.394 [2]
0.568 [2]
0.953*
[1] Underwood et al [1985]
[2] Gross etal[ 1968]
[3] Kappetal[1985]
* using the lefm shape factor, Y = 10.650 from Srawley [1976], as listed by Towers [1981] in the
formK = PY/B/W.
The wide differences in degrees of freedom (DoF) and Central processor units (CPU) times between the
methods are to be noted. They differ roughly by one to two orders of magnitude between the methods. The two
F.E. methods (F.E.(Wh) and F.E.(Ab)) have very similar degrees of freedom but differ in CPU by near 15 fold
for reasons of the method of solution used in this version of WHAMSE. The F.E.(Wh) data have been
generated here by using an existing three-dimensional mesh run in plane strain rather than by generating an
actual two-dimensional mesh.
Table 3
Computational Study No. 3 : Comparison of fine mesh data only
Un-cracked beam; elastic-plastic; plane strain.
Notation used below; (see Fig. 4.3)
P(max) the final load at which a test run was stopped
q displacement of C relative to D
qpl plastic displacement of C relative to D
qel elastic displacement of C relative to D
c(max) maximum bending stress at A.
Results B.E. F.V. F.E(Wh) F.E.(Ab) Expt. [1]
P(max)kN 45.0 45.0 45.0 45.0 44.1*
a(max)MPa . 1296 1176 1247 1214 1310
qmmatP(max) 4.95 3.41 5.45 . 3.65 10.0
qplmm at P(max) 3.55 2.03 4.05 2.25 8.1
qeimmatP(max) 1.40 1.38 1.40 1.40 1.9
qpimmat44kN 3.47 0.84 2.71 0.97 8.0
qpimmat40kN 0.95 0.44 0.55 0.48 2.7
DoF 344 1100 1908 2314
(DoF ratios 1 3.2 5.5 6.7)
CPU time, m 14 130 60 27
(CPU ratios 1 9.3 4.2 1.9)
CPU Case 3/Case 1 2800 130 35 160
[1] Dagbasi [1989]. * A loading roller slipped at this point whilst the load was still increasing. A
maximum estimated by extrapolation is about 45 to 46kN; the test was of course in between plane
stress and plane strain.
Table 4
Computational Study No. 4 : Comparisons of fine mesn data only
Cracked beam; elastic-plastic; plane strain.
Notation used (see Fig. 4.4):-
qej/P elastic compliance of C relative to D
Ve[/P elastic crack mouth opening compliance
q. qpi displacement of C relative to D and plastic component thereof
V, \'pj crack mouth opening displacement and plastic component thereof
Results B.E. F.V. F.E(Wh) F.E.(Ab) Experiment [1]
or Datum [2] [3]
qel/Pmm/MN 6.24 6.25 6.15 6.20 6.60 [1]
6.63*; 6.03 [2]
Vei/Pmm/MN 3.93 4.11 ' 4.00 4.06 4.31 [1]
4.47*; 4.07 [3]
Max q mm 1.35 1.37 1.37 1.37 1.51 [1]
Maxqpimm 0.02 0.02 0.04 0.03 0.03 [1]
Max V mm 0.868 0.905 0.896 0.904 0.955 [1]
MaxVpjmm 0.019 0.028 0.032 0.026 0.024 [1]
J (max) MN/m 0.163 0.133 0.143 0.136 0.159 [1]
DoF 264 1944 6190 4774
(DoF ratios 1 7 23 18)
CPU time, m 5 74 50 63
(CPU ratios 1 15 8 13)
CPU(s)/DoF 1.1 2.3 0.5 0.8
DoF Case4/Case2 1.2 1.5 1.0 0.7
CPU Case 4/Case 2 300 25 1.3 60
[1] Dagbasi, 1989. (Note the four computations here are made in plane strain whereas the beam
tested is probably nearer plane stress: see above discussion and further resuks in Case 6. For the
maximum load used in the computations, P = 216kN, a crack growth, Aa = 0.5mm is estimated from
the experimental data whereas no growth was allowed in these studies).
[2] Underwood ft al, [1985], *computed; plane stress and then plane strain.
[3] Wu Shang-Xian, [1984], "computed: plane stress and then plane strain.
D
W
-L/2-
-t P / 2 Dimensions :S =110 mmW = 13.52 mmB =20.67 mm
L = 120& 140mm
Properties :
E = 2.0E+05 MPa
Poisson's ratio = 0.3
Load. P = 1 kN
L / 2 -
Dimensions :
S =200 mm
W = 50 mm
B =50 mm
L = 250 mm
a/W = 0.5
Propenies :
E = 2.0E+05 MPa
Poisson's ratio = 0.3
Load, P = 1.0 kN
Fig. ,.2
Ti y
4-£— —
I A C
f
<
}
Dimensions :
W = 13.52
S = 110
L = 120
B = 20.67
S/2
W
L/2
Fig. 3
mm
mm
mm
mm
PE]
--"j-^-r
i 1
y
C
S/2 iJP/2
Dimensions :
a = 50.77
W = 94.2
S = 4W = 376
v L =5+30 = 406
W B = 49.8
mm
mm
8 mm
8 mm
mm
L/2
Fis.
I2000
1500 -
1000 -
500 -
500
Fig. 5


Development of methods for predicting large crack growth in elastic-plastic work-hardening materials in fully plastic conditions

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