fem modelling of elastoplastic stress and strain field in

~eries 07 Aerospace Materials 02

FEM Modelling of Elastoplastic Stress and Strain Field in Centre-cracked Plate

I. Kunes

Delft University Press

FEM Modelling of Elastoplastic Stress and Strain Field in Centre-cracked Plate

li' •

2392 327 3

Series 07: Aerospace Materials 02

FEM Modelling of Elastoplastic Stress and Strain Field in Centrecracked Plate

I. Kunes

Delft University Pre ss / 1997

Published and distributed by:

Delft University Press Mekelweg 4 2628 CD Delft The Netherlands Telephone + 31 (0) 15 278 32 54 Fax +31 (0)152781661 e-mail: [email protected]

by order of:

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Cover: Aerospace Design Studio, 66.5 x 45.5 cm, by :

··--~I

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90-407-1588-2

Copyright © 1997 by Faculty of Aerospace Engineering

All rights reserved. No part of the material protected by th is copyright notice may be reproduced or utilized in any farm or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the publisher: Delft University Press.

Printed in The Netherlands

Table of contents

Summary ................................. ........................... .............. .... ..... ...................... ..................... 2

1. Experimental results ........................................................... .................. ......... ................. .3

2. Finite element computations ............................... .... .. ...... ..... ............... ....... ... .. ... ............. 5

2.1 . Finite element model of the CCT specimen ....................................................... 5

2.2. Results of computations with non-propagating cmck ........................................ 8

2.2.1. Development of plastic zones and stresses in the CCT specimen. ...... .l 0

2.2.2. Influence of constraining of the loaded edges ...................................... 18

2.2.3. Stmin changes on the edge ofthe CCT specimen ................................ 18

2.3 . Results of computation with propagating cmck .............................................. 20

3. Buckling ofthe CCT specimen ........... .................. ................................. .. ........ .... ........ 22

4. Conclusions .................................................. ......... .. .. ................... ...... ................... ....... 26

References .......................................................................................................................... 27

Summary

During experiments studying residual strength of a centre-cracked tension (CCT)

specimen from alwninum alloys, unexpected strain changes were measured by strain

gages in the ligament of the specimen [1 J. An elastoplastic FEM research was conducted

to explain this behavior.

The modelling proved that the experimentaIly observed decrease of totaI strain in

the ligament does not occur, when only in-plane deformation takes place, regardless

whether stabie fracture occurs or not. The results of the anaIysis give an interesting

detailed picture of stress and plastic strain changes in the CCT specimen, subjected to

high loads near plastic collapse.

Simple model, explaining the observed strain changes in the ligament, is proposed

for the CCT specimen after its buckling. This model should he verified by a proper FEM

buckling analysis.

2

j ... . , IJ .", Wo, . ! ! ibM' ...... ',- ' X' ' ...... .... ~~ ..... _-

1. Experimental results

Experimental work [1] was performed at Faculty of Aerospace Engineering, TU

Delft, to study residual strength of centre-cracked tension specimen made of 2024-T3 and

7075-T6 alwninwn alloys. During tests, studying the influence of buckling on the

residual strength, the suspicion has arisen that the results are influenced by a bending

moment in the plane of the specimen, caused by the tension machine. To measure this

moment, four strain gages were attached to the specimen (two at each side), as shown in

Figure la. The specimen was loaded with controlled displacements ofthe loaded edge.

0.5 mm

y1' t t l' t t t t

I I

thickness ,

0.9 mm

I n- -28.·100 ~

STRAIN GAGES

w-400

o o

x co I

a)

äi a.. ~ .. .. !!! Oi ., ., e 0

250

200 0 ++f

150 000

~

o~

. 0 '

100 "" "" "" tt'!-++ 50

"" . 0"

0 ." 0.00 0.05 0.10 0.15 0.20

Total strain on Ihe edge [%)

b)

Figure I Position of strain gages (a) and measured total strain on the edge of the specimen (b) in experiments [1] .

3

II •

The results obtained by the strain gages show no bending moment, but surprising

behavior of tbe measured total strain on tbe edge (see Fig.lb). After surpassing certain

load level, tbe rate of strain increase is reduced. After some further increase of load, tbe

total strain on tbe edge even starts to decrease. The displacements of tbe clamping plates

were increasing during whole tbe test. For the otber three strain gages in Fig. la, tbe plot

of tbe results is essentially the same as tbat one in Fig.} b ..

4

2. Finite element computations

2.1. Finite element modelofthe CCT specimen

Elastoplastic fmite element computation of the CCT specimen was perfonned to

find out, whether the experimentally observed behavior of the specimen, described in the

previous paragraph, will occur if only in-plane defonnation without any buckling takes

place.

Program ABAQUS was used for the FEM computations. Finite element mesh was

prepared and the post-processing work was done in PATRAN.

Due to symmetry of the problem, only a quarter of the CCT specimen could be

mode lIed, as shown in Fig.2.

C>

'"

o o

"

C>

'"

I I

,

EXPERIMENT Y'l'

t t t t t t t t t

I ~ ! ~

I -:? g r-r -- x CD 2a,·100 ! ~

\

rigi~PS

mode"ed quartar

~

thicknalS 0.9 mm

+ I ,j, ,j, ,j, ,j, ~ _~~istributad '11

I w-400 loading force

FEM model

Y 200

'''. loaded edga: - straight • no rotation

< ~ - no contractIon

"-0 10

~ l'l

...,

...,

K Ic'

/ 1'\ J x

b:d

Figure 2 Experimental set-up ofthe CCT specimen and its finite element model

5

The loaded edge was enforced to remain straight; aIso its rotation and contraction

were constrained. In this way, very stiff clamping of the reaI specimen was modelled. The

finite element mesh for the whole specimen is shown in Fig.3. Only a quarter ofthe mesh

was reaIly used in the FEM caIcuIation. Eight-nodded quadratic plane stress elements

with reduced integration (CPS8R in ABAQUS) were used.

- -X

Ol!

I 2 ao

Figure 3 Finite element mesh of the whole CCT specimen (only a quarter of the mesh was actually used in the computation)

6

ft .. i .»

Material properties (elastic modulus E and Poisson's ratio v12) of standard

2024-T3 aluminum alloy were prescribed according to [2], as given in Table 1.

Table 1 Properties of 2024-T3 aluminium alloy used in the FEM computations.

E [MPa] vl2 Tensile curve 72400 0.33 see Table 2

To describe the elastoplastic behaviour ofthe aluminum alloy, the isotropic theory

of plasticity and the von Mises yield criterion was used. The large displacements theory

. was assumed.

The stress-strain curve of the 2024-T3 alloy, necessary for the elastoplastic

computation, was prescribed according to an experiment [3]. The stress-strain curve was

constructed as an average curve from 5 experimental tensile curves. Tests were performed

on specimens from bare 2024-T3 alloy sheet, 0.505 mm thick and 12.65 mm wide. Real

stress and real strain from the average curve were used for the construction of the model

curve. The model stress-strain curve is described in a piece-wise linear form in Table 2.

No experimental data were available in [3] for low values of plastic strain

(cl' < 0.2%). Therefore, the stress value for zero plastic strain (cr = 349.0 MPa) in Table 2

was obtained by linear extrapolation from the two points with the lowest plastic strain

measured (cl' = 0.2% and EP = 0.48%). This gives rather sharp transition from elastic to

elastoplastic part of the stress-strain curve, which is similar to often used bilinear

approximation of the stress-strain curve. For higher plastic strains, however, the curve

from Table 2 describes real material behavior better than any bilinear model. If necessary,

7

I '

a more detailed stress-strain measurement or the Ramberg-Osgood equation for standard

2024-T3 aHoy [4] could be used to refine the curve at &1' < 0.2%.

Table 2 Tensile curve of 2024-T3 aluminium alloy used in the FEM computations.

& [%] &1'[%] cr [MPa] 0.0 0.0 0.0 0.48 0.0 349.0 0.7 0.2 361.9 1.0 0.48 380.0 1.98 1.42 407.2 2.96 2.37 429.7 3.92 3.30 450.4 4.88 4.23 467.1 5.83 5.16 483.5 6.77 6.08 498.2 7.7 7.00 510.0 8.62 7.90 521.4 9.53 8.79 532.9 10.44 9.69 542.1 11.33 10.57 550.4 12.22 11.45 558.8 13.10 12.32 563.7 13.98 13.19 568.6

2.2. Results of computations with non-propagating crack

No crack growth was assumed in the frrst phase of elastoplastic computations.

Loading was realised by prescribing displacements of the loaded edge. Maximum

elongation ofthe whole specimen was 8.832 mm (i.e. 4.416 mm for the modelled quarter

of the specimen). This corresponds to the loading force about F= 121.2 kN (or gross

8

, I , i, ;

1

stress cr = 336.6 MPa), acting on the whole specimen. The maximum loading force is

almost double ofthe highest load reached in the experiment (Fmax = 65.49 kN).

The computed eIongation of the whole specimen is plotted by dashed Hne in FigA

for gross stress up to cr = 300 MPa. The crosses in Fig.4 mark the experimentally

obtained values.

300

/'

250 FEM - no crack growth /'./ as

FEM - crack I: Experiment D. :::::i!: 200

(Ij growth ++:;. / (Ij

++ -----G) 150 -----~

\ ------(Ij ++ (Ij + \ (Ij 100 • . N 0 • +

\ +# ~ .+ . C!' .1;+ \

+~

50 •• . ,/ DETAIL

.+ "-

0 0 2 3 4

Total elongation [mm]

Figure 4 Total elongation ofthe CCT specimen

At load about cr = 134 MPa, the crack in experiment starts to grow. Therefore, the

compliance of the specimen increases, while the compliance of the model specimen with

non-propagating crack at the same gross stress almost does not change.

Unstable fracture and failure occur prior to a plastic collapse ofthe specimen in this

experiment. In some cases (for a very smal I crack or very small width W of the

9

specimen), a plastic collapse may, however, occur before reaching the toughness of the

specimen [5]. Therefore, it may be of some interest to have a look at the hehavior of our

model specimen with non-propagating crack when the loads are higher than the

experimental toughness of the specimen. In that case, large plastic deformations take

place in the specimen, leading finally to a plastic collapse. At the gross stress

approximately cr = 250 MPa, visible effect of plastic deformation on the specimen

compliance may be seen in Fig.4.

2.2.1 . Development of plastic zones and stresses in the CCT specimen

Detailed picture of elastoplastic deformation of the specimen is given in several

following figures. Contour plots of three quantities of interest are given in these figures.

Plastic zone size is shown as the contour plot of effective plastic strain

(1)

Here, dt~ is an increment of plastic strain tensor; the integration takes place along the

whole deformation path.

In a multiaxial case, it is reasonable to take the value of E~fI' =0.2% as the limit

marking the beginning of plastic deformation. The plastic zone boundary may he then

defined in the contour plot of effective plastic strain (see the fust plot in Fig.5) as the

contour corresponding to E~ =0.2% , i.e. the contour N. The contour plot of the

longitudinal stress cry (in the loading direction) is given as the second one in Fig.5. The

last plot in Fig.5 shows distribution of the transverse stress crx (in the direction

perpendicular to the loading one).

10

I effective plastic strain [%]

gross stress

235.4 MPa

longitudinal stress [MPa]

transverse stress [MPa]

504 5.0 4.6 4.2 3.8

3.4 3.0 2.6 2.2 ' 1.8 IA 1.0 0.6 0.2

479 446 413 380 347 314 281 248 215 182 149 116 83 50 17

112 88 64 40 16 -8

-32 -56 -80

- 104 -128 -152 -176 -200

-224

A B C 0 E F G H 1

J K L M N

A B C 0 E F G H 1

]

K L M N 0

A B C 0 E F G H I J K L M N 0

Figure 5 Distribution of effective plastic strain, stress in the loading direction and transverse stress in the CCT specimen at gross stress cr = 235.4 MPa.

11

_ R' li jN iiJWSU_S_ j Ir wu

As apparent from Fig.5, at the gross stress a = 235A MPa the plastic zone is still

relatively smaIl when compared with the ligament of the specimen. Therefore, the

compliance of the model specimen in FigA does not almost differ from the compliance of

a fully elastic specimen.

Stress distributions in Fig.5 correspond weIl with the plots published in [6]. Note

the unloaded area next to the free surf ace of the crack, as seen in the plot of longitudinal

stress ay . In the same part of the specimen, the transverse stress a x reaches high

compressive values, which may cause buckling of the material above and below the

crack. Naturally, the buckling limit and the behavior after its surpassing can not be

specified by this type of analysis.

The areas of higher stress (both ay and a x ) in the corners of the specimen are

caused by the rigid clamping, constraining the contraction of the loaded edges (see

Figure 2).

As the gross stress increases (a = 273 .7 MPa in Fig.6 and a = 285.3 MPa in Fig.7),

the plastics zone size increases, too. The plastic deformation is most intensive in the

direction ofthe highest shear stress (about ±45° relative to the crack axis), giving rise to

the typical butterfly-like shape of the plastic zone. Visible change of the specimen

compliance, caused by high ratio of plastification in the ligament, may be already

observed at these loads in FigA.

The contour plot patterns for both the longitudinal and transverse stresses in Fig.6

and Fig.7 are basically the same as those in the corresponding plots for a lower load

(Fig.5); just the magnitude ofthe stresses increases.

12

effective plastic strain [%]

gross stress 273.7MPa

longitudinal stress [MPa]


5.4 5.0 4.6 4.2 3.8 3.4 3.0 2.6 2.2 .

1.8 1.4 1.0 0.6 0.2

479 446 413 380 347 314 281 248 215 182 149 116 83 50 17

112 88 64 40 16 -8

-32 -56 -80

-104 -128 -152 -176 -200 -224

A B C 0 E F G H I

J K L M N

A B C 0 E F G H 1 J K L M N 0

A B C 0 E F G H I J K L M N 0


13

etIective plastic strain [%]

gross stress 285.3 MPa

10ngitudinal stress [MPa]


5.4 5.0 4.6 4.2 3.8 3.4 3.0 2.6 2.2· 1.8 1.4 1.0 0.6 0.2

479 446 413 380 347 314 281 248 215 182 149 116 83 50 17

112 88 64 40 16 -8

-32 -56 -80

-104 -128 -152 -176 -200 -224

A B C 0 E F G H I

1 K L M N

A B C 0 E F G H 1

]

K L M N 0

A B C 0 E F G H 1 J K L M N 0


14

l

As the load further increases, a new phenomenon appears in the development of the

plastic zone. This is shown in Fig.8, containing contour plots of effective plastic strain for

three subsequent gross stress levels. At stress cr = 286.2 MPa (the first plot in Fig.8), four

small plastic zones have developed on the edges of the specimen approximately in the

±45° directions fiom the two crack tips. During further increase of the load

(cr = 287.0 MPa in the second plot in Fig.8), the edge plastic zones keep growing and

fmally they join the main plastic zones growing fiom the crack tips (the last plot in Fig.8).

The whole described process, from the start of creating of the edge plastic zones till

the moment when they can not be distinguished fiom the crack tip plastic zones, takes

place during a load increment about 2.5 MPa (compare Fig.7 and the last plot in Fig.8) .

. This stress change is less than 1 % of the applied gross stress.

After all the plastic zones have joined, the whole ligament is aIready plastified and

the specimen has lost almost all its bearing capacity. If the loading were performed by

controlled force, the plastic collapse of the specimen would now occur after a very small

increase of the load.

Loading by prescribed displacements allows easier study of the specimen behaviour

after the plastification of the ligament. This state is shown in Fig.9 for gross stress

cr = 293.7 MPa. Here, the plastic zones still develop in the ±45° directions fiom the two

crack tips, reaching as far as the edges. The stress plots in Fig.9, however, are not very

different fiom the state at which no plasticity existed on the edges (Fig.7 for

cr = 285.3 MPa).

15

F i iS ".. d§l A 2D , I

gross stress

286.2 MPa

effective plastic strain [%]

gross stress

287.0 MPa

gross stress

287.8 MPa

5.4 5.0 4.6 4.2 3.8 3.4 3.0 2.6 2.2 I.8 1.4 1.0 0.6 0.2

A B C D E F G H I

J K L M N

Figure 8 Distribution of effective plastic strain in tbe CCT specimen at gross stress cr = 286.2 MPa, cr = 287.0 MPa and cr = 287.8 MPa.

16

effective plastic strain [%J

gross stress 293 .7 MPa

longitudinal stress [MPaJ

transverse stress [MPaJ

5.4 A 5.0 B 4.6 C 4.2 D

3.8 E 3.4 F 3.0 G 2.6 H 2.2 . 1 1.8 J 1.4 K 1.0 L 0.6 M 0.2 N

479 A 446 B 413 C 380 D 347 E 314 F 281 G 248 H 215 1 182 J 149 K 116 L 83 M 50 N 17 0

112 A 88 B 64 C 40 · D 16 E -8 F

-32 G -56 H -80 1

-104 J -128 K -152 L -176 M -200 N -224 0


17

2.2.2. Influence of constraining ofthe loaded edges

To find out, whether the creation ofthe edge plastic zones is influenced by the rigid

clamping of the loaded edges (see Fig.2), another computation was performed. All

parameters ofthe computation were the same as in the previous case, only the contraction

of the loaded edge was not constrained.

This time, after the same number of Joading increments (50 increments), only gross

stress cr = 284.2 MPa was reached (corresponding to the tota! elongation ofthe specimen

3.342 mm). This stress is less than the gross stress cr = 286.2 MPa, at which the edge

plastic zones appeared in the previous computation for the fust time (Fig.8).

Nevertheless, four distinct plastic zones on the edges could he observed also in the

second computation. However, to achieve this at least at the highest load applied in this

computation, a little lower plastic limit (E~ff =0.18%) had to be chosen .

According to these results, the phenomenon of the edge plastic zones creation is not

significantly influenced by the rigid c1amping ofthe loaded edges.

2 .2.3. Strain changes on the edge ofthe CCT specimen

The frrst of the two computations without crack growth (the computation with the

constrained contraction ofthe loaded edges) was used for comparison with experimenta!

data from Fig.! b. The strain in the loading direction was taken from the rightmost

element, next to the x-axis in Fig.3. This position was the c10sest approximation of the

strain gages position in Fig.la. The results are plotted by a dashed line in Fig.lO. The rate

18

of strain increases with increasing gross stress, as intuitively expected. This is, however,

just the opposite of the experimentally observed trend in Fig.l b.

300 r-------------------------------------~_.

CiS a..

250

:E 200

Cl) Cl)

! 150 -Cl)

Cl)

Cl) 100 o "-Cl

50

----FEM - no crack growth /' /'

FEM - crack growth

\

\?::~~~ \ \

DETAIL:

o ~~~~~~~~~~~~~~~~~_L~~~~ 0 .00 0.10 0.20 0.30 0.40 0.50

Total strain on the edge [%]

Figure 10 Total strain on the edge ofCCr specimen

19

iiil6t

2.3. Results of computation with propagating crack.

The results of computations of the CCT specimen with non-propagating crack,

described in the previous paragraph, did not show the experimentally observed behavior

in the ligament.

A more distinct decrease ofthe total strain rate occurs in experiment (Figl.b) only

when the crack starts to grow. Therefore, a crack extension was introduced also into the

FEM model. Another computation was performed with the mesh and material described

in the paragraph 2.1. This time, the loading force was increased in three steps, according

to Table 3.

Table 3 Loading steps in the computation of CCT specimen with growing crack.

Step Crack length a Numberof Totalload Gross stress [mm] nodes released atthe end atthe end

in the step ofthe step ofthe step [kN] [MPa]

I 50.0 - 46.54 129.3 2 54.375 I 62.76 174.3 3 38.75 I 65.58 182.2

The maximum loading force in the frrst step is equaI to the last measured load, at

which the experimental crack did not grow yet. In the second and third model loading

step, the crack extension was prescribed. This was done by simple releasing of one node

at the crack tip at the beginning of each step. Thus, the crack length increment in both

steps was equal to half the edge length of the element immediately before the origina!

crack tip (see Fig.3). Current crack length in each step is also given in Table 3.

Corresponding total load to be applied at given crack length in the second and the

third loading step was found by interpolating from the experimentally measured load-

20

,-------- ---

crack length dependence. At the maximwn load applied in the computation, fracture in

the experiment was still stabie (compare the last value in Table 3 and Figure 3). No

attempt was done to model the crack growth during unstable fracture.

The computed load-elongation dependence is plotted by solid line in Fig.4. In the

load range where the model crack propagates, the compliance of the specimen really

increases. That is in an agreement with the experimental data in Fig.4. However, the total

strain on the edge of the specimen, plotted by solid line in Fig.10, has even for the

growing model crack the opposite trend than the experimental data in Fig.l b.

21

t f

3. Buclding of the CCT specimen.

Finite element analysis, described in the previous chapter, showed that the strain

changes in the ligament of the eeT specimen shown in Fig. 1 b do not occur when

buckling of the specimen is constrained and only in-plane deformation takes place.

At frrst, this conclusion seemed to contradict the experimental observations, which

showed the same behavior even if buckling of the specimen was prevented by "buckling

guides" [1]. However, more thorough inspection of the experimental set-up showed a

reason for this behavior [1]. Improved clamping of the specimen then eliminated these

.problems, so that the strain changes from Fig. 1 b are now really observed only on

specimens with buckling.

In many discussions about the problem, a simple model was created, which

explains the strain changes in the ligament after buckling of the eeT specimen. The main

idea of the model is shown in Fig.ll. Here, the behavior of the eeT specimen without

buckling (Fig.lIa) and with buckling (Fig.1Ib) is compared.

Under tensile load applied on the eeT specimen, the material above and below the

crack is compressed by high transverse stress (see Fig.6), but still it remains in the plane

of the specimen and does not allow the outer regions of the specimen to approach each

other. When the buckling limit is surpassed, the compressed areas above and below the

crack move out of the specimen's plane; this is shown very approximately by the two

shaded triangles in Fig.II b. In the plane of the specimen, the stiffness of the buckled

material is very low. The buckled triangles carry therefore very little transverse load and

practically do not constrain the transverse movement of the rest of the specimen. The

outer regions of the specimen may thus get much closer to each other. This behavior we

22

l

can more easily understand, when considering the buckled areas to be completely

"omitted" trom the specimen in Fig.ll b.

a) NO BUCKLING

x

Schematic model of half of the specimen

~1'

rigid clamps

'-J/~~~(J

b) BUCKLING

Stress in loading direction along the crack axis x

schematic stress distribution in front of the crack tip

In-plane bending is reduced by the other half of specimen

Buckled areas do not constrain in-plane bending

schematic stress distribution in front of the crack tip

additional stress trom bending

Figure II Simple model comparing behavior of the CCT specimen before and after buckling

23

The difference between the state without and with buckling may be also more

obvious, if we consider just one half of the specimen, as shown in the rniddle parts of

Fig.ll a and Fig.ll b. In the forrner figure, the influence of the other half of the specimen

is represented by the prescribed boundary conditions along the axis of symmetry. In the

latter case (with buckling), the buckled areas are really "ornitted", so that the transverse

movement of the rniddle part of the modelled half is not constrained.

Stress distribution in the fust case (Figla) can be obtained by elastic or elastoplastic

analysis of the specimen in the state of plane stress. Stress distribution before the crack

tip is schematically shown for the longitudinal stress component (in the loading direction)

in the last part of Fig.ll a. There is, of course, a singularity or at least a concentration of

the stress at the crack tip.

After surpassing the buckling limit, the rniddle parts of the buckled specimen in

Fig.ll b can move rather easily closer to the vertical axis of symmetry. This additional

in-plane bending induces other stresses, which are superimposed on the original stress

distribution in the specimen, as shown schematically in the last part of Fig.11 b. At the

outer edge of the specimen, the stresses from the additional bending are compressive and

will be therefore subtracted from the tensile stresses existing there before the buckling.

Thus, the rate ofthe stress (and strain) increase at the edge win be reduced during further

loading in the same way, as shown in Figlb.

What will happen after an initiation of stabie fracture?

In the case without buckling (Fig.lla), the crack extension win reduce the ligament

of the specimen, increasing thus the tensile stress in the ligament. Since the in-plane

bending is still almost completely constrained, the rate of the stress on the edge win not

be visibly reduced by the compressive stress from the bending. The total stress (and

24

strain) at the edge will therefore increase even faster (the ligament is smaller), as shown

in Fig.l 0 for the specimen with propagating crack.

On the contrary, when some crack extension occurs in the aIready buckled

specimen (Fig.llb), the in-plane bending of each half ofthe specimen wil! he much easier

due to the smaller ligament. Therefore, the compressive stress from the bending wiIl be

increasing at the edge faster then for a shorter crack. During extension of a long crack, the

bending wil! be more and more important. The compressive stress from bending may then

increase at the edge even faster than increases the tensile stress from the tensile load.

Thus, the stress at the edge (and the tota! strain) as a whole wiIl decrease, as shown in

Fig.lb.

Real centre-cracked tension specimen wiIl thus behave in the way described by

Fig.lla, when the load is beIlow the buckling limit or when the buckling guides are used

to prevent buckling. After surpassing the buckling limit, the specimen without buckling

guides wiIl buckle and its behavior wil! be described by Fig.ll b.

The described simple model of specimen behavior after buckling should be verified

by a finite element analysis focused on this problem.

25

4. Conclusions

Elastoplastic finite element modelling of eeT specimen was perfonned. The results

describe in detail the development of stress and et'fective plastic strain distribution in the

specimen up to the loads near a plastic COllapse of the specimen. An interesting

phenomenon of edge plastic zone fonnation is observed at high load levels.

The experimentally observed changes oftotal strain in the ligament ofthe specimen

were studied. The fmite element analysis showed that the strain rate decrease does not

occur when buckling of the specimen is constrained and only in-plane defonnation takes

place, regardless whether the crack extends or not.

Simple model explaining the experimentally observed behavior is suggested.

The strain rate decrease at the edge of the eeT specimen is explained by stress changes

induced by buckling of the specimen. Buckling analysis of the eeT specimen by finite

element method shOuld be carried on in future to verify this model.

26

References

[1] de VRIES, T.: The Influence of Buckling on the Residual Strength of CCT specimens. [Report] . Faculty of Aerospace Engineering, Technical University of Delft, the Netherlands. To be published in 1993.

[2] Metals Handbook, Vo1.2, Properties and Selection: Non-ferrous Alloys and Pure Metals. 9th ed., American Society for Metals, 1979.

[3] Sinhe, J. ; Faculty of Aerospace Engineering, Delft University of Technology, Delft, [personal communication].

[4] Military Standardisation Handbook - 5E; Metallic Materials and Elements for Aerospace Vehicle Structures. Vol.l, June 1987. Departrnent of Defence, USA.

[5] BROEK, D.: Practical Use of Fracture Mechanics. Kluwer Academic Publishers, Dordrecht, the Netherlands, 1989,522 pp.

[6] FUnMOTO, T. - SUMI, S.: Local Buckling of Thin Tensioned Plate with a Crack. Memoirs of the Faculty of Engineering, Kyushy University, Vo1.42, No.4, December 1982. p.355-368.

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Series 01: Aerodynamics

01 . F. Motallebi, 'Prediction of Mean Flow Data for Adiabatic 2-D Compressible Turbulent Boundary Layers' 1997 / VI + 90 pages / ISBN 90-407-1564-5

02 . P.E. Skare, 'Flow Measurements for an Afterbody in a Vertical Wind Tunnel' 1997 / XIV + 98 pages / ISBN 90-407-1565-3

03 . B.W. van Oudheusden , 'Investigation of Large-Amplitude 1-DOF Rotational Galloping' 1998/ IV + 100 pages / ISBN 90-407-1566-1

04. E.M . Houtman / W.J . Bannink / B.H. Timmerman, 'Experimental and Computational Study of a Blunt Cylinder-Flare Model in High Supersonic Flow' 1998 / VIII + 40 pages / ISBN 90-407-1567-X

05 . G.J.D. Zondervan, 'A Review of Propeller Modelling Techniques Based on Euler Methods' 1998/ IV + 84 pages / ISBN 90-407-1568-8

06. M.J. Tummers / D.M. Passchier, 'Spectral Analysis of Individual Realization LDA Data' 1998/ VIII + 36 pages / ISBN 90-407-1569-6

07 . P.J.J. Moeleker, 'Linear Temporal Stability Analysis ' 1998/ VI + 74 pages / ISBN 90-407-1570-X

08. B.W. van Oudheusden, 'Galloping Behaviour of an Aeroelastic Oscillator with Two Degrees of Freedom' 1998 / IV + 128 pages / ISBN 90-407-1571-8

09 . R. Mayer, 'Orientation on Ouantitative IR-thermografy in Wall-shear Stress Measurements' 1998 / XII + 108 pages / ISBN 90-407-1572-6

10. K.J.A. Westin / R.A.W.M. Henkes, 'Prediction of Bypass Transition with Differential Reynolds Stress Modeis' 1998 / VI + 78 pages / ISBN 90-407-1573-4

11. J .L.M. Nijholt, 'Design of a Michelson Interferometer for Ouantitative Refraction Index Profile Measurements' 1998/ 60 pages / ISBN 90-407-1574-2

12. R.A.W.M. Henkes / J.L. van Ingen, 'Overview of Stability and Transition in External Aerodynamics' 1998 / IV + 48 pages / ISBN 90-407-1575-0

13. R.A.W.M. Henkes, 'Overview of Turbulence Models for External Aerodynamics' 1998/ IV + 40 pages / ISBN 90-407-1576-9

Series 02: Flight Mechanics

01. E. Obert, 'A Method for the Determination of the Effect of Propeller Slipstream on a Static Longitudinal Stability and Control of Multi-engined Aircraft' 1997/ IV + 276 pages / ISBN 90-407-1577-7

02. C. Bill / F. van Dalen / A. Rothwell, 'Aircraft Design and Analysis System (ADAS)' 1997 / X + 222 pages / ISBN 90-407-1578-5

03. E. Torenbeek, 'Optimum Cruise Performance of Subsonic Transport Aircraft' 1998 I X + 66 pages / ISBN 90-407-1579-3

Series 03: Control and Simulation

01. J.C. Gibson, 'The Definition, Understanding and Design of Aircraft Handling Qualities' 1997 I X + 162 pages / ISBN 90-407-1580-7

02. E.A. Lomonova, 'A System Look at Electromechanical Actuation for Primary Flight Control' 1997 / XIV + 110 pages / ISBN 90-407-1581-5

03. C.A.A.M. van der Linden, 'DASMAT-Delft University Aircraft Simulation Model and Analysis TooI. A Matlab/Simulink Environment for Flight Dynamics and Control Analysis' 1998/ XII + 220 pages / ISBN 90-407-1582-3

Series 05: Aerospace Structures and Computional Mechanics

01. A.J. van Eekelen, 'Review and Selection of Methods for Structural Reliability Analysis' 1997 / XIV + 50 pages I ISBN 90-407-1 583-1

02. M.E. Heerschap, 'User's Manual for the Computer Program Cufus. Quick Design Procedure for a CUt-out in a FUSelage version 1.0' 1997 / VIII + 144 pages / ISBN 90-407-1 584-X

03. C. Wohlever, 'A Preliminary Evaluation of the B2000 Nonlinear Shell Element Q8N.SM' 1998/ IV + 44 pages / ISBN 90-407-1585-8

04. L. Gunawan, 'Imperfections Measurements of a Perfect Shell with Specially Designed Equipment (UNIVIMP) 1998 / VIII + 52 pages / ISBN 90-407-1586-6

Series 07: Aerospace Materials

01. A. Vasek / J . Schijve, 'Residual Strenght of Cracked 7075 T6 AI-alloy Sheets under High loading Rates' 1997/ VI + 70 pages / ISBN 90-407-1587-4

02. I. Kunes, 'FEM Modelling of Elastoplastic Stress and Strain Field in Centrecracked Plate' 1997 / IV + 32 pages / ISBN 90-407-1588-2

03 . K. Verolme, 'The Initial Buckling Behavior of Flat and Curved Fiber Metal laminate Panels' 1998/ VIII + 60 pages / ISBN 90-407-1589-0

04. P.W.C. Provó Kluit, 'A New Method of Impregnating PEl Sheets for the InSitu Foaming of Sandwiches ' 1998 / IV + 28 pages / ISBN 90-407-1 590-4

05. A. Vlot / T. Soerjanto / I. Yeri / J.A. Schelling, 'Residual Thermal Stresses around Bonded Fibre Metal laminate Repair Patches on an Aircraft Fuselage' 1998 / IV + 24 pages I ISBN 90-407-1591-2

06. A. Vlot, 'High Strain Rate Tests on Fibre Metal laminates' 1998 I IV + 44 pages / ISBN 90-407-1592-0

07. S. Fawaz, 'Application of the Virtual Crack Closure Technique to Calculate Stress Intensity Factors for Through Cracks with an Oblique Elliptical Crack Front' 1998/ VIII + 5,6 pages / ISBN 90-407-1593-9

08. J . Schijve, 'Fatigue Specimens for Sheet and Plate Material' 1998 I VI + 18 pages I ISBN 90-407-1594-7

Series 08: Astrodynamics and Satellite Systems

01. E. Mooij, 'The Motion of a Vehicle in a Planetary Atmosphere' 1997 / XVI + 156 pages I ISBN 90-407-1595-5

02. G.A. Bartels, 'GPS-Antenna Phase Center Measurements Performed in an Anechoic Chamber' 1997 / X + 70 pages / ISBN 90-407-1596-3

03. E. Mooij, 'linear Quadratic Regulator Design for an Unpowered, Winged Reentry Vehicle' 1998/ X + 154 pages I ISBN 90-407-1597-1

rrt "fl'j I j i 11.. I Ir; , 1 • • WEI •• Uil • .. 1 I

During experiments studying residual strength of a centre-cracked tension (CCT) specimen from aluminimum alloys, unexpected strain changes were measured by strain gages in the ligament of the specimen (1). An elastoplastic FEM research was conducted to explain this behavior. The modelling proved that the experimentally observed decrease of total strain in the ligament does not occur, when only in-plane deformation takes place, regardless whether stabie fracture occurs or not. The results of the analysis give an interesting detailed picture of stress and plastic strain changes in the CCT specimen, subjected to high loads near plastic collapse. A simple model, explaining the observed strain changes in the ligament, is proposed for the CCT specimen after its buckling. This model should be verified by a proper FEM buckling analysis.

ISBN 90-40 7 - 15 88-2

DELFT AEROSPi\.~~ 9 799040 715883 ENGI N E U l lN G & TE CHNQlOGY

fem modelling of elastoplastic stress and strain field in

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