Three-Dimensional Elastodynamic Simulation

It appears that the substantial variation observed between the experimentally measured stress-intensity factors from bifocal caustics cannot be explained purely in terms of ( two-dimensional) dynamic effects affecting the caustic patterns. Thus additional reasons must be sought in terms of a) non-linear effects or b) three- dimensional effects under transient conditions. Visual evidence of the plastic de- formation in the fractured specimens indicated that the initial-curves for the ex- perimental caustics were well outside the plastic zone whose maximum extent was seen to be confined to

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Figure 4.8a: Simulated caustics from the computed plane-stress out-of-plane displace- ments.

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Figure 4.8b: Plot of Kea us/ Kj versus

ro/

h from the two-dimensional elastody- namic simulation.

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O'l 00 I

-69-

tic simulations of the current experiments ( reported in Krishnaswamy and Rosakis (1988)), plasticity effects are expected to be negligible. Thus attention will now be directed toward studying the effect of three-dimensionality near the crack-tip by means of a full-field three-dimensional elastodynamic simulation of the drop-weight experiments.

The mesh geometry used had an in-plane layout identical to that used for the two-dimensional cases. This enables the direct comparison of three-dimensional results with the corresponding plane-stress simulations and thereby helps identify the effect of three-dimensionality. Five layers of 8-noded brick elements through half the thickness leading to a total of 1960 elements (2250 nodes) were used to model one-quarter of the three-dimensional body. Recognizing that the largest through-thickness variations in field quantities occur near the free-surface, the mesh was graded in the thickness direction as shown in Figure 4.2. The experimentally obtained tup load histories interpreted as uniform line loads through the thickness were applied as boundary conditions to the appropriate nodes. The uncracked ligament surface and the specimen mid-plane were constrained suitably as dictated by symmetry considerations. The rest of the boundary was left traction-free. Once again, an implicit Newmark predictor-corrector scheme was used in order to be consistent with the algorithm used for the two-dimensional simulation.

The appropriate average dynamic energy-release rate integral here is ( see Ap- pendix Al)

( 4.4) where S is now a tubular surface through the specimen and S -+ 0 symbolically indicates that this surface is shrunk onto the crack-front. In order to compare with experimental results, an 'average' stress-intensity factor can be extracted from lav through relation ( 4.2). The average energy-release rate value obtained from the

-70-

three-dimensional simulation is shown in Figure 4.9 and is not much different from that computed in the plane-stress analysis. Thus, the plane-stress approximation might be adequate if one is interested in integrated energy-release rate type of quantities.

From the point of view of the method of caustics in reflection, the primary issue is the extent of deviation of the near-tip surface out-of-plane displacement field from the corresponding asymptotic plane-stress expression. This is shown in Figure 4.10a for a set of radial lines along 0 0°, 60° and 140° and for one typical time. The surface u3-displacements as obtained from the three-dimensional solution are nor- malized by the corresponding asymptotic values. The two-dimensional counterpart along the radial line 0

=

0° is shown for comparison purposes. Two salient features can be identified from this figure. First, the two-dimensional full-field values seem to be in good agreement with the asymptotic for sufficiently small radial distances.

The asymptotic expression seems to become increasingly inaccurate for larger ra- dial distances. Secondly, it is precisely in those regions where the two-dimensional and asymptotic fields are in close agreement that the three-dimensional structure deviates the most from the asymptotic. In Figure 4.10b, the angular variations of the surface u3-displacements are shown for a range of r / h. Under the normaliza- tion used, the asymptotic u 3-displacements are given by cos(() /2) the solid line - for any radius. Again it is noted that the asymptotic field does not always model the actual three-dimensional structure very well. Indeed, the deviation from the asymptotic is seen to be pronounced toward the crack-tip. For larger r / h, the theta variation of u 3 seems to approach the asymptotic curve in form though not in ac- tual magnitude. Thus it appears that the actual three-dimensional field is not quite captured by the asymptotic expression for the surface out-of plane displacements - toward the crack-tip because of substantial three-dimensional effects and away from

250 . - - - . - - - -

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Figure 4.9: Comparison of the "average" dynamic stress-intensity factor history as com- puted from two- and three-dimensional simulations.

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-1 ' w '

-74-

it because the asymptotic expression becomes increasingly insufficient.

The qualitative effects of the above features insofar as these affect the method of caustics can now be evaluated. As described previously, the three-dimensional sur- face out-of-plane displacements obtained numerically for various times in the simu- lation are mapped using (2.1) to obtain synthetic caustic patterns. A representative sequence of these caustic patterns for one particular time is shown in Figure 4.lla.

These were obtained for the same instant of simulation and for the same set of z₀ s as were used in the two-dimensional case. The role of three-dimensionality is thus clearly brought out. The graphic contrast that emerges between the caustic pat- terns obtained under two-dimensional assumptions and the actual three-dimensional conditions can be seen by comparing Figures 4.8a and 4.1 la. In comparison to the corresponding two-dimensional results, it is noted that for small r0 / h, the shadow spots in Figure 4.lla are less epicycloidal in shape and much smaller in size. This is not surprising considering the angular variations of u3 which deviated markedly from the asymptotic as r -+ 0. Compare this with the results of the plane-stress analysis (see Figure 4.8a) where the qualitative agreement with the asymptotic field appears to be much better over a wider range of r / h. If, now, one were to relate the caustic diameters to the stress-intensity factor through (2. 7), then the result- ing value of Kcaus/ K 1 (shown for two times) is seen to vary quite substantially with increasing initial-curve radius as shown in Figure 4.llb. From the figure, it is seen that as r0 / h -+ 0 the measured stress-intensity factor value becomes sub- stantially less than the value obtained from the domain integral. Further, for larger initial-curve radii, it is _µossible for caustics to provide an overestimate of _{K 1.} More importantly, the almost monotonically increasing Kea us/ K 1 ^{vs r0 /} h curve for the three-dimensional transient simulation case qualitatively captures essentially all the features observed experimentally. Thus, the apparent dynamic stress-intensity fac-

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. . ~:1-.\;. . . . . . .

_{. _..:} ·.

~

· ... ^{: : :} ·... . ^:

. ... .

..

·:.

:-

.. ·. . . . . . . . . . .

··., ...

^_

. . ^: . .

z

0

r

0

/h

3.0m 0.46

• . • ·- •: • .. • • • . 1· :• .. • .

1- • "• • I ••-■ • •••• ,- • • •

. ^., .

^,·

..

^·:

.... ,..

^.-;

. . . . ^. .

• • .: :~ :- • • ~ , _ . A-: i*"-~~ • • • • •

.. --· ,.... ... ..

••••• ., • •ya '-. • •

• • , • • •-::...•L• ~1o\::,. • • •

• i •• .;•-~a • .: .. • •

, . ~ ¥ •

,..- . .~ ..

^{- ~}

. . .

rt;:J · ·. ·. _... ^{. . .}

_~

_'.

i~i,~.

~--••• -:'l

··

••• ^:i

,:,,. .-..4

.

_,

_... ,: · -:.:~-

_~

... -~ ^.

.. . . . -: . .. . .-.r;

^':,.·.•;;

-~-~

^~

••

rr•. . . .

^....

•-. . ,... , -• v: s~ ^{1,-.:- • • • •}

-~.· .. ,:· .- ^{... ;,-.· ..}

^·.::•·

^{.·.: ..}

• • I• •• • • •• J. ~-- ._ .. • • •

zo r

0

/h

9.0m 0.81

: .... ,, r .. . ... . ·.-· .. ::::: ^...

... \-r·•.;c: ... • ••

...

_. .. .. .·

^~

. . . . . . . . .

· · · ~~1-llit;. · · ·

• -: •• :i • • •• • •

:.1 ..

^•J.

·.·1 . ·. · ..

.

^:.~-:~

...

. . .-*\ =·1--': ·

- _-

..

^.. ~ _.

^.

• .-_,: • • • • I

... ..,. ·~ ....

:,.,."'.:··. . ^: : .

• -· ... :l . · ~~~1!!11lllf11i: . ·. ·

-:,· _{._ :_t}

:-

^._.._

.·· .··. · ..

I • • • • • • •

. .,.':',l·,1·«-~ . . : . . ·· .. ·. ·.· .·

... . . .

_{_. '\} :·..

.

^_,,.

. . . . ^. . . . . . . ^. .

z

0

r /h

0

4.0m 0.54

..

• ,!:••••: •.•.. , .. •:.• '• : , ·,.'

.. . . .

r.;.•-:·

..

^I _-

.

^•

. .

> : :..:· ^{' • •}

. .. ..

^~

.

^... l\i<n•

..

^{~ .,}

. ^..

^,.,.,.,.•

^.

..., •"I," • •

^{... ,...}

^{• • •}

^.

. .. · .. : -· .. ""' .1~~ ... :"- \. . . . ::~--:;:·~le . -:,;_ ... -<i.. . ..

. . ...

^{;;:, ,-.~~}

...

r~,~~ -1:i

=-~~. ,, .. ,... I .. .

.

• •••• :••or::

.,· .~ . .... ^,~ . ...- tl- .. .. ^.,

. • :. ··" .;,,;:, :fl •.

;- . : ., · :·;t," ;,).,,,..., .

.. . . ... ..~,,..._ ... , • _•. , · . . . "I>.. _{.• s,~} ..,.,:11•~ .li.-.i,: • .... ~ ; . •

··-c- • -=~ .

• •

. • • • •. • • . . .. ... . J\ ^"J' .. -t " !;, ~/\.-w.t. r• . . . .

• • I •• • • • • •• • -,._.,_ :_-:(•: : •• •

zo r /h

0

. -·

11. Om 0.89

Figure 4.1 la: Simulated caustics from the computed three dimensional free surface u ₃-displacements.

-J '

C.l<

'

...

~ ---.,

.,

c:!

(.)

~

1 . 4 r - - - ~ - - - - 1.2

1.0 .8 .6

I . / /

-♦- t

=

^2.301/cs

I

J _..,. __ t

=

^6.201/cs

. 4 I-

_/

I

/

.2

.0 ; ; - - - ~ - -

.0 .5

r,/h 1.0

Figure 4.1 lb: Plot of Kea us/ Kj versus

ro/

^h from the three-dimensional elastody- namic simulation.

I

~1

°'

^I

-77-

tor as measured by caustics would seem to increase with increasing initial-curve radii. Also, in view of the lack of a sizeable domain of dominance of

Kj,

it would appear that the agreement between the measured dynamic stress-intensity factor history and that computed through the J -integral cannot be expected to be any better than obtained in Figures 4.4a,b.

As a parenthetical note, it is interesting that the variation in

Kj

₁ and

Kj

₂

- the two values for the dynamic stress-intensity factor as measured from pairs of bifocal caustics - turns out to be in quite good quantitative agreement with the results of Figure 4.11 b. That is, the ratio of

Kf

₁ to

Kj

₂ as obtained from experiments seem to be in close agreement with that obtained from Figure 4.11 b for the corresponding initial-curve radii. This can be seen in Figure 4.12 where the numerically generated results of Figure 4.11 b are used to "scale" the results of one particular experiment. To do this, assume that the results of Figure 4.11 b (Kcaus/Kj vs ro/h) hold for the whole duration of the loading. The experimental data (corresponding to the two z₀s) shown in Figure 4.12a can then be scaled, for each time, to the corresponding 'Kj' value by means of Figure 4.llb. The resulting

"scaled" dynamic stress-intensity factor histories corresponding to the two z₀s are shown in Figure 4.12b. It is seen that by this procedure the deviation between the

Kj

values observed in the experiments from pairs of bifocal caustics essentially reduce to within expected experimental scatter for the whole duration of loading.

Thus it would appear that the main reasons for the observed experimental variation are accounted for by the three-dimensional elastodynamic simulation. Note that it is purely to highlight this point that the above "scaling" procedure was adopted. It is not the intention here to offer Figure 4.11 b as some kind of an empirical "correction curve" for experimental data.

Finally, it is instructive to look into some additional features of the near-tip field

-78-

300

,---.----r---,---,---,---,

250 - z _{0 I} = 3.18m

-e- z = 1.80m

02

---

200

~

~ 150 ...

~

---

100

50

0 ,___ _ _ .J...._ _ _ ....1,_ _ _ -1..., _ _ ...L, _ _ __L _ _ _J

0 100 200 300

Time (µs)

(a)

400 500 600

300

1 ---,----r---r---.---.---

250

200

- z _{0 I} = 3.18m

-e- z₀₂ = 1.80m

~ ¹⁵⁰

100

50

0 ~ - - - 1 , . _ _ ---1.,_ _ _ ...l,__ _ _ 1 . . , _ _ - - - - 1 , . _ _ _J

0 100 200 300

Time (µs)

400 500 600

(b)

Figures 4.12a,b: Raw and "scaled" experimental data for specimen (3q).

-79-

quantities. Figures 4.13-14 show the radial variation along the 0

=

45° line of the three-dimensional stresses and displacements for different planes along the thickness direction. This is done for one representative time. Note that in all these plots, the three-dimensional quantities are normalized by the corresponding quantities from the full-field dynamic plane-stress analysis. This is done in an attempt to highlight the effects of three-dimensionality. It also has the added virtue that it minimizes variations due to dynamic effects. One point to note is that the three-dimensional results seem to show maximum deviation from the two-dimensional results as the free-surface is approached. Further, the plane-stress solution seems to be recovered within about one-plate thickness from the crack-front. It is interesting that the deviation in the in-plane displacements u₁ are much less pronounced than in the other quantities. In Figure 4.15, the ratio of the three-dimensional free-surface ur displacement to its two-dimensional counterpart is shown as a contour plot. This again shows that three-dimensional effects seem to be confined to within a radial distance of at most one-plate thickness all around the crack-front. Note that sym- metry requirement forces the two- and three-dimensional u2-displacements to agree exactly along the crack-line. This manifests as a sharp turning in of the contour lines as the crack-line is approached. Figure 4.16 is a plot of the so-called plane- strain constraint which should be unity in regions where plane-strain conditions are obtained and zero where plane-stress conditions prevail. This is shown for one typ- ical time in the simulation. It is seen that, by this measure, plane-stress conditions are obtained at radial distances greater than about one-half plate thickness.

The thickness variations of the plane-strain constraint and also representative stress components ( a₁₁ and a_{22 )} are shown in Figures 4.17 a, b,c for a range of radial distance from the crack-tip along the 0

=

45° line. Once again it can be seen that the deviation in stresses from the two-dimensional fields is largest toward the

"'SIN NN

b

---

"'SIN MN

b

-80-

1. 15 - - - , - - - r - - - ~

---

.95

.85

e

---

- - x /h =

3 - - - · X /h

3 - - - X /h

3

---- ---

---

0.475 0.325 0.075 t

=

^2.30l/c8

. 75 l . . . - - - ' - - - ' - - - '

.0 .5 ^{1. 0 1. 5}

r/h

1. 20 r - - - , - - - . - - - , - - - ~

1.10

1.00

.90

.80

\ '

\ ...

.0

\

\

' _'

'

---...

- --

--·--

' ... __ ---~=-=-·--·--

--=---

...

--- ---

.4

- - x /h = 0.475

3

- - - X /h 0.325

3

---x /h = 0.075

3

.9 r/h

t

=

^2.30l/c8

1. 4 1. 8

Figure 4.13: Radial variation of typical stress-components compared with the corre- sponding plane-stress values.

"<:l N,...,

;:l

----

"<:l M,....

;:l

"<:l NN

;:l

----

"<:l MN

;;:i

-81-

1. 010

r - - - . - - - . - - - , , - - - -

e

⁼ 400 - - X

3/h = 0.500 ---· x/h = 0.275

1.003 ^{·--·- X} ₃ /h = 0 000 _•

r -

,..;_-:;_---.-=...-=..-:-_-.,,::::-:-:;:::'::=.-::...-=..-:-..::-_::

== -:..----· --· --·--. - ---

.997

. 9 9 0 ' - - - ' - - - ' - - - ' - - - _ J

.0 .6 1.3

r/h

1.9 2.5

1 . 2 0 , - - - . - - - , - - - . - - - - . . - - - -

1.00

I

.80

/.y/' /I /I

'

--

^{-- --}

--- --

-- -- -- -- -- ^---- ---

/ ---

? - -

e

^{= 400} - - x

3/h = 0.500 ---· x/h = 0.275

·--·- x/h = 0.000

.60 ..__ _ _ _ -.1---'----....l..---'----__J

.0 .5 1.0 1.5 2.0 2.5

r/h

Figure 4.14: Radial variation of the displacements in comparison with the corresponding plane-stress values.

-82-

crack-flank 2h

Figure 4.15: Contour plot of free-surface u2-displacements normalized by the correspond- ing plane-stress values.

~

"'=lN MN

;;~1+

_b "'<:!,... ,.,,... b

'---' b

;::.

.60

.40 ..

.20

^1-

. 00

^1-

\

\ \

\ \

\ \

\ \

I \

\

\

\

\

\

\

\

\

\

\

\

\

\

I I

e

⁼

45o x

3

/h = 0.475

t = ^2.30l/cs

^X3

/h = 0.325 -

\

\

\

\

\

\

\

\

\

\

\

\

',,, ' ,

^___

'\

'\

'\

...

'\

'-'-

·---- X

3

/h = 0. 075

-

'-

-~~-~---

- - - - · - - - · - - · - - · - - · - - · - - · - - · - -

-

-.20 :----~---L---'----___J

.0 .3 .6

r/h

.9 1.2

Figure 4.16: Radial variation of the plane-strain constraint for various planes through the specimen thickness.

CQ ' c....,

'

,--...,

"'<:!N MN

b

"'<:!M

+

MM

b "'<:!,...

M,...

b

'--'

~

"'<:!,...

N,...

... b

"'<:!,...

M,...

b

1. 2

1.0

,8

.6

.4

.2

.0

-.2 .00

1.20

1.10

1.00

.90

.80

.70 .00

-84-

9 = 450 - r/h = 0.05

---r/h=0.11 -- - r/h = 0 21

r/h = 0. 30 r/h = 0 50

~~

_{- --- ~}

- - ---- -- -- -- -- --

_:---~

.10 .20 .30 .40

- r/h = 0.05 ---· r /h = 0. 40 --·- r/h = 0. 71

.10 .20 .30 .40

.50

.50

1.20 ...---.---,---r---,----7

a = 450 1. 10

1.00

.90

- r/h = 0.05 --- r/h = 0.40

.80 --- r/h = 0.71

.70 L_ _ _ _ ..__ _ _ _ _ _ _ _ _ _ ...._ _ _ _ ..._ _ _ _ __.

.00 .10 .20 .30 .40 .50

Figure 4.17: Through-thickness variation of a) the plane-strain constraint, b) o-₁₁ and c) 0"22,

-85-

free-surface. Also, all these plots underscore the point that the assumption of plane-stress is indeed a good one for regions of radial extent greater than about one plate thickness away from the crack-front. This might conceivably be of use to the experimentalist.