ON THE QUANTIFICATION OF DYNAMIC IN IMPACT LOADING AND THE PRATICAL APPLICATION FOR KId-DETERMINATION

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ON THE QUANTIFICATION OF DYNAMIC IN IMPACT LOADING AND THE PRATICAL APPLICATION FOR KId-DETERMINATION

W. Böhme, J. Kalthoff

To cite this version:

W. Böhme, J. Kalthoff. ON THE QUANTIFICATION OF DYNAMIC IN IMPACT LOADING AND

THE PRATICAL APPLICATION FOR KId-DETERMINATION. Journal de Physique Colloques,

1985, 46 (C5), pp.C5-213-C5-218. �10.1051/jphyscol:1985527�. �jpa-00224757�

JOURNAL DE PHYSIQUE

Colloque C5, suppl6ment au n08, Tome 46, ao0t 1985 page C5 -2 13

ON THE QUANTIFICATION OF DYNAMIC EFFECTS I N IMPACT LOADING AND THE

PRACTICAL APPLICATION FOR

Kid-DETERMINATION

W. BGhme and J.F. Kalthoff

Fraunhofer-Institut fur Werkstoffmechanik W8hZerstrasse 1 2 , 0-7800 Freiburg, F.R. G.

Resume

-

Nous presentons une etude systematique de l ' h i s t o i r e du chargement de l l e x t r e m i t @ de f i s s u r e pour des G c h a n t i l l o n s de f l e x i o n t r o i s - p o i n t s . Nous avons e f f e c t u e une analyse q u a s i s t a t i q u e ti l ' a i d e d'un modele mathematique s i m p l i f i e . Les e f f e t s dynamiques sont q u a n t i f i e s par des experiences. En se basant s u r c e t t e approche nous developpons une procedure q u i permet de p r e d i r e l ' h i s t o i r e du f a c t e u r d 1 i n t e n s i t @ de c o n t r a i n t e dynamique pour des c o n d i t i o n s d ' e s s a i a r b i t r a i r e s .

Abstract

-

A systematic study o f t h e crack t i p l o a d i n g h i s t o r y i s presented f o r impacted three-point-bend specimens. A q u a s i s t a t i c a n a l y s i s i s performed f o r a s i m p l i f i e d mathematical model and t h e dynamic e f f e c t s are q u a n t i f i e d by model experiments. Based on t h i s approach a procedure i s developed which e a s i l y allows t h e p r e d i c t i o n o f t h e dynamic s t r e s s i n t e n s i t y f a c t o r h i s t o r y f o r a r b i - t r a r y t e s t conditions.

I

-

INTRODUCTION

Three-point-bend specimens a r e w i d e l y used f o r determining f r a c t u r e toughness data under s t a t i c as w e l l as dynamic l o a d i n g conditions. I n s t a t i c f r a c t u r e mechanics t h e r e l a t i o n s h i p between t h e l o a d a p p l i e d t o t h e specimen, t h e geometry o f t h e specimen, and t h e s t r e s s i n t e n s i t y f a c t o r KI a t t h e crack t i p i s g e n e r a l l y w e l l known. For impact l o a d i n g c o n d i t i o n s a s i m i l a r simple r e l a t i o n s h i p cannot e x i s t because o f dy- namic e f f e c t s The p r o p o r t i o n a l i t y between t h e a p l l i e d l o a d P ( t ) and t h e crack t i p l o a d i n g Kayn(t) becomes t i m e dependent. The K Yn(t)-behavior, however, must be known f o r \he determination o f t h e impact f r a c i u r e toughness KId by t h e concept o f impact response curves C1,21.

An analytic-experimental approach i s presented f o r determining t h e h i s t o r y o f t h e dynamic crack t i p l o a d i n g d u r i n g t h e impact processes. The procedure s p l i t s t h e problem i n t o two d i f f e r e n t tasks. F i r s t , a rough estimate o f t h e gross crack t i p loading, ~ ? . ~ . ( t ) , i s obtained

85

an a n a l y t i c a l , q u a s i s t a t i c consideration. Se- condly,a c o r r e c t i o n f u n c t i o n , k ' ( t ) , i s e x p e r i m e n t a l l y determined, which quanti- t a t i v e l y describes dynamic e f f e c t s . The combination af both allows a q u a n t i t a t i v e p r e d i c t i o n o f t h e dynamic s t r e s s i n t e n s i t y f a c t o r , K1Yn(t), by t h e r e l a t i o n s h i p :

~fJ'n((t) = ~ P - s - ( t ) ^X kdyn (1

D e t a i l s o f t h e procedure and a p p l i c a t i o n s t o various problems are given i n C3,41 and are summarized i n t h i s paper.

I 1

-

QUASISTATIC CONSIDERATIONS

The t e s t arrangement f o r three-point-bend specimens under impact loading i s shown schematically i n Fig. l a . The l o a d i n g behavior o f t h e crack i s deternined by many parameters, i.e. hammer mass M, impact v e l o c i t y v,, machine compliance C,

, specimen w i d t h W, specimen l e n g t h L, thickness B, crack l e n g t h a, specimen

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1985527

JOURNAL DE PHYSIQUE

~ q - ~ . ( t ) = vOY

iEt'

^{s i n}

(Q

^t.)

compliance CS, Young's-modu-

where C ~ C C C * and S = 4W. The f u n c t i o n Y = Y(a1W) i s t h e known s t a t i c r e l a t i o n - s h i p f o r KI-ietermination w i t h three-point-bend specimens a f t e r Srawley [61:

l u s E, P o i s s o n ' s - r a t i o v, mass d e n s i t y p, and support span S.

For a f i r s t estimate on t h e ge- n e r a l i n f l u e n c e o f these para- meters on t h e gross crack t i p

l o a d i n g behavior, a simple mass- spring-model i s i n t r o d u c e d as shown i n Fig. l b . The hammer i s replaced by a p o i n t mass and t h e specimen by a massless spring.

These assumptions can be made s i n c e t h e hammer mass i s u s u a l l y l a r g e compared t o t h e mass o f t h e specimen. I f necessary, t h e

C* = c:(~/w) = CS/(EB) i s t h e dimensionless specimen compliance f o r plane d r e s s a f t e r Bucci e t a l . C71:

I

a1

I

^bl

I

/

^v.

^l

^@ POINT MASS M

: ^'

SPRING ISPECIMENI:

COMPLIANCE CS

,,,,,.,

^G _I

I

The machine compliance Cm can e a s i l y be taken i n t o account i n eq. ( 2 ) by r e p l a c i n g C< by C:(1

+

Cm/Cs).

machine compliance can be taken Fig. 1

-

Q u a s i s t a t i c a n a l y s i s o f t h e impact event

i n t o account by another spring. by a simple mass-spring-model

This model represents a coarse simpl i f i c a t i o n o f t h e a c t u a l

processes since i t neglects dynamic e f f e c t s w i t h i n t h e specimen, e.g. wave propaga- t i o n processes. But i t leads t o a d i f f e r e n t i a l equation which e a s i l y can be solved.

The s o l u t i o n gives t h e displacement o f t h e specimen as a f u n c t i o n o f time. U t i l i z i n g w e l l known s t a t i c r e l a t i o n s h i p s f o r three-point-bend specimens C5-71 t h e displacement can be expressed i n terms o f s t r e s s i n t e n s i t y f a c t o r s , denoted as t o i n d i c a t e t h e q u a s i s t a t i c c h a r a c t e r o f t h e s o l u t i o n :

The d e r i v e d formula g i v e s a f i r s t overview on t h e g l o b a l crack l o a d i n g behavior and i t s dependence on t e s t parameters, as i l l u s t r a t e d i n Fig.2a-f. Only t h e f i r s t h a l f o f t h e p e r i o d o f t h e r e s u l t i n g s i n e - f u n c t i o n (eq. ( 2 ) ) i s shown. This i s t h e only t i m e i n t e r v a l o f i n t e r e s t i n impact t e s t s , since t h e t e s t s u s u a l l y are performed i n such a manner, t h a t t h e specimens break i n t h e f i r s t i n c r e a s i n g p a r t o f t h e sinusodial loa- d i n g curve. At t h e maximum t h e t o t a l k i n e t i c energy o f t h e hammer would have been completely converted i n t o s t r a i n energy o f t h e specimen. The r e s u l t s a r e presented i n a normalized form. The f u l l curves correspond t o f i x e d t e s t parameters. For lower1 higher values o f t h e t e s t parameter t h e crack l o a d i n g h i s t o r i e s are given by t h e dottedldashed curves. The arrows i n d i c a t e t h e changes obtained by i n c r e a s i n g t h e t e s t parameter.

Two i n t e r e s t i n g observations s h a l l be emphasized, which apply f o r t h e very beginning o f t h e impact process, i.e. t h e t i m e range o f p r a c t i c a l relevance:

F i r s t , q ' S ' ( t ) i s indepenent on t h e hammer mass M and t h e specimen t h i c k n e s s B. Secondly, ~ q . ~ . ( t ) v a r i e s v e r y l i t t l e w i t h c r a c k l e n g t h a/W f o r 0,2Sa/Ws0,5. Only f o r

s h o r t e r and a l s o f o r l o n g e r c r a c k s

..

_m ^aspI

t h e s l o p e o f t h e ~ p ' ~ ' ( t ) - c u r v e -

HAMMER- MASS M

becomes l e s s steep. ^{* - d} - ^C

a

I n o r d e r t o examine t h e accuracy o f

t h e d e s c r i b e d a p p r o x i m a t i o n t h e c a l - i c u l ated, q u a s i s t a t i c c r a c k t i p l o a - ^>

d i n g ~ " ~ ' ( t ) i s compared t o t h e ^t _z r e a l , experime t a l l y d e t e r m i n e d dy-

a

^C

namic values KIYn(t). Data o b t a i n e d _{I V)} w i t h specimens made from t h e model ma-

t e r i a l A r a l d i t e B by means o f t h e sha- dow o p t i c a l t e c h n i q u e C8 9 1 a r e g i v e n i n Fig. 3. O b v i o u s l y Kq.'.(t) r e p r e - s e n t s a good approxima$ion o f t h e g l o - b a l K!Yn(t)-curve, whereas t h e s t r e s s i n t e n s i t y f a c t o r s determined by a

s t a t i c f o r m u l a f

m

t h e measured _{LENGTH CRACK o~~}

hammer1 oad, K:ta'(P show on1 y

poor agreement w i t h ~ i ! ~ ~ ~ ( t ) . D e t a i l s I

as t h e o s c i l l a t i o n s O! t h e ~ ; y n ( t ) - ^{TIME t} r e l a t ~ v e u n ~ t s

c u r v e o f course cannot be d e s c r i b e d F i g . 2

-

I n f l u e n c e o f System parameters by t h e q u a s i s t a t i c approximation. ( q u a s i s t a t i c )

0 5 % .. ..

I . - . IMPACT- .'-,

I VELOCITY V, - \

0 1 .

I 1 1

-

QUANTIFICATION OF DYNAMIC EFFECTS

The dynamic e f f e c t s n e g l e c t e d so f a r i n t h e c a l c u l a t i o n o f q . S . ( t ) a r e d e f i n e d a c c o r d i n g t o eq. (1) by t h e r a t i o :

1.60 1.40 1.20 1.W

0.80 0.60 0.40 0.20 0.00

5 10 15 20 25 30 35

DIMENSIONLESS TlME c , t / W

Fig. 3

-

Q u a s i s t a t i c a l l y c a l c u l a t e d Fig. 4

-

Dynamic c o r r e c t i o n f u n c t i o n kdyn, and measured l o a d i n g determined form d a t a g i v e n i n Fig. 3 h i s t o r i e s ( A r a l d i t e B)

C5-216 JOURNAL DE PHYSIQUE

I n eq. ( 3 ) kdyn i s a t i m e dependent "d namic c o r r e c t i o n f u n c t i o n " which q u a n t i f i e s t h e dynamic e f f e c t s . T h i s f u n c t i o n kdyX i determined f r a n e x p e r i m e n t a l l y o b t a i n e d a c t u a l dynamic s t r e s s i n t e n s i t y f a c t o r s K!yn(t) and t h e v a l u e s

v.'.,

c a l c u l a t e d by eq. ( 2 ) . The c o r r e c t i o n f u n c t i o n determined f r a n t h e r e s u l t s presented i n Fig. 3 i s shown i n Fig. 4 t o g e t h e r w i t h a d d i t i o n a l r e s u l t s . I n t h i s p l o t t h e t i m e i s norma- l i z e d by t h e p r o p a g a t i o n t i m e o f l o n g i t u d i n a l waves across t e specimen w i d t h ; c l i s t h e l o n g i t u d i n a l wave v e l o c i t y f o r p l a n e s t r e s s . The v a l u e kByn = 1 r e p r e s e n t s t h e q u a s i s t a t i c behavior. The o b t a i n e d d e v i a t i o n s from t h i s v a l u e a r e due t o dynamic e f - f e c t s . These e f f e c t s a r e l a r g e i n t h e b e g i n n i n g o f impact event. They decrease w i t h i n c r e a s i n g t i m e and approach t h e

q u a s i s t a t i c s t a t e v i a an o s c i l - l a t i o n w i t h damped amplitude.

Fig. 5 shows kdyn-data o b t a i n e d under d i f f e r e n t impact c o n d i t i o n s . Specimens o f d i f f e r e n t s i z e s W b u t g e o m e t r i c a l l y s i m i l a r shape were impacted a t d i f f e r e n t v e l o c i t i e s by d r o p w e i g h t s o f d i f f e r e n t mass.

D e s p i t e o f t h e s e v a r i a t i o n s i n t e s t c o n d i t i o n s a l l d a t a f o l l o w

a

t h e same curve. h i s r e s u l t im-

p l i e s t h a t t h e k Yn-curve i s i n - _{Oo00 5 10 15 20 25 30 35} dependent o f t h e s e parameters, i.e. DIMENSIONLESS TIME CI~IW

W, vo, and M as f a r as t h e impact

energy i s s u f f i c i e n t l y l a r g e . F i g . 5

-

Dynamic c o r r e c t i o n v a l u e s kdyn D e t a i l e d e x p e r i m e n t a l i n v e s t i g a t i o n s and s t u d i e s on dynamic e f f e c t s l e a d t o a s y s t e - m a t i c o v e r v i e w on t h o s e dynamic events, which s i g n i f i c a n t l i n f l u e n c e t h e dynamic c r a c k t i p b e h a v i o r [41. S p e c i f i c c h a r a c t e r i s t i c s o f t h e kdJn-curves were c o r r e l a t e d t o p h y s i c a l processes. The r e s u l t s o b t a i ed a r e i l l u s t r a t e d i n a c o n c l u d i n g manner by t h e schematic r e p r e s e n t a t i o n o f t h e kayn- c u r v e g i v e n i n Fig. 6. A rough separa- t i o n i s made i n t o a t i m e range which i s s t r o n g l y i n f l u e n c e d by dynamic e f f e c t s and an n e a r l y q u a s i s t a t i c t i m e range. The dynamic range i s d i v i d e d i n t o f i v e r e g i o n s

0-0.

The b e g i n n i n g o f t h e r e g i o n s i s determined by s i g n i f i c a n t dynamic events.

T h e i r a d d i t i o n a l i n f l u e n c e i s i n d i c a t e d by arrows (see Fig. 6 and [41):

a

F i r s t c o n t a c t between t h e i m p a c t i n g hammer and t h e specimen: e l a s t i c waves s t a r t t o propagate i n t o t h e specimen.

@

The f i r s t shear wave generated a t t h e l o a d i n g p o i n t reaches t h e c r a c k t i p : t h e f i r s t s i g n i f i c a n t i n c r e a s e i n c r a c k l o a d i n g i s o b t a i n e d .

a

The f i r s t l o n g i t u d i n a l wave f r o n t _l i n t e r a c t i n g w i t h t h e l o w e r bounda-

I-

dynaml~ally influenced

'

^nearly quasistotc

. . . . . . . .. ,. . .

r y o f t h e specimen e x c i t e s R a y l e i g h ttme range Iood8ng behavlor

waves which p r o p a g a t e a l o n g t h e L c r a c k s u r f a c e s and t h e n reach t h e

6

c r a c k t i p : an a d d i t i o n a l c r a c k t i p

5

l o a d i n g i s obtained. ^L

@

A f t e r r e f l e c t i o n o f t h e f i r s t com-

6

p r e s s i v e waves a t t h e l a t e r a l boun-

3

d a r y l o n g i t u d i n a l t e n s i l e waves

g

r e a c h t h e c r a c k t i p : a f u r t h e r i n c r e a s e i n c r a c k t i p l o a d i n g i s

2

obtained. I I

@

Caused by p r e v i o u s l o s s o f c o n t a c t ^l I

b e h a v i o r ( s e e C101) a d d i t i o n a l im- l I ^I I

p a c t events o c c u r between specimen 0 ; / ' l '

i

^I

and supports, and t h e t h e r e b y ex- ^LIW T&.?S/W .1; ,."S

c i t e d shear waves s i g n i f i c a n t 1 y ^{C, W} DIMENSIONLESS TIME =$/W

i n f l u e n c e t h e c r a c k t i p : an a d d i t i - F i g . 6

-

Dynamics o f impact event onal c r a c k t i p l o a d i n g i s obtained. (schematic)

As can be seen from F i g . 6, t h e onset t i m e s o f t h e c o n s i d e r e d dy- namic e f f e c t s a r e c o r r e l a t e d t o on- l y t h r e e g e o m e t r i c a l parameters a/W, L/W and S/W. These a r e c a l l e d dyna- m i c parameters. Since S/W = 4 h o l d s

f o r most t e s t d e v i c e s t h e p r a c t i c a l - l y r e l e v a n t dynamic parameters a r e o n l y a/W and L/W. These two parame- t e r s a r e used t o c l a s s i f y specimens as d y n a m i c a l l y d i f f e r e n t types. For f o u r t y p e s o f specimen, Type I t o Type I V t h e e x p e r i m e n t a l l y d e t e r - mined kay"curves a r e g i v e n i n

O W o 5 10 15 m s 30 o 5 a 15 m 25 M

DIMENSIONLESS TlME c l t / W m i c e f f e c t s . On1 y reduced dynamic

e f f e c t s a r e o b t a i n e d f o r specimens Fig. 7

-

Dynamic c o r r e c t i o n f u n c t i o n s kdYn w i t h s h o r t e r c r a c k l e n g t h (Type 11)

f o r f o u r d i f f e r e n t specimens and p a r t i c u l a r l y f o r specimens o f l a r g e l e n g t h (Type I and 111).

Because o f t h e n o r m a l i z e d p l o t used i n F i g . 7 t h e d a t a a l s o a p p l y f o r d i f f e r e n t m a t e r i a l s , i n p a r t i c u l a r f o r s t e e l s , as shown i n t h e n e x t chapter.

I V

-

APPLICATIONS

The e v a l u a t e d q a s i s t a t i c f o r m u l a (eq. ( 2 ) ) i n c o m b i n a t i o n w i t h t h e dynam'c c o r r e c - t i o n f u n c t i o n kBy\an be used t o p r e d i c t t h e dynamic c r a c k t i p l o a d i n g K?y"t):

K ? Y ~ ( ~ ) = q - S - ( t ) x kdyn(cl t l ~ ) (4

T h i s i s s c h e m a t i c a l l y i l l u s t r a t e d i n can be c a l c u l a t e d by

specimens o f l a r g e and small s i z e a r e considered. The d a t a o f t h e Charpy- specimens (Fig. 10) a r e n o r m a l i z e d by t h e impact v e l o c i t y v, a c c o r d i n g t o e (2). I n b o t h cases t h e p r e d i c t e d Kg$"t)-curve (dashed l i n e ) i s com- pared t o t h e measured impact response c u r v e ( f u l l l i n e ) . The r e s u l t s i n d f c a - t e t h a t t h e p r e s e n t e d procedure a l l o w s

TIME t a q u a n t i t a t i v e p r e d i c t i o n o f t h e r e a l dynamic c r a c k t i p l o a d i n g w i t h an ac- F i g . 8

-

Procedure t o p r e d i c t h e dynamic

h

c u r a c y which i s s u f f i c i e n t f o r engi-

c r a c k t i p l o a d i n g ~ ~ y " ( t ) n e e r i n g purposes. I n c o m b i n a t i o n w i t h a t i m e t o f r a c t u r e measurement and f o l l o w i n g t h e impact response c u r v e concept [1,21 a s i m p l e and i n e x p e n s i v e de- t e r m i n a t i o n o f impact f r a c t u r e toughness values KId i s p o s s i b l e [41.

C5-218 JOURNAL DE PHYSIQUE

l

100 ZOO 300

TIME t, ps

Fig. 9

-

C a l c u l a t e d and measured dynamic F i g . 10

-

C a l c u l a t e d and measured dyna- c r a c k t i p l o a d i n g f o r l a r g e m i c c r a c k t i p l o a d i n g f o r

s t e e l specimens Charpy-specimens

SUMMARY

v - -

F o r impacted t h r e e - p o i n t - b e n d specimens a s i m p l e mass-spring-model. and q u a s i s t a t i c c o n s i d e r a t i o n s were used t o o b t a i n an overview on t h e g l o b a l i n f l u e n c e o f t h e system parameters on t h e c r a c k t i p l o a d i n g h i s t o r y . The a d d i t i n a l dynamic e f f e c t s a r e quan-

8

t i f i e d by t i m e depend n t dynamic c o r r e c t i o n f u n c t i o n s k Yn

or

d y n a m i c a l l y d i f f e - r e n t specimen t y p e s kgyn-curves have e x p e r i m e n t a l l y been determined and discussed w i t h r e s p e c t t o wave p r o p a g a t i o n phenomena. On t h e b a s i s o f t h e c a l c u l a t e d q u a s i s t a - t i c c r a c k t i p l o a d i n g i n c o m b i n a t i o n w i t h t h e determined s e t o f dynamic c o r r e c t i o n f u n c t i o n s a procedure i s developed t o q u a n t i t a t i v e l y p r e d i c t t h e dynamic l o a d i n g h i s t o r y f o r t e s t c o n d i t i o n s o f p r a c t i c a l relevance. R e s u l t s o b t a t n e d f o r impacted s t e e l specimens demonstrate, t h a t t h e procedure i s s u f f i c i e n t l y a c c u r a t e and t h a t i t g e n e r a l l y a p p l i e s f o r a r b i t r a r y t e s t c o n d i t i o n s .

REFERENCES

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[21 K a l t h o f f , J.F., W i n k l e r , S., Bohme, W., t h i s Proceedings [31 Bohme, W., IWM-Report Z 2/84, F r e i b u r g (1984)

[41 Bohme, W., Ph.D. D i s s e r t a t i o n i n p r e p a r a t i o n , TH Darmstadt (1985) C51 ASTM E 399-74, Annual book o f ASTM-Standards, P h i l a d e l p h i a (1974) C61 Srawley, J.E., I n t . J. o f F r a c t u r e , 12 (1976) 475

C71 B u c c i , R.J., P a r i s , P.C., Landes, J.D., Rice, J.R., Proc. 1971 Nat. Symp. on F r a c t u r e Mech., P a r t 11, ASTM STP 514 (1972) 40

C81 Manogg, P., Ph.D. D i s s e r t a t i o n , F r e i b u r g , Germany (1964)

[g] K a l t h o f f , J.F., i n Handbook o f Exp. Mech., ed. by A.S. Kobayashi, P r e n t i c e H a l l (1985)

[ l 0 1 Bohme, W., K a l t h o f f , J.F., I n t . J. o f F r a c t u r e , 20 (1982) R139