In a previous report a one dimensional model of buckling-delamination in a column was summarized. The main assumption made in that report was that the unbuckled portions of the column can be assumed to remain straight. The purpose of this report is to investigate the

more general problem in which bending effects are taken into account. We deal here with the case of a single off center delamination in a column. (The case of multi-delaminations in a column can be worked out too

under slight modifications.) Following the general procedure outlined in previous reports, consider the column of unit width shown in fig, la, State I represents the unstressed column while state I1 denotes the

axially and uniformally compressed column. State III differs from II by allowing the delamination to buckle. Our aim here is to find an

expression for the strain energy release rate of state III

Babcock, C. D.

Chai, H.

Knauss, W. G.

Caltech Authors - Main

ovember 1979 GALCIT SM REPORT 79-15 H~ Chai,C,D. Balbcock,W.G. ~naBss 
INFORMAL PROGWSS FEPORT NO. 3 
ON 
THE INCORPORATION OF BENDING INTO THE BUCKLING DELAMINATION ANALYSIS 
In a previous reporP a one dimensional model of buckling-delamin- 
ation in a column was summarized. The main assumption made in that 
report was that the unbuckled portions of the column can be assumed 
to remain straight. The purpose of this report is to investigate the 
more general problep in which bending effects are taken into account. 
We deal here with the case of a single off center delamination in a 
column. (The case of multi-delaminaticlns in a column can be worked out too 
under slight modifications.) Following the general procedure outlined 
in previous reports, consider the colunm of unit width shown in fig, la, 
State I represents the unstressed column while state I1 denotes the 
axially and uniformally compressed coltunn. State I11 differs from 11 
by allowing the delamination to buckle. Our aim here is to find an 
expression for the strain energy release rate of state 111. 
In order to do this, we assume that the deformed column of state T T T  
can be divided into three beam-column piortions having the same angle 
of rotation (00) at their ends, as shown in fig. lb, Under this 
assumption, we now proceed as follows: 
The beam-column differential equation is given by 
P is the axial load,E is Young's modulus and I is the cross section 
moment of inertia. 
The general solution of this equation is given by 
7kInformal Progress Report No. 2 
y = A1 cos kx + A2 s i n  kx + A3x + A4 (1  1 
where Ai a r e  constantsof in tegra t ion  
Referring t o  f i g .  l b  we have the  following boundary conditions 
Port ion (1) : yl(0) = y1 (R1) = 0 , y i  (0) = -yi(R1) = 0 0 
Inser t ing  (2) i n  (1) .  we found 
eo 'i 2 ~ .  xi 
- I L Yi 2ui s i n  ui cos (ui - )- cos u i ] . i = 1 , 3  (3a,b) 
'i 
- eoR2 [ 1 - cos - 
Y2 2u s i n  2u2 2 2 
where - 
Now, regarding por t ion ( 3 ) .  we observe tha t  y3(R3/2)> 0 when 
u3 i n and y3 (e3/2) < 0 when 2n < u3 > .rr 
Referring t o  f i g s - l a ,  l b  it i s  c l ea r  t h a t  under physical  consideration 
y3(L3/2) < 0 , i . e . ,  n < u < 2 n  3 
Remembering t h a t  u3 = T i s  the  Euler buckling parameter f o r  clamped- 
clamped beam (8, = O), and having i n  rnind t h a t  we a r e  seeking solut ion 
eo a problem where por t ion ( 3 )  i s  pose buckle, we s e t  u = T + 6, 6 > 0 .  3 
Under the  assumption t h a t  eo i s  small ,  we assume t h a t  6 i s  a l so  small 
(6 << n ) (this assumption w i l l  be confi-rmed l a t e r  on). l u t t i n g  u3 = S + n 
i n  the  expression f o r  y and co l l ec t ing  terms up t o  order 6 , we found 3 
- cos ( 2~ + 28 x - 6) Y 3  = A3 
3 
- 8  R 
0 3 
where A3 = A(l -6/n), A = 27r6 
T - H) H Now we find Bo from the condition that M2 = MI + M3 + P3(T 
- PIZ 
(see fig. lb). Upon using M1 =-EI y" 1 1  ( IO),  M~ = - E I ~ ~ > '  (ap). M~ = - E I ~ ~ ' ;  (0) 
and equations (3) we found 
The distributions of axial strains in t:he three column portions can be 
found by the fact that the column ends are fixed during the passing of 
the system from state I1 to 111. This gives (see figs. la, lb) 
Compatibility between portion (1) and portion (3) is governed by 
Balance of axial forces between the col-umn portions gives 
P2 = PI 3 P3 or, since E is common 
where we made use of the relations P == (T - H)Ecl ,  P2 = TEE 1 2 ' 
P3 = Using the assumption that u3 = n+ 6, we found E~ 
by using (3c) to be 
- E 3 -  - -2 (1 + 26,~). 
%3 
Using (3) and (8), the system (5) - (7) can be expressed as follows 
- - - 3 -2 - 
- 
E - (1 - R)H I R  
o 
(1 - &)a, + a3 
- 
- +-  El (1 - H + aii) (1 - 6 + RH) 
- - 2 - 2  - 
E - H / R  
0 
(1 - R)a2 +. a 
c2 = + - 3 
( L  - R + RH) (1 - R + 1x1) + 
where 
- n2 T~ and E~ - -3 2 (Euler buckling strain. for clamped-clamped column 
L 
of length L and thickness T) . Note tha. t  we have taken el = 2, 
( 2 9  - sin 2ul) - (4u2 - sin 4u2)- 
I sin2 u1 + (1 - R )  2u2 sin 
2u2 1 
with 
and Bo is found from equ. (4) to be 
- L 
u tan ul 1 2u2tan2:u2 Rrrr 
The system ( 9 )  - (12) composed of three basic unknowns, namely E l ,  
- 
E~ and E .  A solut ion of t h i s  system i s  carr ied out by means of a 
numerical i t e r a t i o n  scheme. The method can be summarized as  follows: 
- - Step zero: Choose 8, 8 ,  
- Step one: Solve (9) with al  = a2 - n3 = 0 
Step two: Use these values t o  cal.culate 8 (equ. 12) 0 
Step three : 
Step four:  
Step f i v e :  
Ca 1 cnl a t e  n 1' " 2 '  n3 (eqn. 11) 
Calculate again FL, Z;!, E equ. (9) by using the values 
ju s t  obtained for  e o ,  al ,  a 2 ,  a 3 ,  
Repeat t h i s  process u n t i l  convergency of the system 
variables i s  achieved. It was found t h a t  about f ive  
iterations are needed in order to provide satisfactory 
convergence. 
The assumption tha t  6 << 1 (equ. 10) (so tha t  equ. (3b) could be 
approximated by (3d))  i s  checkedout i n  f i g .  2 and 
found va l id .  Figure 3  shows the dependence of Go on geometrical and 
loading parameter. 
Stra in  energy calculat ions  
The s t r a i n  energy of s t a t e  I I I i s  given by 
where the f i r s t  three terms are  associ.ated with compression, while 
the l a s t  three are  due to  bending. 
Upon making use of equ. (3)  , we found 
(4u2 + s i n  4u2) 
2 - r F  
s i n  2u2 
where again we s e t  R1 = R ,  kg = 2, R2 := IL - 9 , ) / 2  
- - Once the values of E l ,  F2, c ,  8 a re  ohtained using the numerical 
0 
procedure outlined e a r l i e r ,  UIII can b e  calculated,  The s t r a i n  energy 
release r a t e  can be found from 
- 
- a u ~ ~ ~  
a R 
- -T4 a F  
G E ~ - T m  
T h e  der iva t ive  i s  then approximated nunierically a s  fol lows;  
- - n
4  F(T) - F(Z +AT) 
G = ~  AX , AX i s  small 
- 
G has been calculated numerically and t:he r e su l t s  are  shown i n  f i g .  4 
and 5 .  Also shown in  f i g .  ' ; , 5  a re  the r e su l t s  for  the beam-column 
analys is  which does llvt include the  bending e f f e c t .  Note t h a t  t h i s  l i m i t  
case i s  obtained by se t t ing  the number of i t e r a t i o n  to  zero, 
Discussion of r e s u l t s .  
In order t o  in terpre t  the fracture energy c r i t e r ion  using these 
r e s u l t s ,  a plot  of experimentally obtained fracture energy for  graphite 
epoxy composite laminates i s  plot ted i n  f i g  , 5  + o r  t t e  s \ j e c ; a - Q  bs 
- 7 -  
I f  
E = \.oqx\o3 y.s,L , L=7 a n d  T = .24!' Ey considering the  r e s u l t s  of 
f i g .  5 ,  i t  can be concluded tha t  the  a r r e s t i ng  capabi l i ty  of i n i t i a l  
crack i s  grea te r  i n  the  current  ana lys i s  over the  previous one (no 
bending e f f e c t ,  i . e . ,  @o = 0) provided z;.5, roughly. This observation 
i s  important,  s ince  they tend t o  back up our experimental r e s u l t s .  
Furthermore, the  deyia t ion of the  current  analys is  from the  previous 
one (go 5 0) i s  not  a f fec ted  much by the  thickness r a t i o  H/T, but i s  
a f fec ted  l a rge ly  by the  loading parameter E i .  Roughly speaking, both 
analyses produced close r e s u l t s  f o r  To K . 5  and deviate considerably 
A t h in  de l amina t ed  l a y e r  a t t a c h e d  t o  a  hali f -space. 
Consider a uniformly compressed i sotropic-elastic ha1 f-space containing 
a thin delamination of thickness H ,  length R and unit width, a s  shown in 
f i g . 6  . State I represents the unstressed half-sapce, while s t a t e  I1 denotes 
the axia l ly  and uniformally compressed hall' space. State 111 d i f fe r s  from I1 
by a l l o w i n g  t h e  de l amina t i on  t o  s p l i t .  Our- aim he re  i s  t o  f i n d  an expr-esbior~ 
for  the s t ra in  energy of s t a t e s  111. For simplicity l e t  v= 0 ( v  i s  the Poisson's 
r a t i o ) .  Regarding the thin buckle a s  a D . C . B .  h a v i ~ g  zero slope a t  both ends 
i t  i s  easy to show, using strength of material approximation t h a t  
~ ( x )  = A(l + cos   IT x l a )  
where w(x) i s  the deflection and E~~ i s  the Euler buckling s t ra in  of a 
clamped-clamped column of thickness H and 1 ength R .  
- 8 - 
Assuming tha t  a f t e r  buck1 i n g  , the s t a t e  of uniform compression remains 
unal tered in the unbuckled medium, the compatability condition 
before and after buckling of this delamination can be expressed as 
The s t ra in  energy occupied by the buckled column i s  
EH 2  EI a / 2  a2u 2 u = - E  a t -  r ( )  dx 
cr  - a / 2 a x  
by making use of (16) and (I?), we find 
The s t ra in  energy in the unbuckled portion ( the half-space l e s s  a void of 
thickness H and length R )  i s  
Note tha t  Uo i s  the s t ra in  energy of a ha1 1'-space under uniform compression 
E 0 -  (Uo = const.) 
Then the total  s t ra in  energy i n  s t a t e  111 7s 
Introducing f i c t i t i o u s  parameter T and L,  ( 4 b )  can be written as  
- 
w h e r e n =  H / T ,  G = R / L .  Now recall eq. (8)  of P.R. No. 2 with N = 1 
(a column of thickness T and length L containing a single delamination of 
thickness H and length 9.)- This equation can he written as  
Note that  (20) reduces to (19) a s  Ti -t 0, as  might be expected. 
To see the rule of R, we use (19) and (20) to get 
Relation (1gb) i s  plotted in f i g ,  + , r e l a t ion  (21) is p l o t t e d  in f i g . 8 - .  
So f a r  we have made three approximati~ons to the problem of a s ing le  
delqmination i n  a column, The more accurate one i s  tha t  given by eq. (15) 
(a delamination in a column in which the unbuckled portions 
are  allowed to bent,) The second i s  the l e s s  accurate problem given by 
eq. (8) of P , R .  Ko.2 (a delamination in a column i n  which the unbuckled 
portions assumed to  remain s t r a igh t ) .  The third approximation, gl'yen by 
eq. (IYb) o r  (19c) of t h i s  report (a delami~nation in a half-sapce) i s  the 
l imi t  case of the above two as  T -, f i g , 9 '  shows a comparison between the 
three methods. I t  i s  seen that  the error  in the approximation of a real 
problem of delamination in a column as  delamination attached to a half space 
i s  on the order of H, provided the load ils suff ic ient ly  lower than the 
E u l  er buck1 i n g  load of the undel arninated co1 umn. 

FIG. la. THREE STP.TES IN THE DEFORMATION OF COI-UMN 











FIG. 4. BEND IMC AND iXMPRE ION MINTRTBUTXO#ST~ THE STRAIN ENERGY 
RELEASE RATE 

FIG. Ei. OEPENIIENCE QT STRAIN ENERGY RELEA E RATE ON LElAOINC 



















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Informal Progress Report No. 3 on The Incorporation of Bending into the Buckling Delamination Analysis

https://authors.library.caltech.edu/51783/1/SM_79-15.pdf

Informal Progress Report No. 3 on The Incorporation of Bending into the Buckling Delamination Analysis

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