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M E C H A N 1 K A 

T E O R E T Y C Z N A 

I  S T O S O W A N A 

2/3,  21  (1983) 

E X P E R I M E N T A L S I M U L A T I O N O F A N I S O T R O P I C D A M A G E 

A N D R Z E J L  I  T  E  W  К  A 

Politechnika  Poznań ska 

J O A N N A S T A N I S Ł A W S K A 

Politechniko  Poznań ska 

I. Introducion 

Internal cracks developed i n materials due to straining have, as a rule, oriented cha­

racter and do not represent solely sets o f scalar voids distributions. A fissurated material 

becomes anisotropic i n its response as well as anisotropy varies i n the process of conti­

nuing damage. E v o l u t i o n o f continuously regularly distributed cracks in an originally 

isotropic and homogeneous material changes the overall mechanical properties o f the 

material due to the increase in the oriented damage. A formulation o f constitutive relations 

for a damaged material requires specification o f its mechanical characteristics concerning 

both the stress­strain response and the damage evolution. K A C H A N O V [1] proposed to 

formulate the constitutive equation for damaged material as a tensor function including 

an independent variable called the crack density tensor or damage tensor accounting for 

the variation of the mechanical properties of the material due to the microcracks develope­

ment. Accordin g to the definition proposed by R A B O T N O V [2] i n the case o f random dis­

tribution o f the microcracks the damage tensor is the second rank symmetric tensor. 

When the distribution o f the fissuration appears regular the damage tensor should account 

both for the variation of the cracked material strength and for the developement of the 

material anisotropy. Thus the form of the damage tensor depends not only on the micro­

cracks density but also on their arrangement. Some o f the attempts to derive the damage 

tensor and to formulate the constitutive equation were presented by M U R A K A M I [3] and 

B E T T E N [4] but due to the lack o f the experimental results concerning the anisotropy of 

the damaged materials the theories available cannot be verified. Thus it seems worthwhile 

to determine the elastic and plastic characteristics of the material with regular array of 

the microcracks. Because of the complexity o f the suitable analysis o f the real damaged 

materials the special k i n d of modelling is proposed. 

Generally the damage tensor is the function o f the stress history but for the given well 

defined stages o f the damage evolution this tensor depends on the actual mechanical 

Properties of the damaged material only. The subject of this study is the experimental 



362  A .  L i T E W K A ,  J .  STANISŁAWSKA 

simulation of the oriented damage at such stages o f the cracks evolution and the analysis 

o f the overall behavior o f the model o f the cracked material, thus the homogenization o f 

the material response within the continuum mechanics. The attention is purposely given 

to the experimental side o f the question. This results in an experimental homogenization 

for the materials with internal oriented structure. 

T o simulate an oriented damage sets of cracks o f given length, orientation, arrange­

ment and density were cut out in flat metal specimens. The load then was applied produ­

cing an overall uniaxial stress and the material response was recorded regarding the mag­

nitude and direction of the overall strains as well as changes in the fissuration density, 

orientation and evolution. Such a method o f experimental simulation of and oriented 

damage can be applied both to the elastic and plastic response, although the behavior 

on the level of a particular cell is evidently non­homogeneous and elastic as well as plastic­

zones develop i n non­homogeneous manner. 

The present paper deals in particular with the elastic characteristics of the damaged 

material when damage pattern and fissuration length are prescribed. T o determine the 

elastic properties for such a material dairly simple uniaxially loaded models can be used. 

The preliminary tests concerning only one crack length but different crack orientations, 

presented i n [5] enabled us to improve the experimental technique. The aim of the expe­

riments presented i n this note is to establish a modification o f the material constants o f 

the damaged material when the cracks length and arrangement are variable. The tests 

were made on the specimens cut out of the sheets of an aluminium alloy P A 2 . The overall 

length o f the specimens was 400 mm, width 70 mm and thickness 0.7 m m . The cracks 

arranged i n square patterns were cut out in the central part o f the specimen 210 m m long 

by means o f a precise punching device. The details of the cracks geometry is presented 

in F i g . 1. The crack width is 1 m m and their length varies from 2 to 7 m m . The pitch o f 

the square pattern o f cracks equals to 10 mm thus the dimensionless crack length /. =  l/P 

was variable ranging from 0.2 to 0.7. T w o different crack arrangements thus two types 

o f an internal orientation o f the material tested were considered, namely either the l o n ­

2. Experimental technique 

F i g .  1.  Cracks  arrangements. 



A N I S O T R O P I C  D A M A G E  363 

gitudinal axis o f the cracks coincides with the pitch or it makes an angle  л /4 with the 

pitch. The specimens were cut out so as to make the overall principal stress direction and 

the cracks orientation variable. The specimens were subjected to axial loading but for 

various directions with respect to the symmetry axes o f the crack pattern. The direction 

o f loading was defined by the angles  о с  = 0, л / 1 2, л /6,  т с /4, л /3,  5л /\2 and  л /2. In this 

way the oriented character of the induced damage is accounted for. 

The specimens were uniaxially loaded in the testing machine and the longitudinal and 

lateral deformations were measured during the loading process. The strains within an 

elastic range were measured by means o f the electric strain gauges 50 m m long. Then all 

the specimens were loaded to fracture and large plastic deformations were recorded 

employing the mechanical strain gauges. This furnished some information concerning 

the plasticity and fracture o f the models of the damaged materials. 

3 .  Results  of  the  experiments 

3.1,  Elastic  range. The overall mechanical response o f the materials with the oriented 

damage Shown in F i g . 1 corresponds to that observed for an ortholropic solid thus their 

elastic characteristics is described by nine material constants. Accordin g to the nomencla­

ture employed in [6] these constants include three Young's moduli E l s E 2 , E 3 and three 

Poison ratios  v2l,  v32,  vl3 determined for loading i n the directions o f the Cartesian co­

ordinate system axes x , ,  x2,  x3 and three shear moduli G 1 2 , G 2 3 , G 3 1 . The axes x , and 
xi are shown in F i g . 1 and the axis  x3 is perpendicular to the surface o f the specimen. 

A s the specimens were uniaxially loaded only some o f those constants can be determined 

employing the results o f this experiments. 

Restricting the analysis to the plane state o f the stress four constants E ! ,  E2,  v2l and 

G 1 2 must be determined. The three first constants and additionally Poison ratio  vi2 were 

E M * 
E 0 

0 I 1 1 , 
5Г/4 Э Т /2  a 

Fig.  2.  Diagrams  of  the  function  Ј ( a ) . 



364  A .  L I T E W K A ,  J .  S T A N I S Ł A W S K A 

determined directly from the results obtained for the specimens subjected to axial load 

at the directions defined by the angles x = 0 and  л /2. The shear modulus G 1 2 was cal­

culated employing the effective Young's modulus measured for the specimen loaded in 

the direction inclined at the arbitrary angle  x with respect to cracks orientation. The 

diagrams of the function  E(x) for various crack length and orientation is shown in Fig. 2. 

The modification of the elastic constants for increasing crack length presents F i g . 3. The 

values o f the Young's modulus E 3 shown in F i g . 3 were calculated from the relation 

E 3 =  /г Е where  /.t = 1 — Я / 10 is the reduction of the net area in the direction x 3 for the 

cracked material and E = 6 7 7 0 0 M P a is the Young's modulus for the original material. 

' E , / E 

0 0,5 1.0  Л ­ /Р  

1,0 

0,5 

0 0,5 1,0 A ­ l / P 

Fig.  3.  Elastic  constants  versus  the  dimensionless  crack  length:  a)  cracks  in  the  pitch  direction,  b)  cracks 

in  diagonal  direction. 



A N I S O T R O P I C  D A M A U I  365 

3.2.  Plastic  range. The plastic characteristics of the analysed models o f the damaged 

materials includes six constants: three uniaxial yield stresses  Xtl,  X22,  X33 for loading 

i n the directions x 1 ; . v 2 ,  x3 and three shear yield stresses  Xl2,  X23,  X31. The first two 

constants  Xtl and  X22 can be easily determined employing the stress­strain curves for the 

specimens tested. These curves for various loading orientations and various dimensionless 

crack lengths are shown i n F i g . 4 and 5. The results obtained show that the minimal 

strength o f the cracked material models does not correspond to the loading direction 

defined by the angle a =  njl. This is distincly shown i n F i g . 6 where the diagrams o f the 

uniaxial yield stress versus the angle a are presented. The conventional uniaxial yield 

streses were determined as the stress corresponding to the permanent strain equal 0.1%. 

The modification of the uniaxial yield stresses Xt!,  X22 and  А з̂ for increasing crack length 
is shown i n F i g . 7. The yield stress  А ' зз was calculated similarly as E 3 from the relation 
^33  =  f*<To, where  a0 = 132 M P a is the uniaxial yield stress for the original material 
without cracks. 

Л  ­ J 1 * -
0  0,5  1,0  Ј , % 

Fig.  4.  Stress­strain  curves  for  material  with  cracks  in  the  pitch  direction. 

Determining the complete plastic characteristics o f the damaged material even for 

the plane state o f stress requires much more complicated experiments employing the 

tubular specimens subjected to internal pressure and axial load. The specimens should 

possessa given pattern o f fissures and i f the overall anisotropic response has to be determi­

ned the specimens should be loaded in such a way that the principal stress directions are 

inclined at various angles with respect to the symmetry axes o f the material. Such expe­



1366] 



A N I S O T R O P I C  D A M A G E  367 

0  0,5  1,0  X­I/P 

F i g .  7.  Plastic  constants  versus  the  dimensionless  crack  length:  a)  cracks  in  the  pitch  direction,  b)  cracks 
in  diagonal  direction. 

riments  have  not  been  performed  for  the  specific  case  of  the  damaged  material  models 

but  for  this  purpose  can  be  used  the  experimental  technique  established  when  the  plastic 

anisotropy  of  the  perforated  materials  was  analysed  [7, 8].  When carrying the  experiments 

as  was  described  in  [7]  the  material  constants  as  well  as  the  effective  yield  surfaces  can 

be  determined  for  various  loading  orientations  with  respect  to  the  symmetry  axes  of  the 

material. 

3.3.  Fracture of  the  specimens.  A l l  the  models  of  the  cracked  materials  were  loaded  t o 

fracture  what  enabled  to  obtain  some  information  concerning  its  behavior  when  largo 

plastic  strain  occur  in  the  plastic  zones  developed  between  the  cracks.  It  is  seen  from 



368  A .  L I T E W K A ,  J .  S T A N I S Ł A W S K A 

F i g . 4 and 5 presenting the stress­strain curves for the materials tested that the overall ho­

mogenized plastic strains at fracture are relatively small i n comparison with those measured 

for the original material. The permanent strain at fracture for the aluminium alloy P A 2 

measured i n this tests was approximately 10% thus F i g . 4 and 5 show only a part of the 

respective curve for the original material. 

The relative brittleness observed on the macroscale for the cracked material is the 

result o f very narrow plastic zones or rather slip lines appearing i n the cells of the material 

structure. The localization and the shape o f the plastic zones are shown i n F i g . 8 and 9 

where the photographs of the selected specimens after fracture are presented. Considering 

cc =  fT/4 

! : 1 

1 

Щ  r   ~  
4  " " ­ ' i 

i "  •  

Of =  IT/2 

Specimens  after  fracture  (Ź  =  0.5  cracks  in  the  pitch  direction). 

i  i  ! 

t  '  \ 

K x 
t  i 

i  i 
i  i ! 

V . J 
^ V ~ d 

t  'l  \ 

i  i  1 

1  1  i 

r  « 
I I 

i  i 
i  i ! 

V . J 
^ V ~ d 

t  'l  \ 

i  i  1 

1  1  i 

i i 

i  i ! 
V . J 

^ V ~ d 
t  'l  \ 

i  i  1 

1  1  i 

i  i 
1  1; 

i  i . 
"t  i 

Л"  I fi 
• i  l  l 

»L  1  . 
t  i 

1 1 1 
1  '  Г  

• •  .'V  .  j 

1 L .: 

a =7T/2 

Fig. 9.  Specimens  after  fracture  (Я  =  0.5,  cracks  in  diagonal  direction). 



A N I S O T R O P I C  D A M A O F  369 

the problem on the macroscale only it is seen from F i g . 8 that the rupture o f the specimens 

with the cracks arranged i n the pitch direction is regular and occures for all the loading 

orientations in one of the symmetry axes of the material structure. These regularities are 

not observed in F i g . 9 where the models with cracks arranged in the diagonal direction 

are shown. The detailed analysis of the fracture mechanism for the models of the cracked 

materials on macro and microscales as well as the formulation o f the suitable criterion o f 

the fracture requires further experiments. 

4.  Conclusions 

The experiments enable determining the material constants for the models o f the 

damaged materials. It is seen i n F i g . 3 and F i g . 7 that the elastic and plastic constants 

determined for loading in the direction o f the axes  x2 and x 3 are nearly identical. Thus 

the materials tested can be considered as the transversaly isotropic solids with the isotropic 

properties in the plane defined by axes  xt and  x3. 

The mechanical properties i n the elastic range are almost the same for both crack 

arrangements what means that the overall elastic response for given crack pattern depends 

mainly o n the reduction of the net area o f the material but not on the cracks arrangement. 

A s it is seen i n F i g . 6 and 7 this conclusion does not concern the plastic range where the 

yield stresses determined for both cracks arrangements are different especially for increa­

sing crack length. 

References 

»'•  M .  K A C H A N O V ,  Continuum model of  medium with  cracks,  Brown  Univ.,  Porous  Media  Series,  Rep. no. 

16.  1972. 
2 ­  Y . N .  R A B O T N O V ,  Creep  rupture,  12th  Int.  Congr.  Appl.  Mech.  Stanford,  1968.  342­  349. 

3.  S.  M U R A K A M I ,  N .  O H N O ,  A  continuum  theory  of  creep  and creep  damage, 3rd  Ш Т АМ  Symp.,  Creep  in 

Structures,  Leicester  1980,  Springer,  Berlin,  1981,  422*443. 

4.  J . B E T T E N ,  Damage tensors  in continuum mechanics, J .  Mec. Theor.  A p p l . ,  1,  2  (1983). 

5t  A .  L I T E W K A ,  A .  S A W C Z U K ,  Experimental evaluation of  the  overall anisotropic material response  at con­

tinuous damage,  in:  ,,Mechanic  of  material  behavior.  The  D . C .  Drucker  anniversary  volume", G . J . 

Dvorak  and  R ,  T .  Shield,  ed., Elsevier  Sci.  Publ.  C o . ,  1983,  in press. 
6.  S . G .  LEKETNICKU,  Theory of  elasticity for  anisotropic solid, Nauka,  Moscow  1977,  (in  Russian). 
7 ­  A .  L I T E W K A ,  Plastic  anisotropy  of perforated  materials,  Rozprawy  111,  Polit.  Pozn.,  P o z n a ń  1980,  (in 

Polish). 

8.  A .  L I T E W K A ,  A .  S A W C Z U K ,  A  yield  criterion for  perforated  sheets, Ing.­Archiv,  50  (1981),  393  ­400 

P  с 3 ю  M e 

Э К С П Е Р И М Е Н Т А Л Ь Н ОЕ  М О Д Е Л И Р О В А Н ИЕ  М А Т Е Р И А Л ОВ  С  Т Р Е Щ И Н А МИ  

В  р а б о те  п р е д с т а в л я е т ся  . м о д е л и р о в а н ие  м а т е р и а л ов  с  р е г у л я р но  р а с п о л о ж е н н ы ми  т р е щ и­

н а ми  и  э к с п е р и м е н т а л ь н ую  т е х н и ку  д ля  о п р е д е л е н ия  у п р у г их  и  п л а с т и ч е с к их  п о с т о я н н ы х.  П о­

Л у ч е но  з а в и с и м о с ти  п о с т о я н н ых  о т  д л и ны  т р е щ ин  д ля  и х  д в ух  к о н ф и г у р а ц и й.  П р е д с т а в л я ю т ся  
т а к же  н е к о т о р ые  р е з у л ь т а ты  о т н о с и т е л ь но  р а з р у ш е н ия  о б р а з ц ов  .м а т е р и а л ов  с  т р е щ и н а м и. 



370  A .  L I T E W K A ,  J .  S T A N I S Ł A W S K A 

S t r e s z c z e n i e 

D O Ś W I A D C Z A L NA  S Y M U L A C J A  A N I Z O T R O P O W E G O  U S Z K O D Z E N I A  M A T E R I A Ł U 

W  pracy  przedstawiony  z o s t a ł  s p o s ó b  modelowania  m a t e r i a ł ó w  z  regularnie  rozmieszczonymi  zary­

sowaniami  oraz  o m ó w i o n a  jest  technika  b a d a ń  m a j ą ca  na  celu  wyznaczenie  s t a ł y c h  m a t e r i a ł o w y c h  w  za­

kresie  s p r ę ż y s t ym  i  plastycznym.  W  wyniku  b a d a ń  otrzymano  z a l e ż n o ś ci  s t a ł y c h  od  d ł u g o ś ci  szczelin  dla 

ich  d w ó c h  konfiguracji.  Przedstawione  z o s t a ł y  r ó w n i e ż  pewne  wyniki  d o t y c z ą ce  zniszczenia  modeli  ma­

t e r i a ł ó w  z  uszkodzeniami. 

Praca  została  złoż ona  w  Redakcji  dnia  24  lutego  1983  roku