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M E C H A N I K A
TEORETYCZNA
I STOSOWANA

1/2, 22 (1984)

PLASTIC ZONE SIZE OF DUGDALE TYPE CRACKS IN A SELF-STRESSED
TWO-PHASE MEDIUM WITH PARTIALLY PLASTIFIED MATRIX MATERIAL

KLAUS P. H E R R M A N N and IVAN M. M I H O V S K Y 1

Institute of Mechanics
University of Paderborn
4790 Paderborn, Federal Republic of Germany

Abstract

Different boundary-value problems are considered concerning the elastic-plastic and
fracture behaviour of brittle fibres-ductile matrix composites under thermal loading
conditions in both the cases of absence and presence of cracks within the matrix phase.
A model of the plastic deformation process is proposed with regard to a single unit cell
of the fibre-reinforced composite. Numerous details of the deformation process within
this unit cell are investigated by use of the above mentioned model including the possible
failure mechanisms of the fibre-matrix bond. If applied together with the known crack
model of the Dugdale type the proposed model for the plastic deformation process of
a composite unit cell is shown to imply useful conclusions concerning the thermal crack
growth of radial Dugdale type cracks within the matrix phase.

Introduction

The investigation of the interaction between the stress fields caused by the presence
of different inhomogeneities is a problem of great practical importance. This is actually
the basic problem of the mechanics of the composite materials. Of special interest from
the point of view of fracture mechanics of the composite structures are the questions
concerning the interaction between the structural components and existing cracks within
these structures. Both the cases of mechanical and thermal loading of cracked composites
have been since long studied and different models of interaction have been already con-
sidered by means of both micromechanical analysis and macromechanical theories. The
essential features of these two different approaches were characterized in a paper by
SMITH [1]. The fibre-reinforced composites consisting of ductile matrices strengthened
by continuous brittle fibres form a large class of the commonly used composite materials.

1
 On leave from the Department of Mathematics and Mechanics, Sofia University, Sofia 1090,

Bulgaria



26 K. P . H ERRMAN N , I, M . MIH OVSKY

Thereby n um erous investigations concerning th e plastic behaviour of  fibre- reinforced
composites  have  been  perform ed  for  example  in  t h e  papers  of  H I L L  [2],  SPEN CER  [3],
MU LH ERW  et  al.  [4],  C OOP ER  an d  P I G G O T T  [5].  C om prehensive  surveys  abo u t  t h e  state

of  the  art  are  given  in  the  Conference  P roceedings  of  th e  1975  ASM E  Win ter  An n ual
M eeting  [6]  as  well  as  in  th e  books  of  SPEN CER  [7],  K O P I O V  an d  OVC IN SKIJ  [8]  an d  P I G -

G OTT  [9].  F urth er,  a  problem  of  basic  interest  represents  th e  m icrom echan ical  aspect  in
therm al  cracking  of  unidirectionally  reinforced  com posites.  Thereby,  definite  progress  has
been  already  m ade  in  a  series  of  papers  by  H ERRM AN N   [10 - 12]  an d  H ERRM AN N   an d  asso-
ciates  [13- 15]  concerning  the  elastic  an d  viscoelastic  behaviour  of  a  cracked  un it  cell
of  a  low  fibre  con cen tration composite  un der  th e con dition s  of  different  th erm al  loadin g.
I n  a recent work  by  H ERRM AN N   an d  M IH OVSKY  [16] th e plastic  behaviour  of  an  un cracked
un it  cell  an d  the  mechanisms  of  failure  of  the  fibre- matrix  interface  have  been  analyzed
for  the  case  of  isotherm al  longitudinal  extension  of  the  com posite.  Th e  m odel  of  th e
plastic  deformation  process  proposed  in  [16]  is  especially  attractive  for  th e  study  of  th e
behaviour  of  cracks  situated  within  the  m atrix  ph ase.  I t  is  shown  in  the  presen t  paper
th at this model is applicable  to the problem  of therm al loadin g  of  th e com posite.  M oreover,
if  combined  with  the  D ugdale  m odel  solution  of  H ERRM AN N 1  [11]  for  a  crack  situated
within  the  matrix  phase  this  above  m en tion ed  model  of  t h e  plastic  deform ation  process
implies  useful  conclusions  concerning  the  fracture  behaviour  of  th e  considered  un it  cell
of  a unidirectionally  reinforced  composite.

Statem ent  of  tbc  problem

A  unidirectionally  reinforced  fibrous  com posite  with  con tin uous fibres  an d  relatively
small  fibre  volume  fraction  is  considered.  Th e fibre  m aterial  is  linear  elastic  with  You n g's
m odulus Ef,  P oisson's  ratio v

t
  an d  th e  th erm al  expan sion  coefficient a.

f
.  Th e  m aterial

of  the  m atrix  is  elastic- perfectly  plastic  with  correspon din g  elastic  con stan ts E
m
  an d v

m
,

therm al  expansion  coefficient <x
m
 an d  tensile  yield  stress a

y
.  The  therm oelastic  properties

of  the fibre  an d  m atrix  m aterials  as  well  as  the  yield  stress  of  th e  latter  are  assum ed  t o
be  tem perature in depen den t.

A  un it  cell  of  this  fibre- reinforced  com posite  in  th e  sense  of  the  well- known  m odel
of  two  coaxial  fibre- matrix  cylinders  is  studied  in  the  following  where  if  referred  t o  a  cy-
lindrical  coordinate  system (r,Q,z)  the fibre  an d  the m atrix  occupy  th e regions  (0  < r  ^
< ff,  0  <  0  < 2n,  —oo < z <  + c o )  an d (r

{
  <  ;•   *£   /• „,, 0  < 0  < In,  - CO < z <  + o o ) ,

respectively.  Thus,  equation  /•   =   /y is  th e  equation  of  th e  fibre- matrix  interface.

The  following  is  assumed  with  regard  to  the  cracked  com posite  un it  cell.  Th e  crack  is
situated  within  the m atrix  phase  an d  is  presented  in  th e  cross  section  of  th e  un it  cell  (cf.
F ig.  1)  by  a  straight  line  cut  alon g  th e  polar  axis 6 = 0.  Th e  crack  occupies  th e  segm ent
n  < r Ą r

r
  so  th at r

t
  an d r,  den ote  th e  radial  coordin ates  of  th e  left  an d  th e  right  crack

tip,  respectively.  The  crack  length  of  th e  actual  crack  is  2/   =  r,—ri.

The  well- known  elastic- plastic  m odel  of  a  crack  proposed  by  D U G D ALE  [17]  will  be
applied in the following  analysis.  The crack  of length 2L  — R

r
 — Ri  shown  in F ig.  1 presen ts

the  imaginary  crack  in  th e  sense  of  this  m odel.  Th e  segments R
t
  < r  <  r (  an d r,  < r  ^ Rr



PLASTIC  ZONE  SIZE 27

Fig.  1.  Dugdale  type  crack  configuration  in  the  cross  section  of  a  composite  unit  cell

present  the  thin  plastic  zones  at  the tips  of  the actual  crack.  The plastic  zone  lengths  at
the  left  and the  right  crack  tips  are thus si — rt—Ri  and s, = Rr—rr,  respectively.

A quasi-static thermal loading of the unit cell will be considered which implies a tem-
perature distribution over the cross section of the form (cf. Fig. 1 for notation)

0
rm

(1)

The temperatures Tf and Tm do not depend on the axial coordinate z and are constants
at each given instant of the deformation process so that the latter is viewed as a sequence
of stress-strain states of the cell corresponding to a sequence of stationary temperature
distributions of the form (1).

From the viewpoint of a quasi-static crack propagation behaviour such loading con-
ditions are of interest only which result in the appearance of tensile circumferential stresses
within the uncracked matrix phase. According to HERRMAMN [18] these conditions are
satisfied for the elastic state of the unit cell provided the inequality

(1 +vf) oif(Tf ­ To) ­ (1 +vm) am(Tm­ To) > 0, (2)

holds true where TQ is the temperature of the unstressed initial state. Under the simplifying
assumption To — 0 relation (2) is obviously satisfied if, for example, 7/ = 0 and T„ < O.
This case will be actually considered in the following calculations. The thermal loading
process will be thus viewed as a process of monotonous quasi-static decrase of the itself
negative temperature of the matrix phase. The accepted loading conditions provide ob-
viously an axisymmetric state of stress within the uncracked composite cell. The axial
symmetry together with the standard assumptions of perfect fibre-matrix contact and



28  K.  P.  HERRMANN,  I.  M.  MIHOVSKY

generalized  plane  strain  imply  the  evident  result  that  the  normal  stresses  within  the  un-
cracked unit cell are at the same time principal ones and depend on the radial coordinate
only.

The elastic state of the considered unit cell for both cases of absence and presence
of a crack in the matrix phase has been described in detail by Herrmann [11, 18]. These
elastic solutions concern a cell with a traction-free external surface r — rm and traction-
free or partially loaded crack surfaces in the sense of the applied Dugdale model. These
same conditions are supposed to apply in the here considered elastic-plastic problem
as well.

The condition of axial symmetry together with the assumed scheme of loading implies
certain obvious features of the elastic-plastic state of the uncracked unit cell in accordance
with the above mentioned elastic solution [11]. These are that the plastic zone presents
itself an infinitely long cylinder (/y < /• ^ rc, 0 ^ 8 < 2n, — co < z < + o o ; re < rm)
and spreads with developing thermal loading, i.e. with decreasing matrix temperature,
into the matrix coating. The equation of the current elastic-plastic boundary could be
then written in the form rc = rc(Tm).

Finally, it will be assumed that the crack length 2/ is small compared with the radius
rm and that the crack itself is situated relatively far-away from the fibre. This implies
the possibility of neglecting the effect of the crack on the stress-strain state within the
matrix region just surrounding the fibre where the plastic deformation process actually
develops. Then, the latter could be viewed as an axisymmetric one as in the case of an
uncracked unit cell.

With this in mind the thermal stress field within the cracked matrix phase could be
considered to be a superposition of the following two fields. The first one is the elastic-
plastic stress field for the uncracked unit cell while the second one is the field resulting
from the presence of a Dugdale type crack.

The plastic deformation process

The model of the plastic deformation process proposed in [16] will be generalized in
the present paper with regard to the considered thermal loading problem. The possibility
for such a generalization follows from the fact that this model is based in general upon
certain effects of the fibre-reinforcement which are common for both the isothermal [16]
and the here considered thermal problem. Firstly, to these effects belongs the so-called
„shrinkage effect", i.e. the appearance of compressive radial stresses over the fibre-matrix
interface. One comes up with this effect provided relation (2) is satisfied which is actually
the here considered case. Secondly, in accordance with the elastic solution [18] the fibre
acts as a stress concentrator. Because of the local nature of this stress concentration effect
one could expect that especially for the considered composites with low fibre volume
fractions intensive plastic deformation and even fracture processes may develop within
the immediate surrounding of a fibre whereas at a certain distance from the fibre-matrix
interface the matrix material may deform still elastically. Thirdly, it is well-known from
experimental observations that because of the strengthening effect of the fibre the be-



PLASTIC  ZONE  SIZE  29

haviour  of  the  composite  „in  the  fibre  direction"  is  rather elastic-like than perfectly-
plastic. This implies the reasonable assumption that the fibre, consisting itself of linearly
elastic material with a high stiffness, contributes due to the assumed perfect fibre-matrix
contact to the development of a relatively large elastic part el of the total axial strain ss
within the plastificated region and prevents thus the occurrence of a corresponding large
plastic part ef. In other words in the course of the deformation process one should per-
manently account for the current elastic part of the axial strain. It is obvious that for
the considered regime of thermal loading both the eez­ and e!|-strains should be mono-
tonously increasing in absolute value functions in dependence of the absolute value of the
matrix temperature. A reasonable restriction concerning the behaviour of the e^-strain
is associated with the assumption that the matrix material is a perfectly-plastic one and
its elastic response is thus limited. One should expect correspondingly that for the considered
unit cell and type of loading there exists a certain critical value e% of sz such that upon
reaching this value the current increments of the ej-strain become negligible with respect
to the corresponding increments of e?z. Due to the concentration effect of the fibre this
critical value e | should be first achieved over the fibre-matrix interface.

The account for the just introduced limiting characteristic %% implies the following
natural description of the plastic deformation process. The plastic deformations appear
first over the fibre-matrix interface and the plastic zone rf ^ r < rc spreads consequently
into the matrix phase. Within this zone both the e\­ and e£-strains increase simultaneously
up to the instant when el\r=rf = et. At this instant a second plastic zone rf s? r < Rc
where Rc < rc appears within which the relation e* = e

e
z holds true while the e?-strain

further increases. The second plastic zone also spreads into the matrix phase having the
first one, which occupies now the region Rc ^ r ^ rc, at its front r — Rc.

The model of the plastic deformation process just considered implies a simple possible
scheme of an approximate analysis of the elastic-plastic behaviour of the uncracked com-
posite cell.

Analysis of the uncracked unit cell

In accordance with the standard assumption of the plasticity theory the total axial
strain at each instant of the plastic deformation process is a sum of an elastic and a plastic
part. As usually it will be assumed that the matrix material is plastically incompressible
which implies the validity of the following relation within the plastic zone

(temp)

where e% is the deviatoric axial elastic strain and e(slr) and e(lemw are the relative volume
changes associated with the thermal stresses and the thermal expansion respectively, i.e.

.wo = i _ 2 ^ ( f f p + ( r e + ( T i ) , (4)

.-= 3aMrm. (5)



30  K.  P .  H ERRMAN N ,  I .  M.  MIHOVSKY

I n  equation  (4)  as  well  as  in  the  following  analysis  or,-,  /  =  r,0, z  denote  th e  n orm al
stresses  within  the plastic  zon e.

I t will  be  further  accepted th at the  stresses  and  the elastic  strain s  are  as  usually  related
by  the  H ooke's  law  so  th at  one h as  in  particular

a
z
  =  E

m
e

e

z
+v

m
(a

r
 + a

g
).  (6)

In equation  (6)  as  well  as  in the rest  of  the paper  the n otation

^ = e j ł {« ( l l t ) ,  (7)

is  used  so  th at e'  means  (cf.  equation  (3))  the  p art  of  th e  axial  elastic  strain  due  t o  the
therm al  stresses.

Let  the  m atrix  m aterial  obey  the  von  M ises'  yield  con dition ,  i.e.  let  the  stresses a
i7

i  =  r,d,z  satisfy  the relation

(<Tr- tfo)
2  +  ( < r e - ^ )

2  +  (crz- ~ov)
2  -  2a2

y
.  (8)

U pon  substituting  for  the  d ist ress  from  equation  (6)  in t o  the  latter  relation  on e  obtain s

\ ~T~/
 + r

N ow  it is  a  m atter of  simple  com putation to  show  th at equation  (9)  is  identically  satisfied
provided  the stresses  are presented  in  the  form

E,„  e'
z
 a

s

- T^^/ ^- cosO^),  (10)

where  the n otation s are  used

co t an $  =  v_  .  (12)

Equations  (10)  reflect  the  implicit  assumption  th at  cr0  > at  which  implies  in  accordan ce
with  equation  (11) t h at 0  < a>  < n.
Substituting  now  for  the  stresses a,  an d a

g
  from  equation s  (10)  in to  the  equilibrium

equation

<foy +  ar- 'a6  ^  Q^

one  obtains  the equation

E m  del  a,  ,. rfw 2 o v  sin a)
ft- -  — ^ s i n ( c o  +<b)  ~,-•   7 ^  sin(co +<b) ~5

dr  |/ 3sin</ )  H  dr
where  e* is  an  unknown  function  of  th e  radial  coordin ate  r  an d  therefore  the  in tegration
of  equation  (14)  cannot  be  performed.  But  an  approxim ate  solution  of  equation  (14)
can  be  obtained  which  is  valid  at  least  within  th e  im m ediate  surroun din g  of  the  fibre.



PLASTIC  ZONE  SIZE  31

This  solution  is  based  upon  the  assumption  that  within  this region  the  elastic parts  of  the
£,- and efl-strain components due to the thermal stresses are negligible with respect to
the corresponding plastic parts spr and s

p
0. This implies the relation

Equation (15) together with equations (4), (6) and (10) gives now

_JL_—™-cosco = — si (16)

The latter relation applies as assumed in a thin layer surrounding the fibre and over the
fibre-matrix interface r — rs in particular where the condition el = z% is first achieved.
The corresponding value to* s a)(g|) of the angle a> follows from equation (16) to be

(17)
*»(1 +vm)

According to the model of the plastic deformation process proposed in section 3 above
a further increase in thermal loading which corresponds to a further decrease in the matrix
temperature Tm results in the appearance of a second plastic zone /y < r < Rc over the
outer boundary of which equation (17) is valid. Finally, assuming that for the considered
unit cell and the given scheme of loading the quantity s', respectively w* (cf. equation (17))
is approximately constant and introducing the angle o>Ri! as

wK = co(Re), (18)
one obtains

mRc = a>* (19)
where co* is a constant now.
Now the latter assumption makes equation (14) integrable within the whole second plastic
zone r/ < r ^ Rc. The result of this integration with the boundary condition (o\raRc =
= <*>R. reads

(20)

Moreover, it could be easily verified that the set of equations (6), (10), (17), (19) and (20)
defines the stress state entirely within the second plastic zone rf < r < Rc, where Rc has
still to be determined.

The stress state defined above allows certain important conclusions concerning the
fracture behaviour of the considered unit cell. To this end we consider the shrinkage
effect again. It is clear from most general positions that this effect is due to the difference
in the lateral contraction of the fibre and matrix materials. Because of the plastic incom-
pressibility of the matrix material this difference should be expected to increase in the
course of the deformation process. In other words developing plastic deformations should
further contribute to the shrinkage effect or, equivalently, the radial stress orr|P=r/ acting
over the fibre-matrix interface should decrease with increasing loading, i.e. with decreas-
ing temperature of the matrix phase. The latter means in accordance with equations (10)
that the angle cor/ should increase in the course of the deformation process remaining



32  K.  P.  HERRMANN,  I.  M.  MIHOVSKY

obviously  larger  than  the angle co*.  Moreover,  equations  (10) show  that  there  exists  a  na-
tural limitation of the shrinkage effect in the sense that this effect achieves its maximum
at a value of a>rf = n—<f>.
The value R* of the radius Rc at this instant, that is

R* - Re\arf = B - 0 , (21)

follows from equation (20) to be

f - exp \J*­ (n­t -« J • (22)

The model of the process applied here leads thus to the conclusion that further decrease
in 0r\r=rf as well as increase in RC is impossible. Further, it would be of interest to examine
the velocity field corresponding to this limiting state of the plastic deformation process
within a unit cell.

To this regard the known concept of the associated flow rule will be applied with
the yield function (9) serving as a plastic potential. Simple computations show that in
accordance with this concept and the plastic incompressibility condition the plastic strain
rates £*, i — r,6, z satisfy the relation

(23)

within the second plastic zone where
a..

(24)
Of

7= sin

It is easily observed from the latter equations that £r\rmr,~* + ° ° when co^-v n­§.
This result means physically that at this state free plastic flow tends to take place within
a thin layer immediately surrounding the fibre. The behaviour of the composite at this
state will obviously depend upon the interaction between this tendency and the strengthen-
ing effect of the fibre which tends itself to prevent the occurrence of such a singular ve-
locity field. The very nature of these two competing effects implies the reasonable assump-
tion that their interaction results in the occurrence of shearing stresses over the fibre-
matrix interface. Moreover, these shearing stresses should be equal for obvious reasons
to the shear yield stress xy = ay/]/3 of the matrix material.

Let TS be the shear strength of the fibre-matrix interface. If rs < ry then the very reach-
ing of the considered critical state will obviously result in the immediate failure of the
fibre-matrix interface by the so-called debonding effect. If, on the contrary, TS > ry,
then the known mechanism of fibres pull-out (see, for example [9]) will develop, most
probably together with a process of fibre breaking.

Plastic zone size and associated problems

In order to close the solution of the problem for the uncracked unit cell one should
complete the results of the previous section with the temperature dependence of the radius
of the plastic zone. Moreover, when dealing with a given composite material one should
specify the actual value of e* which should be used in the computations.



PLASTIC  ZONE  srzE  33

The model of the plastic deformation  process proposed above implies a simple approach
to  the  latter  problem.  Starting  point  for  this  approach  is the  additional  assumption  that
the  first  plastic  zone  presents  itself  a  thin  layer  and  thus  one  may  consider  the  relation
Rc — rc  to  hold  approximately  true.  This  assumption  appears  as  acceptable  one  for  the
following  reasons.  Firstly,  because  of  the  local  nature  of  the  fibre  concentration  effect
and  since  a  low  fibre  volume  fraction  composite  is  considered.  Therefore,  both  the Rc­
and /^-radii should be small compared with the value of rm. Secondly, because of the low
resistance of the matrix material with respect to the occurrence of intense plastic defor-
mation such as the deformations within the second plastic zone are. Thus, one may expect
that the transition zone between the elastically deformed matrix region and the second
plastic zone is really a thin one. If so, then the first plastic zone could be simply considered
to play the role of an elastic-plastic boundary, the latter having the form of a thin layer.
Further, because of the thin layer shape of the elastic-plastic boundary a softened version
of fulfillment of the standard elastic-plastic transition conditions of continuity of stresses
and displacements could be applied, namely the following.

Firstly, because of the layer thinness one should not expect a substantial change of
the radial stress within the layer itself which implies the relation

ffr\r=Re = a
e
r\r=Rc, (25)

where o*,, i = r,d,z are the stresses acting within the elastic region Rc ^ r < rm of the
matrix phase. Secondly, these stresses should satisfy the yield condition, equation (8),
over the elastic-plastic boundary, that is

[ W - f l ^ + tó-^ + W-oO*]!,.,,. = 2 < (26)

In accordance with the general form of the elastic solution of the problem [11, 18] and
the results of the previous section one may present the latter equations in the form

( ! 1

\ rm Kc l - 2 r m |/3sm^>

(28)

where the constant C has to be determined actually.
The remaining elastic-plastic transition conditions could be now viewed as satisfied

as well in this way that the corresponding stresses and displacements change continuously
within the layer between their values on its „elastic" and „plastic" surface. Thus, the
equations (27) and (28), respectively, present the just mentioned softened version of the
elastic-plastic transition conditions.

If solved for the unknowns Rc and Cthe set of equations (27) and (28) implies as a matter
of fact the temperature dependence of both Rc and C, respectively, in the form

Rc = Rc(Tm, ez; U, E m , vm, am, rm, ay), (29)

C == C(Tm, ex; %l, E m , vm, am, rm, ay), (30)

where ez itself is a still unknown function of the matrix temperature Tm.

3 Mech. Teoret. i Stos. 1—2/84



34  K.  P.  HERRMANN,  I.  M.  MIHOVSKY

It  is important  to  mention at this point  that  with Rc and  C once  determined  from  the
set  of equations  (27)  and  (28)  one may  consider  the  axial  stresses az and a%  acting  within
the  plastic  and  elastic  regions  of the  matrix  phase,  respectively,  to be known  functions
of  the  same  parameters  as those  in the  presentations  (29)  and (30).  The  corresponding
expressions for these  stresses  can  be  easily  given  by

1  *
a,  = - — - — (Em£j+2<jj,cosco), rf < r < Rc, (31)

l-2i>„,

, 2vmEmC
o z ­ p . ! « t + 7 i i .. \~2 > ­tic ^ r ^ rm, (32)

where both Re and C can be considered now as known functions of the form given by the
equations (29) and (30), respectively.

Further, it is a matter of a simple verification that upon satisfying the continuity con-
dition for the radial stresses over the fibre-matrix interface, i.e. the equation

o?|r„r/ = tfUv (33)
one may construct the expressions for the stresses a{, i = r,Q,z acting within t h e fibre.
Thereby the expression for the axial stress a{ r e a d s

= Efsz+2vf \ ­ ^ ~ + ,­°
y. x cosK,+tf)|, 0 < r < rf, (34)

\l~£v ] / 3 s 0 J
where the value of cor/ follows from equation (20) with r — rs and with the quantity Rc
given in the form of equation (29).

It is easily observed that the axial stresses as presented by the equations (31), (32)
and (34) can be considered now as known functions of Tm and ez and the remaining
parameters of the problem, i.e. sz, ay and Et,Vi,oci, rt where / = / , m. These stresses
have to satisfy the equilibrium condition for the forces, acting in the axial direction, which,
in our self-stress problem is given by the following condition of self—equilibrium

. (35)
rs

By substituting for Rc from equation (29) into equation (35) leads to a relation of the
form

ez = eg(Tm; *s
c
z, E j , v,, a,, rt, ay), (36)

where i = / , m. Equation (36) represents in fact the equation of the theoretical ez versus
Tm curve in the framework of the proposed model for the considered two-phase material.

Upon substituting for sz from equation (36) into equation (29) one obtains the de-
sired dependence of the plastic zone radius Rc on the matrix temperature Tm. This depen-
dence is obviously of the form

Rc = Rc(Tm; e§, E,, vt, a(, r,, a,,), (37)

where again / = / , m.
Note that by applying equation (37) to the critical state of the unit cell, i.e. if Rc —

= R* (cf. equation (22)), one obtains the critical temperature T% at which one of the



PLASTIC  ZONE  SIZE  35

failure  modes  of  the fibre-matrix interface described in section 4 occurs. This critical
temperature appears to be of the form

T* = T*(R*, P., E,, vu «i, ru ov)\ i =f,m. (38)

Equation (38) implies itself a simple criterion of failure of the fibre-matrix interface of
the form

Tm - T* (39)

Now it is easily observed that the actual value of e| can be determined by means of
a comparison of the theoretically predicted ez(rm)-curve, equation (36), with a correspond-
ing curve obtained experimentally for the considered composite material.

A simpler approach to the problem consists in the determination of %\ from equation
(38) provided the T%­ and i??-values in this equation are the values which have been ob-
served experimentally.

The scheme described above does not imply a closed form solution for the quantities
Rc and s% but the associated numerical treatment of the problem is not very complicated.
With this in mind one may consider that the whole problem concerning the elastic-plastic
behaviour of the uncracked unit cell has been solved completely.

The cracked unit cell

It should be remembered in the following that as accepted in section 2 the thermal stress
field within the cracked matrix phase can be considered as a superposition of an elastic-
plastic stress field for the uncracked unit cell and a corrective stress field caused by the
presence of a crack situated along the segment Rt ^ r < Rt, 0 = 0 of the symmetry
line of the cross section of the unit cell. Now the first stress field is known from the preced-
ing analysis, sections 4 and 5. The second stress field will be examined as already mentioned
in the framework of the Dugdale crack model [17].

The thermal stress field in the uncracked elastically deformed matrix region is given by
the following expressions

Rc < '• < ra,

o* =

where C = C(r m , ...) and Rc — i ? c ( r m , ...) can be considered as known functions of the
temperature Tm in accordance with the results of the previous section.

The corrective stress firld oft, i,j"— r,d,z can be obtained from the solution of the

following mixed boundary-value problem for a Dugdale type crack with an actual length

2/ = rr — ri and a fictitious length 2L ~ Rr­Rt (cf. Fig. 1 for notation)

­ae\a R <r<r and r < r < R ( 4 0 )

3 *



36  K.  P.  HERRMANN,  I.  M.  MIHOVSKY

<frf)(r, 0)  =  0 , 6 = 0,  V r ,  (41)
ue(r, 0)  =  0,  0 =  0, Rr  <  /• <  i?,.  (42)

The boundary-value problem (40) - (42) has been analyzed in detail by HERRMANN [11].
Applying the results obtained in [11] together with those presented above one immediately
obtains the desired plastic zone lengths si and sr (cf. Fig. 1) for our case from the solutions
of the equation

_ L = 0, (43)
l+s K )

where s is the unknown and r0 — (rr+rt)/2. The quantities st and sr correspond to the
upper and lower signs, respectively, in the brackets in equation (43).

Upon solving equation (43) one obtains the plastic zone lengths as functions of the
temperature Tm and the actual crack length 21, i.e. the relations

St = Si(Tm, 1), i = r, I. (44)

It is well understood that each of the quantities St,t = r,I depends in addition on the
remaining parameters of the problem as well so that the representations (44) are actually
schematic ones. They imply in an obvious way the relations

R t - Rt(Tm, I), i=r,l. (45)

The equations (45) present the dependence of the positions of the left and the right tip
of the imaginary crack on the current matrix temperature. That dependence could be now
considered as known from the solution of equation (43). The latter solution could be itself
obtained by means of a numerical treatment [11].

Crack and cell behaviour

As already accepted (cf. equation (38)) let T„ be the value of the matrix temperature
at which one of the two modes of failure of the fibre-matrix interface (cf. section 4) occurs
and let Rf be the value of Rt corresponding to this value of the temperature (cf. equation
(45)). Then Rf defines in fact the position of the left plastic zone tip at the instant of failure
of the fibre-matrix interface. Further, let Ri be the value of i?, at which the crack begins
to grow in accordance with a certain crack growth criterion and let fm be the correspond-
ing matrix temperature, i.e.

^ = A("t0. (46)
Then equation (46) corresponds to the equation (45) but now applied to the instant of
crack growth initiation. It is assumed implicitly that due to the fibre concentration effect
the crack will start to propagate from its left tip toward the fibre.

Itjbllows from the whole scheme of analysis that the values of both quantities Rf
and Ri, respectively, appear as specific ones for each given cracked unit cell or, equiva-
lently, for a given composite material. Thereby both values could be defined from the
present analysis provided the corresponding crack growth criterion is given. The latter
concerns obviously the determination of i?,. Once evaluated for a certain unit cell with



PLASTIC  ZONE  SIZE  37

a  given  crack  configuration  these Rf­  and ^-values imply immediately the following
evident conclusions concerning the behaviour of the cracked unit cell. If Rf > Rt then
the fibre-matrix interface fails while the crack is still in rest. If on the contrary Rf < i?j
then the crack growth initiation precedes the failure of the fibre-matrix interface. In this
case the whole scheme of analysis remains further valid (up to the possible failure of the
interface) provided the crack propagates quasi-statically and 2/ is its current length upon
which both Rf and i?, depend. Finally, if Rf = R, then the fibre-matrix interface fails
simultaneously with the initiation of the crack propagation. The further behaviour of
the crack and the unit cell in this case as well as in the first one where Rf > Rt needs
a new approach since the present considerations are based upon the assumption of a per-
fect fibre-matrix contact.

It should be mentioned that these simple conclusions are valid under the assumptions
made earlier that both the crack length 2/ and the plastic zone radius Rc are small compared
with the geometrical quantities rm and rQ, respectively. That means the crack should not
influence the stress state within the plastic zone #y < r < Rc where the solution of section 4
is thus expected to apply. Moreover, it has been shown in [19] that the approximate ana-
ytical elastic solution of the considered thermal crack problem in a composite unit cell
lobtained in [11] remains still valid even if the restrictions concerning the quantities 21, r0, rm
and rs (the latter quantity plays the role of Rc in the elastic case) are somehow violated.
One may consequently expect that the results of the present elastic-plastic analysis will
also remain valid when the restrictions mentioned above are somehow softened since the
values of Ri, i = r,l used in our considerations are actually obtained from the same
elastic solution [11]. If so, then one comes easily up with a couple of further implications
of the analysis concerning the crack and cell behaviour.

Let fm be the temperature at which both plastic zones, i.e. the annulus rt < r ^ Rc and
the segment Rt < r < rt join each other, and let Rc = Rc(fm) and Rt = Rt(fm, I) be the
corresponding values of the quantities Rc and i J ; , respectively. Further, let us assume the
validity of the relations fm > T* and Tm > fm (note that the temperatures T*, fm and
Tm are negative) for the considered unit cell. That means that the two plastic zones meet
each other before the conditions for the failure of the fibre-matrix interface and for crack
growth initiation are fulfilled. Upon reaching this instant, that is with the two plastic
zones adjoined, the behaviour of the cracked unit cell will depend essentially on the in-
teraction between the plastic mechanism of failure of the entirely plastificated segment
rf <: r < rx, 0 = 0 and the brittle mechanism of crack growth at the right crack tip r =
= rr. A possible approach to this problem could be based upon the application of the
rigid-plastic body model and the limit load concept associated with this model (see, for
example [20]). When applying the latter concept to the plastificated segment rs < r ^ ru
d = 0 one may expect that further thermal loading, i.e. further decrease of Tm, will re-
sult in the activation of the crack propagation mechanism at the right crack tip.

Another case of interest is the one for which fm «S Tn and fm > T%. In this case both
plastic zones join each other again but the segment Rt ^ r < n, 0 = 0 presents now
the plastic zone at the left tip of the running crack. Depending upon the plastic zone
thickness d = Rc~r/ and the crack velocity the crack may stop before reaching the elastic-
plastic boundary or at the boundary or may traverse partially or entirely the plastificated



38  K.  P.  HERRMANN,  I.  M  MIHOVSKY

aimulus  around  the  fibre.  Depending  in  addition  on  the fibre-matrix contact the crack
may stop at the fibre-matrix interface in order to traverse it or to create an interface crack.
The investigation of all these possibilities, being a problem of definite interest, is associated
with considerable difficulties arising from the necessity of solving boundary-value problems
for cracks partially situated within plastificated regions as well as of applying reasonable
criteria of crack propagation and arrest.

Concluding remarks

The analysis presented above implies certain definite conclusions concerning the be-
haviour of a cracked unit cell of a fibre-reinforced composite material under the conditions
of thermal loading. The analysis could be easily transformed to the more general case
Tf # 0 if the temperature difference T = Tm — Tf will actually play the role of the quantity
Tm used in the preceding calculations. In that case the linear coefficient of thermal expansion
Xf will influence the processes of plastification and fracture as well.

The model of the plastic deformation process proposed leads to both closed form
results and to a relatively simple procedure concerning the numerical treatment of the
problem on the whole. The analysis shows that the entire solution of the considered problem
is associated with the necessity of directed experimental investigations concerning the
determination of the specific measure of elastic response el for the fibrous composite
materials.

Acknowledgement

The support of the Alexander von Humboldt Foundations for one of the authors (I.M.M.)
is gratefully acknowledged.

References

1. C. W. SMITH, Limitations of Fracture Mechanics as Applied to Composites, In C. T. Herakovich (Ed.).
Inelastic Behavior of Composite Materials, AMD-vol. 13, ASME, New York, pp. 157-175, 1975.

2. R. HILL, Theory of Mechanical Properties of Fibre­Strengthened Materials, II. Inelastic Behaviour,
J. Mech. Phys. Solids, vol. 12, pp. 213-218, 1964.

3. A. J. M. SPENCER, A Theory of The Failure of Ductile Materials Reinforced by Elastic Fibres, Int. J. Mech.
Sciences, vol. 7, pp. 197-209, 1965.

4. J. F. MULHERN, T. G. ROGERS and A. J. M. SPENCER, A Continuum Model for Fibre­Reinforced Plastic
Materials, Proc. Royal Society, A 301, pp. 473-492, 1967.

5. G. A. COOPER and M. R. PIGGOTT, Cracking and Fracture of Composites, In D. M. R. Taplin (Ed.),
Fracture 1977, University of Waterloo Press, vol. 1, pp. 557 - 605, 1977.

6. C. T. HERAKOVICH (Ed.), Inelastic Behavior of Composite Materials, AMD-vol, 13, ASME, New York,
pp. 157-175, 1975.

7. A. J. M. SPENCER, Deformations of Fibre­Reinforced Materials, Clarendon Press, Oxford, 1972.
8. I. M. Kopiov and A. S. OVCINSKIJ, Fracture of Fibre­Reinforced Metals, Nauka, Moscow, 1977 (in

Russian).



PLASTIC  ZONE  SIZE  39

9.  M. R.  PIG G OTT, L oad Bearing Fibre Composites,  Pergamon  Press,  Oxford,  1980.
10.  K.  H ERRM AN N ,  Self- Stress Fracture  in  a  T hennoelastic  T wo- Phase  Medium, Mech. Research  Commu-

nications,  vol.  2,  pp.  85 -  90,  1975.
U . K .  H ERRM AN N , Interaction of  Cracks and Self- Stresses in a Composite Structure, In J. W. Provan  (Ed.),

SM   Study  N o .  12  „ C ontinuum  Models  of  D iscrete  Systems",  U niversity  of  Waterloo  Press,  pp. 313 -
338,  1978.

32.  K.  H ERRM AN N ,  Quasistatlc  T hermal  Crack Growth in  the  Viscoelastic  Matrix  Material of  a Brittle
Fiber Reinforced Unit Cell,  M ech. Research  Communications, vol.  8, pp. 97 - 104,  1981.

13.  K.  H ERRMAN N   and  A.  F LE C K,  T hermal  Fracture  in  Compound Materials,  In  D . M. R.  Taplin  (Ed.),
F racture  1977,  U niversity  of  Waterloo  Press,  vol.  3, pp.  1047 - 1054,  1977.

14.  K.  H ERRM AN N ,  H .  BKAU N   and  P .  KEMEN Y,  Comparison  of  Experimental  and N umerical Investigations
Concerning  T hermal  Cracking  of  Dissimilar Materials, I ntern. J.  F racture, vol.  15, R  187- 190,  1979.

15.  H .  BRAU N   and  K.  H ERRM AN N , Analysis of  T hermal Cracking of  Unidirectionally Reinforced Composite
Structures in  the  Micromechanical Range,  In  D . F rancois  (Ed.), Advances  in  F racture  Research,  Per-
gamon  Press,  vol.  1,  pp.  485- 493,  1981.

16.  K.  H ERRMAN N   and  I . M .  MIH OVSKY,  Plastic Behaviour  of  Fibre- Reinforced  Composites  and Fracture
Effects, Proceedings  of  the  F ourth N ational  Congress  of  Theoretical  and  Applied  Mechanics,  Varna/
Bulgaria,  vol.  1  pp.  431- 436,  1981.

17.  D . S.  D U G D ALE,  Yielding  of  Steel  Sheets Containing Slits, J.  Mech. Phys.  Solids,  vol.  8, pp.  100- 104,
1960.

18.  K.  H ERRM AN N ,  Uber Eigenspannungen  im  diskontinuierlich  inhomogenen  Festkorper,  in  K.  Schroder
(Ed.),  Beitrage  zur  Spannungs-   und  D ehnungsanalyse,  Akademie  Verlag, Berlin,  vol.  6,  pp.  21  -  52,
1970.

19.  H .  BRAU N ,  A.  F LE C K  and  K.  H ERRMAN N ,  Finite Element Analysis of  a  Quasistatic  Crack  Extension
in a  Unit Cell of  a  Fiber- Reinforced Material,  I n tern . J. F racture, vol.  14, R  3 -  6,1978.

20.  L.  M.  KACH AN OV,  Foundations  of  the  T heory  of  Plasticity,  N orth  H olland,  Amsterdam,  1971.

P  e 3 10  M  e

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S t r e s z c z e n i e

ZASIĘ G   STREF Y  U PLASTYCZN IEN IA  SZCZELIN   TYPU   D U G D ALE'A  WE  WSTĘ PN IE
N AP RĘ Ż ON YM   OŚ R OD KU   D WU F AZOWYM   Z  CZĘ Ś CIOWO  U PLASTYCZN ION YM

M ATERIAŁEM   MATRYCY

Proponujemy  model  odkształ cenia  plastycznego  dotyczą cy  pojedynczej  komórki  kompozytu  wzmoc-
nionego  wł óknami.  Zbadany  został   proces  odkształ cenia  komórki  przy  uwzglę dnieniu  moż liwego  mecha-
nizmu  pę kania  w  miejscu  poł ą czenia  matrycy  i  wł ókna.

Przy  uwzglę dnieniu  modelu  D ugdale'a  wycią gnię to  wnioski  dotyczą ce  termiczengo  wzrostu  szczelin
promieniowych  w  matrycy.

Praca  został a  zł oż ona  w  Redakcji  dnia  8  czerwca  1983  roku