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Plastic buckling of metal matrix laminated plates
am_L7683 91 s3mt c 1991 pergpmon Ru
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PLASTIC BUCKLING OF METAL MATRIX LAMINATED PLATES M. PALEY and J. ABOUDI Department of Solid Mechanics. hluterials and Structures. Faculty of Engineering. Tel-Aviv University. Ramat-Aviv, Tel-Aviv 69978, Israel (Rewired
17Augustt990 ; in recisedform 2 December 1990)
Abstract-A method is proposed for the dete~ination of plastic bifurcation buckling ioad OF metal matrix composite plates. The metallic matrix behavior is described by an elastic-viscoplastic constitutive law, while the tibers are assumed to be either e&tic or elastic-viscoplastic material. The approach is based on the load level and history dependent instantaneous effective properties of the inelastic plate. which arc established by a micromechanical analysis. An incremental procedure is developed in which a buckling condition has to beestablished and its fultilment must bechecked at euch increment. The method is applied for the prediction of the plastic buckling of boron/aluminum composite plates in various situ;ttions. by employing the classical and higher order shear deformation plate theories.
I
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The dctcrmination of the buckling load of perfectly elastic composite plates has received considerable attention [see Lcissa (1985, 1987) for extcnsivc reviews]. Most of the papers covered in these rcvicws uti1iz.cthe classical theory of plates. The use of a first order theory in the study of bifurcation buckling of elastic plates can bc found in Whitney and Leissa (1969) and Whitney (1987). Analyses of composite plate buckling by employing higher order thcorics can be found in the rcccnt review by Kapania and Raciti (1989). The stability analysis of homogeneous inelastic plates is much more complicated than the pcrfcctly rlastic case, dus to the inherent nonlincnrity of the material constitutive law. The study of bifurcation buckling can be performed by employing total deformation plasticity (c.g.. Bijhmrd. 1949) or incremental theory (e.g.. Hcndelman and Prager, 1948). For extensive reviews of this subject, see Hutchinson (1974) and Bushnell (1982). Recently, the influence of material rate sensitivity on the elastic-plastic buckling of structures has been investigated by Tvcrgaard (198s) Bodnerrral. (1991), and Paley and Aboudi (1991 b). The theory of uniqueness and bifurcation in time-inde~ndent elastic-plastic materials was given by Hill (1958). It was shown by Tvergaard (1989) that Hill’s bifurcation condition can be satisfied for elastic-viscoplastic materials only at the elastic buckling load. This is due to the high strain-rates that occur at the instant of buckling. In practice, imperfections and inertial effects reduce the instantaneous changes of buckling strain rates (Bodner et al., 1991). The use of strain-rates in the pm-buckled state in an appropriate buckling condition of a viscoplastic material provides lower bounds to the actual buckling loads. The prediction of plastic buckling of metal matrix composite structures is di~cult due to the complicated interactions between the inelastic matrix and reinforcement. In previous investigations, the inelastic effects of the matrix were neglected by assuming perfectly elastic behavior for all loading Ievcls. Obviously, this is a restriction since a metal matrix composite deforms plastically at the early stages of loading. For example, in a boron/aluminum composite the aluminum matrix yields at a strain of about 0.002, which is far away from the ultimate strain of the composite. Thus, an efficient utilization of the metal matrix composite up to failure must take into account the plastic flow of the matrix. The plastic buckling analysis of a metal matrix composite structure requires the knowledge of its instantaneous stiffnesses which depend on the history and loading level. The overall instantaneous properties of the inelastic composite can be determined by a suitable micromechanical analysis from the known material parameters of fiber and matrix. In the II39
IIU)
M.
framework
PALEY
of the classical incremental
and J.
&O~Dl
plasticity
theory.
bounds on the overall instan-
taneous elastoplastic moduli of periodic composites were determined by Accorci and NematNasser (1956) and Teply and Dvorak
( 1988).
Explicit expressions of the effective instantaneous properties of metal matrix composites were recently given by Paley and Aboudi (199la). The derivation of the instantaneous properties was based on the micromechanical method of cells (Aboudi, which the metal matrix is assumed to behave as an elastic-viscoplastic by elastic or elastic-viscoplastic
1989) in
material reinforced
fibers. It was shown by Paley and Aboudi (199la)
that at
any stage of loading. the current stiffnesses of the inelastic fiber and matrix material can be employed to generate. in conjunction with the micromechanical analysis, the overall stiffness tensor of the unidirectional
composite. The instantaneous
stiffnesses of metal matrix com-
posite laminates were obtained by using the standard lamination In the present paper, the previous
micromechanical
theory.
method for the derivation
instantaneous properties of metal matrix composites is utilized for the bifurcation analysis of composite plates, where the matrix particular.
the Bodner and Partom
ol
buckling
material could be elastic-viscoplastic.
In
(1975) model is employed in the applications given in
this paper. The method is based on the development of governing buckling equations of the composite inelastic plate. which are expressed in terms of the velocity held, and involve the instantaneous condition.
properties of the laminated plate. Consequently,
the derived buckling
which is obtained from the governing equations, depends on the loading level
and its history.
Due to the existing plastic behavior of an inelastic phase. the method ol
obtaining the critical buckling load Icvcl is incrcmcntal. At each load incrcmcnt the fullilment of the derived buckling condition
is examined. If the condition
is satisfied. then the
buckling load lcvcl of the composite plate has been achieved; othcrwisc procedure is continued. The method is illustratctl
for various types of boron/aluminum
the classical and higher order shear deformation offiber
with the corresponding
composite plates. Roll1
plate theories are employed. The clTccts
volume ratio, plate thickness, material rate sensitivity,
and total numheroflayers
the incremcntai
temperature, ply oricnlalion
on the plastic buckling toad of*the plate arc stutiicd. Comparisons buckling loads obtained by disregarding the plasticity cffccts
ofthe
metal matrix arc prcscntcd.
2. MATERIAL
STII:f:NESS
KEDUCTION
DUE
TO PLASTIC
Consider a homogeneous isotropic elastic-viscoplastic
FLOW
material subjected to a slowly
increasing external loading. At the first stage of loading. the material behavior is essentially elastic and is characterized by its elastic stiffnesscs. develops, leading to anisotropic
At a later stage, plastic deformation
behavior of the material, which is described by its instan-
taneous reduced stiffness properties. This
stiffness reduction, which is caused by the plastic
how of the material, can be determined at any instant
of the loading by the following
considerations. The total strain-rate tensor of the material is assumed to be a sum of two components : reversible (elastic) fjc and irrcversibtc (plastic) q,:, i.e.,
The stress-rrffc tensor s,~ is rclatcd to the elastic part of the total strain-rcrfc by the generalized Hooke’s law (small deformations are assumed) :
where C,,,, is the fourth order elastic stiffness tensor of the material, and the summation convention is employed on repeated italic letters. The plastic components of the strain-rrrlc are controlled by the Prandtl-Reuss Row law as follows :
Plastic buckling of metal matrix laminltted plattts
I Ill (3)
IIP, = b,
where s,, = cr,,- &,,att:3 is the deviatoric part of the stress tensor G,,, ci,, is the Kronecker delta. and A is the How rule function of the adopted viscoplastic constitutive
model which
describes the inelastic behavior of the material. The following constitutiv~
relation for the initially isotropic homogeneous viscoptastic
material can be established (Prtley and Aboudi.
where the instantaneous strcukwte
199lb) :
dependent viscoplastic moduli Cl”;& of the material are
defined by
In (5). the scalar 9 is defined by the following equation :
where 1’ is the elastic rigidity
of the material.
Equations
(4)-(G) were est~~bl~sl~cdafter
various manipulations by employing the flow rule (3) in conjunction with cyns (I) and (2). It should be noted that thssc relations were established without the use of the yictd function concept. In this paper. the superscript ( ) “” is used to dcnotc the various ourrcnt propcrtics which
charactcrizc the viscoplastic material (and structure)
that vary with loading history.
It can hc readily seen that the reduction of matcrinl still’ncss, causctl by plastic Ilow, is a function of the history and the current state of louding. The iftstitr?t~l:l~~~tis ~~)llstitlltiv~ invcstigatc the bifurcation was
pcrformcd by utilizing,
relation (4) was prcviousty cmptoycd by the authors to
buckling of visuoplastic plates (PaIcy and Ahoudi.
199 I h). This
:I( itny stage of loading, the current value of Cf,& of the material
at the buckling condition of the plate. which was dcrivcd from the governing equations in conjunction with the spccificd boundary conditions. It cat1 be observed that the rate-dependent C,$‘, are similar
to the f’orrn of the ratc-
inclcpcndcnt instantaneous moduli used by Hill in the derivation of his bifurcation condition fsec, for example, eyn (3.4) of Tvergsard (19g9)]. One can readily see from eqn (6) that y = 0 when the material behavior is elastic, whereas y is positive when plastic flow takes place. The tatter fotlows from the fact that y is given in terms of the ratio of the plastic work rate (s,,rljj) to the rate of the total work of deformation
(.s~~~/~J which dots not include
volume changes. In the present case of a viscoplustic material the strain-rate at the prc-buckled state is used in computing C,y[,. At the instant of buckling, however, high strain-rates develop in the mitt&al. Consequently, the predicted values of critical loads are lower bounds of the actual values. Thus the obtained bifurcation results are approximate, representing predictions for an analogous time-independent material. Similar
arguments were given by
Bodncr cl ul. ( t 99 I ). By establishing a constitutivc
relation similar to eqn (4). for the unidirectional
lamina
of a composite plate. it will be shown in the sequel that an analogous procedure can bc ndoptrd for the anatysis of the bifurcation
Let us consider a unidir~tional perfectly elastic fibers reinforcing
buckling of metal matrix laminated plates.
mcta! matrix composite material which consists of
an elastic-viscoplastic
matrix. The fibers are oriented in
the s’, direction of a Cartesian coordinate system (Y,.r’&,). For transversely isotropic fibers (.r>-s’, is the plane of isotropy), the material stiffness tensor of the fibers is given by
where
j*(f),
p(
1’1, 11 I.), 8 1’1 2nd
;‘(
I.)
are five independent constants that characterize the fiber material. and can be easily related to its corresponding engineering constants. The above tensor czln be rewritten in a more familiar form using a contracted notation.
The c&tic
stitfncss of the isotropic
matrix
phase is given by
whcrc
with
For perfectly elastic fibers the plastic strain rate qf;“” z 0, implying that y’” = 0 and CVPfI‘) = Cl;;;. ft is readily seen that the present derivation also includes the more general tjkl case of elastic-viscoplastic fibers. In such a case eqns @)-(I I) can be used subject to the requirement that the inelastic fibers are initially isotropic. The overall instantaneous properties of the unidirectional fiber-reinforced material c:m bc determined by a suitable micromechanical analysis in which the detailed interaction bctwcen fiber and matrix phases is considcrcd. Such a micromechanical analysis was dcvclopcd by Paley and Aboudi ( 19r)f a] by employing the method ofcells Aboudi ( f 989). This composite modrt is based on the analysis of a repeating ccl!. and leads to the prediction of the avcrall behavior of various types of composites from the known propcrtics of the fiber and matrix matcriats. It was shown by Paley and Aboudi (1991a) that the knowledge of the: instantaneous properties of the inelastic fiber and matrix phases provides. in conjunction with micromechanical analysis of the method uf ccfls. the overall instantaneous stiffness of the un~direction~ll metal matrix composite. Consequcntfy, from the knowledge of Cf$?’ (4 = j;m) at any stapc of loading, one can dcterminc the etTective instant~kneous behavior of the composite in the form
of
Plastic buckling of metal matrix laminated plates
1113
where f,, and rj,, are the average stress-rate and strain-rate tensors, and Ci$’ is the fourth order tensor of the effective instantaneous reduced stiffnesses of the composite. Thus, the composite constitutive equation (12) is a generalization of relation (4). which was established for a homogeneous viscoplastic material. The explicit expression of C$’ can be found in Paiey and Aboudi (1991a). it should be emphasized that the tensor C$;’ depends on the level and history of loading. It is worth mentioning that a constitutive equation of a similar form was also established by Paley and Aboudi (199la) for composite laminates in which each lamina is a unidirectional metal matrix composite.
3. PLASTIC
BUCKLING
OF LAMINATED
PLATES
In this section. we will utilize the previously discussed instantaneous constitutive relation (I 2) for inelastic unidirectional composites to establish the governing equations for plastic buckling of metal matrix laminates. To this end. consider a rectangular laminated plate which consists of several laminae. The middle plane of the laminate coincides with the .r--)’ plane of the Cartesian coordinate system I&V:). The x and _r axes are parallel to the plate edges. and the origin of the system is taken at a corner. The dimensions of the plate are: N and h in the .r and _rdirections. respectively. and thickness h in the z direction. When it is convenient, we will use the following alternative notation for the axes: .ri = X, .sL = _Vand s1 = z Each layer of the laminate is a unidirectional metal matrix composite which is assumed to be in the plane-stress state. Consequently, the constitutive equation (12) of’thc single laminit in its matcrinl coordinate system (.Y’,x’&) can be rcwrittcn as r; = Q;:“tsi
(i,j=
1.2,4,5,6)
(13)
whcrc i, = F,,,..,. iI = ?clS1. i+ = TX,:,.,.f5 = ix.,..,, i, = 5;.,Xi, and $, = rj,,,r,,, @;= tj,,,.,. )jl = 2rj,.,r.,‘& = 28j,.I..). fjh = ?rj,., Xi’The nonzero elements of Q$” are obtained from C:“’ of the c&rcsponding Iaycr [whcrc CT’-” is the instantaneous cffcctivc material stilTncss tensor C’,$“. eqn ( I?). which is rewritten in the contracted notation] as follows
and Q:““’ = C,:“” (i, j = 4,~).
(14)
The plane-stress instantaneous stiffness Qi’; of the lamina, expressed in the global coordinate system (XJZ), is obtained by the appropriate transformation of the above defined +W.In these plate coordinates, eqn (13) becomes Qt, Sj = 0: I.“$
(i, j = 1,2,4,5,6)
where i, and rj, arc the average stress-rates and strain-rates in the lamina expressed in the plate coordinate system. As usual, the average strain-rates are taken to be equal to the strain-rates in the plate. Assuming smati changes of displacement field during each load increment, the total strain-rates in the plate in the global coordinates become &, =
~(a~j/a.~j+a~j/a.~,)
(16)
where ri, are components of the assumed velocity field associated with adopted laminated plate theory. Both higher order and classicai plate theories will be employed to establish the governing equations of the plastic bifurcation buckling of the considered laminated plate.
itu 3. I. Higher order plutr theor>
The adopted higher order shear deformation
laminate theory (HSDT)
is based on ;i
third order expansion of the displacement field (Reddy and Phan. 19Y5). In the present case of inelastic behavior. the following
third
order expansion of the velocity components is
assumed :
where Ii, k and ii, are the s. _Vand : components of velocity of middle plane points : 4, and (c;Care rotation rates of transverse normals about the J* and s axes. respectively. Substitution total strain-rates
of the above detined velocities into (16) leads to the following
field of
in the laminated plate :
whcrc the following
definitions
arc cmploycd :
The in-plane stress resultants
itrc dcfincd by
s h :
N, =
6, dz
(i = 1,2,6)
(21)
hs ?
where ci, arc the stress components rcfcrrcd to the global coordinate system. The stress-rate resultants
arc dclined by
(iit,.hii*[j,) = and
r”,( I. z. z ‘)dz
(i=
1.2.6)
1145
Plastic buckling of metal matrix laminated plates
J
h/ 2
a
&I
=
(22)
(i = 4.5);
5’,(1,e’)dz
-h,Z
as presiousIy defined,
-2, = f,,* i2 = t’v,., ?a = fy,-, fS = Q,,. r’, = i,, are the components of the stress-rate tensor in the plate coordinate system. i.e., ?il = dd,,;Zt. The equilibrium equations of the plate in terms of the stress-rate resultants become
(‘“Q.&r+;-&jay+ ~ala.~)(~_~r3qS.r) + (alay)~~~allay) =0
-(4~h')(sd,/a.r+a~,/a~)+(4/3h*)(a2~,~ax2+2a'P,jaxay+a2P,/ay2) a,zf,J~s+S‘Zi,Jay-~i)J+(4/h')R,-(4J3h*)(aP,/ax+aP~/ay)
= 0
~l~,l;‘.r+~l~2/~~-_Q5+(4/hZ)Rs-(4/3h*)(dPalax+dP2la~)
= 0.
(23)
Here. S, and N, are the in-plane loads in the x and y directions. respectively. which are acting at time f. It should be noted that eqns (23) are meaningful during the buckling state where the rate of deflection of the plate becomes nonzero. In this instant, the out-of-plane rate resultants (22) become nonzero as well. The in-plane resultants N, (i = 1.2.6) in eqn (21). on the other hand, are assumed to be constants during the instant of buckling, i.e.. & = 0. The instnntaneous stilTnesses of the plate arc defined by
(A:;p.Ef,p.ny. Ef;F.r;f;“Jry-) = (A;;‘, n (y, FY)
=I
J
h/2
&f;p(1,,,:2,=3,=*,;*)d=
(i,j=
1,2,6)
-hi?
‘I2~~(l.c2,z4)dz
f&i = 4,5).
J -h/2
(24)
Substituting eyns (22). in conjunction with (IS) and (IQ-(20), into (23) and using the ~Ie~initi~~ns (24), we obtain the following constitutive equations for the plate:
and
(C&.k,) = (Af;“, Dl’;‘)$ + (DC’, Fy)$ The present approach for the determination two types of composite plates as follows:
(i, i = 4,5).
(25)
of plastic buckling load is illustrated for
(I f symmetric cross-ply laminated plate; (2) anti-symmetric angle-ply laminated plate.
The plates arc initially at rest and subjected to a uniform, biaxiat pressure. The following bounda~ conditions are assumed : lE(X, 0) = k(x, h) = ti(0, y) = ti(a, y) = 0,
P&v,0)
=
P,(.r,hf = P,(O,y)
h-~:12(x.o) = ic’l:!(.r, h)
=
=
ni,(O,y)
P,(a, y) =
it?,(a, y) = 0,
&t-m = &,W) = I(;,(O,Y) = &,(a,y) For cross-ply laminates :
= 0,
= 0.
24~. 0) = ri(.r. bj = 1:(0. F) = i.(~. _b.j= 0.
1$,.(-V, 0) = ii’, (.U. 6) = The boundary
conditions
These boundary
For cross-ply
for anti-symmetric
conditions
laminates
.i-,(O. J) =
_+,(a,J)
= 0.
angle-ply
laminates
are
are exactly satisfied
(27)
by
:
(30) For anti-symmetric
an&ply
laminates
:
Following Kcddy and I’han (IW5). let us substitute (25). in conjunction with (Ic)), (20) and (24). and using (20) and (30) for cross-ply, or (29) and (3 I) for angle-ply Iaminatcs, into (23) leads to the following tivc homogcncous equations for each set of numbers 1~1and I! : [ ~:;fwlnl
_
G]I’“‘]A,
=o
(i,j=
1,...,5)
(32)
where (A,,... , A5) = (U ,,,,,,P’,,,,,,lVn,,,,A’,,,,,,Y,,,,,). The cocllicients K,‘;“““” arc modc-depcndent linear combinations of the instantaneous stitfnesscs of the inelastic plate and can be identified with the combinations of elastic stiffncsses dcnotcd by C,, in the Appendix of Reddy and Phan (1985). To this end, the elastic stiffnesses (A,,, B,,, D,,, . . .) there should be replaced by the corresponding instantaneous stiffnesses (A,?, B,;‘, 07,. . .). For example, the expression for Krr’““’ would therefore be given by (mn/a) ‘A :‘p+ (nnjb)‘A ic. The nonzero element of Gf;ln’ is given by
In order to obtain a nontrivial solution to (32), singularity of the 5 x 5 matrix ofcoetficients must bc required. This leads to the following buckling condition
(33) where K “Pimn)is ;I 5 x 5 matrix formed by the coefficients K,‘;P(n’n), and Y’-p’moJ is a 4 x 4 by deleting the row i = 3 and the column j = 3. From (33), matrix obtained from Kt~P~““‘~ one can see that the bifurcation buckling of the place occurs when the combination of axial loads N,(rrtn/n)‘+N,,(rtn/h)~ for any mode is equal to the above expression of the instantaneous stiffnesses of the laminate (given by the right-hand side of the equation). In the
1147
Plastic buckling of metal matrix laminated plates
process of the determination of plastic buckling. the loads have to be applied progressively in an incremental manner. At each load increment the buckling condition, eqn (33). has to be checked to verify whether it is satisfied. In the case of a perfectly elastic composite plate, where the instantaneous properties are constants, eqn (33) readily provides the buckling load in terms of the laminated plate elastic properties (Reddy and Phan, 1985). An important observation concerning the plastic buckling of inelastic plates is in order. Although the inelastic plate is initially anisotropic, additional anisotropy is developed due to the plastic flow of the inelastic phases (e.g.. the metallic matrix). This follows from the fact that the instantaneous effective stiffnesses of the plate, which describe its anisotropy. vary with loading history. This implies that if buckling of the inelastic plate takes place in accordance with (33) at a certain mode (mn), the corresponding elastic buckling of the same plate (obtained by disregarding the plasticity effects) might occur at a different mode. Thus, the inclusion of plasticity effects of the constituents in the buckling analysis might change the load level and mode of buckling. 3.2. Classical plate theory The following displacement rate field, associated with the classical laminated plate theory (CPT), is assumed : UI =
ti(.r. y) -z d,[email protected]
ri
tqx.y) -z ati/ay
2
=
ti, = k(.r,
y).
(34)
In the framework of this theory, wc will consider the following two types of uniformly, biaxially comprcsscd laminated plates : (a) anti-symmetrical angle-ply plate ; (b) orthotropic plate (unidirectional single-layer or symmetric cross-ply laminate),
In the first cast of an angle-ply laminate. the bifurcation buckling of the plate is governed by the following set of equilibrium equations : L:;‘u, = 0
(i,j=
I,2,3).
(35)
In the above equation, the differential operators Lc are obtained from the well known classical theory of laminated plates. The plate is initially at rest and the following boundary conditions are chosen : lti(.V. 0) = bif.V.h) = 6(0, y) = k(a,
y) =
0,
ti(x, 0) = ti(.r, 6) = ri(0, y) = ti(a, y) = 0, ni,kO)
= ni,(Xh)
= ti,.,(O. y) = Iti,(a, y) = 0,
&(-r.0)
= ri,,(X,h)
= ni,,(O, y) = l”j,,(a, y) = 0.
(36)
The solution is
(37)
II4S
M.
Substituting ofcoefficients
PALL3
and J. ABOCDI
(37), in conjunction with (21). into (35). and requiring singularity of the matrix of the resulting equations. we obtain the following buckling condition for the
considered angle-ply laminate
;L’,(mn~a)‘+‘Zi,.(nn/h)~
1987) :
(Whitney.
= D~pmJ+2(D:8+2D~,P)m’n’(aib)‘+D~pn”(a
h)’
where
J tPtmnl= (.~:‘pnl’+~~,Pn’(a/b)‘)(~~’
6h~~?+rl~PnZ(u,‘b)~)-nl’n~(aib)‘(A:~+A~~)’
J yP’mn’= (,d:~m’+A~,Pn’(a/h)‘)(B:‘,Pm’+3B~,Pn’(Nih)’) -3nr’n?(N,‘h)2(/1~p+A~hP):(B:~nlZ+Br~n’(cl
b)‘)
J :P’n’n’= (A~,Pm’+A~pn’(a/h)‘)(3B:.,Pm’+B~,Pn’((l;h)’) -m?n?(a/h):(il~,P+~~hP)?(BrhPnl?+3B~oPn~((I,ih):). In the CXE of a symmetrically laminated (all B,, = 0). orthotropic plate (D,,, = D:, = 0) that is simply supported, (35) is simplified to the following sin& equation :
For biaxial load, the buckling condition
bccomcs
Having obtained a buckling condition
ofa metal
matrix laminated
plate [c.g. cqn (351,
cqn (38), or cqn (40)] in terms of the instantaneous
propertics of the inelastic plate, wc can
proceed and dctcrminc
load level according
incrcmcntal
the bifurcation
buckling
(a) At a given load level the stress resultants and the strain-rate (b) These qpantities determine the strcsscs and strain-rates (c) The determined mechanical
to the following
proccdurc.
fields of the lamina
analysis, to obtain
ficlcl arc known.
of the single lamin:t.
are used, in conjunction
the instantaneous
with the micro-
effective stiffncsses C,$,“’ of the unidi-
rectional ply [eqn (I 2)]. (d) The instantaneous from the computed
plate stiffnesses [defined by cqn (24)] can be readily detcrmincd
C,:kyPof the laminae.
(e) The use of the instantaneous
plate stifTnesses in a buckling condition
detcrmincs
whether buckling of the plate occurs at the present load level. If the buckling condition
is
not satisfied with the current plate stifrnesses, the load is modified and the above procedure is rcpeatcd. The prcscnt method of the determination for various typos of inelastic laminated If an analytical,
exact solution
methods (e.g., numerical)
of plastic bifurcation
buckling can bc applied
plates, subjected to various boundary
to the governing
equations
can be employed in conjunction
conditions.
is not possible, approximate
with the proposed methodology.
4. APPLICATIONS
The proposed method of bifurcation is illustrated
for the prediction
buckling analysis of metal matrix laminated plates
of the critical load of a boron/aluminum
composite system.
The boron fibers are assumed to behave as a perfectly elastic material, whereas the aluminum matrix is represented as an elastic-viscoplastic work-hardening material. The aluminum
Plastic buckling of metal matrix laminated
I
plates
I49
20X-T4 is an almost rate-insensitive material at room temperature, but its rate sensitivity significantly increases at elevated temperatures. In this paper, the inelastic response of the matrix material is described by Bodner and Partom’s (1975) unified viscoplasticity theory. This theory does not assume the existence of a yield condition. which eliminates the need to specify loading or unloading criteria, and the same equations can be directly used in all stages of loading and unloading. According to these equations plastic deformation always exists. but it is negligibly small when the material behavior should be essentially elastic. The flow rule function A in eqn (3) is given, according to this theory. as follows : alloy
A = Do exp ( -E[Z’/(3Jz)]“)/Ji”
(41)
where n’ = i(n+ 1)/n and J: = $,,s,, is the second invariant of the stress deviator; Do and n are inelastic material parameters, and Z is a state variable that represents the hardened state of the material with respect to resistance to plastic How. In the case of isotropic hardening. the evolution law of this variable is given by
2 = fl(Z, -Z)
ctp/z,
(42)
where ZU. Z, and rrr are additional inelastic parameters of the material. Note that the plastic work W,. whose rate is Ii’, = a,,r$‘, is taken as the mcasurc of hardening. The physical significance of the above inelastic constants is as follows. The paramotcr D, is the limiting strain Wk. Z, is dilttd t0 ttlC “yield stress” of a uniaxial stress-strain curve. and Z, is proportional to the ultinmtc stress. The material constant I)I dctcrmincs the rate of work hardening. and the ralc sensitivity is controlled by the tcmpcraturc-dcpcndcnt paraflltdcr II. In Titblc I, the material paramctcrs of the X34-TJ
aluminum alloy at various tcmpcraturcs arc given in the framework of Rodncr Partom theory. The table prcscnts the ctastic moduli. Eand Y,Young’s modulus and Poisson’s ratio, rcspcctivcly. and five inelastic paramctcrs, namely: D,,. Z,, Z,. nr and N. It should be noted that higher temperatures arc associated with lower values ofn. which results in an increase of the matcriat rate sensitivity. In Fig. I, the stress-strain response of the matcriat is given for various values of temperature. The boron fibers are considrrcd to be pcrfcctly elastic and isotropic with a Young’s modulus of 400 GPa and Poisson’s ratio of 0.3. The present approach is applied to investigate square (U = h) laminated cross-ply and angle-ply plates. In all cases the thickness of each ply is I mm. The effects of the following parameters on the buckling of the plate will be considered : Icngth-to-thickness ratio (u/h), fiber volume ratio (c,). applied strain-rate (rjXK),angle of lamination (0). number of plies (k) and temperature (7). Some of these effects arc examined in the framework of CPT and HSDT. The inelastic buckling of the composite plates is analyzed by the application of a uniaxiat compressive stress loading d,C,X while controlling the strain-rate value G.rX.For convenience the strcsscs and strain-rates arc shown in the figures as positive.
Table I. Material Tcmperaturc ( C)
E (GPa)
‘0.0
72.4
148.9 X4.4 X0.0 371.1
69.3 65.1 58.4 41.5
constams of an aluminum
” 0.33 0.33 0.33 0.33 0.33
IO.000 IO,000 IO.000
IO.800 10,000
340 340 340 340 340
alloy (2024-T4) z, (MPa)
m
I,
435 435 435 435 435
300 300 300 300 300
10.0 7.0 4.0 1.6 0.6
In the elas:ic region : isotropic material with Young’s modulus E and Poisson’s ratio Y; in the plastic region : isotropic work hardening material.
1150
M.
PALEY and J. ARNDI 21.1 lc
v4a9*c 204.4 ‘C
STRAIN Fig. I. Stress-strain
TJ)
Let us consider supported
a square
boundary
symmetric
conditions.
matrix
laminated
prcscnt cast. Also shown
to sustain
Icvcl approaches
of 204.4 ‘C is shown against = IO-' s -’ and ~j,, = IO
i_
rate sensitivity
on plastic
buckling
plate
loading
without
buckling.
be noted. Furthcrmorc, elastic
low load Icvcls bcforc
next an anti-symmetrically
(26).
corresponding
laminatctl.
(2X), and (36) which
buckling between
conditions the buckling
The Jccrcasc
one. This
correspond
arc dctcrmincd loads
under
simply
of this laminate
is
of the present metal
the length-to-thickness
-’ s-‘. it is clearly seen load of the plate when
effects of the aluminum
for a suflicicntly
matrix
dccrcasc the ability
of the buckling
load Lvith
thin plate the plastic buckling
result is cxpcctctl
apprcciahlc
~Y?J-
of the plate is weak in the
of the buckling
elastic (i.e.. when the inelastic
the corresponding
hucklos at rclativcly
;I comparison
condition
This reveals that the plastic elfcuts of the metal matrix
incrcasc ofcr//I should
Consider
laminated
in the figure is the graph
it is assumed to be pcrfcctly arc disregarded).
[Oi90],,
the buckling
stress load. ri,,, for plastic buckling
strain-rates
that the effect of the material
conditions
CPT,
plate at a temperature
a/h for compressive
of the plate
cross-ply,
Within
given by (40). In Fig. 2. the critical ratio
(lb)
curves in simple tension at severs1 temperatures of an ~tlum~nwn lilac ;LScharacterized in Table I. with an upplwd 5tr.lln rate ofO.OI s ‘.
since ;I thin
plate
plastic ctYccts dcvclop. angle-ply
plate subjcctcd
to IISDT
and CPT.
to boundary
rcspcctivcly.
The
by using cclns (33) and (3X). In Fig. 3.
prcdictcd
by CPT and f-ISDT
of a two-layered
[I 301 angle-ply square laminated plate at 204.4’C is shown for an applied strain rate of IO ‘is. The figure also includes the corresponding buckling loads of a perfectly elastic plate. It is readily
seen that
CPT and HSDT values
OfN//J
would
be apprcciablc.
is about
I .3
in the present
it is expected timCs
situation
the buckling
are close in both casts of inelastic
the
that the buckling
For example, corresponding
loads obtained
and perfectly
loads of the plate obtained
assuming
rr/lr = 5, the buckling
buckling
load Isvcl obtained
[OY90*],
T.204.4.C
v,. 50%
CPT
by the two thcorieh
load obtained from
I ISDT
000
6pPd 600
IO
1
t
20
30
I
I
J
40
JO
60
a/h Fig. 2. Buckling load apunst length-to-thickness
by employing
elastic plates. For smaller
ratio of J cross-ply laminate
via CPT when
the
Plastic buckling of metal matrix laminated
700
II51
plates
r
600-
a,, IMPa)
loo20
30
40
30
60
a/h Fig. 3. Buckling load against length-to-thickness
ratio of an angle-ply laminate.
plate is assumed to be perfectly elastic. It is not possible, on the other hand, to predict
a/h in the framework of small strain
plastic buckling of the plate for such a low value of
theory considered in the present investigation. The figure demonstrates again the importance of inclusion of the plastic effects.
except
for thin plates
whcrc buckling of the plate already
occurs in the elastic region. In order to study the cffcct of tcmpcraturc Ict us consider square angle-ply
laminates at
on the plastic buckling of laminated
T = 20°C and T =
plastic buckling loads of scvcral types of two-laycrcd at thcsc tcmpcraturcs.
angle-ply
These buckling loads. normalized
204.4
square
‘C.
plates.
In f:ig. 4a. the
Iaminatcs arc shown
with respect to the corresponding
elastic buckling load (i.e.. when the metal matrix isassumed to bchaveasan
elastic material),
arc shown in f’ig. 4b. It is readily seen that at the elcvatcd tcmpcraturc,
the ability of the
plate to support loading is rcduccd. The amount of this reduction dcpcnds on the value of
at OO
I IO
I 20
I 30
I 40
8 (b) Fig. 4. (a) Critical load against angle of lamination of an angle-ply laminate. (b) Critical load, normalized with respect to the corresponding elastic buckling stress. against angle of lamination of an angle-ply laminate. SAS
20:9-Q
M. P,\LEY and J. ABM_DI
I IS’
the lamination
angle 0. Furthermore.
of the metal matris buckling cation
load.
on the buckling
howc\.cr,
appears
is that the tcmpcraturc llic sinic
c;~scs. by almost
Let us consider laminatctl critical
loads ;lgainsl
;duminum lo~lctl
inelastic
matcly
clatic
note that
mctry Table
0. I3y a comparison
al NJ.4 smell
rcspccl
with
lamination
point
The impli-
loads.
claslic
C;ISC. the ligurc strain-rate.
than that obtainctl the buckling
exhibits,
in the present
stresscs and strains,
by applying loading.
libcr/matrix
by employing
strain
rates:
;L higher
HSDT.
tit, = IO
’
One can
syslcrn. ;I sym-
plates at the instant
prcclictcd
of the
f:or ;I sl~~wly
lost in llw inclaslic
is complctcly
the total strains of the anglc-ply
shows
of the plate is approxi-
it is not atYcctccl by the rate of the applicti buckling
of the resulting
Due to the rate sensitivity
on the applicil
[ + O]
two-laycrcd
cast.
It is
of buckling.
III
of the inulaslic s
’ 2nd IO ” 5 ‘.
seen that apprcciablc strains dcvclop belbre buckling take place for ~~nglc-ply lamination angle 0 at the vicinity of 50’. It should bc nokd that the buckling
strains
;Irc signiticantly
;tnglcs
in
metal
bchnvior
loads, in the present
the variation
the pcrfcctly
is lower
pl;ttcs [ +(I] arc given for two values 0f;kpplicd
str;lin-r:Itc
buckling
5 exhibits
angles 0, howover,
to 0 = 45 ‘, this symmetry
2 thy’ buckling
on the tcmperuture.
of afi anti-symmclric
Figure
C, this tlocrcasc tlcpcnds
the elastic
to rccorrl
It is readily plates with
dqw~dent
dccrcascs the buckling
and thcrcforu
whcrcas
with
intcrosting
by using HSIIT.
Plato’. the buckling
l:or
strain-rate.
to bc slightly
the plastic
on 0 as well. The normalid
proportion.
significantly
m;ilrix
into account
reduces the elastic and plastic
the elastic and plastic buckling
plate prodictctl
that plasticity
the ctfect of taking
load of the plate depends
this
region.
higher This
up to buckling
malris.
for the low loading
rcsull
is cxpcclctl.
is associated
with
since
rate ofanglc-ply loading
apprcciablc:
plates with lamination
of an inclaslic plastic
tlow
plate
al 3 IOW
of the viscoplaslic
Plastic buckling of metal matrix laminated
0.6
0
.30
1153
plates
.sO
Fig. 6. Critical load. normalized with respect to the corresponding elastic buckling stress, against fiber volume ratio.
It is possible
plastic buckling
to study of a metal
the effect of different matrix
composite
values of fiber volume
fraction
of on the
a [ + 301 angle-ply
To this end, consider
plate.
of 204.4”C. In Fig. 6 results for the plastic laminated plate with a/h = 40. at a temperature buckling of the plate using CPT are shown for three values of Us: 0, 0.3 and 0.5. Here, the buckling
levels are normalized
perfectly
elastic
the figure:
plate.
lO^‘s-’
which
buckles
volume
ratio
and 10e6 s-‘.
at the present incrcascs.
due to the increasing loading.
with
respect to the corresponding
Plastic buckling
loads for two different
The case of c, = 0 corresponds
value of n/h = 40 in the elastic
the buckling
ditTcrcnt
occurs in the elastic region
load rate lcvcls is absent.
and plastic
in
plate
As the reinforcement region.
This is
the plate can sustain
higher
flow of the metal matrix.
Since
far away from
the yield
point,
the efl’cct of
values of U, this ctTcct is pronounced
due
to the cxistcncc of plastic flow. For higher values of ul the cffcct of fibers is dominant the rate sensitivity of the composite plate is weak.
and
Let us study the erect angle-ply plastic
composite
buckling
observed
plate.
For modcratc
of the
to a homogeneous
region.
effect of the fibers. as a result of which
loads
rates are shown
of the plate takes place in the plastic
Thcsc high load levels lcad to yielding
for I’, = 0 buckling
buckling
loading
of number
of plies on the plastic
To this end, consider
loads obtained
that, as in the perfectly
within
buckling
a [ 5 301k with
CPT and HSDT
load level of a [ + 01,
k = I, 2.3. The resulting
arc shown
elastic case, the plastic buckling
in Fig. 7. It is readily
increases with increasing
number of plies of the laminated plate. It can be seen that HSDT predicts slightly lower buckling loads as compared with CPT. The small differences between HSDT and CPT predictions
in the present situation
high value of length-to-thickness
are attributed,
as previously
mentioned,
plate (a/h
ratio of the laminated
to the relatively
= 30).
5. CONCLUSIONS
By employing
a micromechanical
erties of metal matrix
composites,
is shown that in the determination
analysis
which can provide
the plastic buckling of the critical
the instantaneous
of laminated
loading
of the plate, the satisfaction
QC% T-204.4.C
prop-
plates is determined.
Q
HSDT
q
CPT
k Fig. 7. Buckling load against number of layers of an angle-ply laminate.
It
of the
115-l
ht.
PhLEY
and
J. ;\tltN
DI
corresponding buckling condition has to be checked at all stages of loading history. The method is illustrated for the plastic buckling analysis of boron Juminum composite plates under various situations. The effect of the elastic-viscoplastic behavior of the aluminum matrix and its rate sensitivity at elevated temperatures on the plastic buckling of the plates is presented.
REFERESCES Aboudi. J. (1989). SltcromechJnical analysis of composites by the method of ceils. .dppl. \I~Y+I RW 12. 19~~ 21. Accorct. XI. L. and Nemat-Nasser. S. ( 1986). Bounds on the overall elastic and ttwt.tnt,tneouz rl,tsrL7-pl.tstJc moduli of pertodic composites. .\lcch. .Iftrrcr. 5. 209 220. BJjlaard. P. P. ‘I.
( 1949).Theory
and nests on the pl.tstJc stabtllty of phtes
Jnd shells. J. ..I~rt~lrtltifi~.(~I.&.I. 42. 193~
Bodner. S. R. and Partom. Y. (1975). Constttutt\e equattons for CI~SIIC~~IISCO~I~~~~~C strrun-hardentng matertals. J. dppl. .Ilech. 42, 385-389. Bodner. S. R.. Naveh. M. and Merzer. A. M. (I991 ). Deformatton and buckling of axisymmetrtc viscoplastic shells under thermomechanical loading. fnf. J. Solids .Sm~cfurr.r 27. I9 I S-1924. Bushnell. D. (1982). Plastic buckling. In Pressure l’essels and Piping: DesignTechnology. .4 Decade of Progress (Edited by S. Y. Zamrik and D. Dietrich). pp. 47-l 17. ASME. New York. Hendelman, G. H. and Praper. W. (1948). Plastic buckling of a rectangular plate under edge thrusts. NACA TN
1530. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phxs. .Sok.b 6.?_%249. tlutchinron. J. W. (19743. Plastic buckling. In ;I&wcc m .4pp/icd ,Mcchmics, Vol. 14. pp. 529541. Academic Press. New York. Kapania, R. K. and I~XIII. S. t 19X9). Rcccnt advances in analysis of IamJnatcd hcam:, and plstcs. Fart I : Shear clfccts and buckltng. .~I/:t.t JI 27, 923 934.
Mf
pk
PLASTIC BUCKLING OF METAL MATRIX LAMINATED PLATES M. PALEY and J. ABOUDI Department of Solid Mechanics. hluterials and Structures. Faculty of Engineering. Tel-Aviv University. Ramat-Aviv, Tel-Aviv 69978, Israel (Rewired
17Augustt990 ; in recisedform 2 December 1990)
Abstract-A method is proposed for the dete~ination of plastic bifurcation buckling ioad OF metal matrix composite plates. The metallic matrix behavior is described by an elastic-viscoplastic constitutive law, while the tibers are assumed to be either e&tic or elastic-viscoplastic material. The approach is based on the load level and history dependent instantaneous effective properties of the inelastic plate. which arc established by a micromechanical analysis. An incremental procedure is developed in which a buckling condition has to beestablished and its fultilment must bechecked at euch increment. The method is applied for the prediction of the plastic buckling of boron/aluminum composite plates in various situ;ttions. by employing the classical and higher order shear deformation plate theories.
I
r
lMRCKNJCTt#N
The dctcrmination of the buckling load of perfectly elastic composite plates has received considerable attention [see Lcissa (1985, 1987) for extcnsivc reviews]. Most of the papers covered in these rcvicws uti1iz.cthe classical theory of plates. The use of a first order theory in the study of bifurcation buckling of elastic plates can bc found in Whitney and Leissa (1969) and Whitney (1987). Analyses of composite plate buckling by employing higher order thcorics can be found in the rcccnt review by Kapania and Raciti (1989). The stability analysis of homogeneous inelastic plates is much more complicated than the pcrfcctly rlastic case, dus to the inherent nonlincnrity of the material constitutive law. The study of bifurcation buckling can be performed by employing total deformation plasticity (c.g.. Bijhmrd. 1949) or incremental theory (e.g.. Hcndelman and Prager, 1948). For extensive reviews of this subject, see Hutchinson (1974) and Bushnell (1982). Recently, the influence of material rate sensitivity on the elastic-plastic buckling of structures has been investigated by Tvcrgaard (198s) Bodnerrral. (1991), and Paley and Aboudi (1991 b). The theory of uniqueness and bifurcation in time-inde~ndent elastic-plastic materials was given by Hill (1958). It was shown by Tvergaard (1989) that Hill’s bifurcation condition can be satisfied for elastic-viscoplastic materials only at the elastic buckling load. This is due to the high strain-rates that occur at the instant of buckling. In practice, imperfections and inertial effects reduce the instantaneous changes of buckling strain rates (Bodner et al., 1991). The use of strain-rates in the pm-buckled state in an appropriate buckling condition of a viscoplastic material provides lower bounds to the actual buckling loads. The prediction of plastic buckling of metal matrix composite structures is di~cult due to the complicated interactions between the inelastic matrix and reinforcement. In previous investigations, the inelastic effects of the matrix were neglected by assuming perfectly elastic behavior for all loading Ievcls. Obviously, this is a restriction since a metal matrix composite deforms plastically at the early stages of loading. For example, in a boron/aluminum composite the aluminum matrix yields at a strain of about 0.002, which is far away from the ultimate strain of the composite. Thus, an efficient utilization of the metal matrix composite up to failure must take into account the plastic flow of the matrix. The plastic buckling analysis of a metal matrix composite structure requires the knowledge of its instantaneous stiffnesses which depend on the history and loading level. The overall instantaneous properties of the inelastic composite can be determined by a suitable micromechanical analysis from the known material parameters of fiber and matrix. In the II39
IIU)
M.
framework
PALEY
of the classical incremental
and J.
&O~Dl
plasticity
theory.
bounds on the overall instan-
taneous elastoplastic moduli of periodic composites were determined by Accorci and NematNasser (1956) and Teply and Dvorak
( 1988).
Explicit expressions of the effective instantaneous properties of metal matrix composites were recently given by Paley and Aboudi (199la). The derivation of the instantaneous properties was based on the micromechanical method of cells (Aboudi, which the metal matrix is assumed to behave as an elastic-viscoplastic by elastic or elastic-viscoplastic
1989) in
material reinforced
fibers. It was shown by Paley and Aboudi (199la)
that at
any stage of loading. the current stiffnesses of the inelastic fiber and matrix material can be employed to generate. in conjunction with the micromechanical analysis, the overall stiffness tensor of the unidirectional
composite. The instantaneous
stiffnesses of metal matrix com-
posite laminates were obtained by using the standard lamination In the present paper, the previous
micromechanical
theory.
method for the derivation
instantaneous properties of metal matrix composites is utilized for the bifurcation analysis of composite plates, where the matrix particular.
the Bodner and Partom
ol
buckling
material could be elastic-viscoplastic.
In
(1975) model is employed in the applications given in
this paper. The method is based on the development of governing buckling equations of the composite inelastic plate. which are expressed in terms of the velocity held, and involve the instantaneous condition.
properties of the laminated plate. Consequently,
the derived buckling
which is obtained from the governing equations, depends on the loading level
and its history.
Due to the existing plastic behavior of an inelastic phase. the method ol
obtaining the critical buckling load Icvcl is incrcmcntal. At each load incrcmcnt the fullilment of the derived buckling condition
is examined. If the condition
is satisfied. then the
buckling load lcvcl of the composite plate has been achieved; othcrwisc procedure is continued. The method is illustratctl
for various types of boron/aluminum
the classical and higher order shear deformation offiber
with the corresponding
composite plates. Roll1
plate theories are employed. The clTccts
volume ratio, plate thickness, material rate sensitivity,
and total numheroflayers
the incremcntai
temperature, ply oricnlalion
on the plastic buckling toad of*the plate arc stutiicd. Comparisons buckling loads obtained by disregarding the plasticity cffccts
ofthe
metal matrix arc prcscntcd.
2. MATERIAL
STII:f:NESS
KEDUCTION
DUE
TO PLASTIC
Consider a homogeneous isotropic elastic-viscoplastic
FLOW
material subjected to a slowly
increasing external loading. At the first stage of loading. the material behavior is essentially elastic and is characterized by its elastic stiffnesscs. develops, leading to anisotropic
At a later stage, plastic deformation
behavior of the material, which is described by its instan-
taneous reduced stiffness properties. This
stiffness reduction, which is caused by the plastic
how of the material, can be determined at any instant
of the loading by the following
considerations. The total strain-rate tensor of the material is assumed to be a sum of two components : reversible (elastic) fjc and irrcversibtc (plastic) q,:, i.e.,
The stress-rrffc tensor s,~ is rclatcd to the elastic part of the total strain-rcrfc by the generalized Hooke’s law (small deformations are assumed) :
where C,,,, is the fourth order elastic stiffness tensor of the material, and the summation convention is employed on repeated italic letters. The plastic components of the strain-rrrlc are controlled by the Prandtl-Reuss Row law as follows :
Plastic buckling of metal matrix laminltted plattts
I Ill (3)
IIP, = b,
where s,, = cr,,- &,,att:3 is the deviatoric part of the stress tensor G,,, ci,, is the Kronecker delta. and A is the How rule function of the adopted viscoplastic constitutive
model which
describes the inelastic behavior of the material. The following constitutiv~
relation for the initially isotropic homogeneous viscoptastic
material can be established (Prtley and Aboudi.
where the instantaneous strcukwte
199lb) :
dependent viscoplastic moduli Cl”;& of the material are
defined by
In (5). the scalar 9 is defined by the following equation :
where 1’ is the elastic rigidity
of the material.
Equations
(4)-(G) were est~~bl~sl~cdafter
various manipulations by employing the flow rule (3) in conjunction with cyns (I) and (2). It should be noted that thssc relations were established without the use of the yictd function concept. In this paper. the superscript ( ) “” is used to dcnotc the various ourrcnt propcrtics which
charactcrizc the viscoplastic material (and structure)
that vary with loading history.
It can hc readily seen that the reduction of matcrinl still’ncss, causctl by plastic Ilow, is a function of the history and the current state of louding. The iftstitr?t~l:l~~~tis ~~)llstitlltiv~ invcstigatc the bifurcation was
pcrformcd by utilizing,
relation (4) was prcviousty cmptoycd by the authors to
buckling of visuoplastic plates (PaIcy and Ahoudi.
199 I h). This
:I( itny stage of loading, the current value of Cf,& of the material
at the buckling condition of the plate. which was dcrivcd from the governing equations in conjunction with the spccificd boundary conditions. It cat1 be observed that the rate-dependent C,$‘, are similar
to the f’orrn of the ratc-
inclcpcndcnt instantaneous moduli used by Hill in the derivation of his bifurcation condition fsec, for example, eyn (3.4) of Tvergsard (19g9)]. One can readily see from eqn (6) that y = 0 when the material behavior is elastic, whereas y is positive when plastic flow takes place. The tatter fotlows from the fact that y is given in terms of the ratio of the plastic work rate (s,,rljj) to the rate of the total work of deformation
(.s~~~/~J which dots not include
volume changes. In the present case of a viscoplustic material the strain-rate at the prc-buckled state is used in computing C,y[,. At the instant of buckling, however, high strain-rates develop in the mitt&al. Consequently, the predicted values of critical loads are lower bounds of the actual values. Thus the obtained bifurcation results are approximate, representing predictions for an analogous time-independent material. Similar
arguments were given by
Bodncr cl ul. ( t 99 I ). By establishing a constitutivc
relation similar to eqn (4). for the unidirectional
lamina
of a composite plate. it will be shown in the sequel that an analogous procedure can bc ndoptrd for the anatysis of the bifurcation
Let us consider a unidir~tional perfectly elastic fibers reinforcing
buckling of metal matrix laminated plates.
mcta! matrix composite material which consists of
an elastic-viscoplastic
matrix. The fibers are oriented in
the s’, direction of a Cartesian coordinate system (Y,.r’&,). For transversely isotropic fibers (.r>-s’, is the plane of isotropy), the material stiffness tensor of the fibers is given by
where
j*(f),
p(
1’1, 11 I.), 8 1’1 2nd
;‘(
I.)
are five independent constants that characterize the fiber material. and can be easily related to its corresponding engineering constants. The above tensor czln be rewritten in a more familiar form using a contracted notation.
The c&tic
stitfncss of the isotropic
matrix
phase is given by
whcrc
with
For perfectly elastic fibers the plastic strain rate qf;“” z 0, implying that y’” = 0 and CVPfI‘) = Cl;;;. ft is readily seen that the present derivation also includes the more general tjkl case of elastic-viscoplastic fibers. In such a case eqns @)-(I I) can be used subject to the requirement that the inelastic fibers are initially isotropic. The overall instantaneous properties of the unidirectional fiber-reinforced material c:m bc determined by a suitable micromechanical analysis in which the detailed interaction bctwcen fiber and matrix phases is considcrcd. Such a micromechanical analysis was dcvclopcd by Paley and Aboudi ( 19r)f a] by employing the method ofcells Aboudi ( f 989). This composite modrt is based on the analysis of a repeating ccl!. and leads to the prediction of the avcrall behavior of various types of composites from the known propcrtics of the fiber and matrix matcriats. It was shown by Paley and Aboudi (1991a) that the knowledge of the: instantaneous properties of the inelastic fiber and matrix phases provides. in conjunction with micromechanical analysis of the method uf ccfls. the overall instantaneous stiffness of the un~direction~ll metal matrix composite. Consequcntfy, from the knowledge of Cf$?’ (4 = j;m) at any stapc of loading, one can dcterminc the etTective instant~kneous behavior of the composite in the form
of
Plastic buckling of metal matrix laminated plates
1113
where f,, and rj,, are the average stress-rate and strain-rate tensors, and Ci$’ is the fourth order tensor of the effective instantaneous reduced stiffnesses of the composite. Thus, the composite constitutive equation (12) is a generalization of relation (4). which was established for a homogeneous viscoplastic material. The explicit expression of C$’ can be found in Paiey and Aboudi (1991a). it should be emphasized that the tensor C$;’ depends on the level and history of loading. It is worth mentioning that a constitutive equation of a similar form was also established by Paley and Aboudi (199la) for composite laminates in which each lamina is a unidirectional metal matrix composite.
3. PLASTIC
BUCKLING
OF LAMINATED
PLATES
In this section. we will utilize the previously discussed instantaneous constitutive relation (I 2) for inelastic unidirectional composites to establish the governing equations for plastic buckling of metal matrix laminates. To this end. consider a rectangular laminated plate which consists of several laminae. The middle plane of the laminate coincides with the .r--)’ plane of the Cartesian coordinate system I&V:). The x and _r axes are parallel to the plate edges. and the origin of the system is taken at a corner. The dimensions of the plate are: N and h in the .r and _rdirections. respectively. and thickness h in the z direction. When it is convenient, we will use the following alternative notation for the axes: .ri = X, .sL = _Vand s1 = z Each layer of the laminate is a unidirectional metal matrix composite which is assumed to be in the plane-stress state. Consequently, the constitutive equation (12) of’thc single laminit in its matcrinl coordinate system (.Y’,x’&) can be rcwrittcn as r; = Q;:“tsi
(i,j=
1.2,4,5,6)
(13)
whcrc i, = F,,,..,. iI = ?clS1. i+ = TX,:,.,.f5 = ix.,..,, i, = 5;.,Xi, and $, = rj,,,r,,, @;= tj,,,.,. )jl = 2rj,.,r.,‘& = 28j,.I..). fjh = ?rj,., Xi’The nonzero elements of Q$” are obtained from C:“’ of the c&rcsponding Iaycr [whcrc CT’-” is the instantaneous cffcctivc material stilTncss tensor C’,$“. eqn ( I?). which is rewritten in the contracted notation] as follows
and Q:““’ = C,:“” (i, j = 4,~).
(14)
The plane-stress instantaneous stiffness Qi’; of the lamina, expressed in the global coordinate system (XJZ), is obtained by the appropriate transformation of the above defined +W.In these plate coordinates, eqn (13) becomes Qt, Sj = 0: I.“$
(i, j = 1,2,4,5,6)
where i, and rj, arc the average stress-rates and strain-rates in the lamina expressed in the plate coordinate system. As usual, the average strain-rates are taken to be equal to the strain-rates in the plate. Assuming smati changes of displacement field during each load increment, the total strain-rates in the plate in the global coordinates become &, =
~(a~j/a.~j+a~j/a.~,)
(16)
where ri, are components of the assumed velocity field associated with adopted laminated plate theory. Both higher order and classicai plate theories will be employed to establish the governing equations of the plastic bifurcation buckling of the considered laminated plate.
itu 3. I. Higher order plutr theor>
The adopted higher order shear deformation
laminate theory (HSDT)
is based on ;i
third order expansion of the displacement field (Reddy and Phan. 19Y5). In the present case of inelastic behavior. the following
third
order expansion of the velocity components is
assumed :
where Ii, k and ii, are the s. _Vand : components of velocity of middle plane points : 4, and (c;Care rotation rates of transverse normals about the J* and s axes. respectively. Substitution total strain-rates
of the above detined velocities into (16) leads to the following
field of
in the laminated plate :
whcrc the following
definitions
arc cmploycd :
The in-plane stress resultants
itrc dcfincd by
s h :
N, =
6, dz
(i = 1,2,6)
(21)
hs ?
where ci, arc the stress components rcfcrrcd to the global coordinate system. The stress-rate resultants
arc dclined by
(iit,.hii*[j,) = and
r”,( I. z. z ‘)dz
(i=
1.2.6)
1145
Plastic buckling of metal matrix laminated plates
J
h/ 2
a
&I
=
(22)
(i = 4.5);
5’,(1,e’)dz
-h,Z
as presiousIy defined,
-2, = f,,* i2 = t’v,., ?a = fy,-, fS = Q,,. r’, = i,, are the components of the stress-rate tensor in the plate coordinate system. i.e., ?il = dd,,;Zt. The equilibrium equations of the plate in terms of the stress-rate resultants become
(‘“Q.&r+;-&jay+ ~ala.~)(~_~r3qS.r) + (alay)~~~allay) =0
-(4~h')(sd,/a.r+a~,/a~)+(4/3h*)(a2~,~ax2+2a'P,jaxay+a2P,/ay2) a,zf,J~s+S‘Zi,Jay-~i)J+(4/h')R,-(4J3h*)(aP,/ax+aP~/ay)
= 0
~l~,l;‘.r+~l~2/~~-_Q5+(4/hZ)Rs-(4/3h*)(dPalax+dP2la~)
= 0.
(23)
Here. S, and N, are the in-plane loads in the x and y directions. respectively. which are acting at time f. It should be noted that eqns (23) are meaningful during the buckling state where the rate of deflection of the plate becomes nonzero. In this instant, the out-of-plane rate resultants (22) become nonzero as well. The in-plane resultants N, (i = 1.2.6) in eqn (21). on the other hand, are assumed to be constants during the instant of buckling, i.e.. & = 0. The instnntaneous stilTnesses of the plate arc defined by
(A:;p.Ef,p.ny. Ef;F.r;f;“Jry-) = (A;;‘, n (y, FY)
=I
J
h/2
&f;p(1,,,:2,=3,=*,;*)d=
(i,j=
1,2,6)
-hi?
‘I2~~(l.c2,z4)dz
f&i = 4,5).
J -h/2
(24)
Substituting eyns (22). in conjunction with (IS) and (IQ-(20), into (23) and using the ~Ie~initi~~ns (24), we obtain the following constitutive equations for the plate:
and
(C&.k,) = (Af;“, Dl’;‘)$ + (DC’, Fy)$ The present approach for the determination two types of composite plates as follows:
(i, i = 4,5).
(25)
of plastic buckling load is illustrated for
(I f symmetric cross-ply laminated plate; (2) anti-symmetric angle-ply laminated plate.
The plates arc initially at rest and subjected to a uniform, biaxiat pressure. The following bounda~ conditions are assumed : lE(X, 0) = k(x, h) = ti(0, y) = ti(a, y) = 0,
P&v,0)
=
P,(.r,hf = P,(O,y)
h-~:12(x.o) = ic’l:!(.r, h)
=
=
ni,(O,y)
P,(a, y) =
it?,(a, y) = 0,
&t-m = &,W) = I(;,(O,Y) = &,(a,y) For cross-ply laminates :
= 0,
= 0.
24~. 0) = ri(.r. bj = 1:(0. F) = i.(~. _b.j= 0.
1$,.(-V, 0) = ii’, (.U. 6) = The boundary
conditions
These boundary
For cross-ply
for anti-symmetric
conditions
laminates
.i-,(O. J) =
_+,(a,J)
= 0.
angle-ply
laminates
are
are exactly satisfied
(27)
by
:
(30) For anti-symmetric
an&ply
laminates
:
Following Kcddy and I’han (IW5). let us substitute (25). in conjunction with (Ic)), (20) and (24). and using (20) and (30) for cross-ply, or (29) and (3 I) for angle-ply Iaminatcs, into (23) leads to the following tivc homogcncous equations for each set of numbers 1~1and I! : [ ~:;fwlnl
_
G]I’“‘]A,
=o
(i,j=
1,...,5)
(32)
where (A,,... , A5) = (U ,,,,,,P’,,,,,,lVn,,,,A’,,,,,,Y,,,,,). The cocllicients K,‘;“““” arc modc-depcndent linear combinations of the instantaneous stitfnesscs of the inelastic plate and can be identified with the combinations of elastic stiffncsses dcnotcd by C,, in the Appendix of Reddy and Phan (1985). To this end, the elastic stiffnesses (A,,, B,,, D,,, . . .) there should be replaced by the corresponding instantaneous stiffnesses (A,?, B,;‘, 07,. . .). For example, the expression for Krr’““’ would therefore be given by (mn/a) ‘A :‘p+ (nnjb)‘A ic. The nonzero element of Gf;ln’ is given by
In order to obtain a nontrivial solution to (32), singularity of the 5 x 5 matrix ofcoetficients must bc required. This leads to the following buckling condition
(33) where K “Pimn)is ;I 5 x 5 matrix formed by the coefficients K,‘;P(n’n), and Y’-p’moJ is a 4 x 4 by deleting the row i = 3 and the column j = 3. From (33), matrix obtained from Kt~P~““‘~ one can see that the bifurcation buckling of the place occurs when the combination of axial loads N,(rrtn/n)‘+N,,(rtn/h)~ for any mode is equal to the above expression of the instantaneous stiffnesses of the laminate (given by the right-hand side of the equation). In the
1147
Plastic buckling of metal matrix laminated plates
process of the determination of plastic buckling. the loads have to be applied progressively in an incremental manner. At each load increment the buckling condition, eqn (33). has to be checked to verify whether it is satisfied. In the case of a perfectly elastic composite plate, where the instantaneous properties are constants, eqn (33) readily provides the buckling load in terms of the laminated plate elastic properties (Reddy and Phan, 1985). An important observation concerning the plastic buckling of inelastic plates is in order. Although the inelastic plate is initially anisotropic, additional anisotropy is developed due to the plastic flow of the inelastic phases (e.g.. the metallic matrix). This follows from the fact that the instantaneous effective stiffnesses of the plate, which describe its anisotropy. vary with loading history. This implies that if buckling of the inelastic plate takes place in accordance with (33) at a certain mode (mn), the corresponding elastic buckling of the same plate (obtained by disregarding the plasticity effects) might occur at a different mode. Thus, the inclusion of plasticity effects of the constituents in the buckling analysis might change the load level and mode of buckling. 3.2. Classical plate theory The following displacement rate field, associated with the classical laminated plate theory (CPT), is assumed : UI =
ti(.r. y) -z d,[email protected]
ri
tqx.y) -z ati/ay
2
=
ti, = k(.r,
y).
(34)
In the framework of this theory, wc will consider the following two types of uniformly, biaxially comprcsscd laminated plates : (a) anti-symmetrical angle-ply plate ; (b) orthotropic plate (unidirectional single-layer or symmetric cross-ply laminate),
In the first cast of an angle-ply laminate. the bifurcation buckling of the plate is governed by the following set of equilibrium equations : L:;‘u, = 0
(i,j=
I,2,3).
(35)
In the above equation, the differential operators Lc are obtained from the well known classical theory of laminated plates. The plate is initially at rest and the following boundary conditions are chosen : lti(.V. 0) = bif.V.h) = 6(0, y) = k(a,
y) =
0,
ti(x, 0) = ti(.r, 6) = ri(0, y) = ti(a, y) = 0, ni,kO)
= ni,(Xh)
= ti,.,(O. y) = Iti,(a, y) = 0,
&(-r.0)
= ri,,(X,h)
= ni,,(O, y) = l”j,,(a, y) = 0.
(36)
The solution is
(37)
II4S
M.
Substituting ofcoefficients
PALL3
and J. ABOCDI
(37), in conjunction with (21). into (35). and requiring singularity of the matrix of the resulting equations. we obtain the following buckling condition for the
considered angle-ply laminate
;L’,(mn~a)‘+‘Zi,.(nn/h)~
1987) :
(Whitney.
= D~pmJ+2(D:8+2D~,P)m’n’(aib)‘+D~pn”(a
h)’
where
J tPtmnl= (.~:‘pnl’+~~,Pn’(a/b)‘)(~~’
6h~~?+rl~PnZ(u,‘b)~)-nl’n~(aib)‘(A:~+A~~)’
J yP’mn’= (,d:~m’+A~,Pn’(a/h)‘)(B:‘,Pm’+3B~,Pn’(Nih)’) -3nr’n?(N,‘h)2(/1~p+A~hP):(B:~nlZ+Br~n’(cl
b)‘)
J :P’n’n’= (A~,Pm’+A~pn’(a/h)‘)(3B:.,Pm’+B~,Pn’((l;h)’) -m?n?(a/h):(il~,P+~~hP)?(BrhPnl?+3B~oPn~((I,ih):). In the CXE of a symmetrically laminated (all B,, = 0). orthotropic plate (D,,, = D:, = 0) that is simply supported, (35) is simplified to the following sin& equation :
For biaxial load, the buckling condition
bccomcs
Having obtained a buckling condition
ofa metal
matrix laminated
plate [c.g. cqn (351,
cqn (38), or cqn (40)] in terms of the instantaneous
propertics of the inelastic plate, wc can
proceed and dctcrminc
load level according
incrcmcntal
the bifurcation
buckling
(a) At a given load level the stress resultants and the strain-rate (b) These qpantities determine the strcsscs and strain-rates (c) The determined mechanical
to the following
proccdurc.
fields of the lamina
analysis, to obtain
ficlcl arc known.
of the single lamin:t.
are used, in conjunction
the instantaneous
with the micro-
effective stiffncsses C,$,“’ of the unidi-
rectional ply [eqn (I 2)]. (d) The instantaneous from the computed
plate stiffnesses [defined by cqn (24)] can be readily detcrmincd
C,:kyPof the laminae.
(e) The use of the instantaneous
plate stifTnesses in a buckling condition
detcrmincs
whether buckling of the plate occurs at the present load level. If the buckling condition
is
not satisfied with the current plate stifrnesses, the load is modified and the above procedure is rcpeatcd. The prcscnt method of the determination for various typos of inelastic laminated If an analytical,
exact solution
methods (e.g., numerical)
of plastic bifurcation
buckling can bc applied
plates, subjected to various boundary
to the governing
equations
can be employed in conjunction
conditions.
is not possible, approximate
with the proposed methodology.
4. APPLICATIONS
The proposed method of bifurcation is illustrated
for the prediction
buckling analysis of metal matrix laminated plates
of the critical load of a boron/aluminum
composite system.
The boron fibers are assumed to behave as a perfectly elastic material, whereas the aluminum matrix is represented as an elastic-viscoplastic work-hardening material. The aluminum
Plastic buckling of metal matrix laminated
I
plates
I49
20X-T4 is an almost rate-insensitive material at room temperature, but its rate sensitivity significantly increases at elevated temperatures. In this paper, the inelastic response of the matrix material is described by Bodner and Partom’s (1975) unified viscoplasticity theory. This theory does not assume the existence of a yield condition. which eliminates the need to specify loading or unloading criteria, and the same equations can be directly used in all stages of loading and unloading. According to these equations plastic deformation always exists. but it is negligibly small when the material behavior should be essentially elastic. The flow rule function A in eqn (3) is given, according to this theory. as follows : alloy
A = Do exp ( -E[Z’/(3Jz)]“)/Ji”
(41)
where n’ = i(n+ 1)/n and J: = $,,s,, is the second invariant of the stress deviator; Do and n are inelastic material parameters, and Z is a state variable that represents the hardened state of the material with respect to resistance to plastic How. In the case of isotropic hardening. the evolution law of this variable is given by
2 = fl(Z, -Z)
ctp/z,
(42)
where ZU. Z, and rrr are additional inelastic parameters of the material. Note that the plastic work W,. whose rate is Ii’, = a,,r$‘, is taken as the mcasurc of hardening. The physical significance of the above inelastic constants is as follows. The paramotcr D, is the limiting strain Wk. Z, is dilttd t0 ttlC “yield stress” of a uniaxial stress-strain curve. and Z, is proportional to the ultinmtc stress. The material constant I)I dctcrmincs the rate of work hardening. and the ralc sensitivity is controlled by the tcmpcraturc-dcpcndcnt paraflltdcr II. In Titblc I, the material paramctcrs of the X34-TJ
aluminum alloy at various tcmpcraturcs arc given in the framework of Rodncr Partom theory. The table prcscnts the ctastic moduli. Eand Y,Young’s modulus and Poisson’s ratio, rcspcctivcly. and five inelastic paramctcrs, namely: D,,. Z,, Z,. nr and N. It should be noted that higher temperatures arc associated with lower values ofn. which results in an increase of the matcriat rate sensitivity. In Fig. I, the stress-strain response of the matcriat is given for various values of temperature. The boron fibers are considrrcd to be pcrfcctly elastic and isotropic with a Young’s modulus of 400 GPa and Poisson’s ratio of 0.3. The present approach is applied to investigate square (U = h) laminated cross-ply and angle-ply plates. In all cases the thickness of each ply is I mm. The effects of the following parameters on the buckling of the plate will be considered : Icngth-to-thickness ratio (u/h), fiber volume ratio (c,). applied strain-rate (rjXK),angle of lamination (0). number of plies (k) and temperature (7). Some of these effects arc examined in the framework of CPT and HSDT. The inelastic buckling of the composite plates is analyzed by the application of a uniaxiat compressive stress loading d,C,X while controlling the strain-rate value G.rX.For convenience the strcsscs and strain-rates arc shown in the figures as positive.
Table I. Material Tcmperaturc ( C)
E (GPa)
‘0.0
72.4
148.9 X4.4 X0.0 371.1
69.3 65.1 58.4 41.5
constams of an aluminum
” 0.33 0.33 0.33 0.33 0.33
IO.000 IO,000 IO.000
IO.800 10,000
340 340 340 340 340
alloy (2024-T4) z, (MPa)
m
I,
435 435 435 435 435
300 300 300 300 300
10.0 7.0 4.0 1.6 0.6
In the elas:ic region : isotropic material with Young’s modulus E and Poisson’s ratio Y; in the plastic region : isotropic work hardening material.
1150
M.
PALEY and J. ARNDI 21.1 lc
v4a9*c 204.4 ‘C
STRAIN Fig. I. Stress-strain
TJ)
Let us consider supported
a square
boundary
symmetric
conditions.
matrix
laminated
prcscnt cast. Also shown
to sustain
Icvcl approaches
of 204.4 ‘C is shown against = IO-' s -’ and ~j,, = IO
i_
rate sensitivity
on plastic
buckling
plate
loading
without
buckling.
be noted. Furthcrmorc, elastic
low load Icvcls bcforc
next an anti-symmetrically
(26).
corresponding
laminatctl.
(2X), and (36) which
buckling between
conditions the buckling
The Jccrcasc
one. This
correspond
arc dctcrmincd loads
under
simply
of this laminate
is
of the present metal
the length-to-thickness
-’ s-‘. it is clearly seen load of the plate when
effects of the aluminum
for a suflicicntly
matrix
dccrcasc the ability
of the buckling
load Lvith
thin plate the plastic buckling
result is cxpcctctl
apprcciahlc
~Y?J-
of the plate is weak in the
of the buckling
elastic (i.e.. when the inelastic
the corresponding
hucklos at rclativcly
;I comparison
condition
This reveals that the plastic elfcuts of the metal matrix
incrcasc ofcr//I should
Consider
laminated
in the figure is the graph
it is assumed to be pcrfcctly arc disregarded).
[Oi90],,
the buckling
stress load. ri,,, for plastic buckling
strain-rates
that the effect of the material
conditions
CPT,
plate at a temperature
a/h for compressive
of the plate
cross-ply,
Within
given by (40). In Fig. 2. the critical ratio
(lb)
curves in simple tension at severs1 temperatures of an ~tlum~nwn lilac ;LScharacterized in Table I. with an upplwd 5tr.lln rate ofO.OI s ‘.
since ;I thin
plate
plastic ctYccts dcvclop. angle-ply
plate subjcctcd
to IISDT
and CPT.
to boundary
rcspcctivcly.
The
by using cclns (33) and (3X). In Fig. 3.
prcdictcd
by CPT and f-ISDT
of a two-layered
[I 301 angle-ply square laminated plate at 204.4’C is shown for an applied strain rate of IO ‘is. The figure also includes the corresponding buckling loads of a perfectly elastic plate. It is readily
seen that
CPT and HSDT values
OfN//J
would
be apprcciablc.
is about
I .3
in the present
it is expected timCs
situation
the buckling
are close in both casts of inelastic
the
that the buckling
For example, corresponding
loads obtained
and perfectly
loads of the plate obtained
assuming
rr/lr = 5, the buckling
buckling
load Isvcl obtained
[OY90*],
T.204.4.C
v,. 50%
CPT
by the two thcorieh
load obtained from
I ISDT
000
6pPd 600
IO
1
t
20
30
I
I
J
40
JO
60
a/h Fig. 2. Buckling load apunst length-to-thickness
by employing
elastic plates. For smaller
ratio of J cross-ply laminate
via CPT when
the
Plastic buckling of metal matrix laminated
700
II51
plates
r
600-
a,, IMPa)
loo20
30
40
30
60
a/h Fig. 3. Buckling load against length-to-thickness
ratio of an angle-ply laminate.
plate is assumed to be perfectly elastic. It is not possible, on the other hand, to predict
a/h in the framework of small strain
plastic buckling of the plate for such a low value of
theory considered in the present investigation. The figure demonstrates again the importance of inclusion of the plastic effects.
except
for thin plates
whcrc buckling of the plate already
occurs in the elastic region. In order to study the cffcct of tcmpcraturc Ict us consider square angle-ply
laminates at
on the plastic buckling of laminated
T = 20°C and T =
plastic buckling loads of scvcral types of two-laycrcd at thcsc tcmpcraturcs.
angle-ply
These buckling loads. normalized
204.4
square
‘C.
plates.
In f:ig. 4a. the
Iaminatcs arc shown
with respect to the corresponding
elastic buckling load (i.e.. when the metal matrix isassumed to bchaveasan
elastic material),
arc shown in f’ig. 4b. It is readily seen that at the elcvatcd tcmpcraturc,
the ability of the
plate to support loading is rcduccd. The amount of this reduction dcpcnds on the value of
at OO
I IO
I 20
I 30
I 40
8 (b) Fig. 4. (a) Critical load against angle of lamination of an angle-ply laminate. (b) Critical load, normalized with respect to the corresponding elastic buckling stress. against angle of lamination of an angle-ply laminate. SAS
20:9-Q
M. P,\LEY and J. ABM_DI
I IS’
the lamination
angle 0. Furthermore.
of the metal matris buckling cation
load.
on the buckling
howc\.cr,
appears
is that the tcmpcraturc llic sinic
c;~scs. by almost
Let us consider laminatctl critical
loads ;lgainsl
;duminum lo~lctl
inelastic
matcly
clatic
note that
mctry Table
0. I3y a comparison
al NJ.4 smell
rcspccl
with
lamination
point
The impli-
loads.
claslic
C;ISC. the ligurc strain-rate.
than that obtainctl the buckling
exhibits,
in the present
stresscs and strains,
by applying loading.
libcr/matrix
by employing
strain
rates:
;L higher
HSDT.
tit, = IO
’
One can
syslcrn. ;I sym-
plates at the instant
prcclictcd
of the
f:or ;I sl~~wly
lost in llw inclaslic
is complctcly
the total strains of the anglc-ply
shows
of the plate is approxi-
it is not atYcctccl by the rate of the applicti buckling
of the resulting
Due to the rate sensitivity
on the applicil
[ + O]
two-laycrcd
cast.
It is
of buckling.
III
of the inulaslic s
’ 2nd IO ” 5 ‘.
seen that apprcciablc strains dcvclop belbre buckling take place for ~~nglc-ply lamination angle 0 at the vicinity of 50’. It should bc nokd that the buckling
strains
;Irc signiticantly
;tnglcs
in
metal
bchnvior
loads, in the present
the variation
the pcrfcctly
is lower
pl;ttcs [ +(I] arc given for two values 0f;kpplicd
str;lin-r:Itc
buckling
5 exhibits
angles 0, howover,
to 0 = 45 ‘, this symmetry
2 thy’ buckling
on the tcmperuture.
of afi anti-symmclric
Figure
C, this tlocrcasc tlcpcnds
the elastic
to rccorrl
It is readily plates with
dqw~dent
dccrcascs the buckling
and thcrcforu
whcrcas
with
intcrosting
by using HSIIT.
Plato’. the buckling
l:or
strain-rate.
to bc slightly
the plastic
on 0 as well. The normalid
proportion.
significantly
m;ilrix
into account
reduces the elastic and plastic
the elastic and plastic buckling
plate prodictctl
that plasticity
the ctfect of taking
load of the plate depends
this
region.
higher This
up to buckling
malris.
for the low loading
rcsull
is cxpcclctl.
is associated
with
since
rate ofanglc-ply loading
apprcciablc:
plates with lamination
of an inclaslic plastic
tlow
plate
al 3 IOW
of the viscoplaslic
Plastic buckling of metal matrix laminated
0.6
0
.30
1153
plates
.sO
Fig. 6. Critical load. normalized with respect to the corresponding elastic buckling stress, against fiber volume ratio.
It is possible
plastic buckling
to study of a metal
the effect of different matrix
composite
values of fiber volume
fraction
of on the
a [ + 301 angle-ply
To this end, consider
plate.
of 204.4”C. In Fig. 6 results for the plastic laminated plate with a/h = 40. at a temperature buckling of the plate using CPT are shown for three values of Us: 0, 0.3 and 0.5. Here, the buckling
levels are normalized
perfectly
elastic
the figure:
plate.
lO^‘s-’
which
buckles
volume
ratio
and 10e6 s-‘.
at the present incrcascs.
due to the increasing loading.
with
respect to the corresponding
Plastic buckling
loads for two different
The case of c, = 0 corresponds
value of n/h = 40 in the elastic
the buckling
ditTcrcnt
occurs in the elastic region
load rate lcvcls is absent.
and plastic
in
plate
As the reinforcement region.
This is
the plate can sustain
higher
flow of the metal matrix.
Since
far away from
the yield
point,
the efl’cct of
values of U, this ctTcct is pronounced
due
to the cxistcncc of plastic flow. For higher values of ul the cffcct of fibers is dominant the rate sensitivity of the composite plate is weak.
and
Let us study the erect angle-ply plastic
composite
buckling
observed
plate.
For modcratc
of the
to a homogeneous
region.
effect of the fibers. as a result of which
loads
rates are shown
of the plate takes place in the plastic
Thcsc high load levels lcad to yielding
for I’, = 0 buckling
buckling
loading
of number
of plies on the plastic
To this end, consider
loads obtained
that, as in the perfectly
within
buckling
a [ 5 301k with
CPT and HSDT
load level of a [ + 01,
k = I, 2.3. The resulting
arc shown
elastic case, the plastic buckling
in Fig. 7. It is readily
increases with increasing
number of plies of the laminated plate. It can be seen that HSDT predicts slightly lower buckling loads as compared with CPT. The small differences between HSDT and CPT predictions
in the present situation
high value of length-to-thickness
are attributed,
as previously
mentioned,
plate (a/h
ratio of the laminated
to the relatively
= 30).
5. CONCLUSIONS
By employing
a micromechanical
erties of metal matrix
composites,
is shown that in the determination
analysis
which can provide
the plastic buckling of the critical
the instantaneous
of laminated
loading
of the plate, the satisfaction
QC% T-204.4.C
prop-
plates is determined.
Q
HSDT
q
CPT
k Fig. 7. Buckling load against number of layers of an angle-ply laminate.
It
of the
115-l
ht.
PhLEY
and
J. ;\tltN
DI
corresponding buckling condition has to be checked at all stages of loading history. The method is illustrated for the plastic buckling analysis of boron Juminum composite plates under various situations. The effect of the elastic-viscoplastic behavior of the aluminum matrix and its rate sensitivity at elevated temperatures on the plastic buckling of the plates is presented.
REFERESCES Aboudi. J. (1989). SltcromechJnical analysis of composites by the method of ceils. .dppl. \I~Y+I RW 12. 19~~ 21. Accorct. XI. L. and Nemat-Nasser. S. ( 1986). Bounds on the overall elastic and ttwt.tnt,tneouz rl,tsrL7-pl.tstJc moduli of pertodic composites. .\lcch. .Iftrrcr. 5. 209 220. BJjlaard. P. P. ‘I.
( 1949).Theory
and nests on the pl.tstJc stabtllty of phtes
Jnd shells. J. ..I~rt~lrtltifi~.(~I.&.I. 42. 193~
Bodner. S. R. and Partom. Y. (1975). Constttutt\e equattons for CI~SIIC~~IISCO~I~~~~~C strrun-hardentng matertals. J. dppl. .Ilech. 42, 385-389. Bodner. S. R.. Naveh. M. and Merzer. A. M. (I991 ). Deformatton and buckling of axisymmetrtc viscoplastic shells under thermomechanical loading. fnf. J. Solids .Sm~cfurr.r 27. I9 I S-1924. Bushnell. D. (1982). Plastic buckling. In Pressure l’essels and Piping: DesignTechnology. .4 Decade of Progress (Edited by S. Y. Zamrik and D. Dietrich). pp. 47-l 17. ASME. New York. Hendelman, G. H. and Praper. W. (1948). Plastic buckling of a rectangular plate under edge thrusts. NACA TN
1530. Hill, R. (1958). A general theory of uniqueness and stability in elastic-plastic solids. J. Mech. Phxs. .Sok.b 6.?_%249. tlutchinron. J. W. (19743. Plastic buckling. In ;I&wcc m .4pp/icd ,Mcchmics, Vol. 14. pp. 529541. Academic Press. New York. Kapania, R. K. and I~XIII. S. t 19X9). Rcccnt advances in analysis of IamJnatcd hcam:, and plstcs. Fart I : Shear clfccts and buckltng. .~I/:t.t JI 27, 923 934.