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Hygrothermoelastic analysis of anisotropic inhomogeneous and laminated plates
J. Me&. /%ys. So&& Vol. 39, No. 1, pp. l-22, I99 1 Printed in Great Britain.
OOZ2-5096j9I $3.a~~O.OO f4? L.<1990 Pergamon Press plc
HYGROTHERMOELASTIC ANALYSIS OF ANISOTROPIC INHOMOGENEOUS AND LAMINATED PLATES S. BASI, T. G. ROGERS and A. J. M. SPENCER Department of Theoretical Mec~nics,
University of Nottingham, NottinghaIn NC7 2RD. U.K
ABSTRACT WE EXTENDthe theory proposed by KAPRIELIANet al. (to be published, 1990) for stress analysis of a moderately thick, inhomogeneous and anisotropic elastic plate to include hygrothermal effects. The inhomogeneities, both in the elastic properties and expansion coefficients, can vary arbitrarily through the thickness of the plate. The applied temperature and/or moisture fields can also vary arbitrarily through the thickness. However, for convenience the inhomogeneities are assumed to be symmetric about the midsurface of the plate, and for much of the paper we consider only thermal or moisture effects, the combined ef%ct requiring only supposition of solution. As in KAPRIELIANet al. (1990) and previous related papers by its authors, the sofntions obtained are for the full equations of three-dimensional linear elasticity, but are expressed in terms of the solution of the approximate, two-dimensional, thin-plate equations governing an ‘equivalent’ homogeneous plate. By considering laminated plates as a special case of inhomogeneous plates, we also derive an ‘accurate’ laminate theory for plates consisting of different homogeneous and anisotropic layers which are perfectly bonded to each neighbouring layer.
1.
INTRODUCTION
IN TWOrecent papers, KAPRIELIAN et al. (1988, 1990) have established a theory which gives solutions of the full three-dimensional elasticity equations for a large class of boundary-value problems involving the stretching and bending of laminated plates under the action of edge forces and couples. A dominant feature of the theory is that the solutions are expressed in terms of solutions of the approximate two-dimensional thin-plate theory (TIMOSHENKO and WOINCJWSKY-KRIEGER, 1959; CHRISTENSEN, 1979) for an ‘equivalent’ homogeneous plate of the same overall geometry; once these ‘equivalent’ solutions are obtained then the corresponding three-dimensional solutions may be derived by straightforward substitution. Furthermore, although the threedimensional solution is expressed in analytical form, there is no restriction on the form of the thin-plate solution. This can be either in analytical or numerical form, and hence existing numerical codes based on the thin-plate equations can be extended in a simple way to give three-dimensional solutions which are also much more accurate than those thin-plate solutions. Indeed, for laminates consisting of layers which are of isotropic materials, these ‘accurate’ solutions (KAPR~ELIAN et al., 1988) are exact, in that they satisfy all the
?
s l~A\;l (‘I
equilibrium
and constitutive
equations.
al-y conditions that
arc satistied
common
the solulion
When
only in an average all plate
to reflection
in the plane
all the conditions the constitutive
together
typic:tl
with
equations
are /70[ satisfied
smaller
in-plant
the shear
stresses
components
of’ powers
dimension
plates
which
are accurate
arc accurale
(see e.g.
traction
point
by point.
Venant’s
Thih
principle
I983) bc valid evcrywhcrc
in i: hcttcr.
theory
and
interfaces
and Ihc
however.
the
involved
arc
the errors
and arc known
to orders
and
01‘
I_ is ;I
For stretching deformations of synimelric about the mid-plant)
the displacement
and
of the same
commenting
remaining
plates
that the cl;~ssical
information
about
( 1990) differs
from
no
(KAIW-
211 is the plate thickness
and for bending
gives
i\
obtained
In this case.
Nevertheless.
by the problem. are ~eonietrically
It is worth
1979)
CHRISTENSEN.
at [he interlaminar
laminate
to O(c)
to O(c’),
Ihe solutions
equations.
of ;: = /I. I_. where
defined
(i.e. plales
at Icast one order
than
and WA&.
exactly.
than in the classical
in torms
symmetric
rather
and relics on Saint
of the plate.
1.1IAN ct u/.. 1990) still satisfy
magnitude
of /cro
layers
pIale
significantly
the condition
whose width is of the order of the plate thickness. are anisotropic. and in particular when their elastic symmetr!
Ihe layers
cquilibriuni
sense.
thcorics.
(GRI:GORY
usually
will then
surfaces.
with
continuity condition5 imposed on the interfacial The only approximation involved is that the cdgc hound-
to nearly
cxtcept in edge boundary onI\, with respect
together
all the
on the la~al surfaces and tractions and displacements. is ;I fcalurc
if/
the
stress
the iiccuracq laminate
shear
is
thcor)
strt’ss
con-
ponenls. The
in KAPIIIELIAN
approach
KAPIUELJAN
ot trl.
the appropriale Gary with applied
( I%#).
in the direction
lo the speci:tl.
of laminated the context
plntcs.
;IS ;I special
obtained
for the bending
extension
of the theory
etfects of specified present
include theory then
would
is linear.
with
The
IS-VI:
modest. only
the same were only ‘teniperaturc
coupling
are then
i.c. the C:I~C
;tpplication
Koc;t:i
in
19X9: Roc+~ts
KAPRII.I.I.A~~ or ~11.
(ROGI~RS, 1989) LV;I\
normal pressure, and an (RO(;IXS and SP~:I\;cIR, 1989) the on such
extension
plates. of the theory. plates.
rruisofuyic thermal
moisture
etiects,
a superposition
For
this
or
time to
simplicity
but we note th:lt
effects considered.
’ in the text.
in ~hc tcmpcrature
is simply
nioduli
obtained moduli.
of
consider\
plate under
being applied
albeit
predeccsaor. first
the claslic
itz successf~~l
WI-C the rcsuIIs
to include
to considering
analysis
except that
f:urlhermore. moisture
of those
the the
the theorb
tiolds, derived
so that
;I
for hygro-
and ttierriloelasticity. inhomogeneity
elastic moduli. plates,
fields
rcpLce
no stress
hygrothermoelasticity elasticity
not only
cfrects for inhomogeneous
would
constant
followed
of an inhomogeneous
temperature
be exactly
‘moisture’
in which
plates (SPIN(‘l:I~.
was presented
bulk of the text is restricted
of its
CLISC, but :IISO the ‘oxact’ solution
paper is ;i further.
hygrothermal
plates
c;~sc of piccewise
isotropic
that
c’t rrl. ( I’M))
to the plate; the solutions
change of approach
19X9). In these papers
rcderived
The
normal
hut important. This
ofinhomogcneoLIs
and SIJ~~I.II. (19xX)
in KAPRII:I.IAS
for i77/7oi77oyc~7~c~olr.s elastic
theory
position
c’r trl.
in that the ;inalysis
Following
and NC consider
through-thickness convenience
:lnd
now applies
to the hygrotherrnal
the previous only
coordinate the extension
tcmpcrature ;
properties
papers we again restrict
(say). to more
fields
The
first
general
that arc spatially of these
iis well
attention
dependent
restrictions
inhomogeneity
;ls to the
to symmetric is
on the
purely
can be ohtaincd
fol in ;t
Hygrothermoelastic
analysis
of laminated
plates
3
reasonably straightforward manner; the details will be presented in a future pubhcation. The restriction to z-dependent temperature fields is more difficult to relax, and is the subject of continuing study. Nevertheless, we believe the work described in this paper is still significant, in that it presents a new class of solutions in three-dimensional thermo- and/or hygro-elasticity with direct application to an area of great current interest, namely that of fibre-reinforced laminates. In Section 2 we present the full governing equations and specified boundary conditions for the general problem. As in the preceding work (KAPRIELIAN et al. 1988, 1990; SPENCER, 1989; ROGERS, 1989; ROGERS and SPENCER, 1989) we seek to express the solution in terms of that given by the simpler equations of thin-plate theory, and these are briefly reviewed in Section 3. In Section 4, the full governing equations are recast in a form which is convenient for the subsequent analysis. The advantage of restricting the present work to symmetric plates is that the problem then completely uncouples into a ‘stretching’ and a ‘bending’ mode. The former is related to the symmetric part of the temperature field, and this case is treated in Section 5. The antisymmetric component of the temperature distribution produces bending of the plate, with zero extension of its mid-plane; this case is considered in Section 6. Section 7 details the combined effects for the simple, but important, case of such a plate deforming under the action of an applied temperature field only, i.e. with zero edge tractions. Finally, in Section 8, we show the implications of these solutions for the special case of laminated plates, in which each layer consists of an orthotropic material with constant elastic moduli. In this case the theory reduces to solving simple recurrence relations for constants describing the solution in each individual layer, and highlights features, such as interlaminar shear stresses, that are particularly significant in the performance of such composites.
2.
INHOMOGENEOUSHYGROTHERMOELASTICPLATES
The theory relates to a plate of uniform thickness 2h whose mid-plane coincides with the plane z = 0 of a rectangular Cartesian system of axes with coordinates x, y, 2. All vector and tensor components are referred to this system. The components of displacement are denoted by U, z’, 11’and the components of the symmetric stress tensor c by
cr=
(LX or! c I’.? g1.j ( 0,x
or. fsVZ .
(2.1)
cJ’-, g3:1
The only material symmetry to be assumed in the analysis is that of reflectional symmetry with respect to the planes z = constant. This symmetry refers not only to the elastic moduli c,, (& j = I,. . , 6), of which there are at most 13 independent nonzero quantities, but also to the hygrothermal expansion coefficients of which there are possibly four non-zero quantities in each set of thermal and moisture coefficients x, and ,8, respectively. Thus the stressdisplacement constitutive relations take the form
7‘
where the commas denote partial difYerentiation with respect to the suffix cariablc~. lfthe material response is orthotropic then the number of independent moduli reduces to nine and the expansion coefficients to three each : in this cast. if the .\- and I’ ;IY~ coincide with the orthotropic axes (CHKISTI~NSEN, 1979). then (’/(, = C’ZI> = (‘I,, = C’J<= 0.
I,, = /i,, ==0.
If furthermore the material is transversely isotropic with the axis oftrnnsvcrsc lying parallel to the .\--axis. then in addition WC haw (‘7, = Clir
(‘12 - (‘1I,
2 ,I = c”-c’J\. _.
(‘5, = c(I(>. 2> = x1.
isotropy
/I: = /il.
leaving just live independent elastic nioduli and two each of the expansion coefficients. The very special case of isotropic behaviour is described by just two independent moduli and one expansion coefficient each for the thermal and moisture effects. with the further relation5
c’, , = (‘” = i+$.
(“; = (‘, I = ;..
C’,, = c ,, = /f.
%, = 2’ = %. /i, = /j2 = /I. The elastic moduli arc functions OF:. as atscj arc the strcsh tcmpcraturc and strcs’\ moisture expansion coefficients 3, and /i>. The moisture field M and the lemperaturc difference 7’ are measured relative to ;I stress-free reference state and are taken to be independent of _xand J’. Furthermore. quasi-static conditions are assumed so that the possible time-dependence of RI and 7‘ (and hence all the other dependent variables) is taken into account by simply trcuting the time I as ;I parameter; accordingly / is omitted from the analysis. Also omitted from now on is the moisture field Al. for reasons already discussed in the previous section. For convenience we cxprcss T as the sum of its symmetric and ~tnti-s~inruetric parts. with 7(:) = T,(z) + T,,(z), where
‘f, is an even function
of z and I’,, an odd function
7’<(z) = 7:(-z) T,,(z) = -T,,(-z)
= ;(T(z-)+T(-r)] = ;(7‘(:)-7--1)].
defined by
5
Hygrothermoelastic analysis of laminated plates The plate geometry is symmetric about the midplane expansion coefficients are all even functions of z : c&) We also introduce
= c,,(-z),
z = 0 so the elastic moduli
q(z) = a,(--2).
and
(2.5)
for later use the quantities Q,, = Q,l = ,-,, -
y,=
:l;;,
T
(2.6)
and note that Q,4 = QY = Qu = Q,5 = Qz5 = Qc
The equations
of equilibrium,
for negligible
~vvr.., + ~‘rv.v +
= 0,
y4
=
‘~5
=
0.
body force, are G.2 =
03
~vr,r+~,,.g+~,;,; =o, ~_\,$+cJ; ,.,. +o,,.,
and the boundary
condition
of zero traction
are defined by
and the bending
moments
(2.7)
on the lateral surfaces z = _th requires
Go._= c?, = cr=: = 0 Stress resultants
=0
onz = *A.
(2.8)
by
In terms of these quantities. the plate equilibrium equations (TIMOSHENKOand WOINOWSKY-KRIEGER, 1959) :
take the integrated
forms
N,,., +N,,,,. = 0, N,,,, fh’,.,.,, = 0
(2.11)
A4X5.X.Y + 2JQfX,~,., + M,., VY= 0,
(2.12)
and
where we have incorporated the boundary averages of u and I’ are defined by
conditions
(2.8). The through-thickness
(2.13)
s.
(1
and the transverse
displacement
3.
and
the in-plane
CI.ASSI~~AI.
displacement
Hw
(‘I
r/i
of’ the middle
aurfitce is denoted
ANISO~KOPK.
THIN-Pm
is approxim~tttxl
rt!
by
THeoKL
by
(3.21
The
remaining
constituti\,c
relalions
in (2.2) Lhcn give. in an obvious
notation.
Hygrothermoelastic
analysis
of laminated
plates
7
(3.7)
Thus Q,, and $, represent average values of Q,, and yI through the plate thickness, and I, involves only the symmetric part T, of the temperature since To is an odd function of z. We note that Q,, are constants defined by the plate properties alone, whereas 7, is a constant which is also dependent on the particular temperature field that is applied. For a homogeneous plate Q,, = Q,, and 7, = (E,-~,~cI,/c~J~~. Similarly substitution of (3.5) into (2.10) yields
where Q,, and y/ are weighted
averages
defined by (3.10)
and (3.1 1) Again Q,, are constants dependent on the plate properties only, and 7, are constants which depend also on the temperature through its anti-symmetric part To(z). For a homogeneous plate p”,, = Q,, and yi = (CI~ - c,+Jc~~) PO. Substitution of (3.6) into (2.11) and of (3.9) into (2.12) leads to the same equations as in isothermal elasticity (KAPRIELIAN et al., 1990; CHRISTENSEN, 1979). Thus the average in-plane displacements 17and tr satisfy ~,I~.\-,+2~,,~.,,.+~(,,~,:, ~rbU,x,~+(~IZ+~6(,)1*,,, whilst the moment
equilibrium
+~,hl,.~+(~12+~hh)i;.u~+~26LT,., fQZ& condition
+~hh2’,r+2~26Llr-,.+~*22;,1~
= 0, = 0,
(3.12)
gives
~,,~~,,,-,,+4&,,~1’.,,,,.+2(~,~+2Q”66)~,xx~~~+4e”~6~r,.~~,~~+~2)M..~,,r
= 0.
(3.13)
These equations (3.12) and (3.13) are the standard thin-plate equations for stretching and bending deformations respectively of a thin anisotropic plate. They uncouple in an obvious way and clearly demonstrate that the stretching deformations are the same as for an equivalent homogeneous plate of the same dimensions and with elastic moduli Q,,, whilst the bending is as for an equivalent plate with moduli Ql,.
s
s.
Fur~herniorc.
by being independent
li.\Sl
(‘I
<,I
of the tcmpcrature
field.
these equations
sho\c
that in the thin plate approximation the ef‘ect of the temperature on the displacement field appears only through the edge conditions through (3.6) and (3.Y). Hence the general solution is as for the isothermal cast and. following KAPKII.I.IA~~ C/ l/1. (IYYO). M’C note that the complex variable formulation described 1-q Lt-.I\;tisI rskfl (lY63) and
ZERNA ( l%O),
GKI,I:N and elaslicity
Iheory
dcvolopcd
(3.12) are obtained
for
example.
in subscqucnt
is convenient sections.
for
the three-dimensioll~ll
In this I‘oi-mul~ilion.
solution<
of
in the form Ii = Ir*c/)(.\-+.Yr).
r- =
r*c/,( \-+.\:I,)
(3.14)
and those for (3.13) in the form
Here II*, I’* and \1.* are complex constants and (b and $ are arbitrary their arguments. Substitution into (3. I?) then gives the relation
so that the complex
with complex
constant
complex
colijugatcs
ciiii be
I ,. I,. 1, and F2. Hence the general solution
written
.C.,.,V1.Also (3.13) and (3.15)
and satisties the quartic
IS?lrJ+402,,t’+7(~,‘+3~,,(,)l~+4~,,l!+~,, with roots
of
.s must bc 8 root of the quarlic
roots .s,. .s’ and their complex
show that t is similarly
functions
= 0. for the combined
(3.1X) deformation
ii5
IT = 2Re[l1,T1~,(_\-+r,~,)+11’~1//~(.\.+t~~’)].
(3.lY)
where UT, cT are the solution of (3.16) for .s = .s,. and similarly for us, PT and .s2: C/J ,. functions of the indicated complex variables and a superfixed dash denotes a derivative with respect to the argument. The stress resultants are then given by (3.6) as
Oz. I), and $ 2arc analytic
4.
GENERAL THREE-DIMENSIONAL THEORY
We now rewrite the full three-dimensional form
governing
p.z = 4,
9.: = BP
equations
in Section
2 in the
(4.1)
where
(4.2)
with
w=
ii‘-
s 0
~(3TlcJ,
d,-
(4.3)
and A and B denote linear operators whose components can involve partial derivatives with respect to x and ~1but, deliberately, not :. As outlined in Sections 5 and 6 the simultaneous equations (4.1) can then be solved recursively to obtain three-dimensional solutions which are significantly more accurate than the thin-plate theory but which are based on the solutions of that theory. By rearranging the g,: and o,.: relations in (2.2) and the equilibrium equation (2.7), it is straightforward to find ,755
A=
s45
- a/&
where
(4.4)
where
(4.6)
Q,,(Z)
and
.Itt\t
as
dcfortnation
y,(z) it1
arc
defined
as
thin-plate
as comprising
cfl’ectively charactcrircd of:
t-cspcctivcly
~ymtnetric l‘or bcnding The
in (2.6).
thcor)
(Section
a strctchins
by Ihe propcrtics
3)
it ih convcnicnt
that
p and q art‘ then
: for ;I bending mode these propcrtics
plate. the stretching only
simultaneous
mode is zxx3ated
the odd. anti-symn~ctric equations
li>r
to consider
mode and ;I bendin, (7 mode.
part
even
arc rwcrscd.
with
the
c\cn
and
For
;I
The
gcncral
for-mcr
c~ki
i\
functions
;I geometricall)
runction
It.(;).
whilst
7‘,,(z) is rclc\ ant.
p and q may be solid
itt
;I mantwr
similar
to that
Hygrothermoelastic analysis of laminated plates
II
given in KAPRIELIAN et ul. (1990), where it is shown that a series solution may be derived in terms of functions of the same complex variables x+s,y, x+sz~, x+ t ,y and x+ tg as are appropriate for the thin-plate theory (Section 3).
5.
For stretching
deformations,
STRETCHING MODE SOLUTION
the W function
is defined by
(5.1) and we note that both w and Ware zero on the midplane KAPRIELIAN rt al. (1990) show that the relevant solution
z = 0. Then the results from can be obtained in the form
U = u,~~(x+.~~~)+u*~“(x+.~1’)+o(&~~z), 2’= tlo~(x+.~~)+Z’Z~“(_Y+S~~)+O(E4d), w = w,@(X+S&Q+
W,@“(Xf.F_Y) +O(Ejd),
0,,; = Cr.,:I @‘(XfS1’)-tO(C3cJ), CrVr= a,.=,~“(X+S~)+O(&3,), CJ.; = ar,2+“‘(X+SJ’)+
O(Pa).
(5.2)
Here E = /z/L,where L is a typical in-plane length defined by the particular boundaryvalue problem, and d and 0 are typical values for the in-plane displacement and stress. It turns out that the complex constant s is defined by precisely the same equation as that given by thin-plate theory, namely (3.17), with ZQ,,u0 being constants which satisfy the same equation (3.16) as I(*. z)*. Hence the thin-plate solution can be used to generate the zeroth order approximation u - uo$(x+sY),
u - vo~(x+sJJ),
(T,, - 0,
GVZ- 0,
w - 0,
fr_._- 0
(5.3)
to the full equations. The dashes denote differentiation with respect to the arguments, so (5.2) further shows that, through the functions 4, the thin-plate solution also generates the higher-order terms. The coefficients u>, z)~, W,, etc. of these higher-order terms are functions only of Z, and are determined by solving the recursive differential equations (given in an obvious notation) :
q;n+ ,(2) = B’(z) ~z,,(;)r
pi,,+ 2(z) = A’(z)q2n+ 1(z),
(n = 0, l),
(5.4)
where A’ and B” are now matrices corresponding to A and B in (4.4) and (4.6) in which each differential operator with respect to x is replaced by unity, and each derivative with respect to J’ by s raised to the appropriate power (e.g. a2/(?y2 replaced by s’). The boundary conditions on each of these coefficient functions are generated by the overall boundary conditions (2.8) of zero traction on the lateral surfaces, and
s.
12
HASI (‘I c/1.
the symmetry conditions on the mid-plane Hence. since cf’ = II’ on I = 0, we rcyuirc
The solutions
(so that nT,.. ci,. and )I’ are Lero there).
ci ./, = (T,., = ci,., = fl._J = 0
on r = if.
II’, = F ,_, = Ti,_, = II’: L-.0
on 3 = 0.
to (5.4) are then expressible
(5.5)
in terms of ZI(~,I’,! through
It should bc noted that (S.3) and (5.6) dolinc ;I valid solution whcnc\~ .s take\ :1n1 one of its i21ur possible values. The compIcte solution for each component (II. t*. 11’. etc.) is then obtained by taking the real part of the sum of the relevant expressions with .v = .s, and .Y= .s?. Fur con\cnicncc. from now on we adopt the convention that any term involving s implics such a siilnln~~~i~~ll ovci- .s! and s2. ~mlcss otherwise stated : thus. for example, the equation (5.2), would represent
The solution is completed by detcrininin an obvious notation for C’) (0 \,.(i,,.fl,,)’ noting
g the in-pianc
stresses from (4.7). with {in
= (C’p,,)f/)‘+(C‘P~)~~“‘~-(;‘,,;‘~.;’,,)’
that only the contribution
+O(i:'O).
(5.7)
of T, is needed for ;‘,. ;lz and ;‘h. The stress resultants
Hygrothermoelastic
analysis
of laminated
plates
13
N,,, N,.,. and N,.,. are then obtained by straightforward quadrature, and we note that comparison with (3.6) shows that, to the given order of approximation,
s h
(N,,, N,., , N.JT
= (N.,.,, m,.,., gVJT + 24”’
0
C'
p,
d,_.
(5.8)
The two complex constants k, and k, in the solution represent the values of u2 and z!~ at z = 0 and are hitherto arbitrary. Accordingly they may be chosen to satisfy further conditions. The simplest choice is to require that the mid-plane displacement coincides with that given by thin-plate theory (Section 3) ; in this case u2 = u2 = 0 at z = 0 so we make k, = k, = 0. Alternatively we could require that the average values of u and u through the thickness coincide with U and 6, in which case k, and kz are chosen so that
In the isotropic case (KAPRIELIAN et al., 1988 ; ROGERS and SPENCER, 1989), yet another possible option is to require that the stress resultants N,,, N,., and N,Y,,coincide with those associated with the thin-plate solution for the equivalent plate. From (5.8) we see that such a condition would require that either 4”‘(x+sy) is zero everywhere or that k, and k2 for each value of s can be chosen such that in each case /r C”p, dz = 0.
(5.10)
s0 The first possibility defines a restricted class of solutions, one of which is treated in Section 7; in such problems the stress resultant condition imposes no constraint on the values of k, and kZ. In general, however, (5.10) would need to be satisfied. Unfortunately, this condition is equivalent to three scalar equations for just two unknowns k, and k,. Whilst it can be shown that for an isotropic material these three equations are linearly dependent, for any anisotropic material this is not the case. A less satisfactory, but possible, alternative is then to require that the traction on a family of parallel planes n,~+n~~’ = constant coincides with that associated with the equivalent thin-plate solution.
6.
BENDING MODE SOLUTION
The anti-symmetric part of the temperature field contributes mation in any problem, and the W function is now defined by w=
J@’ = M’s0
x,Tolc,,
dz;
to the bending
defor-
(6.1)
also note that on z = 0, W = w(x, y, 0) = v?(x, y), the midsurface displacement. Then from KAPRIELIAN et al. (1990) the relevant solution can be written as
we
s
BhSl (‘I cd
q(.\-+ tj,) + ~dy’(.\-+
to’) + o(t:‘(j),
I~ =
I(,
I‘
r.,I//‘(.\-+1~~)+r~,ll/“‘(.\-+/~~)+O(r:’5).
=
Cl. = U’,,I)(.\-+ [.I.) + 12’,$“(.\-+ t?,) + ITI-
Again
i: denotes
displacement (3.
linear
ci_ =
G-_;Il//““(.\-+ /r) + O(t:‘ri-).
/I/L.
exactly
c,,
II’,.
theory
provides,
The recursive ditrercnt
bending
(T,,,
from.
also
stress
through
di@erential those’ (5.4)
Furthwnore
A’ and B’ are given by replacing
IV (, is ;I constant theory.
of approximation
give the next-order
;I
which
and II,
correction
can
and
solution
in (6.2).
typical
equation
and
the
r,
bc are
:
terms
to the clasGal
how that approximate
two-dimen-
[.I,). those corrections. satislicd
for the stretching = A’(:)q,,,(:).
d now represents I satisfies the wnc
mode, the thin-plate
demonstrates
equations
value;
by the thin-plate
order
$(.v+
(6.2)
constant
in the stretching
o,,~ and o,,;
KAPRIELIANrf ul. (1990) as
where
as
theory. in (3. IS)
to the leading
pi,, / ,(-I where
in-plane
Also, OIXX again the solution
theory. sional
II;,
typical
;I
to the plate. The complex
of :. Hence,
corresponds
involving
and (T
the )I.* defined
timctions
Otii'CT).
~,,ll~“‘(.\-+f!‘)+O(i:‘(T).
normal
with
fT,IJI//“‘(.\ + 1.1‘)+
m,_ =
IX) as in thin-plate
identified
=
tl(i:‘d).
by these codlicicnts mode. They
q’z,, / z(r) = B’(r)p,,, .S with
are similar
to.
hut
now take the Ibrm / ,(-I.
t in A’ and B‘. The solution
(6.3) is given in
Hygrothermoelasticanalysisof laminatedplates
g::j =-o’
h
(orz2+ to,v2) dz =
s
dz.
(r~r.2+t~,:J
i
Again the solution is completed by substituting then (4.7) to determine the in-plane stresses as (~,rr 0, II’&Jr
15
(6.7)
for u,, z’,, ui, uj and oZZ3in (6.2) and
= (C’P,)~“+(C’P,)~““-(y,,~2,~,)T+~(~5~),
(6.8)
noting that now only the temperature contribution To is needed in the definition of y,. We note also that, in a similar manner to the stretching case with s, the terms of each over t, and t? and taking the (C’p,)@ and (C’p3)$“” imply summation real part of the sum. The bending moments then follow by simple quadrature, and comparison with (3.9) shows that h (M,,,M,.,.,
M,JT
=
(A?,,,
A?,,.,
A?,JT+2$“”
C’p,zdz. j0
(6.9)
The arbitrary constant K which appears in the typical solution above is another which is at our disposal. If we choose K = 0 then the solution has the useful property that the midplane value of W, and hence of ~7, is the same as in the corresponding classical thin-plate theory. Yet another option is available for an isotropic inhomogeneous plate (ROGERS and SPENCER, 1989) namely to choose the K so that the bending moments M,,, M,.,. and M.,, coincide with the thin-plate values A,,, A,,. and A,,.. In the present more general case, (6.9) shows that either I,P’(x+~,v) must then be zero everywhere, or that K for each value oft can be chosen such that in each case h
C’p,zdz
= 0.
(6.10)
s0 The simple problem treated in Section 7 satisfies the first possibility (I/“’ = 0). It is fairly straightforward to show that for an isotropic material (6.10) is satisfied by again making K zero. However, in general for an anisotropic material (6.10) represents three independent scalar equations, and therefore the bending moment option is no longer available.
7.
DEFORMATIONUNDER ZERO EDGE TRACTIONS
As a very simple example of the preceding theory, we consider an inhomogeneous plate, subject only to an applied temperature field. with its edge I- free from any applied traction. The procedure for solution is straightforward : we first solve the relevant thin-plate equations for the equivalent plates for stretching and bending, and then use the
s.
Ih in the
solutions present and
‘accur~~tc
cxamplc,
HASI C’/ i/l
expressions
gi\:en in the two
the free edge conditions
require
(3. 13) be solved subject to the boundary (iV,,.h’,,
Hence
l’rom
and I//:’ must Then
(3.20)
. N,)’
and (3.21)
be constants.
the ‘stretching’
=
(,v,,.
ae xc
which
.2/,,.
of integration
M,,)
1
0
In the (-3.12)
(7.1)
I‘.
is that C/J’,.C/I:. I,//;’
loss of gcncrality.
is given 1~~ (1,1 =
\-+.s,.r. is mx~.
on
solution
without
(7.2)
.\-$ .! 7I‘.
to xatislj
Hence (i,;’= C/I’;‘= 0 (x = I. 2) 2nd the cxact solution II = 2Rc
=
that the clcmcntary
part of tho solution
sections. equations
conditions
WC make unity
(/I, = \vhere the constant
0.
preceding
that the governing
mm
displaccnicnt
at the origin.
for the stretching
is
(.\-+.\,,I$/:.
C’; i
d:
I ‘.
I
(7.3) and. from
(5.7)
N,ith only
7, contributing
to ;‘,. ;‘: ami ;‘(,.
(7.4)
In (7.3)
and (7.4).
mLlst be xwh
that
ir: and /$ arc related
the zcl-0 traction
the I‘OLII. I-cal wlues
Thus
algebraic
equations.
describing
the arbitrar>
through
condition
r/T
\c;il;Ir
and
(3. 16) in
is wtislied,
II!
an
ob\,ious
i.c. through
have 10 sati~l!,
corrcspondin,
ni;lnncr.
(i.70)
three
0 to ail arbitrary
and
:
simultanwu~ rigid
hi\
rotation. The
‘bonding’
solution
is given bq I///
where slope
the constants
of integration
of the ‘equivalent’
the exact
solution
= g.\- + t / J‘)
thin
.:.
$ 2 = $\-+/:_I.)‘.
are again /cro.
plate at the origin.
for the bending
is
to x\tisfy
Hence
(7.6) XI-O
displacement
I)!,” = I//:“’ = 0 (x =
I .2)
and and
Hygrothermoelastic
u = - 2Re
c (x+ t,?)~,“, 2= 1.2
v = - 2Re
2 t,(x+ 2= I.?
rJ,_
=
o\.r = cirr =
analysis
of laminated
17
plates
t,_~)zw~,
(7.7)
0
and, from (6.8), with only T,, now contributing
to y,, y2 and y6,
(7.8)
The arbitrary constant K has been omitted from the expression for IV, corresponding only to a rigid body translation of the plate. The values for IV: and M$ in (7.7) and (7.8) must be such that the zero bending moment conditions are satisfied, so that, from (3.21),
2Re
QII+2Uz,+~;e”I, 1 IV,* e”,2+2&+t& 1= 1.7 i e”U,+2U&6+-tk~X
7, = -
1
72
(7.9)
ii ?Cl
At first sight, this equation appears to yield just three scalar equations for four unknowns, namely the real and imaginary parts of \vT and ~5. However, by rewriting (7.9) in the form
with
it is clear that X,, Xz and Xh can be uniquely determined. Then since (7.7) shows that w can be expressed in terms of X,, Xz and X6 it follows that w is uniquely determined, and hence so also are all the displacement and stress components. Finally we note that, in principle, the solution for any plate which is subject to a temperature distribution T(z) as well as applied edge loading may be obtained by simple superposition of the above solution with that obtained for the isothermal case for the same boundary-value problem.
8. A \xzry important
special
of ii number
consisting transversely
isotropic
ANISOTROPIC. LAVIIN~TI s
case of an inhomo~en~ous
of bonded
elastic
uniform
niaterials.
Iaycrs
with
plate is that
of ;I Iaminatc.
of honiogcneous
lay-ups
01‘ ditTcrenl
orthotropic
angles
for
ot
difrcrenl
laytxs. and with possibly ditferent materinls and difl?rent thickncsses. It WIS shoa n analysis, that il ih ;t in KAPRII.LIA~01 r/l. (19’90). and is clear fl-oni the preceding straiphtti~rward exercise to express the solution in this GISC in terms of the individu;tl
Ltyer
properties.
In this section. ;IS in the fortnula~ion giwn the solution in terms of constants associated constants
are detcrtnined
interpreted
dircclly
exu11ple. WC consider
the rth
thickness.
orientation
/.th layer
is identified
I’ = 0. the layer from
For
laminated
plntc comprised
other
material
WC dcno[e in some
adopted
quantity
are consecutively
the uniform
thickness
rclatcd
(r = 0) by 2/z,,. but the thickness
by It ,. hI.
, /I,,. Thus
the overall
Ii = t
laniiin
related
to the
nutmbcred
from
of the layers
ROGERSand SPENC’IIR,1989)
is now denoted
bc li)t.
ditferentl>
( KAPRIII.IA’Lc’t r/l.. I9XX :
papers
the thickness
layer
c;tn
acI.jaccnt to the lateral
SPENCER.1980; ROGERS. 1989: of the middle
tractions.
I honiogencoLts
to I’ = ,t:, the layers
previous
and
shear
I = 0 being identical
Any
propwties.
the midplanc.
convenience
of ?:I’+
we express The
boundary.
mtiinw.
and
the mid-plane
by the indcu 1’. and the laycrs
containing
the notation
in ;I simple
displacuncnt
latminae above and below and all
VI r/l. (IWO).
each intcrlaminar
formuhe
of interlaniinar
;I syninictric
nae, with
surfaces.
by recurrence
in terms
in KAPRH:LIA\ wilh
thickness
in that
WC still
of every
is 3
denote
other
layer
with
(X.1)
11,. /I
Wc donote by H, the distance
Ihc (/‘-
I)lh
from
lhe midplane
: = 0 lo the iir/~,./ric~, I = II,
bctu ccn
21nd the ,.th l21yers. so that ,
II, In addition.
= I?,,.
tic define II,,
II, =
= 0 and II,
1 /I,
(r = I. 2..
-= //. ‘IIiub
Fi, s : 6’ I I, The c&tic notation,
and
constants the
define the elastic
tix1y
;tre nou
stress
teniperaturc
coefficients
of the equivalent
dli
hycr
(X.2) ib given
13)
(X.7)
/
of the /.th Iaycr arc dcnotcd
properties
e\xluated
the
. 3).
by c):‘. Q:;’ and .\l;‘. in an ob\,ious
by x)”
The
ulues
plate for stretchin
gti bind o,, \vhich respec-
g and bending
a\ (X.4)
o,,=,;; $,(H,‘, --fI,‘)Yi;’ I
The
cquiwlent
constants
t-dated
to the temperature
field are delernlined
11s
Hygrothermoelastic
analysis
of laminated
plates
19
(8.7) Hitherto all the quantities with index Y, with the exception of H,, refer to constants associated with the rth layer. However, since we are particularly interested in interluminur values of the dependent variables, we now introduce the interluminar constants u(r), 1’o), IV(‘), etc., where these denote the values of the relevant variables at the interface between the (r - I)th and rth layers. Thus, for example, u”) = U(X, J?, H,), We also note that, with this notation,
ui” = u?(H,),
c$‘, = c,;, (H,.).
we have simple recurrence
(8.8)
relations
The stretching solution is given by (5.2), in which the coeffcients (5.6) can now be expressed for the vth layer in the form
such as
as determined
c,.;, (z) = CJ~,+ (z- Hr) (B’,‘;u. + B’,jco), for example, -By’,
in
(8.10)
where B(L’,and B’{: are the values of L?‘,, and B’; 2 in the rth layer : = Q\‘j +2sQ’;k+s2Ql;b,
-B’,‘i
= Q(;~+s(Q~~~+Q(hr~)+~2Q(;~;
hence we obtain &‘, = h,,(B(,q)uo + Bc,‘!vo)
(8.11)
and al’,:“= Similarly
o~~,+h,(B’;~u,+B’;‘lu,),
Y = I,2 ,...,
N.
(8.12)
we can write CJ;;‘~= hO(Bc,02)~0 + B\“z’q,),
(8.13)
cr.!:-: ‘) = o.\‘$+h,(B’;h,+Bi’;z~:,),
(8.14) (8.15)
WC,‘) = h,(B’,~]u, + B\$o), Wl” ‘) = WC;)+II,(B(&~
(8.16)
+ B’;{u,).
Then u7 in the rth layer (r 3 1) is given by uZ(z) = u’;’ + (.Y’;&$‘, +s&r~!;,
- WC;)} (z-H,)
+:((s’;~B:“,+s’,“,B’rZ-B~~)u,+(s:’:B’;:+s’,’:B’z’:-BB’;~)v,} Hence, using (8.1 I), (8.13) and (8. I5), we can immediately
(z-H,)‘. determine
(8.17)
s.
20
d<’
finally.
since CL? is xro I . with
rr!’
WC omit
the detail
for
Thus the solution constants
at I = II. we use’ the ‘rc\mse’
3. For
use
(X.19)
~w.4
to
(X.13). wiLh I’ = (X.22)
with ‘T,,.
recurrence
form
for I. = ?t’--
I.
the classical
(X.1X) wlucs
I, to obtain
correction
straightforward.
and
(X.20)
are then
lo
by using
as described
successively in
determine
(X.12). for
\;I~LIC~
(5.7)
plate’ ;;! from in
.Y = .s, and .Y = .s?. wc (X.14).
and hence. rccurt-ently.
can then bc used to give the
each I/,,. I’,, is coniplctcd
terms
solution with
substituted
(~:f’,.f~:‘\.
‘equivalent
expansion
thin-plate
t/(,, 1‘,, associated
term. The
and the thermal
solution
(X.22)
fly’,,
to dcterniinc
fi:“,. (X.16).
thr r = 2.
The
solution
the in-plane
fli,,. o,,.
bending
however.
(X.4).
determine
(X.15).
and (X.21).
strcsscs
from
These
andr!“.
. ,‘L’. Equation
wxxiatcd The
(X.19)
‘I+ Iv’;‘)).
+.).(oy; “+G:“,);.
is a second-order
is remarkably
each indcpendcnt
(8.11).
(5:!\. IC”,“. U;”
;ff’,‘_, “+d,i!,
IZ’, since this
procedure
Q,, are deterniincd
and these arc
Section
3..
Iv’;’
1 ” = 0:
cii”, = ai’; “+;/I,
then
‘)+ol’,‘,)-(
wc obtain
;&.~~. 3 -.....
(X.5).
oi.
<‘I
” = u’:‘+ i/l, ;.\‘i:(o:‘-; “+rr:‘!,)+.s~:(o~‘_;
Similarly.
and
HkSl
solution
the prwcdure
the interlaminar
shear
can be obtained ia to start constants
in
an analogous
at I’ = :1;. i.c. at the outside successively
with
tmanncr. Inycr.
I‘ = .I’. ,I:-
I..
In
this
UK.
and detertninc
. I. with
all
(6.5)
giving
+(H,+?H, with
rr!!’ ~~1
” = 0. The intcrlaminar values for for I’ = I. 2, . A’ with
successi\cly
/ , ) (d,‘;: + fcr:‘,);
Lb’:. I(; and I.: nia)
.
(X.34)
then bc detertnincd
Hygrothermoelastic
analysis
of laminated
21
plates
together with similar expressions for zly’, r = 1, 2,. , N. Again the solution is completed by using (5.7) to determine o ,.,, c,.,. and c,,. We make two final observations. The first is that classical laminate theory gives no information about g.,._,a,, and or_, and we note that these quantities can be calculated (e.g. (8.11) and (8.22)) without reference to any of the other quantities, both in stretching and bending. The second comment is that if in bending, for example, W, is normalized to unity, then all the interlaminar constants are independent of the deformation, and are therefore characteristics of the given laminate construction; hence they can be calculated once and for all for a given laminate and then applied to the solution
of any boundary-value
eigenvalue
s (through
problem
of the kind treated
in this paper.
The
if the ‘eigenvector’ (u,, vo) corresponding to each (3.16)) is normalized, for example, into a unit eigenvector.
same is true for the stretching
mode,
ACKNOWLEDGEMENTS During this work, one of the authors (S.B.) was supported by an SERC CASE studentship in collaboration with Rolls-Royce plc. Some of this work was also carried out with the support of NATO Collaborative Research Grant No. 0496/87. All the support is gratefully acknowledged.
REFERENCES CHRISTENSEN. R. M
1979
GREEN. A. E. and ZERNA, R. GREGORY, R. D. and WAN. F. Y. M. KAPRIELIAN,P. V., ROGERS, T. G. and SPENCER,A. J. M. KAPRII~LIAN, P. V., ROGERS, T. G. and SPENCER,A. J. M. LEKHNITSKII.S. G.
1960 1983
Mechunics of’ Composite Materials. Wiley, New York. Theoretical Elasticity. Clarendon Press. Oxford. J. Elusticit~~ 14, 27.
1988
Phil. Truns. R. Sot. (Lond.) A324, 565
1990
Submitted
1963
ROC;ERS,T. G.
1989
Theor-), of’Elasticit~~ ofun Anisotropic, Elastic Body. Holden-Day, San Francisco. In Elusticity, Mrrthemutical Methods und Applications, p. 301, (edited by R. W. OGDEN and G. EASON). Ellis Horwood, Chichester.
for publication.
OOZ2-5096j9I $3.a~~O.OO f4? L.<1990 Pergamon Press plc
HYGROTHERMOELASTIC ANALYSIS OF ANISOTROPIC INHOMOGENEOUS AND LAMINATED PLATES S. BASI, T. G. ROGERS and A. J. M. SPENCER Department of Theoretical Mec~nics,
University of Nottingham, NottinghaIn NC7 2RD. U.K
ABSTRACT WE EXTENDthe theory proposed by KAPRIELIANet al. (to be published, 1990) for stress analysis of a moderately thick, inhomogeneous and anisotropic elastic plate to include hygrothermal effects. The inhomogeneities, both in the elastic properties and expansion coefficients, can vary arbitrarily through the thickness of the plate. The applied temperature and/or moisture fields can also vary arbitrarily through the thickness. However, for convenience the inhomogeneities are assumed to be symmetric about the midsurface of the plate, and for much of the paper we consider only thermal or moisture effects, the combined ef%ct requiring only supposition of solution. As in KAPRIELIANet al. (1990) and previous related papers by its authors, the sofntions obtained are for the full equations of three-dimensional linear elasticity, but are expressed in terms of the solution of the approximate, two-dimensional, thin-plate equations governing an ‘equivalent’ homogeneous plate. By considering laminated plates as a special case of inhomogeneous plates, we also derive an ‘accurate’ laminate theory for plates consisting of different homogeneous and anisotropic layers which are perfectly bonded to each neighbouring layer.
1.
INTRODUCTION
IN TWOrecent papers, KAPRIELIAN et al. (1988, 1990) have established a theory which gives solutions of the full three-dimensional elasticity equations for a large class of boundary-value problems involving the stretching and bending of laminated plates under the action of edge forces and couples. A dominant feature of the theory is that the solutions are expressed in terms of solutions of the approximate two-dimensional thin-plate theory (TIMOSHENKO and WOINCJWSKY-KRIEGER, 1959; CHRISTENSEN, 1979) for an ‘equivalent’ homogeneous plate of the same overall geometry; once these ‘equivalent’ solutions are obtained then the corresponding three-dimensional solutions may be derived by straightforward substitution. Furthermore, although the threedimensional solution is expressed in analytical form, there is no restriction on the form of the thin-plate solution. This can be either in analytical or numerical form, and hence existing numerical codes based on the thin-plate equations can be extended in a simple way to give three-dimensional solutions which are also much more accurate than those thin-plate solutions. Indeed, for laminates consisting of layers which are of isotropic materials, these ‘accurate’ solutions (KAPR~ELIAN et al., 1988) are exact, in that they satisfy all the
?
s l~A\;l (‘I
equilibrium
and constitutive
equations.
al-y conditions that
arc satistied
common
the solulion
When
only in an average all plate
to reflection
in the plane
all the conditions the constitutive
together
typic:tl
with
equations
are /70[ satisfied
smaller
in-plant
the shear
stresses
components
of’ powers
dimension
plates
which
are accurate
arc accurale
(see e.g.
traction
point
by point.
Venant’s
Thih
principle
I983) bc valid evcrywhcrc
in i: hcttcr.
theory
and
interfaces
and Ihc
however.
the
involved
arc
the errors
and arc known
to orders
and
01‘
I_ is ;I
For stretching deformations of synimelric about the mid-plant)
the displacement
and
of the same
commenting
remaining
plates
that the cl;~ssical
information
about
( 1990) differs
from
no
(KAIW-
211 is the plate thickness
and for bending
gives
i\
obtained
In this case.
Nevertheless.
by the problem. are ~eonietrically
It is worth
1979)
CHRISTENSEN.
at [he interlaminar
laminate
to O(c)
to O(c’),
Ihe solutions
equations.
of ;: = /I. I_. where
defined
(i.e. plales
at Icast one order
than
and WA&.
exactly.
than in the classical
in torms
symmetric
rather
and relics on Saint
of the plate.
1.1IAN ct u/.. 1990) still satisfy
magnitude
of /cro
layers
pIale
significantly
the condition
whose width is of the order of the plate thickness. are anisotropic. and in particular when their elastic symmetr!
Ihe layers
cquilibriuni
sense.
thcorics.
(GRI:GORY
usually
will then
surfaces.
with
continuity condition5 imposed on the interfacial The only approximation involved is that the cdgc hound-
to nearly
cxtcept in edge boundary onI\, with respect
together
all the
on the la~al surfaces and tractions and displacements. is ;I fcalurc
if/
the
stress
the iiccuracq laminate
shear
is
thcor)
strt’ss
con-
ponenls. The
in KAPIIIELIAN
approach
KAPIUELJAN
ot trl.
the appropriale Gary with applied
( I%#).
in the direction
lo the speci:tl.
of laminated the context
plntcs.
;IS ;I special
obtained
for the bending
extension
of the theory
etfects of specified present
include theory then
would
is linear.
with
The
IS-VI:
modest. only
the same were only ‘teniperaturc
coupling
are then
i.c. the C:I~C
;tpplication
Koc;t:i
in
19X9: Roc+~ts
KAPRII.I.I.A~~ or ~11.
(ROGI~RS, 1989) LV;I\
normal pressure, and an (RO(;IXS and SP~:I\;cIR, 1989) the on such
extension
plates. of the theory. plates.
rruisofuyic thermal
moisture
etiects,
a superposition
For
this
or
time to
simplicity
but we note th:lt
effects considered.
’ in the text.
in ~hc tcmpcrature
is simply
nioduli
obtained moduli.
of
consider\
plate under
being applied
albeit
predeccsaor. first
the claslic
itz successf~~l
WI-C the rcsuIIs
to include
to considering
analysis
except that
f:urlhermore. moisture
of those
the the
the theorb
tiolds, derived
so that
;I
for hygro-
and ttierriloelasticity. inhomogeneity
elastic moduli. plates,
fields
rcpLce
no stress
hygrothermoelasticity elasticity
not only
cfrects for inhomogeneous
would
constant
followed
of an inhomogeneous
temperature
be exactly
‘moisture’
in which
plates (SPIN(‘l:I~.
was presented
bulk of the text is restricted
of its
CLISC, but :IISO the ‘oxact’ solution
paper is ;i further.
hygrothermal
plates
c;~sc of piccewise
isotropic
that
c’t rrl. ( I’M))
to the plate; the solutions
change of approach
19X9). In these papers
rcderived
The
normal
hut important. This
ofinhomogcneoLIs
and SIJ~~I.II. (19xX)
in KAPRII:I.IAS
for i77/7oi77oyc~7~c~olr.s elastic
theory
position
c’r trl.
in that the ;inalysis
Following
and NC consider
through-thickness convenience
:lnd
now applies
to the hygrotherrnal
the previous only
coordinate the extension
tcmpcrature ;
properties
papers we again restrict
(say). to more
fields
The
first
general
that arc spatially of these
iis well
attention
dependent
restrictions
inhomogeneity
;ls to the
to symmetric is
on the
purely
can be ohtaincd
fol in ;t
Hygrothermoelastic
analysis
of laminated
plates
3
reasonably straightforward manner; the details will be presented in a future pubhcation. The restriction to z-dependent temperature fields is more difficult to relax, and is the subject of continuing study. Nevertheless, we believe the work described in this paper is still significant, in that it presents a new class of solutions in three-dimensional thermo- and/or hygro-elasticity with direct application to an area of great current interest, namely that of fibre-reinforced laminates. In Section 2 we present the full governing equations and specified boundary conditions for the general problem. As in the preceding work (KAPRIELIAN et al. 1988, 1990; SPENCER, 1989; ROGERS, 1989; ROGERS and SPENCER, 1989) we seek to express the solution in terms of that given by the simpler equations of thin-plate theory, and these are briefly reviewed in Section 3. In Section 4, the full governing equations are recast in a form which is convenient for the subsequent analysis. The advantage of restricting the present work to symmetric plates is that the problem then completely uncouples into a ‘stretching’ and a ‘bending’ mode. The former is related to the symmetric part of the temperature field, and this case is treated in Section 5. The antisymmetric component of the temperature distribution produces bending of the plate, with zero extension of its mid-plane; this case is considered in Section 6. Section 7 details the combined effects for the simple, but important, case of such a plate deforming under the action of an applied temperature field only, i.e. with zero edge tractions. Finally, in Section 8, we show the implications of these solutions for the special case of laminated plates, in which each layer consists of an orthotropic material with constant elastic moduli. In this case the theory reduces to solving simple recurrence relations for constants describing the solution in each individual layer, and highlights features, such as interlaminar shear stresses, that are particularly significant in the performance of such composites.
2.
INHOMOGENEOUSHYGROTHERMOELASTICPLATES
The theory relates to a plate of uniform thickness 2h whose mid-plane coincides with the plane z = 0 of a rectangular Cartesian system of axes with coordinates x, y, 2. All vector and tensor components are referred to this system. The components of displacement are denoted by U, z’, 11’and the components of the symmetric stress tensor c by
cr=
(LX or! c I’.? g1.j ( 0,x
or. fsVZ .
(2.1)
cJ’-, g3:1
The only material symmetry to be assumed in the analysis is that of reflectional symmetry with respect to the planes z = constant. This symmetry refers not only to the elastic moduli c,, (& j = I,. . , 6), of which there are at most 13 independent nonzero quantities, but also to the hygrothermal expansion coefficients of which there are possibly four non-zero quantities in each set of thermal and moisture coefficients x, and ,8, respectively. Thus the stressdisplacement constitutive relations take the form
7‘
where the commas denote partial difYerentiation with respect to the suffix cariablc~. lfthe material response is orthotropic then the number of independent moduli reduces to nine and the expansion coefficients to three each : in this cast. if the .\- and I’ ;IY~ coincide with the orthotropic axes (CHKISTI~NSEN, 1979). then (’/(, = C’ZI> = (‘I,, = C’J<= 0.
I,, = /i,, ==0.
If furthermore the material is transversely isotropic with the axis oftrnnsvcrsc lying parallel to the .\--axis. then in addition WC haw (‘7, = Clir
(‘12 - (‘1I,
2 ,I = c”-c’J\. _.
(‘5, = c(I(>. 2> = x1.
isotropy
/I: = /il.
leaving just live independent elastic nioduli and two each of the expansion coefficients. The very special case of isotropic behaviour is described by just two independent moduli and one expansion coefficient each for the thermal and moisture effects. with the further relation5
c’, , = (‘” = i+$.
(“; = (‘, I = ;..
C’,, = c ,, = /f.
%, = 2’ = %. /i, = /j2 = /I. The elastic moduli arc functions OF:. as atscj arc the strcsh tcmpcraturc and strcs’\ moisture expansion coefficients 3, and /i>. The moisture field M and the lemperaturc difference 7’ are measured relative to ;I stress-free reference state and are taken to be independent of _xand J’. Furthermore. quasi-static conditions are assumed so that the possible time-dependence of RI and 7‘ (and hence all the other dependent variables) is taken into account by simply trcuting the time I as ;I parameter; accordingly / is omitted from the analysis. Also omitted from now on is the moisture field Al. for reasons already discussed in the previous section. For convenience we cxprcss T as the sum of its symmetric and ~tnti-s~inruetric parts. with 7(:) = T,(z) + T,,(z), where
‘f, is an even function
of z and I’,, an odd function
7’<(z) = 7:(-z) T,,(z) = -T,,(-z)
= ;(T(z-)+T(-r)] = ;(7‘(:)-7--1)].
defined by
5
Hygrothermoelastic analysis of laminated plates The plate geometry is symmetric about the midplane expansion coefficients are all even functions of z : c&) We also introduce
= c,,(-z),
z = 0 so the elastic moduli
q(z) = a,(--2).
and
(2.5)
for later use the quantities Q,, = Q,l = ,-,, -
y,=
:l;;,
T
(2.6)
and note that Q,4 = QY = Qu = Q,5 = Qz5 = Qc
The equations
of equilibrium,
for negligible
~vvr.., + ~‘rv.v +
= 0,
y4
=
‘~5
=
0.
body force, are G.2 =
03
~vr,r+~,,.g+~,;,; =o, ~_\,$+cJ; ,.,. +o,,.,
and the boundary
condition
of zero traction
are defined by
and the bending
moments
(2.7)
on the lateral surfaces z = _th requires
Go._= c?, = cr=: = 0 Stress resultants
=0
onz = *A.
(2.8)
by
In terms of these quantities. the plate equilibrium equations (TIMOSHENKOand WOINOWSKY-KRIEGER, 1959) :
take the integrated
forms
N,,., +N,,,,. = 0, N,,,, fh’,.,.,, = 0
(2.11)
A4X5.X.Y + 2JQfX,~,., + M,., VY= 0,
(2.12)
and
where we have incorporated the boundary averages of u and I’ are defined by
conditions
(2.8). The through-thickness
(2.13)
s.
(1
and the transverse
displacement
3.
and
the in-plane
CI.ASSI~~AI.
displacement
Hw
(‘I
r/i
of’ the middle
aurfitce is denoted
ANISO~KOPK.
THIN-Pm
is approxim~tttxl
rt!
by
THeoKL
by
(3.21
The
remaining
constituti\,c
relalions
in (2.2) Lhcn give. in an obvious
notation.
Hygrothermoelastic
analysis
of laminated
plates
7
(3.7)
Thus Q,, and $, represent average values of Q,, and yI through the plate thickness, and I, involves only the symmetric part T, of the temperature since To is an odd function of z. We note that Q,, are constants defined by the plate properties alone, whereas 7, is a constant which is also dependent on the particular temperature field that is applied. For a homogeneous plate Q,, = Q,, and 7, = (E,-~,~cI,/c~J~~. Similarly substitution of (3.5) into (2.10) yields
where Q,, and y/ are weighted
averages
defined by (3.10)
and (3.1 1) Again Q,, are constants dependent on the plate properties only, and 7, are constants which depend also on the temperature through its anti-symmetric part To(z). For a homogeneous plate p”,, = Q,, and yi = (CI~ - c,+Jc~~) PO. Substitution of (3.6) into (2.11) and of (3.9) into (2.12) leads to the same equations as in isothermal elasticity (KAPRIELIAN et al., 1990; CHRISTENSEN, 1979). Thus the average in-plane displacements 17and tr satisfy ~,I~.\-,+2~,,~.,,.+~(,,~,:, ~rbU,x,~+(~IZ+~6(,)1*,,, whilst the moment
equilibrium
+~,hl,.~+(~12+~hh)i;.u~+~26LT,., fQZ& condition
+~hh2’,r+2~26Llr-,.+~*22;,1~
= 0, = 0,
(3.12)
gives
~,,~~,,,-,,+4&,,~1’.,,,,.+2(~,~+2Q”66)~,xx~~~+4e”~6~r,.~~,~~+~2)M..~,,r
= 0.
(3.13)
These equations (3.12) and (3.13) are the standard thin-plate equations for stretching and bending deformations respectively of a thin anisotropic plate. They uncouple in an obvious way and clearly demonstrate that the stretching deformations are the same as for an equivalent homogeneous plate of the same dimensions and with elastic moduli Q,,, whilst the bending is as for an equivalent plate with moduli Ql,.
s
s.
Fur~herniorc.
by being independent
li.\Sl
(‘I
<,I
of the tcmpcrature
field.
these equations
sho\c
that in the thin plate approximation the ef‘ect of the temperature on the displacement field appears only through the edge conditions through (3.6) and (3.Y). Hence the general solution is as for the isothermal cast and. following KAPKII.I.IA~~ C/ l/1. (IYYO). M’C note that the complex variable formulation described 1-q Lt-.I\;tisI rskfl (lY63) and
ZERNA ( l%O),
GKI,I:N and elaslicity
Iheory
dcvolopcd
(3.12) are obtained
for
example.
in subscqucnt
is convenient sections.
for
the three-dimensioll~ll
In this I‘oi-mul~ilion.
solution<
of
in the form Ii = Ir*c/)(.\-+.Yr).
r- =
r*c/,( \-+.\:I,)
(3.14)
and those for (3.13) in the form
Here II*, I’* and \1.* are complex constants and (b and $ are arbitrary their arguments. Substitution into (3. I?) then gives the relation
so that the complex
with complex
constant
complex
colijugatcs
ciiii be
I ,. I,. 1, and F2. Hence the general solution
written
.C.,.,V1.Also (3.13) and (3.15)
and satisties the quartic
IS?lrJ+402,,t’+7(~,‘+3~,,(,)l~+4~,,l!+~,, with roots
of
.s must bc 8 root of the quarlic
roots .s,. .s’ and their complex
show that t is similarly
functions
= 0. for the combined
(3.1X) deformation
ii5
IT = 2Re[l1,T1~,(_\-+r,~,)+11’~1//~(.\.+t~~’)].
(3.lY)
where UT, cT are the solution of (3.16) for .s = .s,. and similarly for us, PT and .s2: C/J ,. functions of the indicated complex variables and a superfixed dash denotes a derivative with respect to the argument. The stress resultants are then given by (3.6) as
Oz. I), and $ 2arc analytic
4.
GENERAL THREE-DIMENSIONAL THEORY
We now rewrite the full three-dimensional form
governing
p.z = 4,
9.: = BP
equations
in Section
2 in the
(4.1)
where
(4.2)
with
w=
ii‘-
s 0
~(3TlcJ,
d,-
(4.3)
and A and B denote linear operators whose components can involve partial derivatives with respect to x and ~1but, deliberately, not :. As outlined in Sections 5 and 6 the simultaneous equations (4.1) can then be solved recursively to obtain three-dimensional solutions which are significantly more accurate than the thin-plate theory but which are based on the solutions of that theory. By rearranging the g,: and o,.: relations in (2.2) and the equilibrium equation (2.7), it is straightforward to find ,755
A=
s45
- a/&
where
(4.4)
where
(4.6)
Q,,(Z)
and
.Itt\t
as
dcfortnation
y,(z) it1
arc
defined
as
thin-plate
as comprising
cfl’ectively charactcrircd of:
t-cspcctivcly
~ymtnetric l‘or bcnding The
in (2.6).
thcor)
(Section
a strctchins
by Ihe propcrtics
3)
it ih convcnicnt
that
p and q art‘ then
: for ;I bending mode these propcrtics
plate. the stretching only
simultaneous
mode is zxx3ated
the odd. anti-symn~ctric equations
li>r
to consider
mode and ;I bendin, (7 mode.
part
even
arc rwcrscd.
with
the
c\cn
and
For
;I
The
gcncral
for-mcr
c~ki
i\
functions
;I geometricall)
runction
It.(;).
whilst
7‘,,(z) is rclc\ ant.
p and q may be solid
itt
;I mantwr
similar
to that
Hygrothermoelastic analysis of laminated plates
II
given in KAPRIELIAN et ul. (1990), where it is shown that a series solution may be derived in terms of functions of the same complex variables x+s,y, x+sz~, x+ t ,y and x+ tg as are appropriate for the thin-plate theory (Section 3).
5.
For stretching
deformations,
STRETCHING MODE SOLUTION
the W function
is defined by
(5.1) and we note that both w and Ware zero on the midplane KAPRIELIAN rt al. (1990) show that the relevant solution
z = 0. Then the results from can be obtained in the form
U = u,~~(x+.~~~)+u*~“(x+.~1’)+o(&~~z), 2’= tlo~(x+.~~)+Z’Z~“(_Y+S~~)+O(E4d), w = w,@(X+S&Q+
W,@“(Xf.F_Y) +O(Ejd),
0,,; = Cr.,:I @‘(XfS1’)-tO(C3cJ), CrVr= a,.=,~“(X+S~)+O(&3,), CJ.; = ar,2+“‘(X+SJ’)+
O(Pa).
(5.2)
Here E = /z/L,where L is a typical in-plane length defined by the particular boundaryvalue problem, and d and 0 are typical values for the in-plane displacement and stress. It turns out that the complex constant s is defined by precisely the same equation as that given by thin-plate theory, namely (3.17), with ZQ,,u0 being constants which satisfy the same equation (3.16) as I(*. z)*. Hence the thin-plate solution can be used to generate the zeroth order approximation u - uo$(x+sY),
u - vo~(x+sJJ),
(T,, - 0,
GVZ- 0,
w - 0,
fr_._- 0
(5.3)
to the full equations. The dashes denote differentiation with respect to the arguments, so (5.2) further shows that, through the functions 4, the thin-plate solution also generates the higher-order terms. The coefficients u>, z)~, W,, etc. of these higher-order terms are functions only of Z, and are determined by solving the recursive differential equations (given in an obvious notation) :
q;n+ ,(2) = B’(z) ~z,,(;)r
pi,,+ 2(z) = A’(z)q2n+ 1(z),
(n = 0, l),
(5.4)
where A’ and B” are now matrices corresponding to A and B in (4.4) and (4.6) in which each differential operator with respect to x is replaced by unity, and each derivative with respect to J’ by s raised to the appropriate power (e.g. a2/(?y2 replaced by s’). The boundary conditions on each of these coefficient functions are generated by the overall boundary conditions (2.8) of zero traction on the lateral surfaces, and
s.
12
HASI (‘I c/1.
the symmetry conditions on the mid-plane Hence. since cf’ = II’ on I = 0, we rcyuirc
The solutions
(so that nT,.. ci,. and )I’ are Lero there).
ci ./, = (T,., = ci,., = fl._J = 0
on r = if.
II’, = F ,_, = Ti,_, = II’: L-.0
on 3 = 0.
to (5.4) are then expressible
(5.5)
in terms of ZI(~,I’,! through
It should bc noted that (S.3) and (5.6) dolinc ;I valid solution whcnc\~ .s take\ :1n1 one of its i21ur possible values. The compIcte solution for each component (II. t*. 11’. etc.) is then obtained by taking the real part of the sum of the relevant expressions with .v = .s, and .Y= .s?. Fur con\cnicncc. from now on we adopt the convention that any term involving s implics such a siilnln~~~i~~ll ovci- .s! and s2. ~mlcss otherwise stated : thus. for example, the equation (5.2), would represent
The solution is completed by detcrininin an obvious notation for C’) (0 \,.(i,,.fl,,)’ noting
g the in-pianc
stresses from (4.7). with {in
= (C’p,,)f/)‘+(C‘P~)~~“‘~-(;‘,,;‘~.;’,,)’
that only the contribution
+O(i:'O).
(5.7)
of T, is needed for ;‘,. ;lz and ;‘h. The stress resultants
Hygrothermoelastic
analysis
of laminated
plates
13
N,,, N,.,. and N,.,. are then obtained by straightforward quadrature, and we note that comparison with (3.6) shows that, to the given order of approximation,
s h
(N,,, N,., , N.JT
= (N.,.,, m,.,., gVJT + 24”’
0
C'
p,
d,_.
(5.8)
The two complex constants k, and k, in the solution represent the values of u2 and z!~ at z = 0 and are hitherto arbitrary. Accordingly they may be chosen to satisfy further conditions. The simplest choice is to require that the mid-plane displacement coincides with that given by thin-plate theory (Section 3) ; in this case u2 = u2 = 0 at z = 0 so we make k, = k, = 0. Alternatively we could require that the average values of u and u through the thickness coincide with U and 6, in which case k, and kz are chosen so that
In the isotropic case (KAPRIELIAN et al., 1988 ; ROGERS and SPENCER, 1989), yet another possible option is to require that the stress resultants N,,, N,., and N,Y,,coincide with those associated with the thin-plate solution for the equivalent plate. From (5.8) we see that such a condition would require that either 4”‘(x+sy) is zero everywhere or that k, and k2 for each value of s can be chosen such that in each case /r C”p, dz = 0.
(5.10)
s0 The first possibility defines a restricted class of solutions, one of which is treated in Section 7; in such problems the stress resultant condition imposes no constraint on the values of k, and kZ. In general, however, (5.10) would need to be satisfied. Unfortunately, this condition is equivalent to three scalar equations for just two unknowns k, and k,. Whilst it can be shown that for an isotropic material these three equations are linearly dependent, for any anisotropic material this is not the case. A less satisfactory, but possible, alternative is then to require that the traction on a family of parallel planes n,~+n~~’ = constant coincides with that associated with the equivalent thin-plate solution.
6.
BENDING MODE SOLUTION
The anti-symmetric part of the temperature field contributes mation in any problem, and the W function is now defined by w=
J@’ = M’s0
x,Tolc,,
dz;
to the bending
defor-
(6.1)
also note that on z = 0, W = w(x, y, 0) = v?(x, y), the midsurface displacement. Then from KAPRIELIAN et al. (1990) the relevant solution can be written as
we
s
BhSl (‘I cd
q(.\-+ tj,) + ~dy’(.\-+
to’) + o(t:‘(j),
I~ =
I(,
I‘
r.,I//‘(.\-+1~~)+r~,ll/“‘(.\-+/~~)+O(r:’5).
=
Cl. = U’,,I)(.\-+ [.I.) + 12’,$“(.\-+ t?,) + ITI-
Again
i: denotes
displacement (3.
linear
ci_ =
G-_;Il//““(.\-+ /r) + O(t:‘ri-).
/I/L.
exactly
c,,
II’,.
theory
provides,
The recursive ditrercnt
bending
(T,,,
from.
also
stress
through
di@erential those’ (5.4)
Furthwnore
A’ and B’ are given by replacing
IV (, is ;I constant theory.
of approximation
give the next-order
;I
which
and II,
correction
can
and
solution
in (6.2).
typical
equation
and
the
r,
bc are
:
terms
to the clasGal
how that approximate
two-dimen-
[.I,). those corrections. satislicd
for the stretching = A’(:)q,,,(:).
d now represents I satisfies the wnc
mode, the thin-plate
demonstrates
equations
value;
by the thin-plate
order
$(.v+
(6.2)
constant
in the stretching
o,,~ and o,,;
KAPRIELIANrf ul. (1990) as
where
as
theory. in (3. IS)
to the leading
pi,, / ,(-I where
in-plane
Also, OIXX again the solution
theory. sional
II;,
typical
;I
to the plate. The complex
of :. Hence,
corresponds
involving
and (T
the )I.* defined
timctions
Otii'CT).
~,,ll~“‘(.\-+f!‘)+O(i:‘(T).
normal
with
fT,IJI//“‘(.\ + 1.1‘)+
m,_ =
IX) as in thin-plate
identified
=
tl(i:‘d).
by these codlicicnts mode. They
q’z,, / z(r) = B’(r)p,,, .S with
are similar
to.
hut
now take the Ibrm / ,(-I.
t in A’ and B‘. The solution
(6.3) is given in
Hygrothermoelasticanalysisof laminatedplates
g::j =-o’
h
(orz2+ to,v2) dz =
s
dz.
(r~r.2+t~,:J
i
Again the solution is completed by substituting then (4.7) to determine the in-plane stresses as (~,rr 0, II’&Jr
15
(6.7)
for u,, z’,, ui, uj and oZZ3in (6.2) and
= (C’P,)~“+(C’P,)~““-(y,,~2,~,)T+~(~5~),
(6.8)
noting that now only the temperature contribution To is needed in the definition of y,. We note also that, in a similar manner to the stretching case with s, the terms of each over t, and t? and taking the (C’p,)@ and (C’p3)$“” imply summation real part of the sum. The bending moments then follow by simple quadrature, and comparison with (3.9) shows that h (M,,,M,.,.,
M,JT
=
(A?,,,
A?,,.,
A?,JT+2$“”
C’p,zdz. j0
(6.9)
The arbitrary constant K which appears in the typical solution above is another which is at our disposal. If we choose K = 0 then the solution has the useful property that the midplane value of W, and hence of ~7, is the same as in the corresponding classical thin-plate theory. Yet another option is available for an isotropic inhomogeneous plate (ROGERS and SPENCER, 1989) namely to choose the K so that the bending moments M,,, M,.,. and M.,, coincide with the thin-plate values A,,, A,,. and A,,.. In the present more general case, (6.9) shows that either I,P’(x+~,v) must then be zero everywhere, or that K for each value oft can be chosen such that in each case h
C’p,zdz
= 0.
(6.10)
s0 The simple problem treated in Section 7 satisfies the first possibility (I/“’ = 0). It is fairly straightforward to show that for an isotropic material (6.10) is satisfied by again making K zero. However, in general for an anisotropic material (6.10) represents three independent scalar equations, and therefore the bending moment option is no longer available.
7.
DEFORMATIONUNDER ZERO EDGE TRACTIONS
As a very simple example of the preceding theory, we consider an inhomogeneous plate, subject only to an applied temperature field. with its edge I- free from any applied traction. The procedure for solution is straightforward : we first solve the relevant thin-plate equations for the equivalent plates for stretching and bending, and then use the
s.
Ih in the
solutions present and
‘accur~~tc
cxamplc,
HASI C’/ i/l
expressions
gi\:en in the two
the free edge conditions
require
(3. 13) be solved subject to the boundary (iV,,.h’,,
Hence
l’rom
and I//:’ must Then
(3.20)
. N,)’
and (3.21)
be constants.
the ‘stretching’
=
(,v,,.
ae xc
which
.2/,,.
of integration
M,,)
1
0
In the (-3.12)
(7.1)
I‘.
is that C/J’,.C/I:. I,//;’
loss of gcncrality.
is given 1~~ (1,1 =
\-+.s,.r. is mx~.
on
solution
without
(7.2)
.\-$ .! 7I‘.
to xatislj
Hence (i,;’= C/I’;‘= 0 (x = I. 2) 2nd the cxact solution II = 2Rc
=
that the clcmcntary
part of tho solution
sections. equations
conditions
WC make unity
(/I, = \vhere the constant
0.
preceding
that the governing
mm
displaccnicnt
at the origin.
for the stretching
is
(.\-+.\,,I$/:.
C’; i
d:
I ‘.
I
(7.3) and. from
(5.7)
N,ith only
7, contributing
to ;‘,. ;‘: ami ;‘(,.
(7.4)
In (7.3)
and (7.4).
mLlst be xwh
that
ir: and /$ arc related
the zcl-0 traction
the I‘OLII. I-cal wlues
Thus
algebraic
equations.
describing
the arbitrar>
through
condition
r/T
\c;il;Ir
and
(3. 16) in
is wtislied,
II!
an
ob\,ious
i.c. through
have 10 sati~l!,
corrcspondin,
ni;lnncr.
(i.70)
three
0 to ail arbitrary
and
:
simultanwu~ rigid
hi\
rotation. The
‘bonding’
solution
is given bq I///
where slope
the constants
of integration
of the ‘equivalent’
the exact
solution
= g.\- + t / J‘)
thin
.:.
$ 2 = $\-+/:_I.)‘.
are again /cro.
plate at the origin.
for the bending
is
to x\tisfy
Hence
(7.6) XI-O
displacement
I)!,” = I//:“’ = 0 (x =
I .2)
and and
Hygrothermoelastic
u = - 2Re
c (x+ t,?)~,“, 2= 1.2
v = - 2Re
2 t,(x+ 2= I.?
rJ,_
=
o\.r = cirr =
analysis
of laminated
17
plates
t,_~)zw~,
(7.7)
0
and, from (6.8), with only T,, now contributing
to y,, y2 and y6,
(7.8)
The arbitrary constant K has been omitted from the expression for IV, corresponding only to a rigid body translation of the plate. The values for IV: and M$ in (7.7) and (7.8) must be such that the zero bending moment conditions are satisfied, so that, from (3.21),
2Re
QII+2Uz,+~;e”I, 1 IV,* e”,2+2&+t& 1= 1.7 i e”U,+2U&6+-tk~X
7, = -
1
72
(7.9)
ii ?Cl
At first sight, this equation appears to yield just three scalar equations for four unknowns, namely the real and imaginary parts of \vT and ~5. However, by rewriting (7.9) in the form
with
it is clear that X,, Xz and Xh can be uniquely determined. Then since (7.7) shows that w can be expressed in terms of X,, Xz and X6 it follows that w is uniquely determined, and hence so also are all the displacement and stress components. Finally we note that, in principle, the solution for any plate which is subject to a temperature distribution T(z) as well as applied edge loading may be obtained by simple superposition of the above solution with that obtained for the isothermal case for the same boundary-value problem.
8. A \xzry important
special
of ii number
consisting transversely
isotropic
ANISOTROPIC. LAVIIN~TI s
case of an inhomo~en~ous
of bonded
elastic
uniform
niaterials.
Iaycrs
with
plate is that
of ;I Iaminatc.
of honiogcneous
lay-ups
01‘ ditTcrenl
orthotropic
angles
for
ot
difrcrenl
laytxs. and with possibly ditferent materinls and difl?rent thickncsses. It WIS shoa n analysis, that il ih ;t in KAPRII.LIA~01 r/l. (19’90). and is clear fl-oni the preceding straiphtti~rward exercise to express the solution in this GISC in terms of the individu;tl
Ltyer
properties.
In this section. ;IS in the fortnula~ion giwn the solution in terms of constants associated constants
are detcrtnined
interpreted
dircclly
exu11ple. WC consider
the rth
thickness.
orientation
/.th layer
is identified
I’ = 0. the layer from
For
laminated
plntc comprised
other
material
WC dcno[e in some
adopted
quantity
are consecutively
the uniform
thickness
rclatcd
(r = 0) by 2/z,,. but the thickness
by It ,. hI.
, /I,,. Thus
the overall
Ii = t
laniiin
related
to the
nutmbcred
from
of the layers
ROGERSand SPENC’IIR,1989)
is now denoted
bc li)t.
ditferentl>
( KAPRIII.IA’Lc’t r/l.. I9XX :
papers
the thickness
layer
c;tn
acI.jaccnt to the lateral
SPENCER.1980; ROGERS. 1989: of the middle
tractions.
I honiogencoLts
to I’ = ,t:, the layers
previous
and
shear
I = 0 being identical
Any
propwties.
the midplanc.
convenience
of ?:I’+
we express The
boundary.
mtiinw.
and
the mid-plane
by the indcu 1’. and the laycrs
containing
the notation
in ;I simple
displacuncnt
latminae above and below and all
VI r/l. (IWO).
each intcrlaminar
formuhe
of interlaniinar
;I syninictric
nae, with
surfaces.
by recurrence
in terms
in KAPRH:LIA\ wilh
thickness
in that
WC still
of every
is 3
denote
other
layer
with
(X.1)
11,. /I
Wc donote by H, the distance
Ihc (/‘-
I)lh
from
lhe midplane
: = 0 lo the iir/~,./ric~, I = II,
bctu ccn
21nd the ,.th l21yers. so that ,
II, In addition.
= I?,,.
tic define II,,
II, =
= 0 and II,
1 /I,
(r = I. 2..
-= //. ‘IIiub
Fi, s : 6’ I I, The c&tic notation,
and
constants the
define the elastic
tix1y
;tre nou
stress
teniperaturc
coefficients
of the equivalent
dli
hycr
(X.2) ib given
13)
(X.7)
/
of the /.th Iaycr arc dcnotcd
properties
e\xluated
the
. 3).
by c):‘. Q:;’ and .\l;‘. in an ob\,ious
by x)”
The
ulues
plate for stretchin
gti bind o,, \vhich respec-
g and bending
a\ (X.4)
o,,=,;; $,(H,‘, --fI,‘)Yi;’ I
The
cquiwlent
constants
t-dated
to the temperature
field are delernlined
11s
Hygrothermoelastic
analysis
of laminated
plates
19
(8.7) Hitherto all the quantities with index Y, with the exception of H,, refer to constants associated with the rth layer. However, since we are particularly interested in interluminur values of the dependent variables, we now introduce the interluminar constants u(r), 1’o), IV(‘), etc., where these denote the values of the relevant variables at the interface between the (r - I)th and rth layers. Thus, for example, u”) = U(X, J?, H,), We also note that, with this notation,
ui” = u?(H,),
c$‘, = c,;, (H,.).
we have simple recurrence
(8.8)
relations
The stretching solution is given by (5.2), in which the coeffcients (5.6) can now be expressed for the vth layer in the form
such as
as determined
c,.;, (z) = CJ~,+ (z- Hr) (B’,‘;u. + B’,jco), for example, -By’,
in
(8.10)
where B(L’,and B’{: are the values of L?‘,, and B’; 2 in the rth layer : = Q\‘j +2sQ’;k+s2Ql;b,
-B’,‘i
= Q(;~+s(Q~~~+Q(hr~)+~2Q(;~;
hence we obtain &‘, = h,,(B(,q)uo + Bc,‘!vo)
(8.11)
and al’,:“= Similarly
o~~,+h,(B’;~u,+B’;‘lu,),
Y = I,2 ,...,
N.
(8.12)
we can write CJ;;‘~= hO(Bc,02)~0 + B\“z’q,),
(8.13)
cr.!:-: ‘) = o.\‘$+h,(B’;h,+Bi’;z~:,),
(8.14) (8.15)
WC,‘) = h,(B’,~]u, + B\$o), Wl” ‘) = WC;)+II,(B(&~
(8.16)
+ B’;{u,).
Then u7 in the rth layer (r 3 1) is given by uZ(z) = u’;’ + (.Y’;&$‘, +s&r~!;,
- WC;)} (z-H,)
+:((s’;~B:“,+s’,“,B’rZ-B~~)u,+(s:’:B’;:+s’,’:B’z’:-BB’;~)v,} Hence, using (8.1 I), (8.13) and (8. I5), we can immediately
(z-H,)‘. determine
(8.17)
s.
20
d<’
finally.
since CL? is xro I . with
rr!’
WC omit
the detail
for
Thus the solution constants
at I = II. we use’ the ‘rc\mse’
3. For
use
(X.19)
~w.4
to
(X.13). wiLh I’ = (X.22)
with ‘T,,.
recurrence
form
for I. = ?t’--
I.
the classical
(X.1X) wlucs
I, to obtain
correction
straightforward.
and
(X.20)
are then
lo
by using
as described
successively in
determine
(X.12). for
\;I~LIC~
(5.7)
plate’ ;;! from in
.Y = .s, and .Y = .s?. wc (X.14).
and hence. rccurt-ently.
can then bc used to give the
each I/,,. I’,, is coniplctcd
terms
solution with
substituted
(~:f’,.f~:‘\.
‘equivalent
expansion
thin-plate
t/(,, 1‘,, associated
term. The
and the thermal
solution
(X.22)
fly’,,
to dcterniinc
fi:“,. (X.16).
thr r = 2.
The
solution
the in-plane
fli,,. o,,.
bending
however.
(X.4).
determine
(X.15).
and (X.21).
strcsscs
from
These
andr!“.
. ,‘L’. Equation
wxxiatcd The
(X.19)
‘I+ Iv’;‘)).
+.).(oy; “+G:“,);.
is a second-order
is remarkably
each indcpendcnt
(8.11).
(5:!\. IC”,“. U;”
;ff’,‘_, “+d,i!,
IZ’, since this
procedure
Q,, are deterniincd
and these arc
Section
3..
Iv’;’
1 ” = 0:
cii”, = ai’; “+;/I,
then
‘)+ol’,‘,)-(
wc obtain
;&.~~. 3 -.....
(X.5).
oi.
<‘I
” = u’:‘+ i/l, ;.\‘i:(o:‘-; “+rr:‘!,)+.s~:(o~‘_;
Similarly.
and
HkSl
solution
the prwcdure
the interlaminar
shear
can be obtained ia to start constants
in
an analogous
at I’ = :1;. i.c. at the outside successively
with
tmanncr. Inycr.
I‘ = .I’. ,I:-
I..
In
this
UK.
and detertninc
. I. with
all
(6.5)
giving
+(H,+?H, with
rr!!’ ~~1
” = 0. The intcrlaminar values for for I’ = I. 2, . A’ with
successi\cly
/ , ) (d,‘;: + fcr:‘,);
Lb’:. I(; and I.: nia)
.
(X.34)
then bc detertnincd
Hygrothermoelastic
analysis
of laminated
21
plates
together with similar expressions for zly’, r = 1, 2,. , N. Again the solution is completed by using (5.7) to determine o ,.,, c,.,. and c,,. We make two final observations. The first is that classical laminate theory gives no information about g.,._,a,, and or_, and we note that these quantities can be calculated (e.g. (8.11) and (8.22)) without reference to any of the other quantities, both in stretching and bending. The second comment is that if in bending, for example, W, is normalized to unity, then all the interlaminar constants are independent of the deformation, and are therefore characteristics of the given laminate construction; hence they can be calculated once and for all for a given laminate and then applied to the solution
of any boundary-value
eigenvalue
s (through
problem
of the kind treated
in this paper.
The
if the ‘eigenvector’ (u,, vo) corresponding to each (3.16)) is normalized, for example, into a unit eigenvector.
same is true for the stretching
mode,
ACKNOWLEDGEMENTS During this work, one of the authors (S.B.) was supported by an SERC CASE studentship in collaboration with Rolls-Royce plc. Some of this work was also carried out with the support of NATO Collaborative Research Grant No. 0496/87. All the support is gratefully acknowledged.
REFERENCES CHRISTENSEN. R. M
1979
GREEN. A. E. and ZERNA, R. GREGORY, R. D. and WAN. F. Y. M. KAPRIELIAN,P. V., ROGERS, T. G. and SPENCER,A. J. M. KAPRII~LIAN, P. V., ROGERS, T. G. and SPENCER,A. J. M. LEKHNITSKII.S. G.
1960 1983
Mechunics of’ Composite Materials. Wiley, New York. Theoretical Elasticity. Clarendon Press. Oxford. J. Elusticit~~ 14, 27.
1988
Phil. Truns. R. Sot. (Lond.) A324, 565
1990
Submitted
1963
ROC;ERS,T. G.
1989
Theor-), of’Elasticit~~ ofun Anisotropic, Elastic Body. Holden-Day, San Francisco. In Elusticity, Mrrthemutical Methods und Applications, p. 301, (edited by R. W. OGDEN and G. EASON). Ellis Horwood, Chichester.
for publication.