Internat. J. Math. & Math. Sci.
VOL. ii NO. ii (1988) 177-186
177
DYNAMICS OF MULTILAYERED ORTHOTROPIC
VISCOELASTIC PLATES OF MAXWELL SOLIDS
P. PAL ROY
Blasting Department
Central Mining Research Station
Barwa Road, Dhanbad 826 001 India
and
L. DEBNATH
Department of Mathematics
University of Central Florida
Orlando, Florida 32816, U.S.A.
(Received September 9, 1985 and in revised form January 30, 1987)
ABSTRACT.
This paper is concerned with a simplified dynamical analysis of orthotroplc
viscoelastic plates that are made up of an arbitrary number of layers each of which is
a Maxwell type solid.
themselves
constituted
This study includes the case where some or all the layers are
by
thinly
laminated
materials
with
couple
stresses.
The
recurrence equations for the shear stresses are obtained for an arbitrary number of
layers and then applied to plates with two or three layers.
The viscoelastic damping
effect is determined by the process of llnearlzatlon and then illustrated by a plate
composed of one, two or three layers.
anlsotropy and wave number.
KEYWORDS AND PHRASES.
It is found that the damping increases with
These results are shown by graphical representations.
Orthotropic viscoelastic materials, Maxwell solids, Laminated
materials and couple stresses.
1980 AMS SUBJECT CLASSIFICATION CODE:
I.
73F
INTRODUCTION.
Based upon the general
elasticity due to Blot
theory of ultilayered continua in finite anisotroplc
[I-4], Pal Roy [5,6] has
studied the problems of elastic wave
propagation in a thinly layered laminated medium with stress couples under initial
stresses.
In these studies the composite structures are assumed to be made up of
multilayered
elastic
materials
applications to real solids.
and
hence
some
modifications
are
required
for
It is almost impossible to study statics or dynamics of
real solids without suitable approximations and/or assumptions.
However, the Maxwell
solid is generally believed to be one of the best practical examples of real solids.
It is assumed that the Maxwell solid consists of a series of spring with a viscous
(see Biswas [7]), and the main factor which makes this
From a practical point
model different from the elastic one is the relaxation time.
element known as
’dash pot’
P. PAL ROY and L. DEBNATH
178
of view, the study of statics or dynamics of Maxwell solids seems to be important in
its own merit and has physical applications.
In
this
paper,
we
study
simplified
a
dynamical
analysis
of
orthotroplc
viscoelastic plates that are made up of an arbitrary ntnber of layers each of which is
a Maxwell type solid.
This problem Includes the case where some or all the layers are
themselves made of thlnly laminated materials with couple stresses.
The recurrence
equations for the shear stresses are derived for an arbitrary number of layers and
then applied to plates with two or three layers.
is examined with examples.
d. amplng
Results are shown by graphical representations.
and wave number.
2.
The effect of the viscoelastic
The damping is found to increase with anlsotropy
BASIC EQUATIONS AND ASSUMPTIONS
We consider a plane strain deformation of a plate of Maxwell solids of thickness
h.
The x-axls is chosen midway between the boundarles of the two faces and the y-axls
is normal to them.
The stress-straln relations for the Maxwell solid with the effects of relaxation
x
time,
are given by [7]:
d
d
d-"
o
d
xx
d-F
2
e
xx
o
d
xy 2W-exy
_.Y.l
2- eyy
-Y-Y-
d
Oyy
d’--"
3u
e
xx
(2 2)
o
(2 3)
:
t
where the strain components
(2
xx
are
e
}v
eyy
"y
3v
exy -x +
3u
(2.4abc)
The corresponding dynamical equilllbrlum equations are
o
xx+
}y
}o
where
u
and
oij
v
P
}2 u
}o
+
(2.5)
3t 2
}y
xy
}x
o xy
}2 v
yy
(2.6)
3t 2
}y
are stress components,
is the modulus of rigidity,
are displacement components.
(2.6) are functions of
y
P is the density,
The coefficients involved in equations (2.1)
as the plate may be inhomogeneous with continuously or
discontinuously stratified.
We assume that the dlsplacment fields are harmonic function of time
t
and
slnusoldally distributed along the x-dlrectlon so that
u
U(y) sin mx e
(2.7)
v
V cos mx e
(2.8)
MULTILAYERED ORTHOTROPIC VISCOELASTIC PLATES OF MAXWELL SOLIDS
179
is the frequency, V is assumed to be a constant and equal to its average
where
h.
value across the thickness
From the equation (2.2) we obtain
A(y) e
sin mx
-
where M and A(y) are
(I
M
it
(2.9)
-1
and A(y)
(2.10ab)
m V
It follows from (2.1) and (2.7) that
o
Eliminating
o
xx
2Mm U(y) cos mx e
it
"’"z.*i)
and U from (2.11) and (2.5), we obtain
xx
dA
d
y
d-
2m
2
A(y)
2m
3
(2.12)
V
2
where
(I
N(y)
N
2Mm
(2.13)
2
The solution for the shearing stress A(y) is determined from equation (2.12) with
and A for A(y) at the top, y
2
the boundary conditions
-h/2 of he plate.
h/2 and the bottom, y
This solution still contains the unknown constant deflection V
which is determined by integrating the equilibrium equation (2.6) with respect to
y
so that
h/2
f
Mm
S
A(y) my
m
2
v
Pt
(2.14)
-h/2
where the total mass per unit area of the plate face is given by
h/2
PC
f
(2.15)
P(y) dy
-h/2
and
[Oyy]l [yy]2
S cos mx e
represents the total normal load applied to the same unit area.
(2.16)
When this normal load
(2.14), and hence te shear stress
Once we know o
the value of
distribution A(y).
Thus
is known from (2.9).
U(y) is determined by combining (2.5) to obtain
is
V
known, the deflection
d
U(y)
2
N m
Also we obtain
o
xx
o
xx
is obtained from
dy
(2.17)
A(y)
from equation (2.11) in the form
2M
mN
__d A(y)
dy
cos mx
it
(2.18)
P. PAL ROY and L. DEBNATH
180
3.
EXTENSION TO MULTILAYERED PLATES AND VISCOELASTICITY.
The above results can easily be extended to a multllayered plate constituted by a
superposition of
N
Within each layer
solid.
layers each of which is a Maxwell type
thin adherent homogeneous
Considering first a
is supposed to be a constant.
h
single layer of thickness
A
with
and A
as the shear stresses at the top and
2
bottom of the layer, it turns out from differential equation (2.12) that
A
cosh Bray + C
C
p
where
2
2Mm
C
C2
Y
(A +
[
"
)
A2)
(At
U
The displacement amplitude
(3.1)
Bray- mV
2
(3.2)
2
(3.3)
+ mV] cos h
BY
slnh
(3.4)
BY
i
(I
(mh/2), M
sinh
2
-I
(3.Sab)
--)
is given by equation
(2.17).
Its values
U
and U
2
at
the top and bottom of the layer are found to be
a
with
__I
UI
/2N m
U2
2N
tanh
__I
BY +
tanh
+
(Alb
m
1,2...n) of thickness
p
h
A2a)
tanh
b
BY
We consider now a plate composed of
(J
+ 2Vc
(Ala + A2b)
(3.6)
(3.7)
2Vc
BY
tanh
(Uj, Aj)
and
orthotropic homogeneous layers.
n
bj
as it is found for the lower face of the
Aj Dj
Thus
Ej
when
Jth layer
and a mass
The displacement
are respectively
+
2N
(3.7)
(Zj+ Ej+ I) Aj+I+ Aj+ 2 Dj+ I- -2V(cj
+ c
j+i
(3.9)
b
aj
the values
J and (J+l) the displacement amplitude
Jth layer and upper face of the (J+l)th
We obtain from equations (3.6)
layer, must be the same.
where
The
j, Tj
MJth layers
Nj.
(3.8abc)
(Uj+I, Aj+l).
Now, at the interface between the layers
Uj+
BY
tanh
is characterized by coefficients
with the corresponding parameters
cj
aj
j
amplitude and shear stress at the top and bottom of the
density
=
c
BY’
m
of
Dj
the
j
?2N
shear
(3.10ab)
m
stresses
A
and
An+
are
given at
the
outer
boundaries, the recurrence equations (3.9) will lead to the evaulatulon of the (n-l)
A(j--l,3...n) at the interfaces.
normal load
Sj acting on the Jth layer
shear stresses
The
(3.1) and has the form
is obtained from equations (2.14) and
MULTILAYERED ORTHOTROPIC VISCOELASTIC PLATES OF MAXWELL SOLIDS
181
c.
m2V hjMj(l -y.)- m2
+
Aj+ I)
0j V hj
(3.11)
applied to the multi-layered plate is obtained by summing the loads
S
The total load
+
-Mjcj(Aj
Sj
We obtain
S. applied to each layer.
n
>.’
S
S
j--I
-ZjM j
J
c
j
+
Aj+
hj
M (I
(Aj
m2V
+
m-
m
2
t
(3.12)
V
c.
I
Pt
with
I
K
hj
-_3_), j=
Yj
(3.12) will provide the only unknown
The equation
unknown
p
V
the shear stresses
is known,
Aj
V
(m
hi)
/2
(3.13abc)
of the system.
When this
at the interfaces are also completely
kn own.
4.
VISCOELASTIC MATERIALS.
We
next
the
consider
viscoelastic
properties
of
the
layers
for
possible
It follows from (2.4a) and (2.7)
technological applications of the above analysls.
that the strain component is
U(y) cos mx e
m
e
lint
(4.
Hence we find from (2.11) that
2M e
XX
(4.2)
XX
Also it follows from (2.4c) with (2.7)
o
(2.9) that
(4.3)
2M e
It is clear from equations (4.2) and (4.3) that for Maxwell type solid there is only
one
coefficient
of
instead
coefficients
two
the
for
elastic
materials.
Viscoelastic properties of the layers may be taken into account by substituting the
for the coefficient
following general form of operator
f IM(r)
p+r
dr +
M
(see Blot [8]):
(4.4)
M" + M’p
0
where
d
p--.
For harmonic oscillations
p
i and
the operator also becomes
a
complex quantity.
5.
THINLY LAYERED LAMINATED MATERIALS.
We now consider a multi-layered plate in which the layers are themselves composed
of thinly laminated materials of Maxwell solids.
repeated sequence of
of
the
total
coefficient
Mj
thickness
j/(l
For a laminated medium composed of a
thin layers each of which occupies a fraction
n
h’
--i
the equivalent anisotropic
)T
of
the
laminated
medium
The equivalent coefficient
mdium)
is
and
M
j(J--l,2...n)
characterized
by the
i.e. the coefficient of
182
P. PAL ROY and L. DEBNATH
n
This
(5.1)
J
j--1
M
coefficient
equivalent
a
constitute
first
approximation.
[3]
by introducing stress couples
approximation is provided
The
next
i.e. a moment per unit
area equal to
2v
(5.2)
R= b-2
x
where
b
(h’)2..
3
n
I
j--1
(I
The equilibrium equations (2.5)
+
(2.6) are now modified to the following form
y o xy
p
x Oxy +y Oyy
p
--o
Bx
xx
(5.3)
MI
2u
(5.4)
t2
4
2v
--+t b-Vx
(5.5)
4
2
Consequently, equation (3.12) must be replaced by
n
j=l"
S
+
+
Aj+ I) cjMj
m2Vk
a
2
n
Pt
m4V j=ll bjhj
V +
C5.6)
1,2...n) is the couple stress coefficient of the Jth laminated medium.
bj(j
where
(Aj
The result (5.6) can immediately be extended to viscoelastic laminated media.
6.
EVALUATION OF DAMPING.
An important task in the problem of design analysis is the determination of the
We consider a
effect of viscoelastic layers on vibration absorption at resonance.
The span
simply supported homogeneous anlsotroplc plate.
equal to half the wave
p
length is
m
From equations (2.14) and (3.1) we have the expression for
A
0 and A
8
which, after putting
0, becomes
2
S
where
S
m2h MV(I
(I
vf2N
tanh
8
8’.)
2
m
Pt
pa
21
2
For a viscoelastic material
M
is
M + M
replaced by
The Imaginary part of the load
imaginary term.
(6.2)
V
S
obtained from equation (6.2) by linearlzing it with respect to
S
2
m
Vh F M
where
M is the purely
is represented by
S
and is
M. We obtain
(6.3)
MULTILAYERED ORTHOTROPIC VISCOELASTIC PLATES OF MAXWELL SOLIDS
+ I__
2
3 tanhBY
2
BY
F
with
cos
h2By
+
183
l__J___.)
(tanhB_______TT_
2
BY
21B2
cosh
(6.4)
B
Equation (6.3) determines the damping (longitudinal) on vibration attenuation.
F
The variation of
B (anlsotropy)
for different values of
T are shown in
and
It is seen that the damping increases with anisotropy and wave number.
figure I.
The
result (6.3) is also applicable to the laminated plate of the equivalent continuum as
In the case the inclusion of couple stress will contribute an
described in section 5.
4
additional term m Vh Ab where Ab is the imaginary part of equation (5.3).
From equation (2.10a) and (6.3) it follows that
S
with
(6.5)
p. F
2
p
m Vh U
of
The variations
(6.6)
m22
+
P with
are shown
.
the symmetric
2 which confirms
in Figure
As a particular example, we
respect
variation
consider a plate composed of two layers. The material (obviously Maxwell type) of the
and that the
is characterized by the coefficient M
h’
first layer of thickness
and suppose h
second layer of thickness h" is characterized by the coefficient M
the relaxation time
to
with
2
h’ + h".
The
applied
AI=A3ffiO.
shear
stresses
at
the
outerfaces
are
put
equal
to
zero
(i.e.
The shear stress
at the interface is found by equating the interfacial
U
considered as belonging to the first and second layer.
displacement
amplitude
2
Applying equations (3.3) we obtain
u
U2
where
Nland
2(H
/I_
N
2
2-
-2
a
This
2)
2V c
+
(6.7)
2
(6.8)
2v=
N for
M
M
and
M
M
respectively.
2
The total
is given by
cl+ H2c2) (c1+ c2)
Vm
(A2
m
are the values of
ql + q2
q (ffi
load
( aI)
equation determines
2
+
m(h,Ml+h,,.M2)_2(MlCl+M2c2)
V
the deflection
/i
m
(Plh,+P2h:,)
(6 9)
when the normal load applied to the
structure is given.
We next consider a plate composed of three layers.
The first and third layers
are identical in nature and are characterized by the coefficient
one is characterized by the coefficient
shears
are
while
interfacial displacements
U
A
4
M2.Because
0 at
the
M
while the middle
of the symmetry, the interfacial
outerfaces.
considered as belonging to layers
2
If
and
we
equate
2.
the
We obtain
18
P. PAL ROY and L. DEBNATH
+
2(c
A
2
c2)mV
?2N 2
?2N
where
B2
a
is given by
2
The total load q=
q
Vm
2(c +
c2)(2M
c
a
2B2c
#2N
2N 2
+
+ b
2
ql + q2
M2c
(6.10)
2B2c 2
a
+
2B2c2.
in this case is given by
u2
m(2hlM1 + h2M2) 2(2Mici +M2c2)-m
(2P
lh’
The second and
+
P2 h’’)
third terms represent the normal load corresponding to superposed
layers with perfect Interracial slip while the first term represents the effect of
adherence between layers.
1.0
Fig.
Variation of F with 7 for different values of
B (equation 6.3).
MULTILAYERED ORTHOTROPIC VISCOELASTIC PLATES OF MAXWELL SOLIDS
185
,335
.09
Variat:en of
Fi 8. 2
lo0
0
.1
.0
P
over
.
T
Fig. 3
A three-layered symmetric plates composed of a core of
thickness h sandwiched between two identical layers of
2
thickness h
I
7.
CONCLUDING REMARKS.
The foregoing results of a simply supported plate with a loading distributed as a
half
sine
wave
are
obviously valid
expanded in a series along the span.
various
length.
for an arbitrary loading provided
it
can be
In this case, the results are applied to the
Fourier components with a suitable value of
m
corresponding to each wave
The procedures used for the simply supported case can easily be extended to
the case where the plate is built-in
(i.e. when both end points of the plate are
rigidly attached so that the displacement of its faces are both zero.
In this case,
in order to satisfy the boundary conditions we must, instead of sinusoidal solutions,
consider
exponential
solutions.
Such solutions are empirically
trigonometric solutions (2.7) and (2.8) if we replace
m
by ik.
derived
from the
P. PAL ROY and L. DEBNATH
186
AO(NOWLEDGEMENT.
The research of the first author was partially supported by the UGC-
DSA Program of Jadavpur University,
Calcutta, India.
This work was also partially
supported by the University of Central Florida.
REFERENCES
I.
BLOT, M.A.
Mechannlcs of Incremental Deformations, John Wiley and Sons, New
York (196 5).
2.
BLOT, M.A.
3.
BLOT, M.A.
Theory of Stability of Multilayered Continua in Finite Anlsotroplc
Elasticity, J. Franklin Inst. 276 (1963), 128-153.
A New Approach to the Mechanics of Orthotroplc Multilayered Plates,
Int. J. Solids Struct. 8 (1972), 475-490.
4.
BLOT, M.A.
5.
Simplified Dynamics of Multilayered Orthotroplc Viscoelastic
Plates, Int. J. Solids Struct. 8 (1972), 491-509.
PAL ROY, P. A Simplified Approach to the Propagation of Edge Waves in a Thinly
Layered Laminated plate with Stress Couples Under Biaxlal Initial
Stresses, Acta Geoph. Pol. 33 No. 3 (1985), 329-339.
6.
PAL ROY, P.
7.
BI SWAS, A.
Wave Propagation in a Thinly Two-Layered Laminated Medium with
Stress Couples Under Initial Stresses, Acta Mech. 54 (1984), 1-21.
Transmissions of Love
Type Waves Through a Maxwell Solid Layer
Sandwich in Homogeneous Elastic Media, Pure
.and Appl. Geoph.
69
(1968)/I), ?5-79.
8.
BLOT, M.A.
Theory of Stress-Straln Relations in Anlsotroplc Viscoelastic and
Relaxation Phenomena, J. App. Phys. 25
(1954), 1385-1391.