Jtam-A4.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

53, 4, pp. 775-788, Warsaw 2015
DOI: 10.15632/jtam-pl.53.4.775

DYNAMIC BEHAVIOUR OF THREE-LAYERED ANNULAR PLATES
WITH VISCOELASTIC CORE UNDER LATERAL LOADS

Dorota Pawlus
University of Bielsko-Biala, Faculty of Mechanical Engineering and Computer Science, Bielsko-Biała, Poland

e-mail: dpawlus@ad.ath.bielsko.pl

This paper presents the dynamic behaviour of three-layered, annular plates with a linear
viscoelastic core. The annular plate is loaded in plane of facings with variable in time forces
acting on the inner or outer plate edges.The analysedplate structure is symmetric and com-
posed of three-layers: thin facings and a thicker, soft core. The core material is considered
as linear with viscoelastic properties. Analytical and numerical solution will be presented
including the wavy forms of dynamic loss of plate stability. In the analytical and numeri-
cal solution, two approximationmethods: orthogonalization and finite difference have been
applied to obtain the system of differential equations for the analysed problem of dynamic
deflections of the examined plate with the viscoelastic core. Additionally, the suitable plate
model using the finite element method has been built. The numerical calculations carried
out in ABAQUS system show the results which are compatible with those obtained for the
analytical and numerical solution. Presented in figures time histories of plate deflections
and velocity of deflections as well as buckling forms of the plate model show the dynamic
response of the examined plate on time-dependent loads and corematerial properties.

Keywords: sandwich plate, viscoelastic core, dynamic stability

1. Introduction

The examinations of three-layered plates have been undertaken in numerousworks. Rarely, they
concern annular plates under lateral loads variable in time and plate structures where damping
properties of the layer material is taken into account. A wide range of possible applications of
annular layered plates in aerospace industry, mechanical and nuclear engineering presented in
works byDumir and Shingal (1985), Paydar (1990), Chen et al. (2006), Wang and Chen (2004)
causes the evaluation of dynamic behaviour of such complex composite plates still important.
The following, exemplary papers byWang andChen (2002, 2004), Chen et al. (2006) belong

to the group of works, where the plate dynamic problems like vibrations or dynamic instability
havebeenanalysed.Theexaminations concern theplates composedof three layers: the thinouter
constraining layer, the thinmiddle viscoelastic damping layer and the bottomouter host layer of
theplate.The evaluation of the influenceof the viscoelastic core thickness on the improvement of
damping plate properties was presented in work byWang andChen (2002). Inmanyworks, the
problem of the loss of plate stability wss often limited to only the axisymmetric case.Whereas,
full evaluation also requires analysis of circumferentially waved plate buckling forms. In thework
by Krizhevsky and Stavsky (1996), the asymmetric instability problem of a three-layered plate
with a relatively thin core was shown. The results of critical loads were presented.
The evaluation of dynamic behaviour of a three-layered annular plate with a viscoelastic

core loaded in facings plane is presented in this work. The type of the three-layered structure
is characteristic, transversally symmetrical and composed of thin facings and a suitable thicker,
soft core.The asymmetric bucklingmodes of examined, dynamic loaded plates are also observed.



776 D. Pawlus

Including in the description of physical relations the rheological properties of the plate core
made of polyurethane foam correctly approximates the critical real plate behaviour. It could be
supposed that the noticed in the work by Romanów (1995) sensitivity of foammaterials on the
long duration of load could be important particularly in supercritical plate work under a load
constant in time or slowly increasing.
Such observations together with dynamic stability analyses of three-layered annular plates

with viscoelastic cores are the main goal of this work. The presented way of problem solution
and detailed results refer to analyses presented in works by Pawlus (2010, 2011a,b,c).

2. Problem formulation

The three-layered annular plate with the soft foam core and with the symmetric cross-section
structure is the subject of consideration. The scheme of the plate is presented in Fig. 1. The
facings are elastic. The core material is treated as elastic or viscoelastic. The plate is loaded on
the inner or/and outer perimeter of facings with uniformly distributed stress linearly increasing
in time, according to Equation (2.1) or constant in time (see, Eq. (6.1))

p = st (2.1)

where p is compressive stress, s – rate of growth of the plate loading, t – time.

Fig. 1. A scheme of the three-layered annular plate composed of facings – layers 1, 3 and core – layer 2

The plate with bilaterally slidably clamped edges is undertaken in the presented considera-
tions. The form of plate predeflection corresponds with the plate buckling form. The circum-
ferentially waved forms of the loss of plate dynamic stability are analysed, too. The classical
theory of sandwich plates with the broken line hypothesis, presented by Volmir (1967), and the
decomposition of basic stresses to normal and shearing components, carried by facings and the
core, respectively, have been accepted in the mathematical formulation of the problem. In nu-
merical solution, the approximate finite differencemethod and finite elementmethod have been
used.
As the criterion of the loss of plate stability, the criterion presented by Volmir (1972) has

been adopted. According to this criterion, the loss of plate stability occurs at the moment of
time when the speed of the plate point of the maximum deflection reaches the first maximum
value. The essence of the accepted criterion of the loss of plate stability connected with the eva-
luation of values of dynamic critical loads in the region of significant increase in plate deflections
corresponds to the Budiansky-Roth criterion. This criterion is used in the buckling analysis of
laminated shells, presented for example by Tanov and Tabiei (1998).



Dynamic behaviour of three-layered annular plates... 777

3. Basic equations

Solution to the considered problem is based on the relations typical for the edge-initial problem:
dynamic equilibrium equations, geometrical and physical relations and initial and boundary
conditions. Dynamic equilibrium equations have been formulated for each layer of the plate.
Core deformation is described by the angles β and α in the radial and circumferential directions,
respectively

β =
u1−u3
h2

−
1
2
h1+h3
h2

wd,r +wo,r

α =
v1−v3
h2

− 1
2r
h1+h3
h2

wd,θ +
1
r
wo,θ

(3.1)

where wd is additional plate deflection, wo – preliminary plate deflection, u1(3), v1(3) – displace-
ments of the points of themiddle plane of facings (layers 1, 3) in the radial and circumferential
directions, respectively.
Linear physical relations of Hooke’s law for facings and linear viscoelastic relations of the

standard model (see Fig. 2) for the plate core are applied. The physical relations of the visco-
elastic core material subjected to shearing stresses are presented by the equations

τrz2 = G̃2γrz2 τθz2 = G̃2γθz2 (3.2)

where γrz2, γθz2 are shearing strains of the plate core in the radial and circumferential direction,
respectively, expressed by

γrz2 = u
(z)
2,z +wd,r γθz2 = v

(z)
2,z +

1
r
wd,θ (3.3)

where: u(z)2 = u2 − zβ + zwo,r, v
(z)
2 = v2 − zα + zwo,θ/r are the radial and circumferential

displacements of the points in the distance z between the point and the middle surface of the
plate core; u2, v2 – displacements of the points of themiddle plane of the core layer in the radial
andcircumferential direction, respectively; G̃2 –modulusexpressedbythe formulacorresponding
to the form of the constitutive equation of the standard model

G̃2 =
C +D ∂

∂t

E +F ∂
∂t

(3.4)

where C, D, E, F are the quantities formulated by the elastic G2, G′2 and viscosity η
′ constants

of the core material, presented by Skrzypek (1986)

C =
G2G

′
2

G2+G′2
D =

η′G2
G2+G′2

E =1 F =
η′

G2+G′2
(3.5)

and ∂/∂t – differential operator.

Fig. 2. The standardmodel of a viscoelastic layer material



778 D. Pawlus

Using the equations of the nonlinear Kármán’s plate, the sectional forces and moments in
the facings are established. Then, the resultant transverse radial and circumferential forces and
the resultant membrane forces expressed by the introduced stress function are formulated.
The initial conditions, loading and boundary conditions as well as relations connected with

the slidably clamped both inner and outer plate edges are established as follows

w
∣∣
t=0
= wo w,t

∣∣
t=0
=0 wd

∣∣
t=0
=0 wd,t

∣∣
t=0
=0

σr
∣∣
r=ri(ro)

=−p(t)d1(2) σr,t
∣∣
r=ri(ro)

=−(p(t)),td1(2) τrθ
∣∣
r=ri(ro)

=0

w
∣∣
r=ri(ro)

=0 w,r
∣∣
r=ri(ro)

=0 δ = γ
∣∣
r=ri(ro)

=0 δ,r
∣∣
r=ri(ro)

=0

(3.6)

wherew is the total deflection, wd – additional deflection,wo –preliminarydeflection,σr – radial
stress, τrθ – shear stress, d1, d2 – quantities equal to 0 or 1, determining the loading of the inner
or/and outer plate perimeter, δ, γ – differences of radial and circumferential displacements of
the points in middle surfaces of facings, respectively.
Finally, after some calculations the basic differential equation expressed the deflections of

the analysed sandwich plate with the viscoelastic core are determined

N1wd,rrrr +
2N1
r
wd,rrr −

N1
r2
wd,rr +

N1
r3
wd,r +

N1
r4
wd,θθθθ +

2(N1+N2)
r4

wd,θθ

+
2N2
r2

wd,rrθθ −
2N2
r3

wd,rθθ − G̃2
H′

h2r

(
γ,θ + δ+rδ,r +H

′1
r
wd,θθ +H

′wd,r +H
′rwd,rr

)

=
2h′

r

( 2
r2
Φ,θw,rθ −

2
r
Φ,rθw,rθ +

2
r2
w,θΦ,rθ −

2
r3
w,θΦ,θ +w,rΦ,rr +Φ,rw,rr

+
1
r
Φ,θθw,rr +

1
r
Φ,rrw,θθ

)
−Mwd,tt

(3.7)

where N1, N2 are expressed by the material and geometrical parameters of plate layers;
H′ = h′+h2; h′(h1 = h3 = h′) – facing thickness, h2 – core thickness.

4. Problem solution

In the solution, the following dimensionless quantities and the expressions have been assumed

ζ =
w

h
ζ1 =

wd
h

ζo =
wo
h

F =
Φ

Eh2

ρ =
r

ro
δ =

δ

h
γ =

γ

h
t∗ = tK7 K7 =

s

pcr

ζ1(ρ,θ,t)= X1(ρ,t)cos(mθ) ζo(ρ,θ)= Xa(ρ)+Xb(ρ)cos(mθ)

ζ = ζ1+ ζo F(ρ,θ,t)= Fa(ρ,t)+Fb(ρ,t)cos(mθ)+Fc(ρ,t)cos(2mθ)

δ(ρ,θ,t)= δ(ρ,t)cos(mθ) γ(ρ,θ,t)= γ(ρ,t)sin(mθ)

(4.1)

where m is the number of circumferential waves corresponding to the form of plate buckling,
h = h1+h2+h3 is the total thickness of plate, pcr – critical static stress.
Using the orthogonal method to elimination of the angular variable θ, Eq. (3.7) has been

replaced by the approximate one obtained from the following condition

2π∫

0

ψcos(mθ) dθ =0 (4.2)

where ψ is the difference of the left and right side of transformed Equation (3.7).



Dynamic behaviour of three-layered annular plates... 779

Using the physical relations, relations of sectional forces and moments in the facings with
stresses and equations of resultantmembrane forces expressed by the introduced stress function
after elimination of the quantities described by sums of the radial (u1+u3) and circumferential
(v1+v3) facing displacements, the following equation has been obtained

Φ,rrrr +
2
r
Φ,rrr −

1
r2
Φ,rr +

1
r3
Φ,r +

1
r4
Φ,θθθθ +

4
r4
Φ,θθ −

2
r3
Φ,rθθ +

2
r2
Φ,rrθθ

= E
[1
r
wo,rr

(
wo,r +

1
r
wo,θθ

)
− 1
r2

(1
r
wo,θ −wo,rθ

)2
− 1
r
w,rr

(
w,r +

1
r
w,θθ

)

+
1
r2

(1
r
w,θ −w,rθ

)2]
(4.3)

Thequantities δ andγ, unknown inEquation (3.7), have beenobtained determining thedifferen-
ces of the radial and circumferential displacements u1, u3 and v1, v3 of the points of the middle
surface of the plate facings. The equilibrium equations for forces acting on the non-deformed
outer plate layers in the u and v direction are used.
After insertion of Equations (4.1) describing the functions w, wo, Φ and after comparison of

the expressions by the same trigonometric functions, Equation (4.3) has been presented in form
of three equations.
In the solution, the finite difference method has been used for the approximation of the

derivatives with respect to ρ by central differences in the discrete points. The form of the
system of differential equations expressing the deflections of three-layered annular plate with
the viscoelastic core is as follows

PU+Q+PLU̇+QL −W1Ü= W2
...
U

MYY=QY MYẎ= Q̇Y

MVV=QV MV V̇= Q̇V

MZZ=QZ MZŻ= Q̇Z

MDLḊ=MDD+MUU+MULU̇+MGG+MGLĠ

MGGLĠ=MGGG+MGUU+MGULU̇+MGDD+MGDLḊ

(4.4)

where

W1 = K7
h′

h
roh2MEL W2 = K

3
7
h′

h
roh2MFL

and M is the expression: M =2h′µ+h2µ2; µ, µ2 – facing and core mass density, respectively;
ro – outer radius; EL, FL – quantities determined by elastic and viscosity constants of the co-
re standard material; U,Y,V,Z,U̇,Ü,

...
U,Ẏ,V̇,Ż – vectors of plate additional deflections and

components Fa, Fb, Fc of the stress function Fa′ρ = y, Fb = v, Fc = z and their derivatives
with respect to time t, respectively; P, PL – matrices with elements composed of geometric
andmaterial plate parameters, the quantity b (b – length of the interval in the finite difference
method), dimensionless radius ρ and the number m of buckling waves and its derivative with
respect to time t, respectively; Q, QL – vectors of expressions composed of the initial and
additional deflections, geometric and material parameters, components of the stress function,
radius ρ, quantity b, coefficients δ, γ and the number m and their derivatives with respect to
time t, respectively;MY –matrix of elements composed of the radius ρ and quantity b, respec-
tively; MV ,MZ – matrices of elements composed of the radius ρ, quantity b and number m;
QY ,QV ,QZ,Q̇Y ,Q̇V ,Q̇Z – vectors of expressions composed of the initial and additional de-
flections, radius ρ, quantity b and the number m and their derivatives with respect to time t,
respectively; MD,MG,MGG,MGD,MDL,MGL,MGGL,ṀGDL – matrices of elements compo-
sed of geometric and material parameters, quantity b and the number m and their derivatives



780 D. Pawlus

with respect to time t, respectively;MU,MUL –matrices of elements composed of material pa-
rameters andquantity b and their derivatives with respect to time t, respectively;MGU,MGUL –
matrices of elements composed of geometric parameters,material parameters and the numberm
and their derivatives with respect to time t, respectively; D,G,Ḋ,Ġ – vectors of expressions
composed of coefficients δ and γ and their derivatives with respect to time t, respectively.
The system of Equations (4.4) has been solved using Runge-Kutta’s integration method for

the initial state of plate.
Eliminating from the system of Equations (4.4) Equations (4.4)3,5,7 and the expressions con-

nectedwith thedifferential operator∂/∂tapplied inphysical relations of theviscoelasticmaterial
of the plate core, a system of equations for plate with elastic core has been obtained (Pawlus,
2010, 2011b,c). The critical static stress pcr has been calculated solving the eigenproblem for the
problem of the disk state neglecting the inertial components and nonlinear expressions (Pawlus,
2010, 2011b,c).

5. Finite element plate models

In the analysis carried out with use the finite element method, twomodels have been built:

• basic model in form of the complete model of the annular plate, circularly symmetrical
(Fig. 3a),

• simplifiedmodel built of axisymmetrical elements (Fig. 3b).

The facings are built of shell elements but the core mesh is built of solid elements. The grids
of facings elements are tied with the grid of core elements using the surface contact interaction.
The calculations were carried out at the Academic Computer Center CYFRONET-CRACOW
(KBN/SGI ORIGIN 2000/PŁódzka/030/1999) using the ABAQUS system.

Fig. 3. (a) Basic model, (b) simplified model

The viscoelastic properties of the core material have been described by a single term of the
Prony series for the shear relaxation modulus (Hibbitt et al., 2000)

GR(t)= Go
(
1−qp1

(
1− e−t/τG1

))
(5.1)

where qp1, τ
G
1 are material constants, Go – instantaneous shear G2.

The values of material constants qp1, τ
G
1 for the standard model of the plate core have been

calculated from the following equations

q
p
1 =

G2
G2+G′2

τG1 =
η′

G2+G′2
(5.2)

6. Exemplary calculations

Exemplary numerical calculations were carried out for the plate with the following geometrical
dimensions: inner radius ri =0.2m, outer radius ro =0.5m, facing thickness h′ =0.001m, core



Dynamic behaviour of three-layered annular plates... 781

thickness h2 =0.005m, 0.01m or 0.02m. The parameters of steel facing are: Young’s modulus
E =2.1·105MPa, Poisson’s ratio ν =0.3,mass density µ =7.85·103kg/m3. Polyurethane foam
is the core material. Two kinds of foam presented in works by Romanów (1995) and Majewski
andMackowski (1975) have been accepted:

• with the values of constants G2, G′2, η′, µ2 presented in work by Romanów (1995) equal
to: G2 =15.82MPa, G′2 =69.59MPa, η

′ =7.93 ·104MPa·s, µ2 =93.6kg/m3,
• with the data presented in work by Majewski and Mackowski (1975) the Kirchhoff’s mo-
dulus equal to: G2 =5MPa, the creep function ϕ =0.845(2−e−0.36t −e−0.036t) andmass
density equal to: µ2 =64kg/m3.

The creep function presented in the work by Majewski and Mackowski (1975) allows for cal-
culation of the values of elastic and viscous constants of the five-parameters rheological model
composed of two Kelvin-Voigt models and the spring element connected in series. Because the
solution presented in this paper is for the core material described by the three-parameters
standard model (see, Fig. 2), hence the presented characteristic of function ϕ(t) has been ap-
proximated by the function of the standardmodel. The numerical analysis has been carried out
for the following values of standard constants of the plate core: G2 = 5MPa, G′2 = 3.13MPa,
η′ =212.92·104MPa·s, (Pawlus, 2010). According to the standard specification PN-84/B-03230
the value of Poisson’s ratio is equal ν =0.3. The values of Young’s modulus calculated treating
the foammaterial as an isotropic are equal, respectively, to E2 =41.13MPa, E2 =13MPa. The
values ofmaterial constants qp1, τ

G
1 , expressed byEquations (5.2), inProny series (see, Eq. (5.1))

are as follows:

• qp1 =0.615, τG1 =26.19 ·104 s for the foam with G2 =5MPa,
• qp1 =0.185, τG1 =928.46s for the foam with G2 =15.82MPa.

Rapidly increasing loading acting on the edge is expressed by Equation (2.1). The rate of
plate loading growth s is equal for each numerically analysed plate. The value of the rate s is the
result of the following equation: s = K7pcr (seeEq. (4.1)). Thevalue of parameterK7 is accepted
as K7 = 20s−1. Solving the eigenproblem, the value of critical stress pcr is pcr = 217.32MPa
calculated for theplate compressedon the innerperimeterwith the facing thicknessh′ =0.001m,
core thickness h2 = 0.01m and value of core Kirchhoff’s modulus G2 = 15.82MPa and pcr =
46.58MPa for the plate model with N = 26 number of discrete points radially compressed on
the outer perimeterwith the number of bucklingwaves m =7, the facing thicknessh′ =0.001m,
core thickness h2 =0.005m and value G2 =15.82MPa (see Table 1).
The calculations by the finite difference method have been preceded by selection of the

number N of discrete points from numbers N, equal to N = 11, 14, 17, 21, 26. The values
of critical dynamic loads pcrdyn of plates with the viscoelastic core and values of critical static
loads pcr have been evaluated. The numerical calculations show that the number N =14 allows
us to achieve the accuracy up to 5% of technical error. The calculations have been carried out
for this number. Table 1 presents exemplary results of the analysis.
The treating of the polyurethane foam as isotropic is one of the accepted assumptions in

the description of the plate core material. In FEMnumerical analysis, the value of the material
Young’s modulus has been calculated using the following equation: G2 = E2/[2(1+ν)]. Table 2
presents exemplary results of the critical time tcr and the critical additional deflection wdcr for
the basic elastic plate model compressed on the inner edge with the core parameters: thick-
ness h2 = 0.005m, Young’s modulus E2 presented in Table 2, Poisson’s ratio ν = 0.3, Kirch-
hoff’s modulus G2 = 15.82MPa. The axisymmetric form m = 0 of plate buckling has been
examined.
In the wide range of the assumed values of Young’s modulus E2, the results for the critical

time tcr, andafter calculations, also the values of the critical dynamic loads pcrdyn donot change.



782 D. Pawlus

Table 1. Critical loads pcr, pcrdyn of the plates with the viscoelastic core loaded on the outer
edge G2 = 15.82MPa G′2 = 69.59MPa, η

′ = 7.93 · 104MPa·s, h2 = 0.005m, h′ = 0.001m,
µ2 =93.6kg/m3, K7 =20s−1

m

N
11 14 17 21 26

pcr pcrdyn pcr pcrdyn pcr pcrdyn pcr pcrdyn pcr pcrdyn
[MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa] [MPa]

0 73.97 81.47 74.46 81.38 76.23 81.47 76.27 81.47 76.30 81.47
1 70.28 79.42 70.73 79.42 72.51 79.51 72.56 79.51 72.59 79.51
2 61.14 69.54 61.67 69.64 63.44 69.73 63.53 69.82 63.59 69.92
3 53.01 67.96 53.39 68.05 55.10 68.15 55.21 68.33 55.30 68.33
4 48.33 66.28 48.64 66.38 50.02 66.47 50.13 66.47 50.21 66.66
5 46.11 53.43 46.36 52.59 47.45 52.50 47.55 52.21 47.62 52.03
6 45.39 51.01 45.58 49.05 46.45 50.54 46.53 50.26 46.60 50.17
7 45.60 47.84 45.75 47.47 46.45 47.65 46.52 47.93 46.58 47.84
8 46.44 45.69 46.56 45.32 47.14 45.04 47.20 44.86 47.25 44.67

Table 2.Values of the critical time tcr andmaximumdeflection wdcr depending on coreYoung’s
modulus E2

E2 [MPa] 1.0 5.0 13.0 50.0 100.0
tcr [s] 0.019 0.02 0.02 0.02 0.02
wdcr ·103 [m] 4.14 4.48 4.15 3.85 3.73

The values of the plate maximum critical deflection wdcr change according to prediction. The
analysis shows that the accepted corematerial could be treated as an isotropic in the presented
evaluation focused on the critical plate loads.
The calculation results of the plateswith the standard corematerial subjected tomomentary

loads (duration in the range of 0.1s) donotdiffer significantly fromthe results for theplateswith
the elastic core. The time histories of deflections of the plates with the viscoelastic and elastic
core for the number m =0 are presented in Fig. 4. The results are presented using two kinds of

Fig. 4. A comparison of deflection time histories of plates loaded on the inner edge with the viscoelastic
and elastic core

lines grey and black for the pairs: elastic and viscoelastic plate core with the same parameters
h2, G2. The compatibility of deflection time histories is observed for the plates loaded on the
inner perimeterwith the viscoelastic and elastic core and compressed on the outer edge, too (see
Fig. 5). The values of the critical time tcr and critical dynamic loads pcrdyn are practically the



Dynamic behaviour of three-layered annular plates... 783

same. The critical deflections are comparable for the plates with core thickness: h2 = 0.005m
and h2 =0.02m. Of course, the values of the critical time and load increase for the plate with
the stiffer core (higher value of modulus G2 and thickness h2). Then, supercritical vibrations
disappear.
Figure 5 shows characteristic time histories of plate deflections for a selected number of

buckling m =0, m =3, m =5, m =8 in two respective groups. Some differences are observed
in supercritical responses of the plates. Theminimal value of plate critical parameters is for the
waved plate with the number m =5 of bucklingmodes.

Fig. 5. A comparison of deflection time histories of plates (G2 =5MPa, h2 =0.01m) loaded on the
outer edge with the viscoelastic and elastic core for the buckling wave number: (a) m < 4, (b) m > 4

The image of behaviour of the plates with the standard core is enriched by comparison with
the results obtained for the plates with the core expressed by a simpler two-element model.
The accepted in observations Maxwell model built of elastic and viscous elements connected
in series takes attracts attention on the influence of the core material viscosity constant on the
plate time histories of deflection. The numerical calculations have been carried out for quantities
(3.5) assuming G′2 =0. The comparison of deflection time histories of the plates compressed on
the outer perimeter with the standard core (G2 = 15.82MPa) and the core expressed by the
Maxwell model with parameters: G2 = 15.82MPa and viscosity constant η′ = 7.93 ·104MPa·s
is shown in Fig. 6. The results are presented for selected buckling forms, which entirely show

Fig. 6. The time histories of deflections of plates (G2 =15.82MPa, h2 =0.005m) compressed on outer
edge with: (a) standard core, (b)Maxwell core

the character of curves. The standard model of solid body clearly extends the time to the loss
of plate stability. In the case of theMaxwell model, the character of time histories of deflections
of the plates with various numbers of circumferential waves m is comparable – increasing in



784 D. Pawlus

the plate loading time. One can observe that the deflection time histories for the plate with
the standard core model differ for plates with higher and lower numbers of the buckling mode.
Figure 5 shows this character of curves too.
Figure 7 shows the comparison of curves ζ1max = f(t∗) of the plates with the elastic and

viscoelastic core compressed on the outer edge with the number of circumferential waves cor-
responding to the minimal values of the critical dynamic load for different values of material
parameters and core thickness. The same lines (gray and black) are used to present the results
for the plates with the elastic and viscoelstic core. The diagram presents the results for the
plates with the Maxwell core, too. For this simple model, the change of material parameters
(G2 = 15.82MPa, η′ = 7.93 · 104MPa·s and G2 = 5MPa, η′ = 212.92 · 104MPa·s) does not
influence the deflection time histories and the critical time. Whereas, the plate core thickness
has a great importance. The results show the relevant sensibility of the plate structure with
three elements and the standard core on the material and geometrical parameters. It confirms
an obvious andbetter approximation of the structure behaviour of the expected responses of the
actual object. The results obtained for the plate compressed on the outer edge confirm earlier
observations that the critical time prolongs for the plates with the stiffer core. For these plates,
the minimal critical dynamic load is for the mode m =5-7. It depends on the core parameters
G2, h2. The value of the critical additional deflection is about 0.1 of the total plate thickness
(see Eq. (4.1)). It is slightly higher for the plate with the modulus G2 =15.82MPa.

Fig. 7. Time histories of deflections of plates compressed on the outer edge for the elastic standard and
Maxwell core with: (a) G2 =5MPa, (b) G2 =15.82MPa

The viscoelastic properties of the foam material relevantly depend on the viscosity con-
stant η′. The deflection results for the plates with the viscoelastic core of significantly lower
viscosity constant η′ differ. Figure 8 shows the deflection time histories of the plates loaded on
the inner edge with various values of the viscosity constant η′ of the viscoelastic core material.
Large differences could be observed for the core material with very small values of the

constant η′ within the range 7.93 · 10−2(10−3)MPa·s. The time to the loss of plate stability
shortens. The increase in values of the constant η′ does not change the critical or supercritical
deflections of the plateswith the viscoelastic core. They are compatiblewith the results obtained
for the plates with the elastic core.
The numerical FEM calculations of the simplified plate model confirm these observations.

Thecritical dynamic loadsdistributiondependingonvaluesof the constantη′ is presented for two
kinds of the core viscoelasticmaterials inFig. 9. Theplates are loaded on the inner edge. A large
theoretical decrease in values of the number η′ up to a value about 10−8 times lower influences
the contraction of the time to the loss of dynamic stability and decreases critical dynamic loads.
These observations have an important practical meaning. In a wide range of accepted values of



Dynamic behaviour of three-layered annular plates... 785

Fig. 8. The influence of viscosity constant η′ on the deflection time histories of plate (G2 =15.82MPa,
h2 =0.005m) loaded on the inner edge

Fig. 9. Influence of the viscosity constant η′ on the critical dynamic loads pcrdyn of the FEMplate
model loaded on the inner edge

the viscosity constant, the dynamic response of the plate subjected to quickly increasing loads
does not change. Here, it should be noticed that for true materials, precise determination of
viscous material parameters is a great problem. Smaller sensitiblity to fluctuation of values of
the viscoelastic material parameters seems to be the advantageous dynamical behaviour of such
plates.
Exemplary time histories of deflections and velocity of deflections as well as critical de-

formations of the FEM simplified plate model with the elastic and viscoelastic core with
η′ = 212.92 · 10−6 (10−10 times decrease) are presented in Fig. 10. In general, there are no
differences in the character of curves. A slightly less critical time and also dynamic loadwith hi-
gherdeflection arenoticed for theplatewith a strongly reducedvalue of the viscosity constant η′.
For a higher parameter η′, the response of the plate with the viscoelastic core is consistent with
that with the elastic one.
The influence of rheological properties of the corematerial on the increase of plate deflections

is observed for the plates subjected to the load constant in time. The supercritical response has
been analysed for both plates with the viscoelastic and elastic core compressed on the inner or
outer edge in the case of the load constant in time. The assumed load function is expressed as
follows

p(t)=

{
st for t ¬ ntcr
npcrdyn for t > ntcr

(6.1)

where tcr is the critical time of the loss of plate dynamic stability, n – assumed number.



786 D. Pawlus

Fig. 10. Time histories of deflections, velocity of deflections and buckling forms of the plates
(G2 =5MPa, h2 =0.005m) with: (a) elastic core, (b) viscoelastic core loaded on the inner edge

Fig. 11. Time histories of deflections of plates G2 =5MPa, h2 =0.01m, m =5with the elastic and
viscoelastic core subjected to the load constant in time acting on the outer edge

Figure 11 shows the time histories of deflections of the plateswith the elastic andviscoelastic
core compressed on the outer edge with the waved form of predeflection and buckling. The
number m is equal 5. The value of the number n is equal 0.5, 0.8 (undercritical load) and
n =1.01, 1.2.Additionally, Fig. 11presents the curve ζ1max = f(t∗) obtained for a linear (p = st)
increase in the load acting on the plate with the viscoelastic core. Themarked points show the
moment of the loss of plate dynamic stability and static stability after suitablemiscalculations in
the value of the critical static load. The course of deflection time histories of the plates with the
elastic and viscoelastic core under increasing linear load are consistent. The differences could be
observed in the range for the loading constant in time. In a short time (t =2.5s) of the loading,
an increase in the deflection of the platewith the viscoelastic core can be significant. An increase
in the deflection rises in the critical (n = 0.8, 1.01) and supercritical (n = 1.2) range of plate
operation. The sensitivity of the plate structure with the viscoelastic corematerial subjected to
the load constant in time is clear. It is interesting that the course of these curves is observed
for the plates examined in the short loading time. Doubtlessly, a quick increase in the dynamic
loads, which involves the basic part of plate operation, influences such behaviour. One can find



Dynamic behaviour of three-layered annular plates... 787

that the course of curves for the plates subjected to the load constant in time corresponds to
the characteristic rheological curve with zones of the stationary and final creep.
The influence of dynamics of the increasing loading on the course of constant deflections of

the plate with the viscoelastic core compressed on the outer perimeter is shown in Fig. 12. The
rate of loading growth expressed by the parameter K7 is not out of meaning in the response of
the examined plate under the supercritical load (the number n is equal 1.01) constant in time.
The dynamic behaviour with an increase in deflections of the plate initially linearly loaded with
different values of the parameter K7 in a longer time of the loading with a constant value is
observed.The thickness h2 of the platewith the viscoelastic core has here the influence, too.The
results confirm dynamic sensitivity of the sandwich plate with the viscoelastic core. They also
show importance of the general solution with the accepted description of the physical relations
for the core rheological material in the analysed problem.

Fig. 12. Time histories of deflections of plates with the viscoelastic core subjected to the load constant
in time acting on the outer edge depending on the rate of loading growth K7 for different core
thicknesses h2: (a) h2 =0.005m, G2 =5MPa, n =1.01, m =7, (b) h2 =0.01m, G2 =5MPa,

n=1.01, m =6

7. Conclusion

Summarizing the presented results and observations, it could be stated that, in general, the
calculations of plates with an viscoelastic core could be replaced by calculations of similar plates
butwith an elastic core. Then, the solution of the problemcouldbe significantly easier. However,
the presented results show plate responses to its rheological properties. It has been observed
that the plate behaviour differ for plates subjected to short critical time loadings and for plates
loaded longer. The reactions on viscoelastic material parameters of the plate structure under
longer duration of loading are quite clear. The indicate that the use of viscoelastic expressions
in the plate physical relations is relevant. Then, the practical calculations becomemore difficult
and should be carried out solving a more complicated problem formulation. Additional plate
parameters connected with the effects of external factors like higher temperature make the
problemmore complex, which requires the achievement of the generalized solution.
The computational results also showed the influence of dynamics of the preliminary loading

on the growth of rheological deflections of plates loaded constant in time. Both loading time
histories expressed by the assumed function, in this paper by Eq. (6.1), as well as the speed of
loading growth have an important meaning in this problem. A detailed analysis could expand
the knowledge about dynamic behaviour of such a plate structure.
The carried out numerical experiments allow one to formulate the following statements:
• the use of proper constitutive relations for the plate material with rheological properties
is essential in the solution to the dynamic deflection problem,



788 D. Pawlus

• determination of thematerialmodel of the viscoelastic layer and the values of its constants
is a problem in itself. This multiparameter problem requires some compromise between
the attempt to find an exact problem description and the assumption of a rationally
sophisticated one,

• action of loads in longer duration on theplate structure identifies its rheological properties.

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Manuscript received February 21, 2014; accepted for print March 26, 2015