Jtam.dvi


JOURNAL OF THEORETICAL

AND APPLIED MECHANICS

47, 2, pp. 421-433, Warsaw 2009

DYNAMIC STABILITY OF A METAL FOAM CIRCULAR

PLATE

Ewa Magnucka-Blandzi
Poznań University of Technology, Institute of Mathematics, Poznań, Poland

e-mail: emagnucka@poczta.onet.pl

The study is devoted to a radial compressed metal foam circular plate.
Properties of the plate vary across its thickness. The middle plane of
the plate is its symmetry plane. First of all, a displacement field of any
cross-section of the plate was defined. Afterwards, the components of
strain and stress states were found. The Hamilton principle allowed one
to formulate a system of differential equations of dynamic stability of
the plate. This basic systemof equationswas approximately solved.The
forms of unknown functions were assumed and the system of equations
was reduced to a single ordinary differential equation of motion. The
equation was then numerically processed that allowed one to determine
critical loads for a family ofmetal foamplates. The results of studies are
shown in figures. They show the effect of porosity of the plate on the
critical loads. The results obtained for porous plates were compared to
homogeneous circular plates.

Key words: circular plate, dynamic stability, porous-cellularmetal

1. Introduction

There exist many works on the theory and analysis of plates. Most of them
deal with the classical (Kirchhoff) theory, which is not adequate in providing
accurate buckling. This is due to the effect of transverse shear strains. She-
ar deformation theories provide accurate solutions compared to the classical
theory. During the last several years, this problem has been developed by
many authors. Banhart (2001) provided a comprehensive description of va-
rious manufacturing processes of metal foams and porousmetallic structures.
Structural and functional applications for different industrial sectors havebeen
discussed. Awrejcewicz et al. (2001) described regular and chaotic behaviour
of flexible plates. Qatu’s (2004) book documents some of the latest research



422 E. Magnucka-Blandzi

in the field of vibration of composite shells and plates and fills certain gaps in
this area of research. Malinowski and Magnucki (2005) assumed a non-linear
hyphotesis of deformation of a plane cross-section of cylindrical shells. The
buckling problem was described for an isotropic porous shell. Magnucki and
Stasiewicz (2004) presented a problem of elastic buckling of a porous isotropic
beam with varying properties through thickness. They also assumed a non-
linear hypothesis. Instead, Szcześniak (2001) first of all described the problem
of forced vibration of the plate. Forced vibration dependent on impulsive, har-
monic and other loads was analized.

2. Displacements of a porous plate

This work is concerned with two isotropic porous circular plates under radial
uniform compression. The first one has a simply supported edge and the other
one a clamped edge. It is a continuation of the paper by Magnucka-Blandzi
(2008). This kind of material – a metal foam – was described, for example,
by Banhart (2001). Mechanical properties of thematerial vary through thick-
ness of the plate. Minimal value of Young’s modulus occurs in the middle
surface of the plate and maximal values at its top and bottom surfaces. For
such a case, the Kirchhoff and Mindlin plate theories do not correctly deter-
mine displacements of the plate cross-section. Wang et al. (2000) discussed
in details the effect of non-dilatational strain of middle layers on bending of
plates subject to various load cases. Magnucka-Blandzi andMagnucki (2005),
Magnucka-Blandzi (2006) thoroughly described the non-linear hypothesis of
deformation of the plate cross-section. A porous plate (Fig.1) is a generalized
sandwich plate. Its outside surfaces (top and bottom) are smooth,without po-
res. The material is of continuous mechanical properties. The plate is porous
inside, with the degree of porosity varying in the normal direction, assuming
the minimal value in the middle surface of the plate. A polar (cylindrical)
coordinate system is introduced with the z-axis in the depth direction.
Themoduli of elasticity andmass density are defined as follows

E(z)= E1[1−e0cos(πζ)]
G(z) =G1[1−e0cos(πζ)] (2.1)
̺(z)= ̺1[1−emcos(πζ)]

where
e0 – porosity coefficient of elasticity moduli, e0 =1−E0/E1,
em – dimensionless parameter of mass density, em =1−̺0/̺1,



Dynamic stability of a metal foam circular plate 423

Fig. 1. Scheme of a porous plate

E0,E1 – Young’s moduli at z =0 and z =±h/2, respectively,
G0,G1 – shear moduli for z =0 and z =±h/2, respectively,
Gj – relationship between the moduli of elasticy for j =0,1,

Gj = Ej/[2(1+ν)],
ν – Poisson’s ratio (constant for the plate),
̺0,̺1 – mass densities for z =0 and z =±h/2, respectively,
ζ – dimensionless coordinate, ζ = z/h,
h – thickness of the plate.

Choi and Lakes (1995) presented mechanical properties for porous ma-
terials. Taking into account the results of investigations of this paper, the
relation between the dimensionless parameter of mass density em =1−̺0/̺1
and dimensionless parameter of the porosity of the metal foam e0 is defined
as follows em =1−

√
1−e0. The field of displacements (geometric model) is

shown in Fig.2. The cross-section, being initially a planar surface, becomes
a surface (not a flat surface) after deformation. The surface perpendicular-
ly intersects the top and the bottom surfaces of the plate. Magnucka-Blandzi
andMagnucki (2005),Magnucki et al. (2006),Magnucki andStasiewicz (2004)
proposed a non-linear hypothesis of cross-section deformation of the structu-
re wall. Applying this hypothesis to a metal foam circular plate, the radial
displacement in any cross-section is in the form

u(r,z,t) =−h
{
ζ
∂w

∂r
−
1
π
[ψ1(r,t)sin(πζ)+ψ2(r,t)sin(2πζ)cos

2(πζ)]
}
(2.2)

where
u(r,z,t) – longitudinal displacement along the r-axis,
w(r,t) – deflection (displacement along the z-axis),
ψ1(r,t),ψ2(r,t) – dimensionless functions of displacements.



424 E. Magnucka-Blandzi

Fig. 2. Scheme of deformation of a plane cross-section of the plate – the nonlinear
hypothesies

In the particular case ψ1(r,t) = ψ2(r,t) = 0, the field of displacement u
is the linear Kirchhoff-Love hypothesis. Functions ψ1(r,t), ψ2(r,t) extend the
linear classical hypothesis. In the classical theory, the shear force is equal to 0
(it follows from this linear theory), but in the proposed non-linear hypothe-
sis the shear force does not equal 0, what corresponds with the facts. The
geometric relationships, i.e. components of the strain are

εr =
∂u

∂r
=−h

{
ζ
∂2w

∂r2
−
1
π

[∂ψ1
∂r
sin(πζ)+

∂ψ2
∂r
sin(2πζ)cos2(πζ)

]}

εϕ =
u

r
=−h

{1
r
ζ
∂w

∂r
−
1
π

[1
r
ψ1(r,t)sin(πζ)+

1
r
ψ2(r,t)sin(2πζ)cos

2(πζ)
]}

(2.3)
γrz =

∂u

∂z
+
∂w

∂r
= ψ1(r,t)cos(πζ)+ψ2(r,t)

[
cos(2πζ)+cos(4πζ)

]

where εr is the normal strain along the r-axis, εϕ is the circular strain, and
γrz – shear strain.
The physical relationships, according to Hooke’s law, are

σr =
E(z)
1−ν2

(εr +νεϕ) σϕ =
E(z)
1−ν2

(εϕ+νεr)
(2.4)

τrz = G(z)γrz

Moduli of elasticy (2.1) occuring here are variable and depend on the
z-coordinate. The similar porous plate model was presented by Magnucki et
al. (2006).



Dynamic stability of a metal foam circular plate 425

3. Equations of stability

The field of displacements in the above defined problem includes three unk-
nown functions: w(r,t), ψ1(r,t) and ψ2(r,t). Hence, three equations are neces-
sary for complete description of this problem.Theymay be formulated basing
on the Hamilton principle

δ

t2∫

t1

(T −Uε+W) dt = δ
t2∫

t1

T dt− δ
t2∫

t1

(Uε−W) dt =0 (3.1)

where T denotes the kinetic energy

T = πh
R∫

0

1/2∫

−1/2

r̺(ζ)
(∂w
∂t

)2
dζdr

Uε is the energy of elastic strain

Uε = πh
R∫

0

1/2∫

−1/2

r(σrεr+σϕεϕ+ τrzγrz) dζdr

W is the work which follows from the compressive force

W = πN(t)
R∫

0

r
(∂w
∂r

)2
dr

where R denotes the radius of the plate, ̺ –mass density of the plate, t1, t2
– initial and final times, N(t) – compressive force in the following form

N(t)= N0+Nacos(θt)

where N0 is the average value of the load and Na – amplitude of the load.
The kinetic energy is a function of ∂w/∂t, however the total potential

energy (Uε−W) does not depend on it. Taking into account principle (3.1),
a system of three stability equations for the porous plate under compression
is formulated in the following form



426 E. Magnucka-Blandzi

(δw)
∂

∂r

{
r
∂

∂r

{1
r

∂

∂r

[
r
(
c0
∂w

∂r
− c1ψ1− c2ψ2

)]}}
+

+4
1−ν2
E1h3

[
πN(t)

∂

∂r

(
r
∂w

∂r

)
+ c9̺1rh

∂2w

∂t2

]
=0

(3.2)

(δψ1)
∂

∂r

{1
r

∂

∂r

[
r
(
c1
∂w

∂r
− c3ψ1− c4ψ2

)]}
+
1−ν
h2

(
c5ψ1+ c6ψ2

)
=0

(δψ2)
∂

∂r

{1
r

∂

∂r

[
r
(
c2
∂w

∂r
− c4ψ1− c7ψ2

)]}
+
1−ν
h2

(
c6ψ1+ c8ψ2

)
=0

where

c0 =
π3−6e0(π2−8)

3π2
c1 =

8−πe0
π2

c2 =
225π −512e0
300π2

c3 =2
3π −4e0
3π2

c4 =
64−15πe0
30π2

c5 =
3π−8e0
3

c6 =
32−15πe0
30

c7 =
1575π −4096e0
2520π2

c8 =2
315π −832e0
315

c9 = π−2em

Boundary conditions are in the following form:
— for the first case (the plate with a simply supported edge)

w(R,t)= 0 Mr(R,t)= 0

ψ1(0, t)= ψ2(0, t) = 0
∂w

∂r

∣∣∣
r=0
=0

(3.3)

where the radial bendingmoment is in the form

Mr =
E1h

3

4(1−ν2)
[
−c0

∂

∂r
L(w)+

c1
π
L(ψ1)+

c2
π
L(ψ2)

]

L(f)=
df

dr
+
ν

r
f

— for the second case (the plate with a clamped edge)

w(R,t)= 0 ψ1(0, t)= ψ2(0, t) = 0

∂w

∂r

∣∣∣
r=R
=0

∂w

∂r

∣∣∣
r=0
=0

(3.4)



Dynamic stability of a metal foam circular plate 427

The system of differential equations (3.2) may be approximately solved
with the use of Galerkin’s method. Hence, three unknown functions are assu-
med:
— one satisfying boundary conditions (3.3) in the form

ψ1(r,t) =−6ψa1
[2+ν
4+ν

( r
R

)
−
1+ν
4+ν

( r
R

)2]

ψ2(r,t) =−6ψa2
[2+ν
4+ν

( r
R

)
−
1+ν
4+ν

( r
R

)2]

w(r,t) = wa(t)
[
1−3

2+ν
4+ν

( r
R

)2
+2
1+ν
4+ν

( r
R

)3]

—and one satisfying bounadry conditions (3.4) in the form

ψ1(r,t) =−6ψa1
[( r
R

)
−
( r
R

)2]

ψ2(r,t) =−6ψa2
[( r
R

)
−
( r
R

)2]
(3.5)

w(r,t) = wa(t)
[
1−3

( r
R

)2
+2
( r
R

)3]

Because of similarity of the solution in both cases, only the second case
will be considered hereafter. Substitution of above three functions (3.5) into
equations (3.2) and making used Galerkin’s method yields a system of three
equations in the form

[
c0−
4π(1−ν2)R2
15E1h3

N(t)
]
wa(t)− c1Rψa1 − c2Rψa2 +

+c9̺1
4(1−ν2)R4
105E1h2

d2wa
dt2
=0

c1wa− c13Rψa1 − c12Rψa2 =0 (3.6)
c2wa− c12Rψa1 − c11Rψa2 =0

where

c10 =
(1−ν)R2
15h2

c11 = c7+ c8c10

c12 = c4+ c6c10 c13 = c3+ c5c10



428 E. Magnucka-Blandzi

From the second and third equations of system (3.6), the functions ψa1,
ψa2 may be calculated, namely

ψa1 = ψ̃a1
wa
R

ψa2 = ψ̃a2
wa
R

(3.7)

where

ψ̃a1 =
c1c11− c2c12
c13c11− c212

ψ̃a2 =
c2c13− c1c12
c13c11− c212

Substitution of functions (3.7) into the first equation of system (3.6) yields the
Mathieu equation in the followng form

d2wa
dt2
+Ω2

[
1−2µcos(θt)

]
wa =0 (3.8)

where

Ω2 = ω2
(
1−

N0
Ncr

)
µ =
1
2

Na
Ncr−N0

and
ω – the natural frequencies of the plate

ω =

√√√√c0− c1ψ̃a1 − c2ψ̃a2
c9

105E1h2

4R4(1−ν2)̺1

Ncr – the critical force [N/mm]

Ncr =
15
(
c0− c1ψ̃a1 −c2ψ̃a2

)
E1h

3

4πR2(1−ν2)

4. Numerical calculations of unstable regions

TheMathieu equation iswell-knownanddescribed inmamybooksandpapers,
for exampleDoyle (2001), Życzkowski (1988), Gryboś (1980) and others. They
concluded that there are separate regionswhereunboundedsolutions exist and
regions where all solutions are bounded.
Based on Życzkowski (1988), the first unstable region is determined by

2Ω
√
1−µ < θ < 2Ω

√
1+µ (4.1)



Dynamic stability of a metal foam circular plate 429

and the second one by

Ω
√
1−2µ2 < θ < Ω

√
1+
1
3
µ2 (4.2)

Assuming two dimensionless parameters

α1 =
N0+Na
Ncr

α2 =
N0
Ncr

inequality (4.1) for the first unstable region is in the form

2ω

√
1−
1
2
(α1+α2) < θ < 2ω

√
1+
1
2
(α1−3α2) (4.3)

and inequality (4.2) for the second one is in that form

ω

√
(1−α2)2− 12(α1−α2)2

1−α2
< θ < ω

√
(1−α2)2+ 112(α1−α2)2

1−α2
(4.4)

Geometric illustration of constraints of these parameters are shown in Fig.3.
I – determines the parameters α1 and α2 for which only the first unstable
region exists. II – determines the parameters α1 and α2 for which only the
second unstable region exists. If α1 > 1, then the compressive force reaches
the critical value. Only the first unstable region exists if α1 > 1.

Fig. 3. Constraints of the parameters α1 and α2

There are three examples considered below, where the effect of porosi-
ty change is shown. A family of plates of height h = 10mm and radius
R =1500mm are taken into account. Thematerial constants are

E1 =2.05 ·105MPa ̺1 =7.78 ·10−6
kg
mm3



430 E. Magnucka-Blandzi

In the first instance, α1 =1, α2 =1/2 are assumed, which means

N(t)=
1
2
Ncr[1+cos(θt)]

So, the first and the second unstable region have the form

ω < θ <
√
3ω

1
2
ω < θ <

√
78
12

ω

The plots of these two unstable regions are shown in Fig.4.

Fig. 4. Stability regions

The next example is for α1 =1/2 and α2 =1/6, then

N(t)=
1
3
Ncr[2+cos(θt)]

Figure 5 shows two unstable regions. In the above two cases, the compressive
force does not reach the critical load, and it can be observed that two unstable
regions exist.

Fig. 5. Stability regions



Dynamic stability of a metal foam circular plate 431

But in the last example, the compressive force reaches the critical load.
This situation was considered for α1 =5/3 and α2 =1/6

N(t)=
3
2
Ncr
[1
9
+cos(θt)

]

Only one unstable region exists there, which is presented in Fig.6.

Fig. 6. Stability regions

5. Conclusions

The porous-cellular circular plate is a generalization of sandwich or multi-
layer plates. Correct hypotheses of plane cross-sections for homogeneous pla-
tes are useless in the case of porous-cellular plates as elastic constants va-
ry considerably along their depth. The non-linear hypothesis of deforma-
tion of the plane cross-section for porous-cellular plates (structures) inc-
ludes the linear hypothesis for homogeneous plates and the shear defor-
mable effect. The mathematical model of dynamic stability of the porous-
cellular circular plate, under a pulsating compression load, could be re-
duced to the Mathieu equation. Two unstable regions may determined
if the compressive force does not reach the critical load. The influence
of the porosity coefficient of elasticity moduli is small for the unstable
regions.



432 E. Magnucka-Blandzi

References

1. Awrejcewicz J., Krysko V.A., Krysko A.V., 2001, Regular and chaotic
behaviour of flexible plates,Third Int. Conf. Thin-Walled Structures, Elsevier,
349-356

2. Banhart J., 2001, Manufacture, characterisation and application of cellular
metals andmetal foams,Progress in Materials Science, 46, 559-632

3. Choi J.B., Lakes R.S., 1995, Analysis of elastic modulus of conventional
foams and of re-entrant foam materials with a negative Poisson’s ratio, Inter-
national Journal of Mechanical Science, 37, 1, 51-59

4. Doyle J.F., 2001,Nonlinear Analysis of Thin-Walled Structures. Static, Dy-
namics, and Stability, Springer-Verlag NewYork

5. Gryboś R., 1980, Stability of Structures under Impact Load, Warsaw, PWN
[in Polish]

6. Magnucka-Blandzi E., 2006, Vibration of a porous-cellular circular plate,
Proceedings in Applied Mathematics and Mechanics, PAMM, 6, 243-244

7. Magnucka-Blandzi E., 2008,Axi-symmetrical deflection andbuckling of cir-
cular porous-cellular plate,Thin-Walled Structures, 46, 333-337

8. Magnucka-Blandzi E.,Magnucki K., 2005,Dynamic stability of a porous
circular plate,Proc. 8th Conference on Dynamical Systems, Theory and Appli-
cations, J. Awrejcewicz, D. Sendkowski, J.Mrozowski (Edit.), Łódź, Poland,1,
353-360

9. Magnucki K., Malinowski M., Kasprzak J., 2006, Bending and buckling
of a rectangular porous plate, Steel and Composite Structures, 6, 4, 319-333

10. Magnucki K., Stasiewicz P., 2004, Elastic buckling of a porous beam, Jo-
urnal of Theoretical and Applied Mechanics, 42, 4, 859-868

11. Malinowski M., Magnucki K., 2005, Buckling of an isotropic porous cylin-
drical shell,Proc. Tenth Int.Conference onCivil, Structural andEnvironmental
Engineering Computing, B.H.V.Topping (Edit.)Civil-ComputerPress Stirling,
Scotland, Paper 53, 1-10

12. Qatu M.S., 2004,Vibration of Laminated Shells and Plates, Elsevier, Amster-
dam, Boston, Heidelberg, London, New York, Oxford, Paris, San Diego, San
Francisco, Singapore, Sydney, Tokio

13. Szcześniak W., 2001, Selected Problems of Dynamics of Plates,Warsaw,Ofi-
cynaWydawnicza PolitechnikiWarszawskiej [in Polish]

14. Wang C.M., Reddy J.N., Lee K.H., 2000, Shear Deformable Beams and
Plates, Elsevier, Amsterdam, Lousanne,NewYork,Oxford, Shannon, Singapo-
re, Tokyo



Dynamic stability of a metal foam circular plate 433

15. Życzkowski M. (Edit.), 1988,Mechanics Technology, Strenght of Structures
Elements, Warsaw, PWN [in Polish]

Stateczność dynamiczna płyty kołowej wykonanej z piany metalowej

Streszczenie

Przedmiotembadań jest osiowo ściskana porowatapłyta kołowa.Własności płyty
są zmienne na jej grubości. Środkowapłaszczyzna płyty jest jej płaszczyzną symetrii.
Zdefiniowanopole przemieszczeńdla dowolnegoprzekrojupłyty.Ponadtowyznaczone
są składowe stanów odkształceń i naprężeń. Układ równań stateczności dynamicznej
płyty wyznaczono z zasady Hamiltona. Następnie równania te rozwiązano metodą
Galerkina i otrzymano zwyczajne równanie różniczkowe typu Mathieu. Równanie to
rozwiązano metodą Rungego-Kutty i wyznaczono obciążenia krytyczne dla rodziny
płyt kołowych. Wyniki przedstawiono na wykresach. Wskazano wpływ porowatości
materiału na obciążenia krytyczne płyty. Otrzymane wyniki porównano z odpowied-
nimi wielkościami dla płyty jednorodnej.

Manuscript received September 19, 2008; accepted for print February 12, 2009