Analytical and experimental research stiffness characteristics of laminated packet are presented in this work. Package consists of alternating thin elastomeric and metallic layers jointed by vulcanization or gluing. Such packages are used in machine building, shipbuilding, civil engineering as compensating devices. The analytical expression of “compressive force – displacement” dependence is derived for the flat thin layered rubber – metal element on the basis of the variational principle; force is directed flatwise. The analytical expression of “shear force – deformation” is derived for the pre-compressed rubber-metal element. Analytical solution was confirmed by experimental data for flat packet in the shape of rectangular prism

S. Polukoshko

V. Gonca

English

Vibroengineering PROCEDIA

  ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 107 
Research of Compressive and Shear Stiffness of Laminated 
Elastomeric Structures 
V Gonca and S Polukoshko 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
V Gonca1 and S Polukoshko2 
1 Riga Technical University, Institute of Mechanics,  
Ezermalas Str. 6, Riga, LV-1006, Latvia 
2 Ventspils University College, Engineering Research Institute “VSRC”,  
Inzenieru str. 101, Ventspils, LV-3601, Latvia 
 
E-mail: vladimirs.gonca@rtu.lv 
Abstract. Analytical and experimental research stiffness characteristics of laminated packet are presented in this 
work. Package consists of alternating thin elastomeric and metallic layers jointed by vulcanization or gluing. 
Such packages are used in machine building, shipbuilding, civil engineering as compensating devices. The 
analytical expression of “compressive force – displacement” dependence is derived for the flat thin-layered 
rubber – metal element on the basis of the variational principle; force is directed flatwise. The analytical 
expression of “shear force – deformation” is derived for the pre-compressed rubber-metal element. Analytical 
solution was confirmed by experimental data for flat packet in the shape of rectangular prism. 
1. Introduction 
Elastomeric (rubber and rubberlike materials) have very small volume compressibility and are able to 
maintain large elasticity deformation [1-5]. Laminated elastomeric structures consist of a large number 
of alternating thin layers of elastomeric and much more rigid reinforcing layers, usually metal, jointed 
by vulcanization or gluing. This allows to obtain the structures with axial compression stiffness much 
greater than shear stiffness. Packets of thin-layered rubber-metal elements (TRME) are successfully 
used as compensators, shock-absorbers, vibroisolators. These structures are used in machine building, 
shipbuilding, civil engineering, aviation and aerospace due to its unique mechanical properties. In 
practice the TRME packages of different geometric shapes are used: flat, cylindrical, conical, spherical 
and others; number of layers may be different, at least three. In many applications of TRME structures 
it is necessary to know its stiffness characteristics, in particular, if TRME packet is used for vibration 
isolation of the object from vibrating base. In this paper the compressive and share stiffness 
characteristics of flat TRME of rectangular shape are discussed (Fig. 1a). 
TRME packet in Fig. 1 consists of square or rectangular elastomeric and steel layers with 
dimensions 
e
a b h× ×  and 
m
a b h× ×  respectively; further we considered square elements with a b=  
and ,
e m
a h h>> . TRME packet is compressed by the axial force 
z
P  in normal to the layers z-direction 
on Δ-value (Fig. 1b), and then shear force 
x
P  is applied to the margin plate in x-axis direction, causing 
the displacement of non-elastomeric layers on value f in this direction (Fig. 1c).  
For TRME fabrication elastomeric with shear modulus 0.1 1.2MPa
e
G ≈ ÷  and bulk modulus 
1800 3000MPa
e
K ≈ ÷  are used. Taking into account that 
2(1 )
1 2
e e
K G
µ
µ
+
=
−
 the Poisson ratio is equal 
0.485 0.499µ ≈ ÷ .  
 
                                                     
1 Corresponding author 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
108 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
a)  
b)  c)  
d)  
Figure 1. Scheme of testing object: a) flat rectangular TRME packet,  
b) deformation of packet under compression, c) deformation of packet under shear force,  
d) compressive deformation of rubber layer between two metal layers. 
he – thickness of elastomeric layer, hm – thickness of metal layer, h – thickness of margin metal layer. 
2. Analytical study of stiffness characteristics of laminated packet 
The dependences of "axial force – displacement" (
z
P  – Δ) and "shear force – shift" (
x
P  – f) will be 
received in two stages. 
The first stage is axial compression of TRME packet; on this stage the weak compressibility of 
elastomeric layer, effect of deformation non-elastomeric layer, physical nonlinearity (the dependence 
of modules 
e
G  and 
e
K  on the hydrostatic pressure), the geometric nonlinearity (for deformation up to 
50% ÷ 60%) are taken into account. The dependence of "force – displacement» (
z
P  – Δ) for 
deformations up to 10% ÷ 15% may be obtained from the principle  
of minimum total potential energy of deformation Π : 
 ( )( ) 2, , , , , 2
,
1 3 9 1 2
4 1 4 (1 ) k
k
k k
k k k k k k k k k k
k i j j i i j j i i i k i ik k
k e m V F
G u u u u S U S dV P u dF
σ
σ
µ µ
µ µ
=
  − 
Π = + + + − −  
+ +   
∑ ∫∫∫ ∫    (1) 
where: ,  
e m
G G  – shear moduli of the elastomeric and non-elastomeric material layers, k
i
u  – 
displacement function of elastomeric and non-elastomeric layers, kS  – hydrostatic pressure function 
for elastomeric and non-elastomeric layers. 
Sign "," denotes the partial differentiation. The summation is performed over repeated subscripts. 
Compressive stiffness of non-elastomeric layer is by several orders of magnitude larger than of the 
elastomeric layer. Therefore, we only consider non-elastomeric layers stretching in this plane layer, 
which is caused by the layer of elastomeric shear stress in the cross sections of their contacts (Fig. 1d). 
If elastomeric layer is vulcanised to metallic layer, the geometric boundary conditions for the 
functional (1) are: 
 ( , , 0.5 ) ( 0.5 ), ( , , 0.5 ) ( 0.5 ), ( , , 0.5 ) 0.5
e m m e m m e
u x y h u h v x y h v h w x y h± = ± ± = ± ± = ± ∆    (2) 
where ,  ,  u v w  – displacement on ,  ,  x y z  – axis directions. 
The hypothesis of plane sections for the elastomeric layer and uniform tensile strain for 
non-elastomeric layer are used when the displacements functions are selected. In view of introduced 
hypotheses and joint conditions (2) for a flat square TRME required displacements functions it is 
sufficient to choose the form of: 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 109 
 
( )
23
2 2 2 2
2 2
( 0.25 ) , ( 0.25 ) , 2
3 4
0.25 , , , 0
e
e e e e e
e m m m m
zhz
u Ax z h Lx v Ay z h Ly w C Lz
S D z h u Lx v Ly w S
 
= − + = − + = − − − 
 
= − = = = =
   (3) 
From the principle minimum potential energy 
( , , , )
0
( , , , )
A C D L
A C D L
∂Π
=
∂
 for TRME package we obtain the 
dependence: 
 
1
2 2
2 2
4 5 0.5
1 1 , , 1 ,
8 1 (1 2 ) 24
z e m m
e e e e
P h N G hM M a
M
G a M G h h
pi α
υ α
pi υ µ
−
  
∆ = + + = = + =   + −   
   (4) 
where N – the number of elastomeric layers. 
When υ→∞  and 0.5µ→  from (4) the known dependence of 
z
P  – Δ excluding weak 
compressibility of the elastomer layer and deformation non-elastomeric layer follows [6]. At high 
specific axial forces the physical nonlinearity (dependence of 
e
G  and 
e
K  moduli on hydrostatic 
pressure) must be taken into account. The approximate method may be used, for which it is enough to 
have only a linear solution (4), in which the shear modulus 
e
G  is replaced by the relationship ( )
e
G S : 
 2( ) (1 ), ,z
e e
P
G S G S S F a
F
β= + ≈ =    (5) 
β  – coefficient determined from volume compression of elastomer layer experiment [6]. Due to the 
lack of experimental data, it is assumed that: 
 
( ) ( )
(1 )e e
e e
G S K S
S
G K
β≈ ≈ +    (6) 
When considering the average strain (up to 50% ÷ 60%) the delta- method is used [6]. In this case, the 
solution for small strains (4) conveniently to present as the sum of two solutions:  
 
n o
∆ = ∆ + ∆    (7) 
where 
n
∆  – solution for an incompressible material ( 0.5µ = ), 
o
∆  – elastomeric layer volume 
changes. 
For ТРМE 
e
a h>>  and 1
e
a
h
α = >> , in this case from (4) and (7) after simplifying, we have: 
 
1
2 2 2 2
2 2 2
4 5 3(1 2 )
1 1 ,
192 48 2(1 )
z e z e
n o
e e
P h N P h N
G a G a
pi α pi α µ
pi υ µ
−
   −
∆ = + + ∆ =  
+  
   (8) 
Using the technique of the delta method for axial deformation up to 50% ÷ 60%, dependence  
(
z
P  – 
1
∆ ) for one elastomer layer is obtained in the form: 
 ( )
( )2 22
2 2 1
2 2
1
3.6ln 0.25 1 1 , 1
4
z e
e
aa
P G a
h h h
λ
λ λ λ
υ
−
−
 −  ∆
 = − + − + = − 
    
   (9) 
At high specific axial forces 
2
z
P
a
 the shear modulus 
e
G  in (9) must be replaced by 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
110 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
( ) (1 )
e e
G S G Sβ= + . 
From (9) axial displacement of a single layer 
1
∆  is found and displacement for package is equal 
1
N∆ = ∆ . 
At the second stage the shear stiffness of the preliminary compressed by 
z
P  force TRME packet is 
defined.  
The problem is considered as two-dimensional: deformation in z-direction under 
x
P  action is 
negligibly small in compare with the preloading deformation Δ.  Without preloading the dependency 
(
x
P  – 
1
f ) has the form: 
2
1
x e
e
a f
P G
h
= . 
Using the linearized theory of superposition of small deformations [6], we obtain the dependence 
of (
x
P  – 
1
f ) for one elastomeric layer under preliminary axial compression: 
 
2
1 1, where 1
x e
e e
a f
P G
h h
λ λ
∆
= = −    (10) 
It follows from (10) that the shear stiffness of the elastomeric layer decreases continuously with 
preload compression Δ1 increasing. This fact is confirmed by experiments [6, 7].  
If all elastomeric layers are the same, shear displacements of TRME package under the action of 
compressive forces 
х
P  is calculated by the formula:
1
f f N= . 
3. Experimental research of compressive and shear stiffness of TRME packet 
A series of experiments for evaluation of the compressive and shear stiffness stiffness were performed 
for flat TRME of prismatic shape with different thicknesses (from 0.1 to 0.58mm) and number of the 
rubber layers (10, 16 and 32) in scientific – research institute “VNIIMASH” (Moscow). The table 1 
gives the results of share experiments for PRM-35 packet ( )35mm, 0.1mm, 0.1mm
e m
a h h= = =  , and 
PRM-60 ( )60mm, 0.28mm, 0.1mm
e m
a h h= = =  elements with number of rubber layers 16N = , 
made by vulcanization, and analytical solution in accordance with Eq. (10). 
 
Table 1. Experimental results of shear displacement of the 16-layer TRME, compressed with vertical 
force Pz = 15kN. 
Horizontal 
force, Px 
Element PRM-35 Element PRM-60 
Uniform 
pressure,  
Px / A 
Share 
stiffness 
eksperiment 
Displacement 
eksperiment 
Displacement 
teoretical 
Uniform 
pressure,  
Px / A 
Share 
stiffness 
eksperiment 
Displacement 
eksperiment 
Displacement 
teoretical 
kN N/mm2 N/mm mm mm N/mm2 N/mm mm mm 
0.50 0.408 330 1.515 1.499 0.139 460 1.087 1.465 
1.00 0.816 270 3.704 2.998 0.278 415 2.410 2.926 
1.50 1.224 230 6.252 4.498 0.417 375 4.000 4.394 
2.00 1.633 210 9.524 5.997 0.556 350 5.714 5.859 
2.50     0.694 325 7.692 7.324 
3.00     0.833 315 9.524 8.788 
 
Plots of dependence of shear stiffness on horizontal displacement for PRM-35 PRM-60 elements 
are given in Fig. 2; plot of dependence of compression stiffness on vertical displacement for element 
PRM-35 PRM is presented in Fig. 3. 
The shear stiffness may be approximated by the curves for FRM-35: 0.265( ) 194 20 / m6 N mxs x e−= + , 
for  FRM-60: 
0.236( ) 290 220 N/mmxs x e−= + ; compressive stiffness for FRM-35 
( ) 24660 993kN/mmc z z= − . 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 111 
  
Figure 2. Plots of dependence of shear stiffness on 
horizontal displacement for elements:  
FRM-60:  – experimental data,  
 – in accordance with approximating curve;  
FRM-35:  – experimental data,  
 – in accordance with approximating curve. 
 Figure 3. Plots of dependence of compressive 
stiffness on vertical displacement for element 
FRM-35:  – experimental data,  
 – in accordance with approximating 
curve. 
4. Natural frequency of object protected with TRME packet  
Static stiffness of TRME structures may be used for preliminary estimation of the natural circular 
frequency of the single-mass object in case of using TRME packet for protecting it against vibrating. 
Vibroisolator is placed between the object to be protected and a vibrating base. The lower plate of 
vibroinsolator is subjected to kinematic excitation. In Fig. 4 the example of TRME packet applying for 
the protecting the object against against the horizontal vibration of base is shown. 
 
 
a) 
 
b) 
Figure 4. TRME packet for vibroisolation:  
a) scheme of protected object, b) analytical model of vibration object. 
 
Differential equation of protected object motion: ( ) ( ),
b b
mx s x x d x x= − − − −ɺɺ ɺ ɺ  where x – mass 
centre displacement in respect to static equilibrium centre, ( )
b
s f x x= − . Equation of motion may be 
simplified if we denote coordinate of the relative motion of mass centre in respect to the base as 
r
b
x x x= − , then r
b
x x x= −ɺɺ ɺɺ ɺɺ , and r
b
x x x= +ɺɺ ɺɺ ɺɺ : 
 
( )
( )
r
r r r
d s x
z z x t
m m
η+ + = −ɺɺɺɺ ɺ , 
0
( )
r
s x
m
ω =   (11) 
RESEARCH OF COMPRESSIVE AND SHEAR STIFFNESS OF LAMINATED ELASTOMERIC STRUCTURES.  
V GONCA AND S POLUKOSHKO 
112 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
0
ω  – natural circular frequency of vibrating mass, it may be approximately estimated from the 
“stiffness – displacement” analytical dependence. If exitation frequency is known, the resonance 
phenomenon possibility may be estimated.  
5. Conclusion 
The thin-layered rubber-metal structures using in practice are still limited because of the complicated 
theoretical calculations and the lack of simple calculation models. This paper presents a simple model 
of a flat TRME package, which takes into account the physical nonlinearity of elastomeric, provided 
that the task remains geometrically linear. Experimental studies indicate that, under these loads a 
significant non-linearity of the "force-displacement" stiffness characteristics associated with the 
physical nonlinearity of elastomeric materials takes place. Traditional methods of calculation do not 
allow to describe non-linear stiffness characteristics of these elements. 
The analytical expressions for “compressive force – displacement” and “share force – deformation” 
characteristics of the flat TRME in the form of rectangular prism are derived on the basis of the 
variational principle; metallic plates-layers are assumed to be perfectly rigid.  
Analytical solution in accordance with the received formulae shows the good coincidence with 
experimental data. This proves that the posed problem is completely solved and corresponds to the 
research of other scientists. 
Approximating equations for compressive and share stiffness dependence on displacement was 
derived and used in the equation of motion of the object, protected against vibrating by means of 
TRME packet. 
In future investigations it is necessary to clarify the dissipative properties of the rubber and to 
develop a model of dissipative forces; to clarify the limits of application of the formulas for 
calculating of static stiffness of the TRME dampers; to develop a model of dynamic compression 
stiffness and its verifying. 
References 
[1] N N Frolov, S J Moldovanov and S B Lozovoy 2011 Mechanics of thin rubber elements: 
monograph Kuban State Technological University, Krasnodar: Publishing House “Jug” 
[2] J M Kelly and D A Konstantinidas 2011 Mechanics of Rubber Bearings for Seismic and 
Vibration Isolation UK: John Wiley & Sons 
[3] Rivin E I 2003 Passive Vibration Isolation NY: ASME Press 
[4] A N Gent 2001 Engineering with Rubber: How to Design Rubber Components Munich: Carl 
Hanser Verlag 
[5] J T Bauman 2008 Fatigue, Stress and Strain of Rubber Components: Guide for Design 
Engineers Munich: Carl Hanser Verlag 
[6] E E Lavendel 1990 Workpieces from Highly Elastic Materials Design Methods Zinatne, Riga 
[7] N S Gusyatynskaya 1980 Study of TRME machine elements Moscow: Mashinostrojenie 


Research of Compressive and Shear Stiffness of Laminated Elastomeric Structures

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Research of Compressive and Shear Stiffness of Laminated Elastomeric Structures

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