The paper presents a method of identification of physical and mechanical characteristics of the compound (calculated and experimental test method parameters). The idea of the method is that the studied material is connected with other materials, the characteristics of which are well known and differ from the characteristics of the material. Desired physical and mechanical properties should be considered unknown in the proposed calculation model, and the values of the parameters of trial constructions that can be measured accurately with experimental methods in the process of product operation, as well as characteristics of test materials and geometric dimensions of proposed structures are suggested to be placed in the same model as input information

A. Bubulis

A. V. Goroshko

V. P. Royzman

English

Vibroengineering PROCEDIA

  ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 81 
Identification of Physical and Mechanical Properties of 
Compound by Solving Inverse Problems 
A V Goroshko, V P Royzman and A Bubulis 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
A V Goroshko1, V P Royzman2 and A Bubulis3 
1, 2 Khmelnitsky National University, Institutska st. 11, 29016, Ukraine  
3 Kaunas University of Technology, Lithuania 
 
E-mail: algimantas.bubulis@ktu.edu 
Abstract. The paper presents a method of identification of physical and mechanical characteristics of the 
compound (calculated and experimental test method parameters). The idea of the method is that the studied 
material is connected with other materials, the characteristics of which are well known and differ from the 
characteristics of the material. Desired physical and mechanical properties should be considered unknown in the 
proposed calculation model, and the values of the parameters of trial constructions that can be measured 
accurately with experimental methods in the process of product operation, as well as characteristics of test 
materials and geometric dimensions of proposed structures are suggested to be placed in the same model as input 
information. 
1. Introduction 
The object of research was the process of determining physical and mechanical properties of the 
polymer compound that is used for sealing flat micromodules (FMMs). 
Typical structure of flat micrimodule, used in electronic engineering, is shown in Fig. 1. During the 
operation of FMMs the structure “electronic element-compound” is subject to dangerous stresses 
caused by the difference of the coefficients of linear thermal expansion, Poisson’s ratio and elastic 
moduli of these materials, which leads to the destruction of FMMs and the consequent destruction of 
expensive complex systems which contain them. 
The analysis carried out by the authors showed that variations in the physical and mechanical 
properties of compounds in the production of FMMs can reach 300% and depends on many factors, 
such as the place where compound components are manufactured. In addition, the characteristics of 
compounds in the temperature range of 60°C–20°C are unknown, although the FMMs are operated in 
such temperatures. 
Existing methods for determining the physical and mechanical properties of materials have a 
number of drawbacks which significantly narrows the range of applications. Speaking of static 
methods, it should be mentioned that since the studied materials are usually fragile, then, firstly, there 
are difficulties to comply with alignment when attaching samples in testing machines, and, secondly, 
the destruction of these samples occurs under low strain which reduces the accuracy of measurement. 
Moreover, the results of this experiment strongly dependent on the type of load, size of samples and 
their manufacture methods. 
Dynamic methods can be applied only for materials with low absorption of oscillations. They 
depend on the temperature conditions under which the experiment is conducted and in order to be 
effective, they require the information about the speed of propagation of elastic waves in the material, 
the definition of which being a difficult task. 
In addition, both existing static and dynamic methods allow to determine the physical and 
mechanical properties of generally homogeneous materials in which the value of these characteristics 
does not depend on the type of stress state of the material in a particular product. The considered 
compound exists as a sealant only in certain structures, such as FMM, and its physical and mechanical 
                                                     
3 Corresponding author 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
82 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
properties are different from the characteristics of the same compound in a different structure. 
Thus, the problem of finding methods for identification of physical and mechanical properties of 
compounds directly in real structures in the temperature range -60°C ... 20°C is relevant. 
 
 
Figure 1. The design of a 
typical flat Micromodules. 
2. Identification of physical and mechanical properties of the material 
The study [1] is focused on the causes of FMM destruction and provides a mathematical model of 
stress calculation in the system “electronic element-compound”, represented by figures 2 and 3. The 
formula for calculating contact pressure in the materials of resistor and compound at a stable 
temperature looks the following way (1).  
 
( ) ( )
( )
( ) ( )
( )
1 2
1 2 2 2 2
2 1 1 2 2 1 1 2
2 2 2 2
1 2 1 2 3 2
( )
1 1 2 1 1 2
k
t
P
R R R R
E R R E R R
α α
µ µ µ µ
−
=
− + − + + −
+
− −
   (1) 
where 
1 2
,  α α  – is the coefficient of linear thermal expansion in the material of the product and 
compound, respectively; 
1 2
,  E E  – their elastic moduli; 
1 2
,  µ µ  – their Poisson’s ratio.
 
The model proved to be suitable for calculating the strength of parts of the system, and methods for 
determining the physical and mechanical characteristics, based on their identification by solving 
inverse problems of strength, appeared to be promising. 
The authors developed an experimental method of calculation with testing parameters which can 
simultaneously identify the coefficient of linear thermal expansion, elastic modulus and Poisson's ratio 
of a compound. 
 
 
Figure 2. Cross-section of an electronic 
element, compound polymerized with 
compound: 
1 - compound, 
2 – isolated compound cylinder, 
3 - electronic element. 
 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 83 
 
Figure 3. The electronic element surrounded by a layer of compound. 
 
The idea of the method is that the studied material is connected with other trial materials, the 
properties of which are well known and differ from the properties of the material. Whereas the 
connecting parts should have specific forms that make it possible to describe the stress-strain state of 
the samples in real structures with the same equations. 
In order to determine the physical and mechanical characteristics it is offered to consider the values 
of the identified parameters to be unknown in the calculation model. It is suggested that the values of 
the parameters of trial structures, which can be measured accurately by experimental means in the 
process of product operation, and characteristics of trial materials and geometric dimensions should be 
inserted in the same model as input information. Producing a number of test samples with values of 
physical and mechanical properties that are set so that the measured deformation values would be 
different, it is possible to write the number of linearly independent equations with the identified 
parameters based on the computation model which is required for their determination. 
To identify the abovementioned physical and mechanical properties of the compound that seals and 
surrounds resistor we have made double-layer cylindrical structures “trial material - compound”, 
choosing the trial material with thoroughly investigated properties, different from the properties of 
compound. In these structures contact pressure occurs at the interface of materials at temperature 
extremes.  The parameters that are included in the selected mathematical model and can be accurately 
measured by experimental methods are strains emerging under contact pressure on the outer surface of 
the testing cylinder, their values being connected with stress values by Hooke's law. The ratio 
determined by Hooke's law and the ratios specified by the authors of the developed model make it 
possible to find the value of contact pressure for known radial sizes of structures according to the 
experimentally determined values of strain. 
According to the authors' model, the contact pressure can be considered as a structure parameter, 
which is determined experimentally and whose value can be substituted in the calculation model as 
input information in the form of 
 
2 2
1 1 2 1
1 2 2 2 2
2 1 1 2 1 1
1 1 2
k t
R R
P
R R E R R E
µ µ
ε
 + −
= − ⋅ + ⋅ 
− − 
   (2) 
where 
t
ε  is relative circumferential deformation of the contact surface. 
Let’s explore the expression (1) for the contact pressure and turn it into the form suitable for the 
determination of the identified parameters. The analysis of the expression on the right side of (1) 
shows that its denominator is different from zero, since the ratio 
1 2 3
R R R< <  is always preserved 
between the radial size of the studied structure, and the value of Poisson's ratio for any material lies 
within 0 0,5µ< < . 
Let’s transform the right-hand side of (1) into the following 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
84 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
 1 2
1 22
31 2 1 1 2
2 2 2 2
1 2 1 2 1 2 3 2
( )
1 2 1 2 2 2
k
t
P
RR
E E E R R E R R
α α
µ µ µ µ
−
=
− − − −
− + ⋅ +
− −
   (3) 
After introducing new variables ( )1 2 21 2 Eχ µ= − , ( )2 2 22 Eχ µ= − , expressed by parameters 2E , 
2
µ , subject to identification, we’ll transform equation (3) in such a way that all the parameters that 
need definition, should be located in the left side of the resulting equation. After performing the 
transformation and introducing additional notation, we obtain 
 
1 2 1 1 2 1 1
( )
k k k
P B P t E CD P tχ χ α α− + = − + +   (4) 
where 
2 2 2
3 3 2
( )B R R R= − , 
1 1
(2 )C Eµ= − , 
2 2 2
1 2 1
( )D R R R= − , 
2 1
(1 2 )E Eµ= − . What we get is a 
linear equation with respect to the unknown 
1
χ , 
2
χ , 
2
α . For their determination it is necessary to 
write a system of three linear algebraic equations. For this purpose, we’ll firstly assume that the 
sample is made of three different sets of values of the entire set of primary factors and the 
corresponding measured values of contact pressures. This will add the following system of equations 
 
0 2 1 1 2 1 1
( ) ,
1,2,3,
ki i ki i i i ki i
P B P t E C D P t
i
χ χ α α− + = − − +

=
   (5) 
where the indices of coefficients 
i
B , 
i
C , 
i
D , 
i
E  and contact pressure 
1ki
P  correspond to the vector of 
parameters { }1 1 1 1 2 3, , , , ,i i i i i i ix E R R Rα µ=  of the “i” trial sample ( 1,2,3i = ). 
Solution of equations (5) can find the properties of the material being identified. 
However, the method of test parameters doesn’t require setting various values of all parameters 
vector in order to achieve different values of contact pressures. It's enough to change one or more 
parameters in various combinations. Further we will explore such options or their combination, which, 
firstly, are most easily changed, and secondly, which ensure the existence, uniqueness and stability of 
solutions of equations (5). 
The condition for the existence and unity of solutions of the mentioned equations is that its 
determinant ∆ , should be different from zero, i.e. 
 [ ]2 1 1 1 2 2 3 1 1 1 3 1 3 1 2 1 3 3 2( ) ( ) ( )k k k k k kt P P B B P P B B P P B Bα∆ = − + − + −    (6) 
Analysis (6) shows that it cannot be met, for example, if the three studied samples have 
simultaneously equal constant surface radii and outer radii of compound. This conclusion can 
immediately exclude from consideration such common case as the use of different trial materials in 
three samples with preservation of the radial size of structures or changing contact pressure just by 
changing the internal diameter of the cylinder of the testing material. Further study of the system 
(3.12) shows that changing only the outer radius of the cylinder made of trial material with the 
preservation of the same values remaining coordinate vector of parameters in three samples leads to 
the fact that the equations have trivial solutions. 
To ensure the stability of the solution (5), and thus given accuracy, it is necessary to approximate 
angles between the straight lines described by equations of the system (5), and 90°. From the analysis 
(5) it is clear that the most effective way is to change the geometric dimensions of devices included in 
the equation system. 
Here are the most appropriate options for implementing the method of test parameters. 
In the three studied samples only the values of the contact surface radius are variable. Indeed, 
2
R B↑⇒ ↑ , which improves the stability of solution. In this case, to calculate the identifiable 
characteristics it is necessary to put 
i
C C= , 
i
E E= , 
1 1i
α α= , 
2 2 2
3 3 2
( )
i i
B R R R= − , 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533 85 
2 2 2
1 2 1
( )
i i
D R R R= − , 1,2,3i =  in (5). The condition 
2
0R ≠  should also be met here. 
In the three studied samples simultaneous alteration occurs only in the value of internal and 
external radii of the complex cylinder. To implement this option, 
i
C C= , 
i
E E= , 
1 1i
α α= , 
2 2 2
3 3 2
( )
i i i
B R R R= − , 
2 2 2
1 2 1
( )
i i i
D R R R= − , 1,2,3i =  must be substituted into (5). 
3. Results and discussion 
Practical implementation of the method of test parameters for determining the physical and mechanical 
properties of compounds requires, as mentioned earlier, the correct use of design model of complex 
bilayer cylinders. 
This implies meeting the following conditions: cylinders in complex design should be thick; 
double-layer composite cylinders must be long enough to prevent the influence of the edge effects at 
the point where the deformation occurs. 
For practical implementation of the method a special device was developed and manufactured 
(Fig. 4). The main elements of adaptation is hollow cylinder (trial element), hollow steel cylinder cut 
according to two diametrically opposite generatrices, cylindrical bearing plate with two coaxial 
cylindrical grooves designed for fixation of trial and steel cylinders. 
Three devices of this type have been made, differing from one another by cylinder diameters and 
the respective sizes of fixing anchor plate surfaces. 
The sizes of cylinder diameters of these devices are shown in Table 1.  
To ensure the stability of solutions averaged values of deformation were used for calculation.  
According to the measured values of strain, the amount of contact pressure was determined, using 
formula (1), and then – the coefficient of linear thermal expansion, elastic modulus and Poisson's ratio 
of the compound in different ranges of negative temperatures (Table 2). 
 
 
Figure 4. Scheme of the 
device for determining 
physical and mechanical 
properties of the compound: 
1 – hollow copper cylinder; 
2 – hollow steel cylinder; 
3 – compound; 
4 – bearing plate. 
 
IDENTIFICATION OF PHYSICAL AND MECHANICAL PROPERTIES OF COMPOUND BY SOLVING INVERSE PROBLEMS.  
A V GOROSHKO, V P ROYZMAN AND A BUBULIS 
86 ©VIBROENGINEERING. VIBROENGINEERING PROCEDIA. SEPTEMBER 2013. VOLUME 1. ISSN 2345-0533  
Table 1. Dimensions of cylinder diameters for determining the 
physical and mechanical properties of the compound. 
Device # Diameters, mm 
d1 d2 d3 
I 10 10.4 40 
II 20 20.4 40 
III 30 30.4 60 
 
Table 2. The value of physical and mechanical properties of 
resistor ceramic. 
Temperature 
range 
Young's 
modulus Е, 
N/mm2⋅104
 
Poisson's 
ratio µ
 
Local thermodynamic 
equilibrium α, 
1/degree 10-6
 
-20°C…-30°C 13.794 0.294 7.0023 
-30°C…-40°C 13.658 0.282 6.376 
-50°C…-60°C 12.95 0.276 5.898 
4. Conclusions 
The method for calculation and testing of trial parameters to identify physical and mechanical 
properties of compound has been developed. A program in the package of MATLAB applications 
which realizes the identification has been created. Using the program, the authors identified the 
physical and mechanical properties of EZK-25 compound, specified in the selected mathematical 
model. 
The value of the developed identification method is that it makes it possible to accurately 
determine the physical and mechanical properties for a particular batch of material, temperature, and 
especially the design scheme that describes the stress-strain state of “electronic element-compound” 
system. This is due to the fact that any mathematical model of the real product contains various 
assumptions and substituting it with inaccurate coefficients can compromise even the most accurate 
calculation scheme. Whereas the use of identification method contributes to the effectiveness of the 
offered mathematical model, since it will be substituted with identifiable characteristics that derived 
from the same mathematical model. 
References 
[1] А М Lokoschenko, S А Petraschuk and V P Royzman 2012 Evaluate and ensure the strength of 
the system of "electronic element-compound" Journal of Khmelnitsky National University 5 
193–197 


Identification of Physical and Mechanical Properties of Compound by Solving Inverse Problems

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Identification of Physical and Mechanical Properties of Compound by Solving Inverse Problems

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