=Paper=
{{Paper
|id=Vol-3896/short16
|storemode=property
|title=Mathematical and numerical modeling of nonlinear deformation processes
|pdfUrl=https://ceur-ws.org/Vol-3896/short16.pdf
|volume=Vol-3896
|authors=Мykhailo Мykhailyshyn,Roman Mykhailyshyn,Halyna Semenyshyn
|dblpUrl=https://dblp.org/rec/conf/ittap/ykhailyshynMS24
}}
==Mathematical and numerical modeling of nonlinear deformation processes==
https://ceur-ws.org/Vol-3896/short16.pdf
Mathematical and numerical modeling of
nonlinear deformation processes
Мykhailo Мykhailyshyn1,*,†, Roman Mykhailyshyn2,†, Halyna Semenyshyn1,†
1
Ternopil Ivan Puluj National Technical University, 56 Ruska St, Ternopil, UA46001, Ukraine
2
Osaka University, Osaka 560-8531, Japan.
Abstract
To solve the problems of thermal elastic-plastic deformation of structural elements, a method of using
the deformation theory of plasticity, generalized to the possibility of taking into account unloading
with the development of plastic deformations during unloading, or repeated loading with the
development of repeated plastic deformations, is proposed. Dependencies between the intensities of
excess (differences between current values and the corresponding values recorded at the moment of
unloading) stresses and excess deformations are built on the basis of Mazing's principle. An algorithm
for solving problems based on the method of successive loads has been developed. The method of
additional deformations was used to linearize the indicated stress-strain dependences..
Keywords
deformation theory, thermal plasticity, unloading, mathematical modeling
1.Introduction
In many technological processes, structural elements are subjected to significant force and
temperature loads, as a result of which irreversible plastic deformations occur in some areas of
the structure. After complete removal of the load, residual stresses and deformations occur in
such structures, which can have a significant impact on the operational properties of such
structures. Therefore, the problem of quantitative assessment of residual stress fields and
deformations that occur in some heat treatment processes, during welding, restoration of
operational properties by surfacing, is very relevant. Currently, approximate methods based on
the use of unloading theorems, computational and experimental methods [1, 2, 3], as well as
methods based on the theory of plastic flow [4, 5] are used to solve similar problems. The latest
mathematical models are quite complex and do not always satisfy the required accuracy when
tracking the load surface in the process of plastic deformation. Therefore, the paper proposes a
mathematical model based on the deformation theory of plasticity, which is generalized for the
case of taking into account unloading.
---------------------------------------------------------
ITTAP’2024: 4th International Workshop on Information Technologies: Theoretical and Applied Problems, October
23–25, 2024, Ternopil, Ukraine, Opole, Poland
∗
Corresponding author.
†
These authors contributed equally.
mms000@ukr.net (M. Мykhailyshyn); roman.mux.mux@gmail.com (R.Мykhailyshyn); halyna-sem@ukr.net
(H.Semenyshyn)
0009-0001-9173-9032 (M.Мykhailyshyn); 0000-0002-1203-3446 (R.Мykhailyshyn); 0009-0001-6991-7701
(H.Semenyshyn)
© 2023 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
To solve such problems, you can use commercial packages of computer programs such as
ANSYS, SYSWELD, etc. Renting commercial packages for the appropriate term is quite
expensive, and therefore, in practice, the creation of original problem-oriented mathematical
support for solving problems in the field of welding and related technologies is widely used in
practice. This way allows synthesizing working programs from ready-made modeling blocks
and information bases, and it is much cheaper than renting a commercial package.
2.Methodology
To simulate the processes of elastic-plastic deformation we have suggested to use the theory
of small thermal elastic-plastic deformations generalized for the case of unloading taken into
account [6].
Stress-strain relations of small elastic-plastic deformations can be written as [6]
(1)
(2)
where
(3)
here are the components of stresses and deformations
deviators. Average stress and deformation are as follows connected
σ0 T 2G (1+v )
by the dependence ε 0 = +ε , where K = – volume compression modulus ,
K (1−2 v )
E T ¿
G= – shear modulus, ε =α t T =α t ( T −T 0 ) – average temperature deformation, α t -
2(1+v )
coefficient of thermal linear expansion of the material, – components of stresses and
deformations deviators which were reached in the specified point of the environment at the
moment of unloading start. The last values are equal to zero, if any unloading wasn’t observed in
the specified point. is the value of temperature which was fixed in the specified point at the
moment of unloading start. Values і are the stresses intensities and , which are
calculated by formulae
(4)
(5)
.
Apparently, if any unloading isn’t observed in the specified point yet, the values and
are transformed into ordinary intensities of the stresses and .
In the formula (1) is the parameter of plasticity determined by formula (2). Moreover, they
consider that there is a unique dependence between the intensities and , which is not
influenced by the kind of stressed state and can be found on the basis of experimental data for
the simplest homogeneous stressed states.
At the stage of initial deformation from stress-free and deformation-free state in the points
where some active loading is taking place the intensity of total deformation is equal to the sum
p e
of intensities of elastic and plastic deformations components ε i =ε i +ε i .
Stress-strain relations (1) in this stage look like
(6)
,
(7)
.
The relationship between the intensities of stresses and deformations in this stage for the
most structural materials can be written as
(8)
.
where – material plasticity limit which depends on the temperature. The identical
relationship can be obtained on the basis of Mazing principle [7], if it is
generalized on isothermal processes of deformation. After such generalization we have found
[8]
(9)
.
Stress-strain relations (2) can be presented as the solved ones relative to the deformation
tensor component
ε ij =
−~
[
ψ G (1)
2G G 1
σ ij −σ ij −
( 1+v ) ~
ψ−( 1−2 v )
~
( 1+v ) ψ
δ ij
G (1)
Gm ( )]
σ 0 −σ 0 −δ ij ( ε T ( 1 )−ε T ) +ε (1)
ij , (10)
or relative to the stress tensor component
~
G (1) 2G ( 1 )
[
σ ij = σ ij − ~ ε ij −ε ij +
G1 ψ
ψ ( 1+ν )
1−2 ν
[ ] ]
δ ij ( ε (01 )−ε 0 )−( ε T ( 1 )−ε ) −δ ij ( ε (01 )−ε 0 ) .
T
(11)
Here the plastic deformation can be determined by formulae
(~
ψ−1 ) (12)
ε ijp =ε ijp(1)− ~
ψ
[ ε −ε +δ ( ε −ε )].
(1)
ij ij ij
(1)
0 0
Now we use the same symbols for ordinary stress and deformation tensors component which
have been introduced earlier for deviator components
~ G
σ ij = σ (ij1 )−σ ij , ~ε ij =ε (ij1 )−ε ij ,
G1 (13)
~ G
σ 0 = σ (01 )−σ 0, ~ε 0 =ε (01 )−ε 0, ~ε T =ε T ( 1 )−ε T .
G1
Then the dependence (10) is written as
~ (1+v )~
~ε = ψ ~
ij
2G
σ
(
ij −
ψ−(1−2 v ) ~
(1+v )~
ψ
δ ij σ 0 +δ ij ~ε T .
) (14)
Having introduced the symbols ~ p p(1) p
e ij =ε ij −ε ij , the formula (10) will look like
~
~e p = ψ−1 ~e .
ij ~
ψ ij (15)
We can also show that
~e e = 1 ~s ,
ij
2G ij (16)
where ~ e e(1) e
e ij =e ij −e ij.
To linearize the specified stress-strain relations the method of additional deformation (MAD)
is used. We will demonstrate this method for the case when the unloading is taking place with
the development of plastic deformation.
Figure 1: Picture of using the additional deformation method on the unloading stage with
plastic deformation development
We assume that in the beginning of some k step we have ~ ε ijp( k−1) and point Q corresponds to
these deformations (fig.1). We must admit that for visual clarity we have superposed the
εi ,~
beginnings of axes of references ~ σ i for two different temperatures.
The process of successive approximations by the method of additional deformation is carried
out by formulae
~e( k )=~e
ij 1 ~¿( k ) ~ p( k−1)
ij∗¿e ( k ) + ~e ijp( k−1)= s + ε ij ¿
2 G ij (17)
Having solved the problem under stress-strain relations conditions (17), we will find the
e(ijk ) , ~
solution ~ σ ¿( k)
ij which point P corresponds to on fig.1. By the known values of component
~
e ij we have calculated the intensity of total deformations ~ε i . Using the surface equation
(k ) (k )
~ ~
σ i = Φ( ~ε i ,T ) for the specified temperature value T for the specified stage and value ~ε (i k ) we
have found the intensity of stresses ~ σ ( k ) (point N on the figure). It has enabled us to find by
i
~ε ( k )
~
formula ψ =3 G ~( k ) the value of plasticity parameter ~
(k ) i
ψ ( k ) for the specified approximation,
σi
~
ψ ( k )−1
and by the formulae ~ e ijp( k )= ~( k ) ~e(ijk ) we have calculated the components of plastic residual
ψ
deformation of this k approximation which can be used in formulae of the method of additional
deformations (17) in the next approximation.
The formulae of the method of additional deformations in this case can be written as
~ε ( k )= 1 ~
ij
2G (σ ij¿( k )−
3v
δ ~
1+v ij 0 )
σ ¿( k ) +δ ij ~ε T + ~ε ijp( k−1), (18)
~( k )
~ε p( k )= ψ −1 ( ~ε ( k )−δ ~ε ( k ) ) , (19)
ij ~
ψ( k )
ij ij 0
~ε ( k ) (20)
~
ψ ( k )=3 G ~i( k ) .
σ i
The whole process of loading (heating, cooling) is divided into separate stages. Specifying the
values of the deformation plasticity component for zero approximation in (18) equal to these
components which were reached for the previous stage of loading (at deformation from initial
undeformed state they are accepted as zero) the elastic problem with additional deformations is
solved. according to the found total deformations in approximation the intensities
and are calculated. Then according to the formula (20) for each point of the structure
approximation of the plasticity parameter is calculated and by formula (20) the
components of the deformation plasticity which will be further used in formulae (18) in the next
approximation to find the component . Iteration process lasts till its complete
coincidence, after that the transition to the next stage of loading is taking place.
We must admit that initially on every iteration in each point of the structure the above-
mentioned deformation has been assumed that occurred in it during the previous stage of
loading, i.e. initial elastic or plastic deformation, elastic unloading or unloading with the
development of further plastic deformation. After coincidence of the iteration process the
examination is conducted in every point of the structure to find out if such deformation was
taking place in fact. If in some points the deformation behavior does not correspond the
accepted one on the basis of information from the previous stage of loading, then the stage is
fully recalculated with the previous replacement of the deformation behavior to the opposite
one in such points.
3.The results
Due to the above-mentioned technique a number of practical problems have been solved,
namely the welding of thin-walled structural parts, building-up welding aimed at strengthening
or restoring the operational characteristics.
In this way, for instance, the problem of welding procedure simulation of two cylindrical
shells by circular joint providing the welding is taking place along the whole length of the
welding seam simultaneously [9]. The obtained results have completely correlated with the
similar results found in the paper [5] using more complicated theory of plastic flow. The results
of modelling have made possible to find the fields of residual stresses, deformations and
displacements, study the kinetics of stress-and-strain state of the welding process, study the
diagram of deformation in different points of the structure. As an example, we will show the
distribution of residual welding stresses, elastic deformations, and also residual deflection of a
shell.
Figure 2: Distribution of residual welding stresses in cylindrical shell.
Figure 3: Distribution of elastic residual deformations.
Figure 4: Residual deflection.
X
In the figures, x= −¿ is a dimensionless coordinate along the length of the shell, the
L
stresses are related to the yield point of the material at some initial temperature.
4.Conclusions
It is shown that to solve the complex problems of thermal elastic-plastic deformation of
structural elements, it is possible to use the theory of the deformation theory of thermal
plasticity deformation, which is much simpler than the flow theory, generalized to consider the
possibility of unloading with the development of plastic deformations, or repeated loading with
the development of repeated plastic deformations.
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