Numerical analysis of strain and stress state in cylindrical notched tensile specimens
The paper presents the numerical results of stress and strain distribution in specimens with various sizes of notches in monotonic uniaxial tensile tests. The adopted method involves the finite element method and the elastic-plastic model of material with hardening. The distribution of the princi...
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Фізико-механічний інститут ім. Г.В. Карпенка НАН України
2013
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| Cite this: | Numerical analysis of strain and stress state in cylindrical notched tensile specimens / L. Derpenski, A. Seweryn // Фізико-хімічна механіка матеріалів. — 2013. — Т. 49, № 2. — С. 105-108. — Бібліогр.: 10 назв. — англ. |
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nasplib_isofts_kiev_ua-123456789-1357902025-02-23T17:16:23Z Numerical analysis of strain and stress state in cylindrical notched tensile specimens Числовий аналіз напружено-деформованого стану циліндричних зразків з надрізом за розтягу Числовой анализ напряженно-деформированного состояния цилиндрических образцов с надрезом при растяжении Derpenski, L. Seweryn, A. The paper presents the numerical results of stress and strain distribution in specimens with various sizes of notches in monotonic uniaxial tensile tests. The adopted method involves the finite element method and the elastic-plastic model of material with hardening. The distribution of the principal strain, principal stress and maximum plastic shear strain in the plane of notch symmetry is shown in figures. Методом скінченних елементів отримано числовий розв’язок пружно-пластичної осесиметричної задачі про розтяг циліндричного зразка з поверхневою U-подібною канавкою. Числові результати одержано для зразка з алюмінієвого сплаву EN-AW 2024. Проаналізовано розподіл напружень і деформацій у симетричному поперечному перерізі зразка залежно від відносного радіуса закруглення у вершині канавки, коли розтягальна сила досягає критичного значення перед руйнуванням зразка. Методом конечных элементов получено числовое решение упругопластичной осесимметричной задачи о растяжении цилиндрического образца с поверхностной U-подобной канавкой. Числовые результаты получены для образца из алюминиевого сплава EN-AW 2024. Проанализировано распределение напряжений и деформаций в симметричном поперечном сечении образца в зависимости от относительного радиуса закругления в вершине канавки, когда растягивающая сила достигает критического значения перед разрушением образца. This study was carried out in compliance with the plan of the N-N501-120536 research project of Ministry of Science and Higher Education, Poland. 2013 Article Numerical analysis of strain and stress state in cylindrical notched tensile specimens / L. Derpenski, A. Seweryn // Фізико-хімічна механіка матеріалів. — 2013. — Т. 49, № 2. — С. 105-108. — Бібліогр.: 10 назв. — англ. 0430-6252 https://nasplib.isofts.kiev.ua/handle/123456789/135790 539.3 en Фізико-хімічна механіка матеріалів application/pdf Фізико-механічний інститут ім. Г.В. Карпенка НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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English |
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The paper presents the numerical results of stress and strain distribution in specimens with
various sizes of notches in monotonic uniaxial tensile tests. The adopted method involves
the finite element method and the elastic-plastic model of material with hardening. The
distribution of the principal strain, principal stress and maximum plastic shear strain in the
plane of notch symmetry is shown in figures. |
| format |
Article |
| author |
Derpenski, L. Seweryn, A. |
| spellingShingle |
Derpenski, L. Seweryn, A. Numerical analysis of strain and stress state in cylindrical notched tensile specimens Фізико-хімічна механіка матеріалів |
| author_facet |
Derpenski, L. Seweryn, A. |
| author_sort |
Derpenski, L. |
| title |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens |
| title_short |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens |
| title_full |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens |
| title_fullStr |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens |
| title_full_unstemmed |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens |
| title_sort |
numerical analysis of strain and stress state in cylindrical notched tensile specimens |
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Фізико-механічний інститут ім. Г.В. Карпенка НАН України |
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2013 |
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https://nasplib.isofts.kiev.ua/handle/123456789/135790 |
| citation_txt |
Numerical analysis of strain and stress state in cylindrical notched tensile specimens / L. Derpenski, A. Seweryn // Фізико-хімічна механіка матеріалів. — 2013. — Т. 49, № 2. — С. 105-108. — Бібліогр.: 10 назв. — англ. |
| series |
Фізико-хімічна механіка матеріалів |
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2025-11-24T02:33:15Z |
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2025-11-24T02:33:15Z |
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1849637308582592512 |
| fulltext |
105
Ô³çèêî-õ³ì³÷íà ìåõàí³êà ìàòåð³àë³â. – 2013. – ¹ 2. – Physicochemical Mechanics of Materials
UDC 539.3
NUMERICAL ANALYSIS OF STRAIN AND STRESS STATE
IN CYLINDRICAL NOTCHED TENSILE SPECIMENS
L. DERPENSKI, A. SEWERYN
Bialystok University of Technology, Poland
The paper presents the numerical results of stress and strain distribution in specimens with
various sizes of notches in monotonic uniaxial tensile tests. The adopted method involves
the finite element method and the elastic-plastic model of material with hardening. The
distribution of the principal strain, principal stress and maximum plastic shear strain in the
plane of notch symmetry is shown in figures.
Keywords: numerical analysis, fracture of specimen with notches, ductile fracture.
The process of material fracture is the result of physical processes on a microsco-
pic scale [1–3]. At the macroscopic scale the values of state variables, which are com-
ponents of stress and strain tensors and their variation during loading are responsible
for fracture [4]. These values are determined with the help of numerical calculations,
e.g. using the finite element method (FEM). The presence of stress and strain concen-
trations, for example notches, has an influence on the elements fracture [5, 6]. The ex-
perimental research does not provide the possibility of determining the values of local
state variables, such as components of stress and strain tensors, or the components of
displacement vectors at each specimen point, and especially inside the material. With
this goal in mind the use of a numerical method is essential. This paper presents the re-
sults of numerical modeling of stress and strain fields in tensile (to the point of critical
load) specimens with different shapes of notches, made of aluminum alloys EN-AW 2024
(the results of the experimental research are shown in paper [7]). Taking into account
nonlinear analysis and the material nonlinearity (elasto-plastic material model with
work-hardening) of the modeled problem, the finite element method was used.
Numerical analysis. The distribution of stresses and strains in the specimens with
notches, and the unnotched specimens made of aluminum alloy EN-AW 2024, was
studied by the MSC.MARC software, based on the finite element method. In the calcu-
lations the axial symmetry of the specimen and the notch symmetry (Fig. 1) were taken
into consideration. The four-node axis-symmetric finite elements were applied. Nonli-
near analysis with material non-linearity was taken into consideration. The dimensions
of the specimens used in the analysis are shown in the Table. Also the maximal stress
value σmax and critical load Fc for different shape of a notch are shown in this table. In
the analysis eight radii rK of the U-type notch were used: 0.3, 0.5, 1, 2, 4, 8, 15, 30 mm.
The boundary conditions (Fig. 1c) were applied in the calculation: the axial sym-
metry of the geometric model (ur = 0 on the specimen axis), notch symmetry (uz = 0 on
the plane of the notch symmetry). The loading of the numerical calculation was done
with the help of the set displacement uc of the measurement base (see Table), which
were calculated for each specimen type directly as a result of the experiment [7]. In or-
der to describe the relationship between the stress and strain in the specimens, an elas-
tic and plastic material model with isotropic hardening was used. The Huber–von Mis-
ses plasticity yield criteria was applied. The curve of material hardening was obtained
Corresponding author: L. DERPENSKI, e-mail: l.derpenski@pb.edu.pl
106
from paper [7]. Fig. 2 shows the actual hardening σ–ε curves received in the complete
range for aluminum alloys.
Fig. 1. The 3D specimen (a), the shape of model used in numerical calculation (b),
boundary conditions and loads (c).
The dimensions of specimens and value of critical load (2H = 25mm, D = 10 mm)
№ of specimen 1 2 3 4 5 6 7 8 9 10 11 12
rK, mm 0.3 0.5 1.0 2.0
D
im
en
-
si
on
s
φK, mm 6 7 8 6 7 8 6 7 8 6 7 8
uc, mm 0.25 0.38 0.58 0.35 0.46 0.73 0.48 0.64 1.07 0.67 1.02 1.63
Fc, kN 19.1 23.5 26.7 19.8 23.7 27.5 19.7 24.1 28.6 18.2 24.1 29.5
Lo
ad
s
σmax, MPa 244 300 339 253 302 350 251 307 365 232 307 375
№ of specimen 13 14 15 16 17 18 19 20 21 22 23 24
rK, mm 4.0 8.0 15.0 30.0
D
im
en
-
si
on
s
φK, mm 6 7 8 6 7 8 6 7 8 6 7 8
uc, mm 0.73 1.10 2.00 1.11 1.23 2.06 1.57 1.71 2.14 2.10 2.29 2.62
Fc, kN 16.3 22.3 28.5 15.2 20.1 27.3 14.5 19.6 26.0 13.7 19.3 25.1
Lo
ad
s
σmax, MPa 207 283 363 193 256 348 184 250 330 174 245 320
Fig. 2. Stress-strain curve for aluminum alloy
EN-AW 2024: E = 69.65 GPa, ν = 0.34,
Re = 260 MPa, Ru = 658 MPa.
The analysis determined the influence of
the notch shape on stress and strain fields in the
whole loading range, the development of plastic
zone and the position of the maximum stress
and plastic strain value in the specimen. Some
of them have already been presented in [7]. In
the tested specimen, under the tensile load in-
fluence in the plastic zone, the notch generated
107
a tri-axial stress state. At the moment of fracture determined by the uc and the Fc taken
from experimental research, the largest value was noted in σzz. Therefore, it was assu-
med that this component has the greatest influence on fracture initiation. Attention was
focused on the location of the maximum value of σzz in the specimen, depending on the
value of rK.
Based on the results it was found that the distribution of σzz depends on the rK
size. For notches with larger radii (rK ≥ 2 mm) these values are located on the specimen
axis. In the case of notches with smaller radii (rK < 2 mm) the maximum value of σzz is
found at the notch root (Fig. 3). The zone of σzz maximum values is very concentrated
and it changes with the increase of the notch radius. Increasing the radius of the notch
from rK = 0.3 mm to rK = 1 mm, the zone increases without changing its location and
remains in the vicinity of the notch root. When rK exceeds 1 mm, this zone shifts to-
wards the specimen axis. Further increases of rK do not change the zone shape.
Fig. 3. The distribution of stress (a, c, e) and plastic strain (b, d, f) in the symmetry plane
of the notch at the moment of maximal load: rK = 0.3 mm (a, b); 4 mm (c, d); 30 mm (e, f).
– φK = 6 mm; – 7 mm; – 8 mm.
It should be noted that in the case of specimens with rK = 0.3 and rK = 0.5 mm
plasticization did not take place in the entire cross-section of the notch plane symmet-
ry. In specimens with rK = 1 and rK = 2 mm under the critical load Fc the cross-section
in the specimen plane symmetry a complete plasticization occured. Still the difference
in values of stresses at the notch root and at the axis were very wide. Almost equal
plasticization took place in the whole cross-section in the remaining specimens with
notches (rK = 4, 8, 15, 30 mm). The above refers to the specimens made of both alumi-
num alloys. From the experimental research it is known that the fracture surface de-
pends on the notch radius and partially or fully covers the notch plane of symmetry.
Therefore, the distributions of stress and strain fields were subjected to detailed ana-
lysis in the notch plane of symmetry where the fracture initiated. Figure 3 presents the
selected distributions: stress and plastic strain tensor components, the maximal plastic
shear strain under critical load. The values of r (distance from the specimen axis) were
normalized by φK.
The numerical calculations showed the place where the maximal normal stress and
the maximal plastic strain depends on the shape of the notch, and that rK has a greater
108
influence on the stress distribution than φK. Comparing the location of the maximum
value of σzz and p
zzε and max
pγ it is important to note that for the notches with a larger
radius (rK ≥ 2 mm) these values are found on the specimen axis, therefore at this point
fracture initiation should be expected. In the case of notches with smaller radii (rK < 2 mm),
the maximum stress values are found near the notch root, while the maximum strain
values are placed exactly at the notch root. Therefore, the location of crack initiation in
this case is not clearly defined. The calculation shows that the fracture of specimens
with rK bigger than 2 mm occurs in a similar way as for smooth specimens. For these
radii fracture initiation occurs in the same place on the axis of specimen symmetry [7].
CONCLUSION
On the basis of the numerical calculation of stress and strain fields and the earlier
experimental researches conducted on the specimens with notches made of the aluminum
alloy EN-AW 2024 one can state that normal stress vector component on the critical
plane determines the fracture. This plane, in the case of tensile specimens with notches,
is perpendicular to the load direction. The value of the critical normal stress may be de-
pendent on the maximal plastic shear strains, assuming that the accumulated damages
(and material weakening) occurs faster on the free surface than inside the material.
РЕЗЮМЕ. Методом скінченних елементів отримано числовий розв’язок пружно-плас-
тичної осесиметричної задачі про розтяг циліндричного зразка з поверхневою U-подібною
канавкою. Числові результати одержано для зразка з алюмінієвого сплаву EN-AW 2024.
Проаналізовано розподіл напружень і деформацій у симетричному поперечному перерізі
зразка залежно від відносного радіуса закруглення у вершині канавки, коли розтягальна
сила досягає критичного значення перед руйнуванням зразка.
РЕЗЮМЕ. Методом конечных элементов получено числовое решение упругоплас-
тичной осесимметричной задачи о растяжении цилиндрического образца с поверхностной
U-подобной канавкой. Числовые результаты получены для образца из алюминиевого
сплава EN-AW 2024. Проанализировано распределение напряжений и деформаций в сим-
метричном поперечном сечении образца в зависимости от относительного радиуса за-
кругления в вершине канавки, когда растягивающая сила достигает критического значе-
ния перед разрушением образца.
This study was carried out in compliance with the plan of the N-N501-120536
research project of Ministry of Science and Higher Education, Poland.
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Received 21.02.2013
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