Dynamic Response of Gradient Foams
The Voronoi-type density-gradient foams with three layers are numerically simulated, in order to study their dynamic response. The focus of the study is not only on the energy absorption and the distal stress of the gradient foam, but also the impact stress. The results obtained show that reduc...
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Інститут проблем міцності ім. Г.С. Писаренко НАН України
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Hu, L.L. Liu, Y. 2017-01-26T18:29:40Z 2017-01-26T18:29:40Z 2014 Dynamic Response of Gradient Foams / L.L. Hu, Y. Liu // Проблемы прочности. — 2014. — № 2. — С. 172-177. — Бібліогр.: 9 назв. — англ. 0556-171X https://nasplib.isofts.kiev.ua/handle/123456789/112705 539.4 The Voronoi-type density-gradient foams with three layers are numerically simulated, in order to study their dynamic response. The focus of the study is not only on the energy absorption and the distal stress of the gradient foam, but also the impact stress. The results obtained show that reduction of both the initial impact peak stress, and the early energy absorption of the gradient foam can be privided by reducing density of the first layer. The undesirable effect on the energy absorption can be alleviated by diminishing the thickness of the first layer. The difference between densities of the first two layers density should be limited to a certain range to avoid the peak stress appearing in the second layer. A weak distal layer can reduce the distal stress of the foam under high-velocity impact, while a high density gradient between the last two layers will result in the early increase of the distal stress under moderate-velocity impact С помощью численного метода исследованы динамические характеристики градиентных трехслойных пеноматериалов типа Вороного. Исследованы поглощение энергии, дистальные напряжения градиентного пеноматериала и напряжения при ударе. Установлено, что начальное максимальное напряжение при ударе и преждевременное поглощение энергии градиентного пеноматериала уменьшаются при низкой прочности первого слоя. Нежелательное влияние на поглощение энергии можно смягчить путем уменьшения толщины первого слоя. Различие между плотностью первых двух слоев необходимо контролировать в рамках предельного диапазона, чтобы избежать возникновения максимального напряжения во втором слое. Слабый дистальный слой может способствовать снижению дистального напряжения пеноматериала при высокой скорости удара, тогда как значительный градиент плотностей последних двух слоев приводит к преждевременному увеличению дистального напряжения при средней скорости удара. За допомогою числового методу досліджено динамічні характеристики градієнтних тришарових піноматеріалів типу Вороного. Досліджено поглинання енергії, дистальні напруження градієнтного піноматеріалу і напруження під час удару. Установлено,що початкове максимальне напруження під час удару і передчасне поглинання енергії градієнтного піноматеріалу зменшуються за низької міцності першого шару. Небажаний вплив на поглинання енергії можна зм’якшити шляхом зменшення товщини першого шару. Різницю між щільністю перших двох шарів необхідно контролювати у межах граничного діапазону, щоб запобігти виникненню максимального напруження у другому шарі. Слабкий дистальний шар може сприяти зниженню дистального напруження піноматеріалу за високої швидкості удару, в той час як значний градієнт щільностей останніх двох шарів призведе до передчасного збільшення дистального напруження при середній швидкості удару. The authors would like to thank the support from the National Natural Science Foundation of China under Grant Nos. 11172335 and 10802100. The support from the Fundamental Research Funds for the Central Universities No. 13lgzd02 is also gratefully acknowledged. en Інститут проблем міцності ім. Г.С. Писаренко НАН України Проблемы прочности Научно-технический раздел Dynamic Response of Gradient Foams Динамическая характеристика градиентных пеноматериалов Article published earlier |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| title |
Dynamic Response of Gradient Foams |
| spellingShingle |
Dynamic Response of Gradient Foams Hu, L.L. Liu, Y. Научно-технический раздел |
| title_short |
Dynamic Response of Gradient Foams |
| title_full |
Dynamic Response of Gradient Foams |
| title_fullStr |
Dynamic Response of Gradient Foams |
| title_full_unstemmed |
Dynamic Response of Gradient Foams |
| title_sort |
dynamic response of gradient foams |
| author |
Hu, L.L. Liu, Y. |
| author_facet |
Hu, L.L. Liu, Y. |
| topic |
Научно-технический раздел |
| topic_facet |
Научно-технический раздел |
| publishDate |
2014 |
| language |
English |
| container_title |
Проблемы прочности |
| publisher |
Інститут проблем міцності ім. Г.С. Писаренко НАН України |
| format |
Article |
| title_alt |
Динамическая характеристика градиентных пеноматериалов |
| description |
The Voronoi-type density-gradient foams with
three layers are numerically simulated, in order
to study their dynamic response. The focus of the
study is not only on the energy absorption and
the distal stress of the gradient foam, but also the
impact stress. The results obtained show that reduction
of both the initial impact peak stress,
and the early energy absorption of the gradient
foam can be privided by reducing density of the
first layer. The undesirable effect on the energy
absorption can be alleviated by diminishing the
thickness of the first layer. The difference between
densities of the first two layers density
should be limited to a certain range to avoid the
peak stress appearing in the second layer. A
weak distal layer can reduce the distal stress of
the foam under high-velocity impact, while a
high density gradient between the last two layers
will result in the early increase of the distal stress
under moderate-velocity impact
С помощью численного метода исследованы динамические характеристики градиентных
трехслойных пеноматериалов типа Вороного. Исследованы поглощение энергии, дистальные
напряжения градиентного пеноматериала и напряжения при ударе. Установлено, что начальное максимальное напряжение при ударе и преждевременное поглощение энергии градиентного пеноматериала уменьшаются при низкой прочности первого слоя. Нежелательное
влияние на поглощение энергии можно смягчить путем уменьшения толщины первого слоя.
Различие между плотностью первых двух слоев необходимо контролировать в рамках предельного диапазона, чтобы избежать возникновения максимального напряжения во втором
слое. Слабый дистальный слой может способствовать снижению дистального напряжения
пеноматериала при высокой скорости удара, тогда как значительный градиент плотностей
последних двух слоев приводит к преждевременному увеличению дистального напряжения при
средней скорости удара.
За допомогою числового методу досліджено динамічні характеристики градієнтних
тришарових піноматеріалів типу Вороного. Досліджено поглинання енергії, дистальні напруження градієнтного піноматеріалу і напруження під час удару. Установлено,що початкове максимальне напруження під час удару і передчасне поглинання
енергії градієнтного піноматеріалу зменшуються за низької міцності першого шару.
Небажаний вплив на поглинання енергії можна зм’якшити шляхом зменшення товщини першого шару. Різницю між щільністю перших двох шарів необхідно контролювати у межах граничного діапазону, щоб запобігти виникненню максимального
напруження у другому шарі. Слабкий дистальний шар може сприяти зниженню
дистального напруження піноматеріалу за високої швидкості удару, в той час як
значний градієнт щільностей останніх двох шарів призведе до передчасного збільшення дистального напруження при середній швидкості удару.
|
| issn |
0556-171X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/112705 |
| citation_txt |
Dynamic Response of Gradient Foams / L.L. Hu, Y. Liu // Проблемы прочности. — 2014. — № 2. — С. 172-177. — Бібліогр.: 9 назв. — англ. |
| work_keys_str_mv |
AT hull dynamicresponseofgradientfoams AT liuy dynamicresponseofgradientfoams AT hull dinamičeskaâharakteristikagradientnyhpenomaterialov AT liuy dinamičeskaâharakteristikagradientnyhpenomaterialov |
| first_indexed |
2025-11-24T11:37:43Z |
| last_indexed |
2025-11-24T11:37:43Z |
| _version_ |
1850845613170622464 |
| fulltext |
UDC 539.4
Dynamic Response of Gradient Foams
L. L. Hu1 and Y. Liu
Department of Applied Mechanics & Engineering, School of Engineering, Sun Yat-sen University,
Guangzhou, China
1 hulingl@mail.sysu.edu.cn
ÓÄÊ 539.4
Äèíàìè÷åñêàÿ õàðàêòåðèñòèêà ãðàäèåíòíûõ ïåíîìàòåðèàëîâ
Ë. Ë. Õó1, É. Ëèó
Ôàêóëüòåò ïðèêëàäíîé ìåõàíèêè è ìàøèíîñòðîåíèÿ, êàôåäðà ìàøèíîñòðîåíèÿ, Óíèâåðñèòåò
èì. Ñóíü ßòñåíà, ×óàí÷æîó, Êèòàé
Ñ ïîìîùüþ ÷èñëåííîãî ìåòîäà èññëåäîâàíû äèíàìè÷åñêèå õàðàêòåðèñòèêè ãðàäèåíòíûõ
òðåõñëîéíûõ ïåíîìàòåðèàëîâ òèïà Âîðîíîãî. Èññëåäîâàíû ïîãëîùåíèå ýíåðãèè, äèñòàëüíûå
íàïðÿæåíèÿ ãðàäèåíòíîãî ïåíîìàòåðèàëà è íàïðÿæåíèÿ ïðè óäàðå. Óñòàíîâëåíî, ÷òî íà÷àëü-
íîå ìàêñèìàëüíîå íàïðÿæåíèå ïðè óäàðå è ïðåæäåâðåìåííîå ïîãëîùåíèå ýíåðãèè ãðàäèåíò-
íîãî ïåíîìàòåðèàëà óìåíüøàþòñÿ ïðè íèçêîé ïðî÷íîñòè ïåðâîãî ñëîÿ. Íåæåëàòåëüíîå
âëèÿíèå íà ïîãëîùåíèå ýíåðãèè ìîæíî ñìÿã÷èòü ïóòåì óìåíüøåíèÿ òîëùèíû ïåðâîãî ñëîÿ.
Ðàçëè÷èå ìåæäó ïëîòíîñòüþ ïåðâûõ äâóõ ñëîåâ íåîáõîäèìî êîíòðîëèðîâàòü â ðàìêàõ ïðå-
äåëüíîãî äèàïàçîíà, ÷òîáû èçáåæàòü âîçíèêíîâåíèÿ ìàêñèìàëüíîãî íàïðÿæåíèÿ âî âòîðîì
ñëîå. Ñëàáûé äèñòàëüíûé ñëîé ìîæåò ñïîñîáñòâîâàòü ñíèæåíèþ äèñòàëüíîãî íàïðÿæåíèÿ
ïåíîìàòåðèàëà ïðè âûñîêîé ñêîðîñòè óäàðà, òîãäà êàê çíà÷èòåëüíûé ãðàäèåíò ïëîòíîñòåé
ïîñëåäíèõ äâóõ ñëîåâ ïðèâîäèò ê ïðåæäåâðåìåííîìó óâåëè÷åíèþ äèñòàëüíîãî íàïðÿæåíèÿ ïðè
ñðåäíåé ñêîðîñòè óäàðà.
Êëþ÷åâûå ñëîâà: ãðàäèåíòíûé ïåíîìàòåðèàë, óäàð, ìàêñèìàëüíîå íàïðÿæåíèå,
äèñòàëüíîå íàïðÿæåíèå, ïîãëîùåíèå ýíåðãèè.
Introduction. Metallic foams have been developed in recent years and are growing in
use as new engineering structural materials. The advantages of these ultra-light metal
materials are not only the high relative stiffness and strength, but also the effective energy
absorption during accidental impacts while limiting the crushing force [1–3]. It has been
recognized that the mechanical properties of the foams are dominated by the relative
density [1]. Moreover, the impact velocity has also proved to play an important role in
controlling the mechanical properties of cellular materials [3–5].
In the past few years, research interest has been triggered in studying various graded
cellular materials. It is shown that a density gradient could significantly change the
deformation mode and the energy absorption of cellular structures [6]. Experimental results
show that placing the hardest layer as the first impacted layer and the weakest layer as the
last layer has some benefits in terms of maximum energy absorption with a minimum force
level transmitted to the protected structures under the high-velocity impact [7]. The energy
absorption capacity of the double-layer foam cladding under blast load is analytically
derived based on a rigid-perfectly plastic-locking (RPPL) foam model [3], in which the
gradient foam is assumed to deform in the “shock wave” mode until entirely absorbing the
blast energy [8].
© L. L. HU, Y. LIU, 2014
172 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 2
In the previous studies, insufficient attention was paid to the impact peak stress of
gradient foams, which is important in case of such car accidents, as the car-to-passenger or
car-to-car collisions. The damage of the bumped object depends on the peak stress during
impact and on the energy absorption value: the former is required to be low and the latter –
high. However, both of these values increase with the foam density.
The present paper is focused on the impact peak stress, the energy absorption and the
distal stress of the gradient foam. A method is proposed to relief the conflict between the
impact peak stress and the energy absorption in designing the gradient foam. The results on
the distal stress alert the deficiency of the design method on the gradient foams reported in
literature.
Numerical Models. To study the dynamic properties of a graded cellular material, a
finite element model is constructed using the ANSYS/LS-DYNA software. Three layers of
the Voronoi-type foam with random cells are constructed with the different density of each
layer. It is known that both the impact stress, and the distal stress of cellular materials
increase with the material density [5, 9]. Thus, four kinds of a gradient foam with the
strongest foam in the middle layer are considered in the present study, as listed in Table 1.
The average relative density � � �r s�
* of the four kinds of gradient foams is 0.098,
where �
* and �s are the density of the foam and the base material, respectively.
Moreover, a nongraded foam (N) with the relative density of 0.098 is also studied for
comparison.
The cell wall material is assumed to be elastic/perfectly plastic with E � 68 GPa,
� ys � 130 MPa, �s � 2700 kg/m3, and � � 0 3. , where E and � are the Young modulus
and Poisson’s ratio of the base material, respectively. In simulations the foam block is
placed on a fixed rigid base at one end (the distal end) and crushed by a rigid plate with a
constant crushing velocity V (75 or 120 m/s) at the other end (the impact end). The first
layer of the foams is always at the impact end and the third layer is always at the distal
end.
T a b l e 1
Physical Parameters and the Peak Stress of Foams
Foam Layer Thickness
(mm)
Relative
density
Peak stress (MPa)
120 m/s 75 m/s
A 1st
2nd
3rd
20
20
20
0.099
0.119
0.075
11.28
7.80
6.30
3.83
B 1st
2nd
3rd
20
20
20
0.075
0.119
0.099
9.82
8.57
3.62
2.91
C 1st
2nd
3rd
10
40
10
0.055
0.119
0.055
5.57
11.34
1.74
6.18
D 1st
2nd
3rd
10
40
10
0.075
0.114
0.055
7.77
9.56
3.81
4.57
N 60 0.098 11.16 6.19
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 2 173
Dynamic Response of Gradient Foams
Results and Discussion. Impact Stress. Due to shock wave and inertia effects, the
cells of the foam will collapse layer-by-layer from the impact end to the distal one under
the high-velocity impact, such as 120 m/s, which is similar to propagation of a shock wave
in a continuum bar. Here we refer to this deformation mode as a “shock wave” mode.
Under the moderate-velocity impact, such as 75 m/s, the foams collapse in this “shock-
wave” mode in the early crushing stage, while the third layer usually begins to deform
during the compression of the second layer, and even is crushed to densification before the
second layer deforms completely, especially for the foams with a high density gradient
between the last two layers, such as foams A, C, and D.
A typical stress–strain curve of the gradient foam is shown in Fig. 1. The stress before
densification (i.e., the plateau stress) is important for the energy absorption of the foam,
which is dominated by the foam density [1]. Thus, the plateau stage of the gradient foam
consists of three parts, as shown in Fig. 1, while the stress level in each part depends on the
density of the crushed foam layer.
There is a peak of the stress at the initial impact, which may cause the destruction of
the crushing objects in the collision accidents. For the gradient foams, the peak stress is
expected to occur at the initial instant of the impact or during the crushing of the foam layer
with the largest density. Since the strongest foam is placed in the middle layer in the present
design, Table 1 lists the peak stress of the foams during the crushing of the first and the
second layers, respectively. It is obvious that the initial impact peak stress (the first peak
stress) can be obviously reduced by decreasing the foam density in the first layer. However,
provided the difference between densities of the first and the second layers is large enough,
the peak stress occurring in the second layer will exceed the initial one, which occurs in
foams C and D, as listed in Table 1. This means that the density of the first layer should be
reduced within a limit range for the purpose of reducing the peak stress of the gradient
foam.
Energy Absorption. Cellular materials are frequently used in the energy absorption
devices. Figure 2 plots the variety of the absorbed energy per volume (specific energy) of
the foams under study versus strain. It is obvious that the energy absorption of the foams is
enhanced by the impact velocity, which is exhibited by the two clusters of the energy
curves according to the impact velocity, as shown in Fig. 2. This is due to the inertia effects
of the foam microstructure, as discussed in [2, 9].
It is shown in Fig. 2 that the specific energy of the nongraded foam linearly increases
with strain due to the steady plateau stress, while the energy absorption of the gradient
foams undulates depending on the distribution of the foam density. As compared to a
nongraded foam, there is a drop at the early stage of the energy curve for the gradient
foams with a weak first layer, such as foams B, C, and D, as shown in Fig. 2. The weaker
Fig. 1. A typical stress–strain curve of gradient foam B (impact velocity 120 m/s).
174 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 2
L. L. Hu and Y. Liu
the first layer, the steeper the drop. The absorbed energy recovers to the average level, i.e.,
that of the nongraded foam, at the strain of about 20% for foams C and D with the 10mm
thickness of the first layer, and at the strain of about 40% for the foam B with the 20 mm
thickness of the first layer, which indicates that the early energy absorption of the gradient
foam can be improved by decreasing the thickness of the first weak layer while conserving
the buffer function of this layer for reducing the impact peak stress.
Distal Stress. When the foams perform as the protecting devices, the stress transferred
to the protected object is concentrated in the engineering applications, the level of which
can be reflected with the distal stress of the foams during crushing, as shown in Fig. 3. The
horizontal axis in Fig. 3 denotes the compressed strain of the foams. It is shown that the
curve of the distal stress includes three stages, i.e., the elastic stage, the plateau stage and
the densification stage, which is similar to that of the stress at the impact end. It is observed
that the response of the distal stress manifests a delay in reference to strain, i.e., it begins at
a certain (nonzero) strain, which is related to the propagation of the stress wave within the
foam from the impact end to the distal one.
By comparing the densities of the gradient foams listed in Table 1 and plotting them
in Fig. 3, it is shown that the distal stress level of the gradient foams is dominated by the
the last layer density. A weak distal layer can effectively reduce the distal stress of the foam
under impact. Under high-velocity impact, such as 120 m/s, the distal stress of the foams
increases rapidly after the strain exceeds about 0.8, which is close to the densification strain
for the impact stress of the foams, as shown in Fig. 1.
It is seen from Fig. 3b that, under the impact of 75 m/s, the densification strain for the
distal stress of foams A, C, and D is about 0.6 or much less. By comparing the density of
the last layer with that of the second one, as listed in Table 1, it is found that there is a high
gradient for the three foams A, C, and D (the density of the last layer is by 37% lower than
that of the second one in foam A, by 54% in foam C, by 52% in foam D, and only by 17%
in foam B). Under the impact with a moderate velocity, such as 75 m/s, the weak distal
layer will be deformed before the second layer is compressed completely due to a high
gradient between the last two layers, which leads to the early increase of the distal stress.
This is more pronounced for foam C with the highest gradient between the last two layers,
the distal stress of which increases at strain about 0.4. More attention should be paid to the
early increase in the distal stress in the industry applications, otherwise the protected object
may be destroyed.
Fig. 2. Dependence of the foam specific energy on strain.
ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 2 175
Dynamic Response of Gradient Foams
It has been already stated that the RPPL foam model [3] was used in work [7] to
design the gradient foam under blast load, in which the gradient foam is assumed to deform
in the “shock wave” mode until entirely absorbing the blast energy. However, at the later
stage of the foam deformation the impact velocity has dropped down since a part of the
blast energy has been absorbed. Consequently, the “shock wave” mode ceases to exist with
subsequent premature compression of the distal layer, which means that the RPPL model
becomes no longer applicable to this case.
Conclusions. The initial impact stress of the gradient foam can be diminished by
reducing the density of the first layer. However, the difference in the densities of the first
and the second layers should be limited to a certain range; otherwise, the higher peak stress
will appear during the crushing of the second layer with a higher density, which is even
larger than the one at the initial impact.
Adopting a lower density for the first layer is beneficial for the impact stress
reduction, but it leads to reduction of the energy absorption of the foams. This undesirable
effect on the energy absorption can be effectively reduced by diminishing the thickness of
the first layer.
A weak distal layer can effectively reduce the plateau level of the foam distal stress
under impact, which means a lower stress is transferred to the protected object when the
foam is used for the protection devices. However, a high density gradient between the last
two layers will cause the distal layer to deform before the front layer is compressed
completely under the moderate- or low-velocity impact, which would finally result in an
early increase in the distal stress. More attention should be paid to this phenomenon in the
industry design, since the early increase in the transferred stress may destroy the object
being protected.
Acknowledgments. The authors would like to thank the support from the National
Natural Science Foundation of China under Grant Nos. 11172335 and 10802100. The
support from the Fundamental Research Funds for the Central Universities No. 13lgzd02 is
also gratefully acknowledged.
Ð å ç þ ì å
Çà äîïîìîãîþ ÷èñëîâîãî ìåòîäó äîñë³äæåíî äèíàì³÷í³ õàðàêòåðèñòèêè ãðà䳺íòíèõ
òðèøàðîâèõ ï³íîìàòåð³àë³â òèïó Âîðîíîãî. Äîñë³äæåíî ïîãëèíàííÿ åíåð㳿, äèñòàëü-
í³ íàïðóæåííÿ ãðà䳺íòíîãî ï³íîìàòåð³àëó ³ íàïðóæåííÿ ï³ä ÷àñ óäàðó. Óñòàíîâëåíî,
ùî ïî÷àòêîâå ìàêñèìàëüíå íàïðóæåííÿ ï³ä ÷àñ óäàðó ³ ïåðåä÷àñíå ïîãëèíàííÿ
åíåð㳿 ãðà䳺íòíîãî ï³íîìàòåð³àëó çìåíøóþòüñÿ çà íèçüêî¿ ì³öíîñò³ ïåðøîãî øàðó.
a b
Fig. 3. Distal stress of the foams versus strain under the impact of 120 (a) and 75 m/s (b).
176 ISSN 0556-171X. Ïðîáëåìû ïðî÷íîñòè, 2014, ¹ 2
L. L. Hu and Y. Liu
Íåáàæàíèé âïëèâ íà ïîãëèíàííÿ åíåð㳿 ìîæíà çì’ÿêøèòè øëÿõîì çìåíøåííÿ òîâ-
ùèíè ïåðøîãî øàðó. гçíèöþ ì³æ ù³ëüí³ñòþ ïåðøèõ äâîõ øàð³â íåîáõ³äíî êîíòðî-
ëþâàòè ó ìåæàõ ãðàíè÷íîãî ä³àïàçîíó, ùîá çàïîá³ãòè âèíèêíåííþ ìàêñèìàëüíîãî
íàïðóæåííÿ ó äðóãîìó øàð³. Ñëàáêèé äèñòàëüíèé øàð ìîæå ñïðèÿòè çíèæåííþ
äèñòàëüíîãî íàïðóæåííÿ ï³íîìàòåð³àëó çà âèñîêî¿ øâèäêîñò³ óäàðó, â òîé ÷àñ ÿê
çíà÷íèé ãðà䳺íò ù³ëüíîñòåé îñòàíí³õ äâîõ øàð³â ïðèçâåäå äî ïåðåä÷àñíîãî çá³ëü-
øåííÿ äèñòàëüíîãî íàïðóæåííÿ ïðè ñåðåäí³é øâèäêîñò³ óäàðó.
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Received 22. 11. 2013
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Dynamic Response of Gradient Foams
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