V O ~ .41
SCIENCE IN CHINA (Series E)
NO. 3
June 1998
Theoretical prediction for Young's modulus and yield
strength of high-density foamed plastics *
LU Zixing (+?%)
and GAO Zhentong
('&@m )
(Research Center of Solid Mechanics, Beijing University of Aeronautics and Astronautics, Beijing 100083, China)
Received November 28, 1997
Abstract
The mechanical properties of high-density foamed plastics under uniaxial load are investigated and the relations of shear modulus and yield strength with porosity are derived from the self-consistent model. The results show
that not only can the same shear moduli with the ones derived from the three phase spheroidal model be obtained with
the present model, but also the theoretical yield strength of high-density foamed plastics can be obtained. By mmparing the present results with those of other empirical models as well as the available experimental data, it is shown that
the yield strength of high-density foamed plastics can be predicted quite well with this model.
Keywords:
foamed plastics, yield strength, self-consistent model.
The high-density foamed plastics are important light structural material, and the investigation of their mechanical properties has attracted much attention'']. The previous studies are mainly concentrated on the modulus of high-density foamed plastics, while work on the theoretical investigation of strength problems is insufficient. Although there are some theoretical predicting
formulae, they are generally based on empirical description[21. Because the high-density foamed
plastics with spherical cell structure can be considered as a kind of composite composed of two
phase material, the method of determining the mechanical properties of composite with the same
spherical reinforced particles appliesE3]
. The effective modulus of foamed plastics has been investigated by means of differentiation and the three-phase spheroidal model. The results all agree well
with the experimental results[4s51.However, the effective prediction for yield strength of highdensity foamed plastics is lacking. According to the method of dealing with the composite reinforced by spherical particles provided by ref. [ 6 ], the mechanical properties of high-density
foamed plastics under uniaxial load (tension) is investigated with self-consistent model. The relations between the effective shear modulus as well as the yield strength of foamed plastics and the
porosity of material are derived, which*are compared with the previous model and the available
experimental data. The results show that the shear moduli predicted by the present model are basically the same with those by three phase spheroidal model. In addition, the prediction for yield
strength of high-density foamed plastics within a Iarger porosity range (O< f G 0 . 6 ) is in agreement with the empirical model, and at the same time, in good agreement with the experimental
results given by reference [ 71 .
1 Self-consistent model
In our discussion, we consider a foamed plastic specimen under uniaxial loading, and assume
* Project supported by the National Natural Science Foundation of China (Grant No.
19672005).
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SCIENCE IN CHINA (Series E)
272
that all of the cell structure of foamed plastics has spherical shape with the same radius R , but the
cells are randomly distributed. In order to analyze them
f f
f f
conveniently, we divide the foamed plastics into three parts: the
spherical cavity at its center, the matrix material spherical shell
(its outer radius is R o ) and the equivalent homogeneous media
(its properties represent the mechanical properties of foamed
plastics). Based on these, we construct a model to simulate a local region of foamed plastics as shown in fig. 1. When the
spherical cavity and the matrix spherical shell are used as a cell of
foamed plastics, its effective properties should be the same as
1
1 1 1n-- those of equivalent homogeneous media out of it. In order to
~ i 1.~ ~h~
.
self-consistent model of make the model represent real foamed plastics, it is required to
foamed plastics.
define the porosity of foamed plastics as follows:
imf
I
I
I
1
I I I
I
Thus, the elastic properties of imaginary equivalent homogeneous media can be determined by
self-consistent conditions. The detailed discussion will be given in sections 2 and 3.
2
Basic equations
Under uniaxial loading, the local model will be deformed axisymmetrically. If the spherical
coordinate system ( r , 8, 9 ) as shown in fig. 2 is
used to describe the deformation, then we have the
geometric relations :
't
E,
a ur
= a,.,
€0
=-
d B
r
in which u,, us denote the displacements, e,, €0,
E ~ ,E,+ denote the strain. Obviously, formula ( 2 )
is valid for both matrix material and equivalent homogeneous media. In order to make a distinction,
Fig. 2. The spherical coordinate system.
in the following formulae we use superscript * to
represent the physical quantity of equivalent homogeneous media. Supposing matrix material to be
incompressible for foamed plastics, we have
E, + Eg + egi = 0.
(3)
If the matrix is elastic, the stress and strain comply with the Hooke's law; that is
oe = 2 G ~ g+ 0,
a, = 2Ge, + a ,
(4)
up = 2Ge9 + a ,
a,+ = 2GE,+,
in which a is a hydrostatic stress to be determined, and G stands for shear modulus of the matrix
material. In the spherical coordinate system, the stress in matrix should satisfy the following equilibrium equations :
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YOUNG'S MODULUS & YIELD STRENGTH OF HIGH-DENSITY FOAMED PLASTICS
273
However, we should consider the compressibility of equivalent homogeneous media (or foamed
~lastics),so the strain in equivalent homogeneous media should satisfy the following relation
+ E; + E;
DO*
= -
(6)
K * '
Here, we aassume that the hydrostatic stress in equivalent homogeneous media equals a constant
a; , and K * is the bulk modulus of foamed plastics. Under the supposition of matrix being incompressible, we have
Ef
In addition, for the compressible equivalent homogeneous media (or foamed plastics), the Hooke's
law should be written in such a way:
c$ = ~ G * E $ .
Furthermore, the stress in equivalent homogeneous media should also satisfy the equilibrium equations in formula (5) at the same time.
3
The solution of problem
According to the derivation in ref. [ 6 ] , the displacement field in matrix material can be obtained by solving the basic equations (2)-(5) under uniaxial loading as follows:
The hydrostatic stress to be determined in matrix can be written as
Finally, the stress in matrix can be given as follows:
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Vol. 41
In the above formulae (9), ( l o ) , a , b, c , d and a0 are the constants to be determined. These expressions are valid for matrix material, but not suitable for equivalent homogeneous media. For
equivalent homogeneous media, if assuming that the hydrostatic stress to be determined is constant, i. e. a * = a t , we can prove that the strain field obtained from the following displacement
field satisfies formula ( 6 ) :
us.
In addition, in the above formula, the expressions of u, and ue are the same as in formula ( 9 ) .
Thus, the strain field given by formula (12) is
ue^ =
*
Ere
=
Ed.
In formula (13), the components of strain without * should satisfy formula ( 3 ) , and under the
supposition of the hydrostatic stress being constant, the stress determined by formula (4) should
satisfy the equilibrium equation (5) . For the compressible equivalent homogeneous media, substituting formula ( 13) into formula (8 ) , we have
'3,;) = Or@,
in which a,, as, a,, are have the same stress form as given in formula (11) . Considering the hydrostatic stress in equivalent homogeneous media to be assumed as a constant, it required that the
corresponding undetermined constants a * = 0 and d * = 0 in the expressions of displacement and
stress field. Thus, the displacement and stress field in equivalent homogeneous media can be written as
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YOUNG'S MODULUS & YIELD STRENGTH OF HIGH-DENSITY FOAMED PLASTICS
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In order to determine the undetermined constants in the above expressions, it is necessary to consider the continuity conditions of stress and displacement on the interface of differenet phase materials. In the case of foamed plastics, on the interface of gas and matrix material ( r = R ), the
stress should satisfy
6, = PO,
Cd3 = 0,
(17)
in which po is the gas pressure in spherical cell. From the above conditions, a0 = po, and the following equations can be derived:
According to the stress and displacement continuity conditions of interface between matrix materi3K"ao
a1 and equivalent homogeneous media ( r = R o ) , we have 0; =
and the following e3K* +4G*
quations :
a
b
l
dRi - 6 "
1
-+,--cRo----* Ro,
R:
2Ro 3
5
2R;f
TC
Assuming that the size of foamed plastic specimen is much larger than the cell size of foamed plastics, the following supplementary boundary conditions hold at the infinite distance (far from the
local model) :
C;
= umCOS2 e,
c , ~= -
umsinOcos8,
6;
= dmsin2 6 ,
6;
=o,
(20)
in which om is the uniaxial stress applied on the distant end of the specimen. From the above conditions, we have
Considering the above results, from formula (19), we can get
276
in which
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SCIENCE IN CHINA (Series E)
8 = (?G - 1). If substituting the expressions of
a,
following linear homogeneous equation group about b * and c *
b, c , d into ( 18), we obtain the
:
Because the above equation group has non-zero solution, its coefficient determinant should equal
zero. This condition can provide a formula for determining effective shear modulus of foamed plastics, that is
A@+BP+C=O,
in which
The relation between shear modulus and porosity of
foamed plastics is given by the above result. If we use
formula ( 3 ) to calculate the bulk modulus of foamed
plastics, the Young's modulus of foamed plastics can be
predicted theoretically (as shown in table 1) . It is not
difficult to find from the calculation of shear modulus for
foamed plastics that the present model can give the same
shear modulus prediction as the three phase spheroidal
model. This result indicates that the magnitude of shear
modulus of foamed plastics has nothing to do with the
0.2
0.4
0.6
0.8
1 O
stress state. Furthermore, under the supposition of ma3
trix being
- incomvressible, it can be totally determined by
Fig. 3 . The comparison and prediction of Young' s
the porosity of foamed plastics. In fig. 3, the comparimoduli for foamed plastics. 1, Present model; 2.
son of Young's modulus of foamed plastics, predicted by
differentiation; 0 , experimental data.
the present model with the one predicted by differentiation is given, and they are also compared with the experimental data. They all are in agreement
with the experimental data.
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YOUNG'S MODULUS & YIELD STRENGTH OF HIGH-DENSITY FOAMED PLASTICS
Table 1 Relations for mechanical properties of foamed plastics with porosity
In order to predict the yield strength of foamed plastics, the stress distribution in material
should be determined. Obviously, according to formula (21 ) , the undetermined constant c * can
be found by using the distant field stress and the effective shear modulus of equivalent homogeneous media. But, the determination of constant b * needs a self-consistent condition. This condition required that the average a, in a cell equals the nominal stress a, of distant field; that is
Substituting corresponding stress expressions in formula ( 16) into the above integration (let r =
Ro), and considering the relations in formula (21), we have
Substituting c * of formula (21) and b
of formula (27) into formula (22) again, we obtain
Thus, the stress distribution in matrix under uniaxial loading in a local model of foamed plastics is
determined, and the equivalent stress strength at any point in matrix is also determined, that
is
1
- a0)' + (no - up)' + (up - u,)'+ 6 2~ ~ 1 .
(29)
a: =
In the above formula, the stress components can be calculated by (11) and (28). It is not diffiX
cult to prove that on the interface of matrix and equivalent homogeneous media, 6 = f - corre2
sponds to the largest equivalent stress. Therefore, if the foamed plastic is yielding, we can consider that the matrix nearby it is also getting to yield. Thus, taking the equivalent stress of matrix equal to the yield stress u, of matrix as a yield condition of foamed plastics, we can obtain the ratio
of yield stress of foamed plastics and matrix, that is
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Vol. 41
in which
or it can be written in a rather simple form:
In the course of deriving the above result, we have
used formula ( 7 ) . Therefore, the yield strength ratio
of foamed plastics under uniaxial loading can be expressed by the shear modulus ratio and porosity of
material. By means of prediction for shear modulus of
foamed plastics, the yield strength of foamed plastics
can be predicted by formula (31). The predicting results are also listed in table 1. They are compared
with experimental results as shown in figure 4.
4
0.01
1
I
0.2
0.4
I
0.6
I
Discussion and conclusions
The shear modulus and Young' s modulus of
.t
foamed ~lasticsare calculated bv formulae (.24 ..
1,
(25) and (7) within the whole porosity range (table
1 ) . According to the shear modulus ratio given in
table 1 and formula ( 31 ), the yield strength of
foamed plastics is calculated as shown in table 1. By comparing ours with the previous results, we
find that the moduli are the same as the ones calculated by three phase spheroidal model (but they
have different expressions). They agree better with the experimental data as shown in fig. 3.
The prediction for yield strength of foamed plastics given by the present model shows that the results predicted by formula (31) are in agreement with the experimental data as well as Masi and
Nicolais' empirical curve which is used to fit the experimental data (fig. 4 ) . The empirical curve
in fig. 4 corresponds to the following expression:
Fig. 4 . The comparison and prediction of yield strength
for foamed plastics. 1, Present model; 2, empirical
model; , experimental data.
in which afis the tensile yield strength of foamed plastics, a; is the modified yield strength of matrix. Masi and Nicolais think a',= a,/K,, in which a, is the ideal yield stress of matrix, K , is a
modified constant. The modified constant reflects the effects of a smaller gas hole in matrix on the
yield strength of material. So, in the formula of predicting yield strength of foamed plastics we
take a, = a', for calculation.
It is necessary to point out that the present model can predict modulus and yield strength of
No. 3
YOUNG'S MODULUS & YIELD STRENGTH OF HIGH-DENSITY FQAMED PLASTICS
279
foamed plastics within the whole porosity (or density)range. But, when porosity becomes large enough, the supposition of spherical cell does not hold so that the results given by self-consistent
model are no longer conceivable. In this case, it is only used as a reference. Therefore, the present model can effectively predict modulus and yield strength of high-density foamed plastics with
spherical cell structure.
References
1
2
3
4
Hilyard, N. C. (ed. ), Mechanics of Cellular Plastics. London: Appl. Sci. Publishers, 1982.
Lu Zixing, Wang Ren, Huang Zhuping et al. , A review of studies on the mechanical properties of foam plastics, Adwnces in
Mechanics (in Chinese), 1996, 26(3) : 306.
Progelhof, R. C . , Throne, J . L . , Young's modulus of uniform density thermoplastic foam, Ploym . Eng. Sci., 1979, 19
(7): 493.
Lu Zixing, Huang Zhuping, Wang Ren, The determination for elastic constants of foam plastics having spherical cell structure,
Chinese J . M a t . Res. (in Chinese). 1995, 9(3) : 268.
5 LU Zixing, Huang Zhuping, Wang Ren, Determination of effective moduli for foam plastics based on three phase spheroidal
model, Acta Mechanics Sol& Sinica (in Chinese), 1995, 8(4) : 294.
6 Gao, Y. C., Lelievre, J . , A theoretical analysis of the strength of composite gels with rigid filler particles, Polym Eng.
Sci., 1994, 34(18): 1369.
7 Masi, P. , Nicolais. L. , MazzoIa. M. et al. , Tensile behavior of high-density thermosetting polyester foams, Polym . Eng.
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% . , 1984, 24(7) : 469.