The dynamic-elastic behaviour of foamed plastics

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The dynamic-elastic behaviour of foamed plastics

Becker, G. W.; National Research Council of Canada. Division of Building

Research

In designing floors for minimum transmission of

mechanical vibrations and impact sounds such as footsteps, a "floating" floor construction is sometimes resorted to. It is believed that the technique presented in this paper would be useful in obtaining design parameters for the sandwich

material in floating floors. This article describes a technique for measuring the dynamic-€lastic modulus of plastic foam

materials. Presumably, this technique could be applied to measure the dynamic properties of other types of resilient materials as well.

The Division of Building Research is grateful to D. A. Sinclair of the National Research Council's Translation Section for preparing this translation and to D. Olynyk of the Building Physics Section of the Divison who checked the translation.

Ottawa June, 1968

R. F. Legget Director

Technical Translation 1328

Title: ｔｾ･ dynamic-elastic behaviour of foamed plastics

(Uber das dynamisch-elastische Verhalten geschaumter Stoffe)

Author: G. W. Becker

Reference: Acustica, 9: 135-143, 1959

Abstract

Different methods are described for determining the

dynamic Young's and dynamic shear moduli of foamed plastics in the range 0.1 to 100 cps - for harder substances up to 1000 cps. In estimating the elastic modulus on short cylindrical specimens with attached end surfaces the influence of the shape of the specimen for a given Poisson ratio is considered theo-retically and tested experimentally, The results obtained on a series of different materials can be explained in terms of a simple model for closed and open mesh foamed material.

1. Introduction

Foamed plastics are materials with a cellular structure characterized

*

by a very low bulk density. Depending on the choice of initial products and the method of expansion, the cells can be closed or open, with all possible intervening stages. The size of the cells can also be varied within wide limits. In the various expanding methods, nitrogen is mainly used as an expansion gas, and in the case of closed-cell plastics it remains trapped in the cells(l).

The development of foamed plastics has advanced considerably in recent years. It is possible today, by using certain foaming processes, to produce foamed materials from most natural and synthetic high polymers. Moreover, with suitable finishing methods special foamed plastics have been produced for broader applications.

Besides the familiar heat-insulating applications, foamed plastics are gaining ever greater importance in the field of acoustics on account of the wide possibilities of varying their elastic properties. For example, they are suitable as wall coatings for sound absorption in rooms and vehicles, as soft inter layers in "floating" floors to reduce noise from footsteps, for the damping of the vibrations of large surfaces by the use of non-extensible external coats, as well as vibration-damping pedestals for measuring instruments and manchines(2-6).

In these applications it is necessary to know not only the special acoustic coefficients, but also the dynamic-elastic coefficients (moduli and damping values) of the foamed plastics in particular. If we know the dynamic Young's modulus of a material and the corresponding damping

*

For the definitions compare DIN 7726 (Foamed plastics, terms,

values as a function of the frequency and, where applicable, of the

temperature, we are then in a position to appraise the sUitability of the material for its various possible applications.

It was the purpose of the research hereinafter described to develop simple measuring processes (suitable also for industrial testing labo-ratories) for determining the dynamic-elastic moduli of foamed plastics and

*

to test these methods on a number of different materials

2. Measuring Procedures 2.1. General

Figure 1 reviews the measuring principles applied. The values

determined are the Young's modulus E and the shear modulus G as well as the corresponding loss factors

n

E and

n

G of the dynamic (complex) moduli:

£- .. £(1+1'18)

C--C(l+I'1a).

For their determination in the frequency range of approximately 0.1 to 100 Hz (up to a maximum of 1000 Hz in the case of harder materials), the following measuring set-ups were chosen: two pendulum procedures (bending and torsion pendulums, respectively), in which the measured specimens produce the directional moment, and two vibrometer methods (extension and shear vibrometer, respectively) in the form of mass-spring systems with the specimens representing the springs.

The same cylindrical, short specimens are used for all four kinds of stress. Their dimensions are small compared with the length of the

different elastic waves. Cylindrical shape was chosen on account of ease of production and mechanical stability; moreover, simple evaluation

formulae can be derived from this form of specimen for all measuring

procedures. For unstressed support the specimens are cemented at both end faces between metal plates. In the pendulum method the behaviour of the damped-fading natural vibrations is observed. Given known moments of inertia

ｾＨ｢･ｮ､ｩｮｧＩ or GT(torsion) the moduli E and G, and the corresponding loss

factors E and G are obtained from the frequencies f

B and fT, respectively,

*

Among the described procedures, the tensile vibrometer was copied from the blueprints of the Physikalisch-Technischen Bundesanstalt by several firms in the chemical industries and has been introduced for testing purposes.

*

and from the fading times T

B and TT' respectively, of the free-damped vibrations of the system. For cylindrical specimens of height hand diameter d the following equations can be deduced (f

B ｡ｾ､ fT measured in Hz, T B and TT in sec): Bending: (la) (lb) Torsion: 2.2 110= - .

,TrT

(2a) (2b)

The bracketed expression in equation (la), in which m

B is the mass of the corresponding moment of inertia ｾ applies a correction which must be taken into account for specimens with comparatively large values of h. Given a suitable choice of specimen dimensions, however, the bracketed expression can be put, in good approximation, equal to unity.

In the vibrometer methods we investigate the resonance behaviour of mass-spring systems under forced vibration. The (reduced) masses M of this system are made up of two individual masses in each case, between which the specimens being measured are secured (cf. Figure 1); therefore, denoting the individual masses by M

l and M2, we may write:

.L __

I ...

-1....

(3)

AI AI. AI:

If we know the (reduced) mass M

D (tensile stress) or MS (shearing stress), then the moduli E and G, as well as the corresponding loss factors

n

E and

n

G, are obtained from the resonance frequencies fD and fS and the half-widths ｾｦｄ and ｾｦｓ of the resonance curves of the forced vibrations of the systems. Using the nomenclature already introduced for the specimen

*

i.e. the time in which the oscillation amplitude drops to 1/1000 (corresponding to -60 db) of its initial value.

dimensions, we then get the equations:

Tensile stress:

(4a)

(4b) Shear: no> 8nhMs/RI cfl

A/

R

/s

(5a) (5b)

If equation (5a) is to hold, the heights h of the two individual specimens must be practically identical.

Arrangements similar in principle to the pendulum methods described here and the tensile vibrometer have already been described on several occasions for the investigation of unfoamed plastic materials(7-9). 2.2. Description of apparatus

2.2.1. Pendulum method

Figure 2, by way of example, shows the bending pendulum method

employed. The arrangement corresponds to a short beam fixed at one end and subject to bending stress, with a weight applied to the unfixed end.

To support its weight the swinging mass is suspended from a thin steel wire (diameter = 0.4 mm). It comprises a centre piece into which rods can be screwed. The rods carry sliding weights. By using rods of different lengths and weights of different value the moment of inertia can be varied in a ratio of approximately 1 : 4000.

The cylindrical foamed plastic specimen is placed horizontally between the centre piece of the pendulum and a chuck that can be locked in position. Two disks of duraluminum 2 mm thick cemented to the end faces of the

specimen, and which can be screwed tightly into the centre piece of the weight and into the supporting chuck, are used for securing purposes. The position of the supporting chuck is freely adjustable so that it can

receive cylindrical specimens of different length.

By suitable shaping of the centre piece it was assured that the mass of the pendulum, in all combinations with different moment of inertia would be situated at the centre of gravity, and further that the centre of gravity

*

would be always situated at the centre of the end face of the specimen . For adjustment of the pendulum mass, and in order to compensate for the elongation of the suspension wire when larger masses are used (up to a

*

This is an important assumption for the applicability of equation (la).

maximum of

6.9

kg), the point of suspension of the wire can be adjusted appropriately.

Free vibrations are induced in the pendulum by means of a mechanical horizontal deflection. In order to record the vibrations, there is a

light steel plate secured to the suspension wire which, on deflection of the pendulum, changes the inductance of two coils which are differentially connected to form parts of a Wheatstone bridge, to which a

5

kc alternating current voltage is fed. The diagonal voltage of the bridge which is

balanced at the position of rest of the pendulum is amplified, filtered, and fed to a logarithmic level recorder with rectification for recording purposes; bridge, amplifier and filter are parts of a commercially

obtainable apparatus.

During the oscillation of the bending pendulum, strictly speaking, a translation of the centre of gravity along a circular arc is superimposed

*

on the simple rotation of the pendulum mass about its centre of gravity . However, as long as the pendulum ?eflection'remains small enough so that the tangent of the angle of rotation can be ｲ･ｰｾ｡｣･､ with sufficient accuracy by the angle itself, the indicator of the measuring bridge is proportional to the angle of rotation of the pendulum. By way of example, Figure

3

shows the curve of a damped, fading vibration of the bending pendulum produced on the level recorder.

The torsion pendulum is constructed by a simple modification of, the bending pendulum. The centre piece of the pendulum mass here is suspended horizontally by the steel wire and the specimen is mounted vertically

between the centre piece of the pendulum mass and the holding chuck, which is now also placed horizontally; the vibrations are excited and recorded in the same way as for the bending pendulum.

In establishing the dimensions of the specimens it was considered first of all tQat foamed plastics are generally used in the form of slabs and in most applications are stressed in the direction of their thickness. Therefore, if the specimens are cut out with their cylindrical axes in the direction of the thickness of the slab, then the dependence of the

measured values on the orientation of the specimen, as may frequently be observed in the case of foamed plastics, can be disregarded in transferring the result to practical applications. For the specimen height h we

*

As can easily be shown, the correction term in equation CIa) takes this translation into account.

expanded in the direction of higher frequencies Figure 4 shows the longitudinal vibrometer employed the normal slab thickness corresponding to about 2 cm; for the diameter d values of about 2 to 3 cm proved satisfactory.

With the two pendulum methods a frequency range of about 0.1 to 10 Hz can be covered by suitable variation of the moments of inertia, if the measurements are taken close enough together. This range is limited in the direction of ｨｩｾｨ･ｲ frequencies by vibrating string effects of the suspension wire, and in the direction of lower frequencies by the moment of direction of the wire, the influence of which is still negligible at 0.1 Hz.

Measurements at lower frequencies with consideration of proper corrections, were not undertaken.

2.2.2. Vibrometer methods The measuring range is by the vibrometer methods.

*

employed

The test specimen is secured without any prestressing to two weights suspended from thin steel strips by means of duraluminum plates cemented to the specimens. The mountings for the ｳｾ･･ｬ strips are rigidly joined to a heavy block (weight approximately 40 kp) that,rests on a softly elastic base and therefore acts practically as an inert mass. This largely prevents any accompanying vibrations on the part of the suspension mounts and keeps the boundary damping small; for the same reason the suspension strips were furnished with a vibration-damping coating.

The system is excited electrodynamically at the heavier of the two weights (M

l) to sinusidal vibrations (in Fig. 4 only the shielded electro-magnet of the system can be seen), These vibrations are received from the lighter of the weights (M

2) by a barium titanate transverse vibrator; the "voltage reception" is amplified and fed to a vacuum tube voltmeter for reading purposes. As a frequency generator, an RC-generator with continuous frequency change is employed in conjunction with a low resistance output. At the receiver end the output is fed to an impedance converter with high-resistant input followed by a regular commercial voltage amplifier.

The mechanical impedance of the excitor system, which above the

natural frequency of the system is given almost exclusively by the inertia of M

l, was chosen as large as possible (Ml approximately 1600 g). In this way the reaction of the resonant vibrating mass-spring system on the

electro-*

The longitudinal vibrometer was developed jointly by Dr. H. Oberst and the author. Its final design for routine tests was worked out by Mr. K. Frankenfeld.

dynamic transmission system is kept small. To change the resonance frequency only the mass M

2 has to be varied within a range of about 30 to 1200 g (including vibration absorbers). For unstressed mounting of the test specimen the position of M

2 can be changed arbitrarily. To make the instrument portable, a lock is provided on

MI'

The shear vibrometer is obtained by a simple modification of the longitudinal vibrometer (cf. Fig.

5).

For this purpose the weight M

l is fitted with an accessory for securing two test specimens subject to shearing stress. The rest of the measuring apparatus is the same as for the

longitudinal vibrator.

With the two vibrometer methods a frequency range of about 10 to 100 Hz is covered by suitably changing the weight M

2 (in the case of hard materials up to a maximum of 1000 Hz) and by employing test points sufficiently close together. This range is limited in the direction of low frequencies by the natural frequencies of the excitor system and the pendulum comprising weight M

2 and its suspension strip. In the direction of higher frequencies the range is limited by the combination of the smallest weight M

2 with specimens of extreme rigidity. An upper limit is placed on the investigation of

extremely hard materials by the strength of the cement employed.

3.

Coefficient of Elasticity of Cemented Specimens

The defining equations given in Section 2.1 for the dynamic Young's modulus E, equations (la) and (4a), hold true only under the condition that the transverse contraction of the specimens is unrestricted over its entire height. However, on account of the cementing of the specimens at the end faces, this condition is not satisfied; a cylindrical specimen under compression thereby receives a "barrel shape", Although cementing of the specimens had no appreciable effect on the shearing strength, on account of the constancy of volume associated with shearing deformations, the

coefficient of elasticity is increased by cementing; accordingly the Young's modulus obtained by the expressions given in Section 2.1 is ｾｯｯ large. The shorter the specimen in relation to its diameter, i.e. the

*

smaller the "form factor" hid, the more marked this influence .

By way of example, Figure 6 shows the frequency curves of E

f, G, and the corresponding loss factors n

E and nG measured with the two-pendulum method on specimens of different form factors; the index f refers to the

*

Differing from the usage here, the form factor is often defined as the quotient d/4 h (cf. e.g. ref. 10).

form factor dependence of the Young's modulus determined by equation (la). All the specimens were taken from the same slab of closed-cell (medium-soft) PVC foam. It is evident from Figure

6

that the values of

0, nO

and

n

E measured with different form factors are in approximate agreement, if we overlook the scatters that are unavoidable in foam plastics owing to the inhomogeneities resulting from the foaming process. On the other hand

*

the values of E

f increase in monotone fashion with decreasing form factor. The dependence of the coefficient of elasticity on the form factor has already been investigated experimentally by various authors on specimens of rubber where the effect is particularly marked on account of the high

transverse contraction coefficient

Ｈｾｾ

0.5)(10-1

3 ) ;

in these cases, also, it was found that the measured internal damping is practically independent of the form factor of the specimen.

Deserving of particular attention are the extensive investigations of C.

W.

Kosten who finds an empirical relationship whereby the

form-dependent modulus increases linearly with the reciprocal of. the form factor (d/h). The problem has also been dealt with theoretically by Y. Rocard(14) for the special case of incompressible material; the calculation, however, yields a formula in which the form-dependent modulus varies with (d/h)2. In these two studies different basic assumptions are made; while Kosten assumed that the zones of the two opposite ends of the specimens where

transverse deformation is prevented do not influence each other reciprocally, in Rocard's work a reciprocal influencing of these regions is not ruled

out. The two formulae are in approximately quantitative agreement for h ｾ d; considerable differences result in the vicinity of h « d, where Kosten's assumption is most definitely not satisfied.

Since the dimensions of foamed plastic specimens generally conform to the case h < d (cf. Section 2.2 and page

8)

we attempted, on the basis of Rocard's considerations, to derive a relation between the form factor and the specimen rigidity which would also apply to compressible materials. We shall Just indicate the calculation procedure here.

Consider a cylindrical specimen (height h, diameter d) secured at both

*

Since for stability reasons the investigation of soft materials is possibly only on slab-shaped or short rod-shaped specimens, this effect may be

expected to show up in all similar processes, regardless of whether the specimens are assembled by cementing or by stacking.

ends, e.g. by cementing, the axis of which coincides with the z axis of a system of cylindrical coordinates (p, ｾＬ z) and the centre of which lies at the origin of the system of coordinates. If the specimen is deformed by external forces, a work of deformation A is performed which must be equal to the "potential" energy density w integrated over the volume of the specimen:

A= fWdv.

"

For a forced K

z applied in the z direction and producing a deformation ｾｨＯｨＬ the work is

and taking into account the symmetry conditions present in the case in question, we get the following for the energy density

K. Ah=

Ef [(

aW)2

_ 1 _1 (

ar)2J

dv

az

2(1 +11)

az

I

"

where wand r are the components of the displacement vector in z or direction, as the case may be; they are linked together by the relation

and are sUbject to the boundary conditions

By applying a Fourier series it is possible to solve the energy integral (cf. ref. 14).

From this we get the "form-dependent " Young's modulus

E,

=

E(1

+

iJ

_llf_

(!!-)2] ,

for which, employing the relation

it follows that:

E=2G(l+Il) ( 7)

( 8)

It is useful to use equation

(8)

as a basis for verifying the calcu-lation, since this relation includes the ratio of measured values E

f/2G which, given similar measuring procedures and similar specimens, depends only on the quotient of two frequencies and moments of inertia or masses which can be determined with extreme accuracy (cf. the pairs of equations

(la) and (2a) or (3a) and (4a)); at the same time it is assumed that the bracketed expression in equation (la) can be put equal to 1. In this way the properties of individual specimens which affect the values of E

f and G can be eliminated in a similar manner; this is an advantage in the calcu-lation of the transverse contraction coefficient ｾ for which otherwise great measuring accuracy would have to be required in the determination of the two individual measured values E

f and G.

The relationship between the quotient E

f/2G and the form factor hid given by equation

(8)

is represented in Figure

7

in the form of a family of curves with transverse contraction coefficient ｾ as a parameter. Figure

7

shows that. for h < d and at higher values of ｾ the modulus E

f decreases

rapidly with decreasing form factor; for ｾ = 0, E

f , as expected, is

independent of the shape of the specimen.

For experimental verification of the calculated relation

(8)

the values obtained from Figure

6

are plotted for various specimens of (medium soft) PVC foam in Figure

7.

The measured values fit satisfactorily into the pattern of the family of curves, when it is taken into account that the diameter chosen for the plotted poi.nts corresponds to a maximum deviation of approximately ± 2% of the stated values E

f/2G.

The modulus values reported in the next section are in all cases based on measurements from several specimens with different form factors; at the same time, in all cases the dependence of the measured values on the form factors, as required by equation

(8),

could be confirmed.

4. Test Results In Figure 8 the values of E,

n

E, G and

n

G, as measured by the four methods described on a closed-cellular silicon-rubber foam, are plotted against the frequency (see legend beneath the figure for designation of

measured by the different methods. The slowly rising frequency curves of E and G run practically parallel to each other other and the transverse

contraction coefficient is therefore independent of the frequency, in accordance with equation

(7);

the values n_E and n

G almost coincide. Similar frequency curves were obtained by measurement of the other foam plastics investigated. We therefore omit the presentation of these curve diagrams and discuss the varying behaviour of foam plastics from the standpoint of the values of the moduli and loss factors measured at a given frequency (1 Hz).

The characteristic dynamic-elastic values obtained by measurement at this frequency are given separately in Table I for closed- and open-cell

*

materjals. The nine plastics selected are representative of a large number of commercial materials, some of which are very similar to each other, which have been investigated during recent years by the elastic vibrometer method

**

at the Physikalisch-Technische Bundesanstalt Comparison of the measured values for G and E entered in the columns shows that the foamed plastics extend over a very broad range of moduli; the values for the softest and hardest materials differ by a factor of almost 103

•

In column

3,

Table I, contains the values of the bulk density

determined by weighing measured volumes and taking the buoyancy into account, and Table IV gives the ratio C of the cell volume to the total volume. On the condition, always satisfied, that the density of the gas in the cells in

negligible compared with that of the skeletal material (ps)' the value C is obtained from ref. 15:

C= 1-

ol«.

₍₉₎

The density p usually agrees only within certain limits with the s

density determined on the compact material; the possible errors in the

determination of C as a consequence of this, however, are so slight that they have no effect on the values presented in column

4.

*

The materials "vulkollan" and "moltopren" are registered trademarks of the Farvenfabriken Bayer A.G., Leverkusen.

**

Joint investigations of Dr. H. Oberst and the author of about 50 different materials, unpublished.

In column 5 the values of the length-specific resistance to flow

=

*

are entered for the open-cell materials For the closed-cell materials the value of

=

is practically infinite.

Columns 6 to 9 contain the characteristic values G, E,

n

G and

n

E for the frequency 1 Hz. The values of G vary from 1.1 . 105 to 7.7 . 107 _dyn/cm

2, those of E from 3.2 '10 5 to 1.96 . 108 _dyn/cm2

• The values of

n

G and

n

E are determined essentially by the proportion and properties of the high polymer material used for foaming. The loss factors

n

G and

n

E differ but slightly; in general

n

G is somewhat larger than

n

E.

The values of ｾ as determined from E and G by equation (7) are given in column 10. They show that the values ｾ ｾ 0.5 for closed cell materials and

ｾ ｾ 0 for open cell materials, as frequently assumed, by no means hold true. Values of ｾ ｾ 0.5 (cf. in this connection columns 11 and 12) are observed only for very soft closed-cell foamed plastics in which the modulus of compression agrees substantially with that of the enclosed gas; in the case of harder foamed plastics ｾ is smaller and for Hart PVC foam it obtains a value of only 0.27.

As the values of ｾ : 0.2 obtained from open-cell foamed plastics show, the transverse contraction of these materials may always be different from 0

if the deformation amplitudes remain small enough so that a "staggering" of

**

the ｦｲ｡ｭ･ｷｾｲｫ occurs Agreement of the values determined on different materials in this connection might be purely accidental. The error in the open cells for the case of the low stressed frequencies under consideration

and the relatively small flow resistances had no effect on the measured values, as measurements at different atmospheric pressure between about 760 to 1 mm Hg showed.

For comparison, values of the compression modulus determined in various ways are also entered in columns 11 and 12 of the tables for the materials under investigation. The values of column 11 were determined from the measured values E and G by the general equation

K= EG__ ,

3(3G-E) (10 )

hence, they too can be regarded as "measured values". The values of column 12, however, were calculated from a simple "mixed model" in which the spring rates of the structure and the cell gas are added together for a large slab stress in the thickness direction (for similar consideration c.f. ref. 17,18).

*

Measured by the ｾ｡｢ｯｲ｡ｴｯｲｹ for space acoustics of the

Physikalisch-Technischen Bundesanstalt (Director: Dr. G. Venzke).

**

The behaviour of the tensile stress as a function of the deformation under static stress also indicates this(16).

Assuming that only the material of the structure makes a contribution to the reaction force of a foamed plastic under shearing stress, the modulus of compression for this model is given by

K=CK(J+;C(I+31'.). (11 )

In equation (11) KG is the modulus of compression of the cell gas and ｾ the transverse contraction coefficient of the structural material. In

s

the case of the closed cell materials the modulus of compression of air under isothermal stress (106 dyn/cm2) was taken as an approximation for KG;

for the opened cell materials KG

=

O. For ｾｳ the value of 0.4 was assumed for all materials. Since the transverse contraction coefficient of solid high polymer plastics is largely in the range of 0.3 to 0.5, this assumption

yields a maximum error of ±15% of the value of K calculated according to equation (11). However for the calculation of the compression modulus by equation (11), which is intended only as a rough estimate, this error can be disregarded.

Comparison of the moduli of compression determined according to

equations (10) and (11) yields a satisfactory agreement for all materials. It can be concluded from this that the additivity of the spring rates of the skeleton and of the cell gas, as assumed for the "mixed model", is

approximately correct. On the other hand, the agreement can also be regarded as confirmation of the measured values. For the closed cell materials K varies between 1.06 . 106 and 1.42 . 108 dyn/cm2; as may be expected, the modulus of compression of very soft foams agrees with that of the enclosed gas. The values of K given for opene-cell materials hold, presumably, only for the structure itself; in considering large specimens from which the air cannot escape during deformation, the compression modulus of air should logically be added proportionally to the stated values.

In closing, it may be stated that the accuracy of measurement attained by the described method in ､･ｴ･ｲｭｩｮｩｮｾ the dynamic-elastic characteriptic values G and E for foamed plastics permits a sufficiently accurate calcu-lation of the other moduli. This makes it possible to expand the frequency range of approximately 0.1 to 100 Hz (for harder materials up to a maximum of 1000 Hz) in the direction of high frequencies by the application of

suitable methods in which other kinds of stress are employed, e.g. impedance tube methods, cf. ref.

4,

since the corresponding modulus can be determined at low frequencies by calculation from the two moduli E and G.

The author thanks Mr. K. Frankenfeld for his cooperation especially in the construction of the mechanical apparatus, and Mr. H. Sauer, for carrying out the measurements and evaluating them.

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loid.·Z. 127 11%2],65.

K""I'I':I,MANN,J., Neuere phyHikoliHme

Priilmetho-(len Iiir Kunststofle, Kunststnfle 47 [1957], 416.

KIMMICU,E.G., Ruhhcr in compression. Ind. Rub·

her World 103 11910],ＴｾＮ

Rm:J,Jl;, II., Dynamiedrc Dewertung tier Diimplung und Doucrfcstigkcit ven Vulknnlsnten. Kautsdmk 15 [1939], 7.

KOSTEN, C. W., Over de clustidre elgensdrappen von gcvulconiseerde rubber, Dissertation, Dellt 1942.

KOSTEN. C. W., Bcrcdmung von

Fcdcrungselemen-ten ous Gummi. VDI·Z. 86 [1942], 535.

ROCARlJ, Y., Note sur 10 eoleul des proprietcs clo·

stiques des supports en caoutdioue adherent. J.

Phys, HodiumII [1937],197.

Vgl. hierzu DIN 53120, Ilostimmung der

Roh-dichte von Schaumstoflen. Kunststoffe 47 [1957], 31.

CAn.:v, R. H. uud R"':EIlS,E.A., Mechanismes

Verhalten von wciehcn Sdmumstoflen. Modern

Plostics :13 fl956J, 139.

ｋｏｓｔｉｾｎＬ C. W. und ZWIKKEIl, C., Quolities or Spongc-Huhbcr as a material for vihratien- and shockdamping. Physico" [1937], 643.

DKeRlln,G.W. UlldOmlnST,ll., Ubcr dos dyna.

misch·elostisdlc Verholtenlinearer, vernetzter und

Summary of dynamic-elastic characteristic values of foamed plastics 1 2 3 4 5 6 7 8 9 10 11 12 Type of K . 10-6 Foamed cellular _plastic C G 10-6 E . 10-6

"o

.10 n . 10 deter. from

structure p

-

E ]J _{equ. (10) lequ. (11)}

g/cm3 _{% Rayl/cm dyn/cm}2 _dyn/cm2 _dyn/cm2 _dyn/cm2

Closed PVC (soft) 0.071 94

-

0.11 0.32 1. 00 0.95 0.45 1. 06 1.1 Silicon-cells rubber 0.336 71

-

0.42 1.16 1. 05 1. 05 0.38 1. 61 1.4 PVC (medium 0.134 89

-

0.96 2.45 1. 35 1. 25 0.275 1. 82 2.3 soft) Rubber 0.204 82

-

1. 45 3.95 0.52 0.60 0.36 4.70 3.0 Vulkollan 0.341 73

-

4.90 12.9 0.42 0.40 0.32 11. 9 9.0 PVC (hard) 0.043 97

-

77 .0 196 0.30 0.25 0.27 142 114 Latex 0.095 91 18 0.23 0.55 0.37 0.47 0.20 0.31 0.34 Open _{Rubber 0.150 86 45 0.28 0.69 0.80 0.72 0.23 0.43 0.41} cells _{Moltopren 0.032 98 25 1. 77 4.25 1.14 1.10 0.20 2.36 2.60} I f-' --.J I

Modulus

[

G

ox=u

L )

Pendulum

G=Z;U

methods Free

oscil- Bending Torsion

lation (Plan) (elevation)

4 ｾ

DIJJ

Vibrometer

I

ｾ

methods Forced oscil- _Elongation Shear latlon _(Plan) Figure 1

Principle of measurement for the investigation of foamed plastics

2.

3.

Figure 2

View of the bending pendulum; sensor plate (A) elevated

1. Point of suspension 2. Carrier frequency bridge

•••••••••••••••••••••

Fo am Rubber

Paper Speed rate: 1 mm/s

-5dB

f

•

f = 0.222 Hz I.

ｾｴｬｮｮａａ

• • • • • • • • • • • • • • • • • • • • • 1 Figure 3

Records of a damped fading oscillation (bending pendulum)

Figure

4

View of tensile vibrometer 1. Measurement vibrometer

2. Power amplifier

3.

Vacuum tube voltmeter

4.

Voltage amplifier

Figure

5

Specimen mount in shear vibrometer

? HI 0.4 0,6 O,? 2 Hz a,' f _ 0.4 0,6 O,? I I

I-I ... n - - . j - - ..• -I !

.=e

t=o-

0 0 -? _, I

a

I i I i

t,

I

Ｚｾｾｲﾷ

I:> r---La.9 a n d

...

f

-·'" 1

00;91 ｾＧＭ _-.L I t. ' - - - • O,S4 -hId .z _{ＭＭｔＭｾ} ｾ •ｯ｣Ｎｯｾ ._.. " 0:385 1 . . . -

-

+0,31 ｾｾｾ 'lG I

ＨＮｾ

ｾｾＢｏＧ

..

•

｟ｲＮＭｯｈｾ 0 ｾｯＢ ｾ ,1

.t-

ｾ ' . . . 0 08f-- --- - -- _'1£ f -06 a 0, 0. 0,' Figure

6

Frequency curves of the value G,

n

G, Ef and

n

_E of PVC foam (medium soft) measured

from cylindrical specimens with different form factor hid

0.4 0,& ---,---,----,-.,...,

I

- 0)75 -4 & () ｾｩｧｵｾ The quotient E f/2 0 as a function of the form factor hid for cylindrical specimens

for different values of the transverse contraction coefficient (according to equation (8)). For comparison measured values (0) for PVC foam (medium soft),

obtained from the curves of Figure

6

ｏＬＱｾ . , ,

ｾｾｯｾｊｾｾｊ｝ｊｩ

;- m-

I HI

f _ _ ₁₀₀

ｾｵｲ･

8

Frequency curves of the values E,

n

E, J and

nO

of silicon-rubber foam, measured by various

methods: obending pendulum, x tensile vibrometer, otorsion pendulum + shear vibrometer

*

Pendulum methods