The Shore hardness tester is used extensively throughout industry to determine the static modulus of materials. The new apparatus described here extends the capability of an indentor-type tester into the dynamic regime, and provides a measurement of the dynamic shear or Young's modulus and loss factor as a function of frequency. The instrument, model and data of typical rubber samples are given and compared to other dynamic measurements

Fiorito, Ralph B.

Madigosky, Walter M.

English

NASA Technical Reports Server

DYNAMIC HARDNESS TESTER AND CURE METER 
Walter M. Madigosky and Ralph B. Fiorito 
Naval Surface Warfare Center, MS R31 
10901 New Wa~llpshire Avenue 
Silver Spring, MD 209035000 
ABSTRACT 
1342 Shore hardness tester is used extensively throughout industry to determine the static modulus of materials. 
73e new apparatus described here extends the capability of an indentortype tester into the dynamic regime, and 
provldes a measurement of the dynamic shear oryoung's modulus and loss factor as a function of frequency. The 
mstmrnenl, model and data of typical rubber samples are given and compared to other dynamic measurements. 
INTRODUCTION 
73,e standard durometer has been used to measure the static "hardness" or resistance of materials to 
mdemaeatlon. Vanous static models have been developed to  quantify these measurements by relating the static 
Young's n-~odulws and Poisson ratio of a viscoelastic material to the force of indentation, the penetration depth, 
and s;ze (rad~ws) of the indentor. 
To our knowledge, an equivalent satisfactory dynamic model in which the Indentation force IS tlmc 
dcpcndent has not been developed analytically. Various empirical models, however, have been exammed to 
1nodc9 thc ~nteractlon of transducers placed in contact with a viscoelastic slab to excite shear and compressional 
waves m the rmatenal. R. von ~ierke* e.g. has produced such a model to examine the acoustic properties of living 
flssue He assumes the transducer behaves as a hemispherical radiator whose diameter equals that of the 
cyhndncal transducers used in his experiments. In this model, the radiation impedance of a vibrating sphere is 
related to the complex dynamical shear and compressional moduli of the material. This model, under certaln 
appro~matlons, can be used to infer the dynamic Young's modulus, E', and loss factor, 6. To perform these 
measdrements the amplitude and phase of the driving force and acceleration of the transducer which is in contact 
wth the rnatenal are measured. 
We are examining the validity and applicability of such models experimentally by comparing the 
predzcteed results for E' and 6 to those obtained by more conventional methods such as those developed by 
Mad~goslcy and ke2,or other type of dynamic mechanical testing apparatus such as the dynamic mechanical 
thermal ariaiyser (13 MTA). 
The goal of this study is to develop a model and simple apparatus which will allow a dynamical 
;neasunlment of E'and 6 of a material in a manufacturing environment. 
https://ntrs.nasa.gov/search.jsp?R=19930016419 2020-03-24T07:27:12+00:00Z
THEORY 
The theoretical model we will first examine for measuring the dynamic response of a vlss;oelastsc matena'i 
to intrusion by a perturbing force is that developed by von Gierke to examine the acoust~c propc~aes of bung 
tissue. The theory considers the medium to be homogeneous, isotropic, compressible, and viscoelashc probe 
perturbing the medium is a cylindrical transducer placed in contact with the surface of the rnednum. The 
transducer-medium interaction is modeled as a hemispherical radiator whose diameter equals that of thc 
transducer. 
The complex impedance of the medium which is defined as the ratio of force over velodty I s  related 
analflcally to the complex shear and compressional moduli of the material which are defined as: 
where op2/pl = 6S is the shear loss factor and wh2/hl = 2jLis the compressional loss factor. 
The Young's Modulus is related to the shear modulus by 
so that 
The radiation impedance ZS of the sphere is 
3 2 2 2 2 2 2 2 2 2 2 
Z s  = -4npwa j/3- ((1 - 3j/ah - 3/a h ) - 2(j/ah + l / a  h ) - [ 3  - a k /(jak+ l)l)/((l/a h + j/ah)a k /(pie + 1) 
where a = radius of the sphere, 
2 2 
and 
We have examined several limiting cases of ZS 
This limit is achieved when ak << 1, then 
b. high loss materials (cop2 >> vl) 
c low loss materials (cop2 << pl) 
Z2 and Z4 will be of primary interest for the materials with properties in the frequency range of interest 
(50 - l @ O O  HZ). 
A comparison of the real part of ZS, Z2 and Z4 as a function of frequency when the sphere radius a = 
1 mm shows that the formulas for Z2 and Z4 are in excellent agreement with ZS over a wide frequency range, and 
alverge from ZS only at very high frequency. 
Z4 since it is straightforward to interpret and use it as the basis for an experimental method to obtain 
and p2. 
The impedance Z4 can be interpreted as that of a simple oscillator. To see this, consider the equation of a 
forced hamnasnic oscillator 
mx" + k% = F (11) 
I Sere, kC 1s complex to account for the case of viscous damping. The impedance of the oscillator Z = F / x  can be 
jot jot 
cbtarned by assurnlng x = %e and F = Foe so that x. = jox and 
3 
Comparing Z to the equation for Z4 we see that Z4 behaves like an oscillator with mass = m = 2na p/3 
2 1 / 2 (which is half the mass of a sphere), stiffness = k' = 6napl and frictional resistance = k" = 6na p - [(pl / p) + 
p2/ apl as observed by von Gierke. 
If then, a hemispheric impedance head of real mass M is placed in contact with the viscoelastic material, 
and 11 ns assumed that the impedance of the material is ideally modeled by that of the radiating sphere model 
under the approximation ZS = Z4 for the frequency range of interest, the acceleration measured by such an 
~rnpedance head is given by: 
Thi~n, in view of the interpretation of k' and k" gven above, we may write: 
We can then use these equations to calculate E' = 3pl and 6 = wp2/p1. 
S1rn11arI.y the expression for Z2 can be inverted to produce these values by solving for the variable 1 / h  
:n eqwatlon (8). 
EXPERIMENT 
We have performed experiments on a variety of viscoelastic materials to test the validity and 
applicability of the mode1 grven above. 
To do this we employ a compact Wilcoxon piezoelectric shaker- impedance head to which we can attach 
indentors of various diameters, a Hewlett-Packard noise generator /spectrum analyzer which drives the shaker 
with white noise and measures the real and imagnary parts of the acceleration (amplitude and phase of the 
driving force). With the help of a computer, the complex impedance of the excited material is calaiated and the 
real and imaginary parts are used to produce the Young's modulus and loss factor as described above. 
Tables I. and 11. show a comparison of experimental results on generic nitrile and urethane rubbcr 
samples using the method described above and those obtained using a dynamic mechnaical themai analyser 
(DMTA) made by Polymer Laboratories. The DMTA was used to measure the Young's modulus and loss factor 
as a function of temperature using the frequencies 0.3,1, 3, 10 and 30 Hz. The data was then shfted l o  the 
frequency domain by using the standard Williams-Landel-Ferry (WLF) technique. 
Table I. Comparison of Results on Nitrile Rubber 
Dynamic Durometer Method 
E (m Pa) Loss Factor 
4.3 0.3 
5.3 0.47 
7.0 0.34 
7.5 0.31 
7.8 0.25 
8.1 0.27 
DMTA Method 
E (m Pa) Loss Factor 
7.5 0.2 
7.7 0.25 
7.9 0.3 
8.5 0.33 
8.9 0.35 
9.5 0.4 
Table 11. Comparison of Results on Urethane Rubber 
Dynamic Durometer Method DMTA Method 
E (m Pa) Loss Factor E (m Pa) Loss Factor 
54 0.1 5 63 0.085 
59 0.1 6 66 0.090 
6 1 0.1 0 68 0.094 
62 0.1 1 70 0.097 
70 0.1 7 74 0.100 
These results indicate the good agreement between the two types of measurement. Furlher studies are 
being done to test the effect of indentor surface area and the effect of static contact pressure of the indentor 
In addition to the dynamic modulus measurements on cured materials, a study of the change in dpamlc 
modulus as a function of time was made on a two component polyurethane (Techthane 13, Seaward 
International) as i t changed from a viscous mixture to a cured solid. The results are given in Table 111. A small 
disc, 0.8 cm diameter, was used. As can be seen from the data, this technique may be used to accuratcly 
determine the rate of cure and the state of cure in materials. 
Table 111. Dynamic Cure Test of Polyurethane 
%me ( hours ) 
1 
2 
3 
5 
6 
7 
19 
31 
43 
53 
3 4 
182 ( 8 days 4 
Young's Modulus (kPa) 
4 
51 
320 
61 0 
770 
850 
1150 
1620 
1850 
1890 
1980 
21 50 
Loss Factor 
1.7 
0.9 
0.68 
0.52 
0.42 
0.41 
0.38 
0.37 
0.37 
0.36 
0.34 
0.33 
Similar rate and state of cure measurements can be obtained on epoxies and vulcanized rubbers, thus 
replabrag ci~n-ent methods which provide only relative data. 
REFERENCES 
1. H. von Gierke et. al., Physics of Vibrations in Living Tissues, J.Appld. Physiology, 4, p. 886-900, 1952. 
2. W, Madigos-y and 6. Lee, Improved Resonance Technique for Materials Characterization, J. Acoust. Soc Am. 
73, p. 1374-:VP 1985. 


Dynamic Hardness Tester and Cure Meter

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